This monograph focuses on the mathematical and numerical analysis of simplicial partitions and the finite element method

*397*
*72*
*5MB*

*English*
*Pages 203
[197]*
*Year 2020*

*Table of contents : PrefaceContentsGlossary of Symbols1 Introduction 1.1 Motivation 1.2 Preliminaries2 Simplices: Definitions and Properties 2.1 The Triangle 2.2 The Tetrahedron 2.3 Higher-Dimensional Simplices3 Simplicial Partitions 3.1 Triangulations 3.2 Tetrahedral and Simplicial Partitions 3.3 Coloring Simplicial Partitions 3.4 The Basic Idea of the Finite Element Method4 Angle Conditions 4.1 Zlámal's Minimum Angle Condition 4.2 Volumic Regularity Conditions 4.3 Minimum Angle Conditions in Higher Dimensions 4.4 The Maximum Angle Conditions 4.5 The FEM on Highly Distorted Partitions 4.6 Superconvergence and Post-processing5 Nonobtuse Simplicial Partitions 5.1 Preliminaries 5.2 Acute Partitions 5.3 Nonobtuse Partitions and Path-Simplices 5.4 Applications in Numerical Mathematics 5.5 Further Applications6 Nonexistence of Acute Simplicial Partitions in R5 6.1 Preliminaries 6.2 Auxiliary Lemmas 6.3 The Proposed Proof Technique for d=3 6.4 The Proposed Proof Technique for d=4 6.5 The Nonexistence of Acute Partitions in R5 6.6 Extension to Higher Dimensions7 Tight Bounds on Angle Sums of Simplices 7.1 Preliminaries 7.2 Discussion of Gaddum's Dihedral Angle Bounds 7.3 Dihedral Angle Bounds for Nonobtuse Simplices8 Refinement Techniques 8.1 Refinements of Unstructured Partitions 8.2 Red and Green Refinements of Tetrahedra 8.3 Properties of Red Refinement Techniques 8.4 Refinements into Path-Tetrahedra 8.5 Partitions into Well-Centered Simplices 8.6 Local Nonobtuse Refinements 8.7 The Longest-Edge Bisection Algorithm 8.8 Refinements of Triangular Prisms9 The Discrete Maximum Principle 9.1 The Maximum Principle for an Elliptic Problem 9.2 Finite Element Discretization 9.3 Sufficient Algebraic Conditions 9.4 Associated Geometrical Conditions 9.5 Typical Problems with Standard Conditions 9.6 Less Severe Conditions Based on Stieltjes Matrices10 Variational Crimes 10.1 What Are Variational Crimes? 10.2 Efficient Quadrature Formulae on Simplices 10.3 Isoparametric Quadratic Elements 10.4 Setting the Problem 10.5 Approximate Solution 10.6 Slice and Hat Elements 10.7 A Convergence Result11 0/1-Simplices and 0/1-Triangulations 11.1 Congruence Versus 0/1-Equivalence 11.2 Representation as 0/1-Matrices 11.3 Counting and Enumeration of 0/1-Equivalence Classes 11.4 Orthogonal 0/1-Simplices 11.5 Acute 0/1-Simplices 11.6 Neighbor Theorems for Nonobtuse Simplices 11.7 The Hadamard Conjecture 11.8 Triangulations of Id Using 0/1-Simplices12 Tessellations of Maximally Symmetric Manifolds 12.1 Regular Polytopes 12.2 Regular Triangular Tessellations 12.3 Regular Tetrahedral Tessellations 12.4 Regular Simplicial TessellationsAppendix ReferencesIndexAuthor Index*

Springer Monographs in Mathematics

Jan Brandts Sergey Korotov Michal Křížek

Simplicial Partitions with Applications to the Finite Element Method

Springer Monographs in Mathematics Editors-in-Chief Isabelle Gallagher, Département de Mathématiques et Applications, Ecole Normale Supérieure, Paris, France Minhyong Kim, School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea; Mathematical Institute, University of Warwick, Coventry, UK Series Editors Sheldon Axler, Department of Mathematics, San Francisco State University, San Francisco, CA, USA Mark Braverman, Department of Mathematics, Princeton University, Princeton, NJ, USA Maria Chudnovsky, Department of Mathematics, Princeton University, Princeton, NJ, USA Tadahisa Funaki, Department of Mathematics, University of Tokyo, Tokyo, Japan Sinan Güntürk, Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, New York, NY, USA Claude Le Bris, CERMICS, Ecole des Ponts ParisTech, Marne la Vallée, France Pascal Massart, Département de Mathématiques, Université de Paris-Sud, Orsay, France Alberto A. Pinto, Department of Mathematics, University of Porto, Porto, Portugal Gabriella Pinzari, Department of Mathematics, University of Padova, Padova, Italy Ken Ribet, Department of Mathematics, University of California, Berkeley, CA, USA René Schilling, Institut für Mathematische Stochastik, Technische Universität Dresden, Dresden, Germany Panagiotis Souganidis, Department of Mathematics, University of Chicago, Chicago, IL, USA Endre Süli, Mathematical Institute, University of Oxford, Oxford, UK Shmuel Weinberger, Department of Mathematics, University of Chicago, Chicago, IL, USA Boris Zilber, Mathematical Institute, University of Oxford, Oxford, UK

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Jan Brandts Sergey Korotov Michal Křížek •

•

Simplicial Partitions with Applications to the Finite Element Method

123

Jan Brandts Korteweg-de Vries Institute for Mathematics University of Amsterdam Amsterdam, The Netherlands Michal Křížek Institute of Mathematics Czech Academy of Sciences Prague, Czech Republic

Sergey Korotov Department of Computer Science, Electrical Engineering and Mathematical Sciences Western Norway University of Applied Sciences Bergen, Norway

ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-3-030-55676-1 ISBN 978-3-030-55677-8 (eBook) https://doi.org/10.1007/978-3-030-55677-8 Mathematics Subject Classiﬁcation: 65N50, 65N30 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to the memory of Prof. Miloš Zlámal (1924–1997). Photograph Ó Ludmila Zlámalová

Preface

This monograph arose from our 90 works [19, 34–55, 64, 92–96, 129–136, 139, 140, 155, 157, 160, 161, 167–209, 223, 291], that we published between the years 1982 and 2020 mostly in international journals such as SIAM Journal on Numerical Analysis, SIAM Review, Mathematics of Computation, Numerische Mathematik, IMA Journal of Numerical Analysis, Linear Algebra and its Applications, ESAIM: Mathematical Modelling and Numerical Analysis, and Applications of Mathematics. Its contents are also based on research and survey papers on simplicial partitions of other authors (see the References). Some of the presented results have not been published anywhere else. This book is intended for numerical analysis students and researchers. The reader is assumed to be familiar with only a few basic results from linear algebra, geometry, and mathematical and numerical analysis. The ﬁrst three chapters form a necessary basis for the rest of the book. The other chapters can be read more or less independently of the previous exposition. The monograph is organized in the following way. We start with several deﬁnitions and statements concerning simplices and triangulations. In the second part, we will discuss various geometrical properties of simplicial partitions. We present a wide range of applicable results, especially involving the solution of partial differential equations by the ﬁnite element method. We are well aware that their presentation is oriented more theoretically than practically. Finally, we introduce some more special results on simplicial tessellations of the unit hypercube and maximally symmetric manifolds. The contents of this exposition were improved by interesting and helpful debates with Ivo Babuška, Wei Chen, Jan Chleboun, István Faragó, Miloslav Feistauer, Miroslav Fiedler, Antti Hannukainen, Ivan Hlaváček, Robert Horváth, Zoltán Horváth, János Karátson, Oldřich Kowalski, Václav Kučera, Petr Kůrka, Qun Lin, Dalibor Lukáš, Victor Podsechin, Ángel Plaza, Karel Segeth, Jakub Šístek, Bedřich Šofr, Jakub Šolc, Lawrence Somer, José Suárez, Jon Eivind Vatne, Tomáš Vejchodský, Hehu Xie, Alexander Ženíšek, Shuhua Zhang, and Miloš Zlámal. Their help is greatly appreciated. We would like to thank Mrs. Ludmila Zlámalová for giving us permission to publish the photo of her husband. vii

viii

Preface

Furthermore, we are deeply grateful to Jana Grünerová and Eva Ritterová for proofreading, Antti Hannukainen, Pavel and Filip Křížek for drawing some ﬁgures, Jarmila Štruncová for ordering expert literature, and Hana Bílková for her technical assistance in the ﬁnal typesetting of the manuscript. Finally, we are indebted to Ms. Elena Griniari from Springer-Verlag for her helpful cooperation in the preparation of this book. Our great thanks go to the referees for their valuable suggestions and also to our families for their patience and understanding. The work on this book was supported by grants 18-09628S and 20-01074S of the Grant Agency of the Czech Republic and RVO 67985840 of the Czech Republic. This support is gratefully acknowledged. Amsterdam, The Netherlands Bergen, Norway Prague, Czech Republic May 2020

Jan Brandts Sergey Korotov Michal Křížek

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Simplices: Deﬁnitions and Properties 2.1 The Triangle . . . . . . . . . . . . . . . 2.2 The Tetrahedron . . . . . . . . . . . . 2.3 Higher-Dimensional Simplices . .

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Simplicial Partitions . . . . . . . . . . . . . . . . . . . . . . . 3.1 Triangulations . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Tetrahedral and Simplicial Partitions . . . . . . . 3.3 Coloring Simplicial Partitions . . . . . . . . . . . . 3.4 The Basic Idea of the Finite Element Method .

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Angle Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Zlámal’s Minimum Angle Condition . . . . . . . . . . . 4.2 Volumic Regularity Conditions . . . . . . . . . . . . . . . 4.3 Minimum Angle Conditions in Higher Dimensions . 4.4 The Maximum Angle Conditions . . . . . . . . . . . . . . 4.5 The FEM on Highly Distorted Partitions . . . . . . . . 4.6 Superconvergence and Post-processing . . . . . . . . . .

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Nonobtuse Simplicial Partitions . . . . . . . . . . 5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . 5.2 Acute Partitions . . . . . . . . . . . . . . . . . . 5.3 Nonobtuse Partitions and Path-Simplices 5.4 Applications in Numerical Mathematics . 5.5 Further Applications . . . . . . . . . . . . . . .

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Contents

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Nonexistence of Acute Simplicial Partitions in R5 6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Auxiliary Lemmas . . . . . . . . . . . . . . . . . . . 6.3 The Proposed Proof Technique for d ¼ 3 . . . 6.4 The Proposed Proof Technique for d ¼ 4 . . . 6.5 The Nonexistence of Acute Partitions in R5 . 6.6 Extension to Higher Dimensions . . . . . . . . .

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Tight 7.1 7.2 7.3

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Reﬁnement Techniques . . . . . . . . . . . . . . . . . . 8.1 Reﬁnements of Unstructured Partitions . . . 8.2 Red and Green Reﬁnements of Tetrahedra 8.3 Properties of Red Reﬁnement Techniques 8.4 Reﬁnements into Path-Tetrahedra . . . . . . . 8.5 Partitions into Well-Centered Simplices . . 8.6 Local Nonobtuse Reﬁnements . . . . . . . . . 8.7 The Longest-Edge Bisection Algorithm . . 8.8 Reﬁnements of Triangular Prisms . . . . . .

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Bounds on Angle Sums of Simplices . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion of Gaddum’s Dihedral Angle Bounds Dihedral Angle Bounds for Nonobtuse Simplices

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The Discrete Maximum Principle . . . . . . . . . . . . . . . . . 9.1 The Maximum Principle for an Elliptic Problem . . . 9.2 Finite Element Discretization . . . . . . . . . . . . . . . . . 9.3 Sufﬁcient Algebraic Conditions . . . . . . . . . . . . . . . 9.4 Associated Geometrical Conditions . . . . . . . . . . . . 9.5 Typical Problems with Standard Conditions . . . . . . 9.6 Less Severe Conditions Based on Stieltjes Matrices

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11 0=1-Simplices and 0=1-Triangulations . . . . . . . . . . . . . . . . . 11.1 Congruence Versus 0=1-Equivalence . . . . . . . . . . . . . . 11.2 Representation as 0=1-Matrices . . . . . . . . . . . . . . . . . . 11.3 Counting and Enumeration of 0=1-Equivalence Classes . 11.4 Orthogonal 0=1-Simplices . . . . . . . . . . . . . . . . . . . . . .

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10 Variational Crimes . . . . . . . . . . . . . . . . . . . . . . 10.1 What Are Variational Crimes? . . . . . . . . . . 10.2 Efﬁcient Quadrature Formulae on Simplices 10.3 Isoparametric Quadratic Elements . . . . . . . 10.4 Setting the Problem . . . . . . . . . . . . . . . . . . 10.5 Approximate Solution . . . . . . . . . . . . . . . . 10.6 Slice and Hat Elements . . . . . . . . . . . . . . . 10.7 A Convergence Result . . . . . . . . . . . . . . .

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Contents

11.5 11.6 11.7 11.8

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Acute 0=1-Simplices . . . . . . . . . . . . . . . . . . . Neighbor Theorems for Nonobtuse Simplices . The Hadamard Conjecture . . . . . . . . . . . . . . . Triangulations of I d Using 0=1-Simplices . . . .

12 Tessellations of Maximally Symmetric Manifolds 12.1 Regular Polytopes . . . . . . . . . . . . . . . . . . . . 12.2 Regular Triangular Tessellations . . . . . . . . . 12.3 Regular Tetrahedral Tessellations . . . . . . . . . 12.4 Regular Simplicial Tessellations . . . . . . . . . .

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Glossary of Symbols

N ¼ f1; 2; . . .g d Rd Sd S; Si T ^S; T^ C, C i G ða; bÞ ½a; b bac dae gcdðm; kÞ X @X X vold X n dist vjK Di v @[email protected] D grad div hes conv e

The set of natural numbers Dimension d-dimensional Euclidean space d-dimensional hypersphere d-simplex Triangle or tetrahedron Reference element Generic constants (different at each occurrence) Center of gravity Scalar product in Rd Open interval in R ¼ R1 Closed interval in R Integer part of a real number a Smallest integer greater than or equal to a Greatest common divisor of integers m and k Problem domain (nonempty connected open set in Rd ) Boundary of X Closure of X d-dimensional Lebesgue measure of X Outward unit normal to @X Distance Restriction of a function v to a set K ith generalized derivative of v (i multi-index) First order generalized derivative of v, j 2 f1; . . .; dg Laplace operator Gradient Divergence Hessian Convex hull Euler number xiii

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det A A> AH i; j; k dij 9 8 Pk ðXÞ CðXÞ k C ðXÞ Lp ðXÞ; p 2 ½1; 1Þ L1 ðXÞ H k ðXÞ H 10 ðXÞ W kp ðXÞ kkk;X j jk;X ð; Þ0;X kkk;p;X j jk;p;X V ! v u k kV jj k k að; Þ FðÞ dim Pk ðSÞ hS ; diam S h Th F ¼ fT h gh!0 Vh …h v fvi gNi¼1

Glossary of Symbols

Determinant of the matrix A Transpose of the matrix A Adjunct matrix to A Integer indices (subscripts) Kronecker’s symbol: dij ¼ 1 for i ¼ j, dij ¼ 0 otherwise There exist(s) For all Space of polynomials of degree at most k deﬁned on X Space of continuous functions deﬁned on X Space of functions whose classical derivatives up to order k belong to CðXÞ Lebesgue space of measurable functions v deﬁned on X for R which X j vðxÞjp dx is ﬁnite Lebesgue space of measurable essentially bounded functions deﬁned on X Sobolev space of functions whose generalized derivatives up to order k belong to L2 ðXÞ Space of functions from H 1 ðXÞ whose traces vanish on @X Sobolev space of functions whose generalized derivatives up to order k belong to Lp ðXÞ Norm in H k ðXÞ Seminorm in H k ðXÞ Scalar product in ðL2 ðXÞÞd Norm in W kp ðXÞ Seminorm in W kp ðXÞ Banach space of test (or trial) functions Convergence symbol Test function Classical or weak (variational, generalized) solution Norm in V Absolute value Euclidean norm Bilinear form Linear form Dimension of Pk ðSÞ Diameter of a simplex S Discretization parameter Triangulation (partition) with discretization parameter h Family of partitions Finite element space(s) Vh -interpolant of v Basis functions in V h

Glossary of Symbols

OðÞ ; x2A x 62 A fx 2 A jPðxÞg AB A\B A[B AnB f :A!B x 7! f ðxÞ h

xv

Landau’s symbol: f ðaÞ ¼ OðgðaÞÞ; if there exists C [ 0 such that jf ðaÞj CjgðaÞj as a ! 0 or a ! 1 Empty set Element x belongs to set A Element x does not belong to set A Set of all elements x in A which possess property PðxÞ A is a subset of set B Intersection of sets A and B Union of sets A and B Subtraction of B from A A function f acting from A to B A function which assigns value f ðxÞ to x Halmos symbol

Chapter 1

Introduction

1.1 Motivation Numerous scientific and engineering problems that have arisen in the third millennium require the simulation of various phenomena and physical fields over complicated three-dimensional structures, and even in dimensions higher than three. The construction of simplicial partitions is an important task, since not all structures can be decomposed into simpler objects such as d-dimensional rectangular blocks. Therefore, this monograph will focus on mathematical and numerical analysis and the interplay between simplicial partitions and the finite element method (FEM). This is now a very active research area. Richard Courant (1888–1972) is generally considered to be the founder of the finite element method. On June 16, 1942, he submitted a paper [75] to the Bulletin of the American Mathematical Society, which was a crucial contribution to the development of the finite element method. On p. 21, he considers a triangulation of a two-dimensional multiply connected rectangular domain with four holes. He looks for an approximate solution of a certain variational problem in the form of a continuous piecewise linear function over this triangulation.1 This problem requires solving a large system of linear algebraic equations. However, at that time there were no computers and thus Courant’s idea was forgotten for a while. Ten years later, the finite element method was rediscovered by American engineers while analyzing aircraft structures using the first commercial computers. One of the first monographs on this method is [275] by John L. Synge, published in 1957. He gives a thorough mathematical analysis of the approximation properties of continuous piecewise linear functions over a triangulated domain. In 1968, Miloš Zlámal (see [306]) presented the first mathematical theory of the convergence of the FEM.

1 Already

in 1696, Gottfried Wilhelm Leibniz approximated the solution of a one-dimensional problem (the brachistochrone problem) by a continuous piecewise linear function.

© Springer Nature Switzerland AG 2020 J. Brandts et al., Simplicial Partitions with Applications to the Finite Element Method, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-55677-8_1

1

2

1 Introduction

From that time thousands of papers and hundreds of monographs (see e.g. [2, 6, 68, 150, 269, 276, 301]) on this method have appeared. A significant advantage of the FEM is that it is very flexible in approximating complicated domains, which was not possible using classical methods (e.g. finite difference methods or collocation methods). Another benefit is that the FEM is based on variational principles. This enables systematic use of effective tools from functional analysis to prove the existence and uniqueness of the true and approximate solutions, and the convergence of the sequence of approximate solutions with less stringent smoothness assumptions on the true solution than would be the case for classical methods. At present the finite element method seems to be the most efficient numerical method for solving problems of mathematical physics and engineering that are described by partial differential equations, algebro-differential equations, integrodifferential equations, systems of these equations, and variational inequalities. Typical examples are the heat conduction equation, linear elasticity equations, semiconductor equations, Maxwell’s equations, and partial differential equations describing magnetic, electric, or gravitational fields. The main advantage of the finite element method is that it enables one to simulate the above mentioned problems on a computer. It thus replaces the creation of expensive technical models (prototypes) or performing complicated measurements. For example, in designing and manufacturing electrical devices we can sufficiently accurately compute their electromagnetic, temperature, or stress fields by the finite element method. The whole computational process based on the finite element method can be essentially automated, including the following steps: (1) (2) (3) (4) (5) (6) (7)

preprocessing of input data, generation of finite element partitions, assembling finite element matrices, solving the corresponding discrete problems, post-processing of output data, a posteriori error estimation, and graphical illustration of results.

We aim to concentrate especially on point (2), which is difficult to implement and which typically demands a significant amount of computational time because of its large algorithmic and combinatorial complexity. For example, the construction of simplicial partitions of bodies with certain desired geometrical properties (acuteness, nonobtuseness, etc.) is a very challenging problem, both practically and theoretically, in the three-dimensional case. However, if all dihedral angles are nonobtuse (i.e. not greater than 90◦ ), then the matrices produced by the finite element method have some useful mathematical properties for certain specific equations and their discretizations, as we shall see in Sect. 5.4. The universality and flexibility of the FEM led to a dramatic development of software for a wide range of problems. Real-life technical problems are usually

1.1 Motivation

3

solved by standard open-source codes and software packages, including for instance Abaqus 2019, ALBERTA, ALGOR, ANSYS, COMSOL, COSMOS, DUNE-FEM, ELLPACK, FEniCS, FreeFEM, GT STRUDL, I-DEAS FEM, LUSAS, Maple, Marc, Mathematica, MATLAB, Mechanica, Modulef, MSC/PROBE, MSC Nastran, Nektar++, NISA, PAFEC, Patran, SYSTUS, TetGen.

1.2 Preliminaries Throughout this monograph we will use the standard Sobolev space (named after Sergey L. Sobolev) notation [68]. Let Ω be a nonempty bounded domain in the d-dimensional Euclidean space Rd , d = 1, 2, . . . , with Lipschitz boundary ∂Ω, see [203, p. 17]. Recall that W pk (Ω) is the Sobolev space of functions whose generalized partial derivatives up to order k ∈ {0, 1, . . . } belong to the Lebesgue space L p (Ω), p ∈ [1, ∞]. For p < ∞ they are equipped with the norm ⎛ vk, p,Ω = ⎝

⎞ 1p

|m|≤k

|D m v| p dx ⎠ , v ∈ W pk (Ω),

Ω

and seminorm ⎛ |v|k, p,Ω = ⎝

⎞ 1p

|m|=k

Ω

|D m v| p dx ⎠ , v ∈ W pk (Ω),

where m = (m 1 , . . . , m d ) is the so-called multiindex, m 1 , . . . , m d are nonnegative integers, |m| = m 1 + · · · + m d , and Dm v =

∂ |m| v . · · · ∂ xdm d

∂ x1m 1

For p = ∞ we set vk,∞,Ω = max D m v0,∞,Ω , |v|k,∞,Ω = max D m v0,∞,Ω . |m|≤k

|m|=k

Sometimes we omit the index Ω, i.e. · k, p = · k, p,Ω , | · |k, p = | · |k, p,Ω .

4

1 Introduction

Fig. 1.1 A geometric illustration of the finite element method for a self-adjoint linear elliptic problem

If kp > d then the following Sobolev imbedding holds [258] (see also [121, 237]) W pk (Ω) ⊂ C(Ω).

(1.1)

· k = · k,Ω = · k,2,Ω , | · |k = | · |k,Ω = | · |k,2,Ω ,

(1.2)

If p = 2 then we write only

and set H k (Ω) = W2k (Ω). This is a Hilbert space. Let V = H01 (Ω) = {v ∈ H 1 (Ω) | v = 0 on ∂Ω}, where the equality v = 0 on ∂Ω is in the sense of traces (see [237]). Let (·, ·)0 = (·, ·)0,Ω be the scalar product on L 2 (Ω) = H 0 (Ω) or in [L 2 (Ω)]d . The space of continuous functions over the closure Ω of Ω will be denoted by C(Ω). The FEM for a self-adjoint linear elliptic problem has a very nice geometric interpretation (see Fig. 1.1). The finite element solution is usually sought in a subspace Vh ⊂ V of continuous piecewise polynomial functions. It is a projection of the true solution u ∈ V of some elliptic problem onto Vh with respect to a suitable (problemdependent) scalar product a(·, ·), as we shall demonstrate in Eq. (3.12).

Chapter 2

Simplices: Definitions and Properties

2.1 The Triangle Consider a triangle T ⊂ R2 with vertices A = (a1 , a2 ), B = (b1 , b2 ), and C = (c1 , c2 ) that do not lie on one line. Denote its interior angles at A, B, and C by α, β, and γ, respectively. Let a, b, and c be the sides opposite to A, B, and C, respectively, see Fig. 2.1. Denote by r the radius of the inscribed circle and by R the radius of the circumscribed circle of the triangle T . Recall the well-known formulae for the area of T , vol2 T =

abc 1 ab sin γ = r s = = s(s − a)(s − b)(s − c), 2 4R

(2.1)

with the last expression being Heron’s formula, where s=

a+b+c 2

is the semiperimeter (half-perimeter) of T and vold T stands for the volume in the Euclidean space Rd . The area of T can also be expressed as vol2 T = where

|δ| , 2

⎡ ⎤ 1 a1 a2 b1 − a1 b2 − a2 = det ⎣1 b1 b2 ⎦ . δ = det c1 − a1 c2 − a2 1 c1 c2

(2.2)

© Springer Nature Switzerland AG 2020 J. Brandts et al., Simplicial Partitions with Applications to the Finite Element Method, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-55677-8_2

(2.3)

5

6

2 Simplices: Definitions and Properties C

a

r

b r r

A

B

c

Fig. 2.1 The area of the triangle ABC is equal to 21 r (a + b + c) = r s

The next formula is also well known vol2 T =

ava , 2

where va is the length of the altitude of T emanating from the vertex A orthogonally to the straight line containing the opposite side a. We further recall the following fundamental relations: b c a = = (= 2R) sin α sin β sin γ

(the Sine theorem)

and a 2 = b2 + c2 − 2bc cos α

(the Cosine theorem).

The centroid (the center of gravity) G of T has the coordinates G=

1 (a 3 1

+ b1 + c1 ), 13 (a2 + b2 + c2 ) = 13 (A + B + C).

2.2 The Tetrahedron Let A = (a1 , a2 , a3 ), B = (b1 , b2 , b3 ), C = (c1 , c2 , c3 ), and D = (d1 , d2 , d3 ) be points in R3 that are not contained in one plane. We denote by T the tetrahedron with vertices A, B, C, and D (see Fig. 2.2). It is the simplest closed convex threedimensional polyhedron and it has 4 triangular faces and 6 edges. The volume of T can, for example, be calculated by the following formula (cf. (2.2)): |δ| , (2.4) vol3 T = 6

2.2 The Tetrahedron

7

Fig. 2.2 Tetrahedron T with denotation

D l4

l3 l5

C

l2

A

l6

l1

B

where

⎡ ⎤ 1 a1 b1 − a1 b2 − a2 b3 − a3 ⎢1 b1 ⎢ δ = det ⎣c1 − a1 c2 − a2 c3 − a3 ⎦ = det ⎣ 1 c1 d1 − a1 d2 − a2 d3 − a3 1 d1 ⎡

a2 b2 c2 d2

⎤ a3 b3 ⎥ ⎥, c3 ⎦ d3

(2.5)

see [249, p. 200]. The rightmost expression in (2.5) is not sensitive to rounding errors if a Q R-factorization of the matrix is used to compute it [278]. Other formulae for the volume, that are not sensitive to rounding errors (which is especially important for flat tetrahedra, see Fig. 4.1), are presented in [151]. Further, 3 vol3 T r= (2.6) vol2 ∂T is the radius of the inscribed ball of T , where ∂T denotes the boundary of T . Formula (2.6) can be derived similarly to the two-dimensional case sketched in Fig. 2.1. By [101, p. 316], the radius of the circumscribed ball about T can be computed as √ Z , (2.7) R= 24 vol3 T where Z = 2l12 l22 l42 l52 + 2l12 l32 l42 l62 + 2l22 l32 l52 l62 − l14 l44 − l24 l54 − l34 l64 .

(2.8)

In the above formula, li and li+3 are the lengths of opposite edges of T for i = 1, 2, 3: l1 = |AB|, l2 = |AC|, l3 = |AD|, l4 = |C D|, l5 = |B D|, l6 = |BC|, see Fig. 2.2. The center of gravity of T is G = 41 (A + B + C + D).

8

2 Simplices: Definitions and Properties

Fig. 2.3 Number of acute interior angles for various positions of the orthogonal projection of the vertex D on the plane ABC

1

C 2

2 3

1

A B 2

1

The dihedral angles of a tetrahedron are the six angles between each pair of faces of T . They are defined as the complementary angles of the angles between the outward unit normals to the corresponding faces and can thus be calculated by means of the inner product (2.9) cos α = −n i · n j , where n i and n j , i = j, are outward unit normals of particular faces. Lemma 2.1 For any face of an arbitrary tetrahedron there exists at least one acute interior angle adjacent to this face. Proof Let ABC D be an arbitrary tetrahedron. Consider its arbitrary face, say ABC. We cut the plane containing the triangle ABC by the three straight lines AB, BC, and AC. Thus, we get seven open areas in the plane. In Fig. 2.3, we indicate the number of acute interior angles adjacent to the face ABC with respect to the position of the orthogonal projection of D onto the plane ABC. If this projection lies on a line or at some vertex then the number of acute interior angles is equal to the minimum among all the numbers corresponding to the areas adjacent to the associated side or vertex. Lemma 2.2 All dihedral angles of a tetrahedron T are nonobtuse if and only if all of its spatial altitudes are contained in T , i.e., the orthogonal projection of each vertex onto the plane containing the opposite face lies on this face. The proof is a straightforward consequence of Lemma 2.1 and therefore omitted.

2.3 Higher-Dimensional Simplices

9

2.3 Higher-Dimensional Simplices Let A0 , A1 , . . . , Ad be points in Rd , d ∈ {1, 2, . . . }, not lying in one hyperplane. Then the convex hull of these points S = conv{A0 , A1 , . . . , Ad } is called a d-simplex or shortly a simplex. Throughout the book, simplices will sometimes also be called elements. The points A0 , A1 , . . . , Ad are called the vertices of S. If d = 2 or d = 3, then the letter T is used instead of S. The set Fi = conv{A0 , . . . , Ai−1 , Ai+1 , . . . , Ad } is called the facet of S opposite to vertex Ai for i ∈ {0, 1, . . . , d}. Let k ≤ d be a positive integer and let {B0 , . . . , Bk } ⊂ {A 0 , . . . , Ad }. Then S = k-facets. conv{B0 , . . . , Bk } is called a k-facet of S. Obviously, S has d+1 k+1 Let V = (A1 − A0 , . . . , Ad − A0 ) and let W = V V be the corresponding Gramian matrix. Then the volume of S can be calculated as follows (see [20]) 1 1√ det W , | det V | = d! d!

(2.10)

vi vold−1 Fi , i ∈ {0, 1, . . . , d}, d

(2.11)

vold S = or vold S =

where Fi is an arbitrary facet of S and vi is the length of the corresponding spatial altitude (height) emanating from Ai . For example, for the regular simplex in Rd we have d +1 a (2.12) vi = 2d and

√ d +1 vold S = √ a d , d! 2d

where a is the length of its edge. Note that all d + 1 spatial altitudes do not intersect in one point, in general (except for the case d = 2). If all d + 1 spatial altitudes of a simplex S intersect in one point, this point is called the orthocenter of S, and S is called orthocentric. Only if d = 2 are all simplices orthocentric. Each simplex has a unique circumscribed ball and a unique inscribed ball. The center of the circumscribed ball is usually called the circumcenter and its radius R the circumradius. A formula for the calculation of R will be presented later, see (2.14).

10

2 Simplices: Definitions and Properties

Theorem 2.1 The circumcenter, the center of gravity, and the orthocenter of each orthocentric d-simplex lie on a line. For the proof, see [105, p. 66]. Denoting the circumcenter by O and the center of gravity of any d-simplex by G, then OG is called the Euler line if O = G. The radius of the inscribed ball of S, also called its inradius, is given by r=

d vold S . vold−1 ∂ S

The proof is similar to the two- and three-dimensional case, see Fig. 2.1 and (2.6). By [28, 148], or [261, p. 125], the volume of S can be computed in terms of lengths of its edges using the so-called Cayley–Menger determinant of size (d + 2) × (d + 2), Dd = (−1)d+1 2d (d!)2 (vold S)2 ⎡

1 1 ··· 1 2 2 0 a0,1 · · · a0,d−1 2 2 a1,0 0 · · · a1,d−1 .. .. .. . . . 2 2 2 ad,1 · · · ad,d−1 1 ad,0

0 ⎢1 ⎢ ⎢ = det ⎢1 ⎢ .. ⎣.

1

⎤

2 ⎥ a0,d ⎥ 2 ⎥ a1,d ⎥ , .. ⎥ . ⎦

(2.13)

0

where ai j is the length of the edge Ai A j for i = j. The radius R of the circumscribed ball B around S satisfies (see [21]) R2 = − where

⎡

1 Δd , 2 Dd

2 2 0 a0,1 · · · a0,d−1 2 2 ⎢a1,0 0 · · · a1,d−1 ⎢ Δd = det ⎢ . .. .. ⎣ .. . . 2 2 2 ad,1 · · · ad,d−1 ad,0

(2.14) ⎤ 2 a0,d 2 ⎥ a1,d ⎥ .. ⎥ . . ⎦

(2.15)

0

The following formula for the volume of a d-dimensional ball b with radius r holds [145] d/2 ˜ d , where C˜ = π vold b = Cr , (2.16) d Γ ( 2 + 1) and Γ denotes the Gamma function. We say that b0 , b1 , . . . , bd are the barycentric coordinates of the point x ∈ S if x = b0 A0 + b1 A1 + · · · + bd Ad ,

2.3 Higher-Dimensional Simplices

where bi ≥ 0,

11

d

bi = 1.

i=0

The barycentric coordinates are uniquely defined for each x ∈ S. In Chap. 10, we will use special kinds of barycentric coordinates for d = 2, 3, the so-called triangular and tetrahedral coordinates.

Chapter 3

Simplicial Partitions

3.1 Triangulations A polygon Ω is the closure of a nonempty bounded planar domain Ω whose boundary can be expressed as a union of a finite number of straight line segments. By a triangulation of a polygon Ω we shall mean a finite set of triangles such that any two different triangles from this set have one common side or one common vertex or they have no common point and the union of all these triangles is Ω (see Fig. 3.1 for some typical real-life practical triangulations and also Refs. [61, 114]). Such triangulations are called conforming or face-to-face, i.e., we shall not allow the situation illustrated in Fig. 3.2. Let us point out that a triangulation is not uniquely determined only by vertices. To see this, it is enough to consider a square divided into two right triangles in two different ways—either the diagonal with positive or with negative slope. Triangulations are used not only in the finite element method for numerically solving partial differential equations, for instance, but also in geodesy, meteorology, crystallography, chemistry, electrical engineering, computer graphics, various visualization techniques, etc. (see e.g. Sect. 5.5 for several references in this respect).

3.2 Tetrahedral and Simplicial Partitions A polyhedron is the closure Ω of a nonempty bounded domain Ω ⊂ R3 whose boundary can be expressed as a finite union of polygons. Recall that a convex polytope in Rd is usually defined as a convex hull of a finite set of points in Rd with nonempty interior. Now, we shall generalize the notion of triangulation to higher dimensions (cf. Fig. 3.3). In numerical analysis, combinatorics, computer graphics, etcetera, the following terms are often used: a face-to-face partition, a simplicial complex, © Springer Nature Switzerland AG 2020 J. Brandts et al., Simplicial Partitions with Applications to the Finite Element Method, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-55677-8_3

13

14

3 Simplicial Partitions

Fig. 3.1 Examples of planar face-to-face triangulations: a triangulation of a wrench (left) and a triangulation of the Oder lagoon shared by Germany and Poland (right—courtesy of V. Podsechin [245])

Fig. 3.2 Two triangles that do not form a face-to-face triangulation. The bullet indicates a so-called hanging node

Fig. 3.3 A tetrahedral finite element mesh, generated by Netgen, of a DC electromagnet after shape optimization of pole heads. The optimization aim was to minimize the variance of magnetic flux density in a central cubic area among the pole heads while preserving the flux magnitude above a given value (courtesy of D. Lukáš [226])

3.2 Tetrahedral and Simplicial Partitions

15

decomposition, dissection, division, grid, lattice, mesh, net, network, space discretization, subdivision, tessellation, tetrahedralization, tiling, triangulation, etc. In what follows, we shall mainly use the terms triangulation or partition. Definition 3.1 A finite set of tetrahedra is a partition of a polyhedron Ω if (i) the union of all the tetrahedra is Ω, (ii) the interiors of the tetrahedra are mutually disjoint, (iii) any face of any tetrahedron from the set is either a face of another tetrahedron in the set, or a subset of ∂Ω. The third condition (iii) guarantees that every partition is face-to-face. Theorem 3.1 For any polyhedron there exists a partition into tetrahedra. The main idea of the detailed constructive proof, presented in [189, 203], is the following. Denote the faces of a given polyhedron Ω by F1 , . . . , Fm . Consider the planes P1 , . . . , Pm ⊂ R3 such that Fi ⊂ Pi , i = 1, . . . , m. It can be shown that all components of the set Ω\

m

Pi

(3.1)

i=1

are open convex polyhedra. Their closures can be decomposed into tetrahedra as follows. First, we triangulate each of its convex polygonal faces in one of two manners sketched in Fig. 3.4. Second, we take e.g. the convex hull of the center of gravity of the convex polyhedron (which, obviously, lies in the interior of this polyhedron) with each of the triangles on its surface. If all common faces involved are triangulated in the same way, a partition of Ω into tetrahedra satisfying the conditions of Definition 3.1 results. For a given partition Th of Ω the discretization parameter h (also called the mesh size) is the maximum length of all edges in the partition, i.e., h = max h T , T ∈Th

where h T = diam T. To prove various convergence statements in numerical analysis, we unavoidably have to work with infinite sequences of partitions whose discretization parameter tends to zero. For this reason we introduce the following definition.

16

3 Simplicial Partitions

Fig. 3.4 Two partitions of a convex polyhedron into tetrahedra. Each polygonal face of its surface is divided into triangles in two different ways

Definition 3.2 A set of tetrahedral partitions F is called a family of partitions if for every ε > 0 there exists a Th ∈ F such that h < ε. Definition 3.3 A family of tetrahedral partitions F of a polyhedron Ω into tetrahedra is said to be regular (strongly regular) if there exists a constant κ > 0 such that for any partition Th ∈ F and for any tetrahedron T ∈ Th there exists a ball bT ⊂ T with radius ρT such that κh T ≤ ρT (κh ≤ ρT ), see [68, p. 132]. For example, we can take ρT = r T , where r T is the radius of the inscribed ball about T . The above definition was introduced in [68] to treat also non-simplicial elements. Remark 3.1 Obviously, any strongly regular family is regular. A strongly regular family of triangulations of any polygon is easy to construct due to the fact that any triangle can be subdivided into four congruent triangles, similar to the original one, by connecting the midpoints of its edges. However, this simple idea cannot be directly employed in higher dimensions, see [177, 303]. Nevertheless, the following statement holds. Theorem 3.2 For any polyhedron there exists a strongly regular family of partitions into tetrahedra. For a detailed proof see [189]. It consists of three parts: (a) First, the Sommerville tetrahedron Tˆ with vertices (see Fig. 3.5) Aˆ = (−1, 0, 0), Bˆ = (1, 0, 0), Cˆ = (0, −1, 1), Dˆ = (0, 1, 1)

3.2 Tetrahedral and Simplicial Partitions

17

Fig. 3.5 The lengths of edges of the Sommerville tetrahedron

D

3

3 2 2

A

B 3

C

3

is partitioned into 8 congruent subtetrahedra that are similar to the reference tetrahedron Tˆ (see Remark 5.8 and Theorem 8.4). Repeating this refinement process, we obtain a strongly regular family of partitions of Tˆ into subtetrahedra. (b) Second, an arbitrary tetrahedron T with vertices A = (a1 , a2 , a3 ), B = (b1 , b2 , b3 ), C = (c1 , c2 , c3 ), D = (d1 , d2 , d3 ) is partitioned into 8 subtetrahedra by means of the affine one-to-one mapping FT : Tˆ → T given by ˆ = Q xˆ + q, xˆ ∈ Tˆ , FT (x) where q = 21 (A + B) and Q = 21 (B − A, D − C, C + D − A − B) ˆ = A, . . . , FT ( D) ˆ = D. Using part is a nonsingular 3 × 3 matrix, since FT ( A) (a) one can easily show that this leads to a strongly regular family of partitions of T into tetrahedra, since the mapping FT is fixed. (c) Third, an arbitrary polyhedron Ω can, by Theorem 3.1, be partitioned face-toface into a finite number of tetrahedra. Further, we apply the construction from part (b) to each of them, and thus we get a strongly regular family of tetrahedral partitions of the whole polyhedron Ω. The face-to-face property is, obviously, preserved at every refinement step. The above definitions and theorems can be naturally generalized to higher dimensions for d > 3. In Chaps. 5 and 6 we shall consider simplicial partitions of unbounded domains, including the whole space Rd . In Chap. 8 we will deal with refinement techniques producing families of simplicial partitions of polytopic domains.

18

3 Simplicial Partitions

3.3 Coloring Simplicial Partitions The visualization of simplicial partitions is an important and difficult problem, which has potential applications in computer graphics and various software packages. One way is to paint adjacent simplices1 with different colors to emphasize their positions, which can be vital especially in three-dimensional space. We meet a similar problem in domain decomposition methods, where adjacent subdomains are also painted by distinct colors. Let us point out that there are fast iteration methods that perform calculations on subdomains having the same color in parallel processors (see e.g. [257]). The number of processors has to be equal to an integer multiple of the maximum number of subdomains painted with the same color. We now highlight several standard definitions and statements from graph theory. A coloring of a partition Th is an assignment of colors to its simplices such that no two adjacent elements have the same color. An n-coloring of a partition Th uses n colors. A partition is said to be n-colorable if there exists a coloring of Th that uses n colors or fewer. Remark 3.2 It was not until 1976 that Kenneth Appel and Wolfgang Haken proved, with the help of computers, that every planar map (e.g. a political map) is 4-colorable (see [3]). In contrast to the coloring of a general map, it is very easy to find an algorithm for a 4-coloring of any planar triangulation. We can proceed, for instance, by induction. Assume we have a triangulation with k triangles. Remove an arbitrary triangle and assign a coloring to the remaining triangulation of k − 1 triangles. Then add the kth triangle again and color it differently than its (max. 3) neighbors. Now we show that the number of colors can be reduced from 4 to 3 for any triangulation (cf. Fig. 3.6). Our proof is constructive and thus it can also be used as a simple coloring algorithm [192]. The key point is the avoidance of colorings containing a triangle surrounded by three triangles already colored with three different colors. Theorem 3.3 Any planar triangulation is 3-colorable. Proof Let Th be a triangulation of a bounded polygon Ω consisting of k triangles T1 , T2 , . . . , Tk . First, number the triangles inductively as follows. Let M1 = Ω and let i successively increase from 1 to k. Choose an arbitrary Ti ∈ Th which has at least one side on the boundary ∂ Mi and then set Mi+1 = Mi \ Ti . We observe that Mk = Tk . Second, let i successively decrease from k to 1. Since each Ti has at most two neighbors with higher indices, we may assign to Ti any color different from its at most two neighbors. 1 Adjacent

d-simplices have a common (d − 1)-dimensional facet.

3.3 Coloring Simplicial Partitions

19

Fig. 3.6 A 3-coloring of a given triangulation

For example, we can define the color c(Ti ) of the ith triangle Ti by c(Ti ) = min(bi ),

(3.2)

where bi = {1, 2, 3} \ di , and di ⊂ {1, 2, 3} is the set of colors of those adjacent triangles of Ti ⊂ M i that were already colored. Remark 3.3 The function min in (3.2) can obviously be replaced by max, or in visual applications by rnd which (pseudo)randomly chooses an element from the set bi . Theorem 3.4 Any simplicial partition in Rd is (d + 1)-colorable and this number cannot be reduced, in general. For a constructive proof see [192]. Its main idea is similar to that of the proof of Theorem 3.3. Corollary 3.1 (The four color theorem for tetrahedral partitions) Any tetrahedral partition is 4-colorable.

20 Fig. 3.7 A 4-coloring of a given tetrahedral partition that cannot be reduced to a 3-coloring

Fig. 3.8 A 2-coloring of a given tetrahedral partition

3 Simplicial Partitions

3.3 Coloring Simplicial Partitions

21

Remark 3.4 Although any tetrahedral partition is 4-colorable, the associated graph is not planar, in general. Thus Corollary 3.1 is not a consequence of the classical four color theorem [3]. Remark 3.5 Figure 3.7 illustrates a partition of a tetrahedron which cannot be reduced to a 3-coloring. On the other hand, in Fig. 3.8 we see an example of a uniform tetrahedral partition which is only 2-colorable and whose associated graph is not planar.

3.4 The Basic Idea of the Finite Element Method A classical (smooth) solution of an elliptic boundary value problem may not exist if the domain in question has concave (i.e. re-entrant) corners or if some coefficients have jumps or if mixed boundary conditions are prescribed, etc. Therefore, the following concept of a weak solution is usually employed [215]. Let V be a Hilbert space. A mapping a(·, ·) : V × V → R is called a bilinear form if for any fixed v ∈ V the mappings a(v, ·) : V → R and a(·, v) : V → R are linear. Lemma 3.1 (Lax–Milgram lemma) Let V be a Hilbert space equipped with the norm · V and let F : V → R be a continuous linear form. Let a(·, ·) be a continuous bilinear form, i.e., there exists a constant C1 > 0 such that |a(v, w)| ≤ C1 vV wV ∀v, w ∈ V.

(3.3)

Further, assume that there exists a constant C2 > 0 such that a(v, v) ≥ C2 v2V ∀v ∈ V

(the V -ellipticity condition).

(3.4)

Then the problem: Find u ∈ V such that a(u, v) = F(v) ∀v ∈ V,

(3.5)

has exactly one solution. The proof can be found in [68, p. 8] and [215]. Problem (3.5) is usually called the variational formulation of an elliptic problem. A standard procedure for rewriting some boundary value problems in the form (3.5) is illustrated in the following example.

22

3 Simplicial Partitions

Example 3.1 Let Ω ⊂ Rd , d ∈ {1, 2, 3, . . . }, be a bounded polytopic domain with Lipschitz boundary ∂Ω. Consider the Poisson equation with homogeneous Dirichlet boundary conditions −Δu = f in Ω, u = 0 on ∂Ω,

(3.6) (3.7)

where f ∈ L 2 (Ω). We look for the solution u in the Sobolev space V = H01 (Ω) equipped with the norm · 1 . Multiplying (3.6) by an arbitrary function v ∈ V , integrating by parts, and using the boundary conditions (3.7), one transforms the above problem into problem (3.5) with corresponding bilinear and linear forms (see [68, 203] for details) grad v · grad v dx, F(v) = f v dx. (3.8) a(v, w) = Ω

Ω

It is easy to show that any classical solution of problem (3.6)–(3.7) is a weak solution and that the assumptions of the Lax–Milgram lemma 3.1 are satisfied [68, 203]. However, when Ω is for example L-shaped, a classical solution does not exist, in general, but the problem (3.6)–(3.7) has exactly one weak solution. Definition 3.4 A function u ∈ V satisfying (3.5) is said to be a weak solution of problem (3.6)–(3.7). Now let Vh ⊂ V be a non-empty finite-dimensional subspace and let us look for u h ∈ Vh so that (3.9) a(u h , vh ) = F(vh ) ∀vh ∈ Vh . By the Lax–Milgram lemma 3.1 we again observe that there is precisely one solution u h ∈ Vh . It is called the Galerkin solution and can be considered as a discrete approximation of u. We often assume that Vh consists of continuous piecewise polynomial functions over some partition of Ω into simplices, see [276, p. 103]. In this case u h is called the finite element solution. For computing the finite element solution of problem (3.6)–(3.7) we can choose, for instance, Vh = {v ∈ H01 (Ω) | v| S ∈ P1 (S) ∀S ∈ Th }, where P1 (S) stands for the space of linear polynomials over the simplex S. By [202, p. 27], a piecewise polynomial function is an element of H 1 (Ω) if and only if it is continuous. Seeking u h ∈ Vh as a linear combination of basis functions v 1 , . . . , v N of the space Vh (N = dim Vh ), N cjv j, uh = j=1

3.4 The Basic Idea of the Finite Element Method

23

we obtain by (3.9) the following system of linear algebraic equations for the unknowns c1 , . . . , c N ∈ R, N

a(v j , v i )c j = F(v i ), i = 1, . . . , N ,

(3.10)

j=1

whose matrix

N A = a(v i , v j )

i, j=1

is positive definite (i.e., ξ Aξ > 0 for all vectors ξ ∈ R N , ξ = 0) due to the V ellipticity condition (3.4). The solution u h is, of course, independent of the choice of basis functions. The main idea of the finite element method is that we may choose the basis functions v i having small supports. Then the matrix A is sparse, which means that only O(N ) entries are nonzero, in general. This enables us to solve the system (3.10) efficiently and to store A with much less computer memory than in the case of a full matrix with N 2 nonzero entries, in general. The true solution u is usually not known. Thus, a natural question arises: How can we estimate the difference u − u h ? One possibility is to use the following lemma (see [63, 64]). Lemma 3.2 (Céa’s lemma) Let the assumptions of the Lax–Milgram lemma 3.1 be satisfied. Then there exists a constant C > 0 such that for any subspace Vh ⊂ V , Vh = 0, we have (3.11) u − u h V ≤ C inf u − vh V , vh ∈Vh

where u and u h are the unique solutions of (3.5) and (3.9), respectively. Proof Let wh be an arbitrary element in Vh . Then by subtracting (3.9) from (3.5), we find that (3.12) a(u − u h , wh ) = 0. From this, the V -ellipticity (3.4), and the continuity condition (3.3) it follows that C2 u − u h 2V ≤ a(u − u h , u − u h ) = a(u − u h , u − u h ) + a(u − u h , u h − vh ) = a(u − u h , u − vh ) ≤ C1 u − u h V u − vh V . Hence, (3.11) holds with C = C1 /C2 .

The problem of estimating the discretization error u − u h V is, thus, by the simple inequality (3.11), reduced to the following problem in approximation theory: Evaluate the distance

24

3 Simplicial Partitions

d(u, Vh ) = inf u − vh V vh ∈Vh

between the function u ∈ V and the subspace Vh ⊂ V , see Fig. 1.1. Notice that this distance is independent of u h . Now we show how to estimate the above infimum. It turns out that this issue is directly related to the preservation of certain geometric characteristics (also called the regularity property, see Definition 3.3) of partitions during refinement steps. Further, we shall present the interpolation theorem. For simplicity we shall consider only linear elements. For u ∈ C(Ω) and Vh = {v ∈ H 1 (Ω) | v| S ∈ P1 (S) ∀S ∈ Th } we define the so-called Vh -interpolant πh u belonging to Vh such that πh u(x) = u(x) for all nodal points x of Th , i.e., all vertices of all simplices S from Th . The following typical interpolation error estimate in Sobolev norm (1.2) holds. Theorem 3.5 Let F = {Th } be a regular family of simplicial partitions of Ω and let u ∈ H 2 (Ω) ∩ C(Ω). Then there exist positive constants C and h 0 such that for any Th ∈ F with h ∈ (0, h 0 ) the following estimate holds u − πh u1 ≤ Ch|u|2 . For the proof see [202, p. 48]. This theorem can also be generalized to higher order finite elements (see [68, p. 124]). If u ∈ H 2 (Ω) ∩ C(Ω) is the weak solution of a second order elliptic problem, then the convergence of the corresponding finite element approximations u h follows directly from Céa’s lemma 3.2 and the above interpolation Theorem 3.5, since we can take vh = πh u. The infimum on the righthand side of Céa’s lemma can then be estimated by the interpolation error,

u − u h 1 ≤ C inf u − vh 1 ≤ Cu − πh u1 ≤ C h|u|2 vh ∈Vh

(3.13)

as h → 0. Remark 3.6 There are many interpolation and convergence results in various norms, see e.g. [11, 12, 68, 202, 212, 230, 248]. For a regular family of simplicial triangulations the finite element method converges without any additional smoothness assumptions than that the true solution u of a second order elliptic problem is an element of H 1 (Ω), see [68, p. 134], [202, p. 50], even though the convergence could be slower than O(h). In the next chapter we show how to reduce the regularity assumptions on families of simplicial partitions while still keeping the finite element approximations convergent.

Chapter 4

Angle Conditions

4.1 Zlámal’s Minimum Angle Condition The regularity of finite element partitions is important both in analysis and practical applications of the finite element method, as it greatly influences interpolation properties of finite element spaces and through Cea’s Lemma 3.2 also the convergence rate of the finite element method. The initial step in FEM implementations is to establish an appropriate partition (triangulation) of the solution domain. For a number of applications and complicated geometries, simplicial partitions are preferred over the others (e.g. block, hexahedral, prismatic, etc. partitions) due to their flexibility. However, from both theoretical and practical requirements, such partitions and their refinements cannot be constructed arbitrarily. Thus, first of all we must ensure, at least theoretically, that the finite element approximations converge to the exact (weak) solution of the mathematical model under consideration when the discretization parameter h of the associated partitions tends to zero [68]. Mainly due to this reason the concept of regular families of partitions appeared. Second, the regularity is also important for real-life computations, because partitions that contain large angles may yield ill-conditioned stiffness matrices [253, 282], which may lead, in turn, to a loss of accuracy and require more computer time. In 1968, Miloš Zlámal [306] (see also [107]) introduced the so-called minimum angle condition that ensures the convergence of finite element approximations for solving linear elliptic boundary value problems of the second and fourth order on planar triangulations. This condition requires the existence of a constant α0 > 0 such that the minimal angle αT of each triangle T in all triangulations involved satisfies the bound (see [68, p. 130]) αT ≥ α0

(Zl´amal’s condition).

To be more precise, Zlámal used the equivalent condition © Springer Nature Switzerland AG 2020 J. Brandts et al., Simplicial Partitions with Applications to the Finite Element Method, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-55677-8_4

25

26

4 Angle Conditions

sin αT ≥ sin α0 . The same condition was also introduced by Alexander Ženíšek [300] for the finite element method applied to a system of linear elasticity equations of second order. Later, the so-called inscribed ball condition was introduced, see e.g. [68, p. 124], which uses a ball contained in a given element (cf. (4.2)). Thus, it can also be used for nonsimplicial elements in any dimension. There we require that the ratio of the radius of the inscribed ball of any element and the diameter of this element must be bounded from below by a positive constant over all partitions from a given family. This condition has an elegant geometric interpretation: no element of any partition should degenerate to a hyperplane as the discretization parameter h tends to zero. This property is called the regularity of a family of partitions (cf. Definition 3.3). For a family of planar triangulations it is, obviously, equivalent to Zlámal’s condition. This condition has a large number of applications, see e.g. [68, 107, 130, 150, 203, 300]. In [220] the inscribed ball condition (4.2) was replaced by a simpler equivalent condition on the volume of every element (cf. (4.1)). Another equivalent circumscribed ball condition for simplices (cf. (4.3)) was first introduced in [50].

4.2 Volumic Regularity Conditions Recall that a simplex S in Rd , d ∈ {1, 2, 3, . . . }, is the convex hull of d + 1 vertices A0 , A1 , . . . , Ad that do not belong to the same (d − 1)-dimensional hyperplane, i.e., S = conv{A0 , A1 , . . . , Ad }. We denote by h S the length of the longest edge of S. Definition 3.2 for tetrahedral partitions can be naturally generalized for general simplicial partitions as follows. Definition 4.1 The set of simplicial partitions F = {Th }h→0 of Ω is called a family of simplicial partitions if for every ε > 0 there exists a Th ∈ F with h < ε. Throughout the chapter, all constants Ci are independent of S and h, but can depend on the dimension d. In what follows, we discuss the equivalence of three volumic regularity conditions used for simplicial partitions in Rd (cf. [47]). Condition 4.1 There exists a C1 > 0 such that for any Th ∈ F and any S ∈ Th we have (4.1) vold S ≥ C1 h dS . Condition 4.2 There exists a C2 > 0 such that for any Th ∈ F and any S ∈ Th we have (4.2) vold b ≥ C2 h dS ,

4.2 Volumic Regularity Conditions

27

where b ⊂ S is the inscribed ball of S. Condition 4.3 There exists a C3 > 0 such that for any Th ∈ F and any S ∈ Th we have (4.3) vold S ≥ C3 vold B, where B ⊃ S is the circumscribed ball about S. Lemma 4.1 For any simplex S and any of its facets Fi , i ∈ {0, 1, . . . , d}, we have vold S ≤ h dS ,

vold−1 Fi ≤ h d−1 S .

(4.4)

Proof Relations (4.4) follow from the fact that the distance between any two points of S is not larger than h S . Thus, S and any of its facets Fi are contained in hypercubes of the corresponding dimensions d or d − 1 with edges of length h S . We denote by r = r S and R = R S the radii of the inscribed and circumscribed ball of a given simplex S, respectively. Theorem 4.1 Conditions 1, 2, and 3 are equivalent. Proof (1) =⇒ (2): Let o be the center of the inscribed ball b of S. We decompose S into d + 1 subsimplices Si = conv{o, Fi }, i ∈ {0, 1, . . . , d}. All of them have the same height r above the facet Fi . Using the formula 1 r vold−1 Fi d

vold Si =

for the volume of each subsimplex Si , we find that r

d

vold−1 Fi = d

i=0

d

vold Si = d vold S.

i=0

Hence, by (4.4) and (4.1) we obtain ≥ d vold S ≥ C1 d h dS , r (d + 1)h d−1 S which implies that r≥

C1 d hS. d +1

From this and the formula for the volume of a d-dimensional ball (2.16), we finally get vold b = C6r d ≥ C6

C d d π d/2 1 h dS = C2 h dS , where C6 = . d +1 Γ ( d2 + 1)

(4.5)

28

4 Angle Conditions

(2) =⇒ (3): From (2.14), (2.13), (2.15), and (4.2) we find that |Δd | 1 Δd |Δd | R = = d+1 < d+1 2 Dd 2 (d!)2 (vold S)2 2 (d!)2 (vold b)2 2

≤

(d + 1)! h 2d+2 d +1 2 S = d+1 h . 2 d!C22 S 2d+1 (d!)2 C22 h 2d S

(4.6)

Thus, there exists a C7 > 0 such that for any S from any Th ∈ F we have R ≤ C7 h S .

(4.7)

Using (4.2) once again, (4.7), and (4.5), we immediately see that vold S > vold b ≥ C2 h dS ≥ C2

Rd C2 = d vold B, d C7 C7 C6

(4.8)

where C6 is given in (4.5). (3) =⇒ (1): Since B ⊃ S, we obtain 2R ≥ h S . Hence, in view of (4.3) and (4.5) we observe that C3 C6 d h , (4.9) vold S ≥ C3 vold B = C3 C6 R d ≥ 2d S which implies Condition 4.1 taking C1 =

C3 C6 . 2d

Definition 4.2 A family of simplicial partitions is called regular if Condition 4.1 or 2 or 3 holds. Note that Definitions 4.2 and 3.3 are equivalent for tetrahedral elements. Remark 4.1 For a triangle T with sides a, b, c we get by (2.15) and (4.6) ⎡

⎤ 0 a 2 c2 1 a 2 b2 c2 R2 = 3 2 det ⎣a 2 0 b2 ⎦ = 2 , 2 2 · 2 (vol2 T ) 4 (vol2 T )2 c2 b2 0 compare with (2.1). Similarly, for a tetrahedron we obtain (2.7) using (2.15) and (4.6) for d = 3. Remark 4.2 Condition 4.2 is a lower bound of the volume of the inscribed ball in terms of h. Similarly, Condition 4.3 represents an upper bound of the volume of the circumscribed ball in terms of h vold B ≤ C3 −1 h dS , which follows if we use (4.4) for (4.3).

4.2 Volumic Regularity Conditions

29

Remark 4.3 Condition 4.1 is, in practice, easier to verify than a condition that involves an inscribed or circumscribed ball, since it avoids their construction and computation altogether. Remark 4.4 Any of Conditions 1, 2, and 3 guarantees the optimal order of the interpolation error of simplicial finite elements, which is employed in various convergence proofs of the finite element method, see Sect. 3.4.

4.3 Minimum Angle Conditions in Higher Dimensions Now we present another notion of regularity for families of simplicial partitions, which is based on angle-type conditions. It directly generalizes Zlámal’s condition, see [48]. An appropriate definition for the d-dimensional sine of angles in Rd was introduced by Folke Eriksson in [89] (see also [17]). In terms of the simplex S, for any of its vertices Ai , the d-dimensional sine of the angle of S at Ai , denoted by Aˆ i , is defined as follows (see (3) in [89, p. 72]): sind ( Aˆ i |A0 A1 . . . Ad ) =

d d−1 (vold S)d−1 ,

(d − 1)! dj=0, j =i vold−1 F j

(4.10)

where F j is the facet opposite to A j . Remark 4.5 For d = 2, sin2 ( Aˆ i |A0 A1 A2 ) is the standard sine of the angle Aˆ i in the triangle A0 A1 A2 , due to the following well-known formula (see (2.1)), e.g. for i = 0, 1 (4.11) vol2 (A0 A1 A2 ) = |A0 A1 ||A0 A2 | sin Aˆ 0 . 2 In fact, one can similarly define a sine for any k-dimensional (vertex) angle of any k-dimensional facet of S for k ∈ {2, . . . , d}. Namely, let us denote the k-dimensional facet of S spanned by the k + 1 (distinct) vertices Ai0 , Ai1 , . . . , Aik by Fi0 ,i1 ,...,ik . Then for any index i ∈ {i 0 , . . . , i k } we set sink ( Aˆ i |Ai0 Ai1 . . . Aik ) =

k k−1 (volk Fi0 ,i1 ,...,ik )k−1 ,

i (k − 1)! volk−1 Fi0j,i1 ,...,ik

(4.12)

i j ∈{{i 0 ,...,i k }\{i }}

i

where Fi0ji1 ...,ik denotes the (k − 1)-dimensional facet of Fi0 i1 ...,ik (which is clearly itself a (k − 1)-dimensional simplex) lying opposite to the vertex Ai j . Remark 4.6 Notice that in the above notation we have S ≡ F0,1,...,d and F j ≡ j F0,1,...,d for j = 0, 1, . . . , d. The above definition of d-sine inspires the following:

30

4 Angle Conditions

Condition 4.4 There exists a C4 > 0 such that for any Th ∈ F and any S = conv{A0 , . . . , Ad } ∈ Th we have sind ( Aˆ i |A0 A1 . . . Ad ) ≥ C4 > 0

∀ i ∈ {0, 1, . . . , d},

(4.13)

where sind is defined in (4.10). From Remark 4.5 we observe that (4.13) really presents a generalization of Zlámal’s condition to higher dimensions, i.e. for d ≥ 3. First, we present some useful auxiliary results. Theorem 4.2 For any k ∈ {1, . . . , d}, for any simplex S and its k-dimensional facet Fi0 ,i1 ,...,ik , we have (4.14) volk Fi0 ,i1 ,...,ik ≤ h kS . The proof is similar to that of Lemma 4.1. Theorem 4.3 Let condition (4.1) hold for some family F. Then there exists a C8 > 0 such that for any lower dimensional facet Fi0 ,i1 ,...,ik of any simplex S from any Th ∈ F we have the following lower estimate volk Fi0 ,i1 ,...,ik ≥ C8 h kS .

(4.15)

Proof First, we demonstrate how to prove (4.15) for (d − 1)-dimensional facets F j . Write v j for the spatial altitude of the vertex A j to the face F j . Then, since v j ≤ h S , we observe that (i.e. for k = d − 1) C1 h dS ≤ vold S =

1 1 v j vold−1 F j ≤ h S vold−1 F j , d d

from which (4.15) follows immediately. By induction, we can easily prove (4.15) for the other (lower-dimensional) facets of S in the same manner. Theorem 4.4 Conditions 1 and 4 are equivalent in Rd for any d ≥ 2. Proof Condition 4.1 =⇒ Condition 4.4: Let i ∈ {0, 1, . . . , d} be given. Then from (4.10), (4.1), and (4.4) we immediately observe that d d−1 (vold S)d−1

(d − 1)! dj=0, j =i vold−1 F j d−1 d d−1 C1 h dS d d−1 C1d−1 = C4 > 0. ≥ = (d − 1)! (d − 1)! h (d−1)d S

sind ( Aˆ i |A0 A1 . . . Ad ) =

4.3 Minimum Angle Conditions in Higher Dimensions

31

Condition 4.4 =⇒ Condition 4.1: By [89, p. 76] and (4.13), we get for any i ∈ {i 0 , i 1 , i 2 } that sin2 ( Aˆ i |Ai0 Ai1 Ai2 ) ≥ sin3 ( Aˆ i |Ai0 Ai1 Ai2 Ai3 ) ≥ . . . ≥ sind ( Aˆ i |A0 . . . Ad ) ≥ C4 ,

(4.16)

where i 0 , i 1 , i 2 , i 3 , . . . are distinct indices from the set {0, 1, . . . , d}. From this and (4.12) we have for k = 2, 3, . . . , d the following relation (volk Fi0 ,i1 ,...,ik )k−1 (k − 1)! ij sink ( Aˆ i |Ai0 Ai1 . . . Aik ) = vol F k−1 i ,i ,...,i 0 1 k k k−1 i ∈{{i ,...,i }\{i }} j

(k − 1)! C4 ≥ k k−1

0

k

i

volk−1 Fi0j,i1 ,...,ik

(4.17)

i j ∈{{i 0 ,...,i k }\{i }}

for any i ∈ {i 0 , . . . , i k }. Now we apply (4.17) several times in order to estimate vold S from below by a product of some constant which only depends on k and the given constant C4 from (4.13) with the lengths of some d edges of S. Further, in order to finally prove (4.1), it remains to show that there exists a constant C8 > 0 such that for any simplex S from any Th ∈ F and any edge Am An of S we have |Am An | ≥ C8 h S . In order to show that, let us take any edge in S. Without loss of generality, let it be A0 A1 and assume that h S = |Ai A j | for some i and j. We consider first the triangle A0 A1 A2 , where by the Sine theorem from Sect. 2.1 one has |A0 A1 | ≥ |A0 A1 | sin Aˆ 0 = |A1 A2 | sin Aˆ 2 ≥ C4 |A1 A2 |. Further, we consider the triangle A1 A2 A3 , where we show that |A1 A2 | ≥ C4 |A2 A3 |, etc. Finally, using all the inequalities obtained, we get that |A0 A1 | ≥ C8 |Ai A j | = C8 h S , where C8 depends only on the dimension d and the given constant C4 . Recently, in [186] a more natural minimum angle condition was introduced: Condition 4.5 There exists a constant C5 > 0 such that for any partition Th ∈ F, any simplex S ∈ Th and any subsimplex S ⊂ S (2 ≤ dim S ≤ d) with vertex set contained in the vertex set of S, the minimum dihedral angle α S in S satisfies α S ≥ C5 . Also in [186] the following result was proved. Theorem 4.5 Conditions 4 and 5 are equivalent in Rd for any d ≥ 2.

(4.18)

32

4 Angle Conditions

Corollary 4.1 Conditions 1, 2, 3, 4, and 5 are equivalent for any d ≥ 2. Remark 4.7 A three-dimensional version of condition (4.18) was presented earlier in [46].

4.4 The Maximum Angle Conditions In the previous section we introduced a regularity concept which can roughly be described as a general requirement on simplicial elements to guarantee that their shape does not degenerate in the course of refinement (see [48, 68, 300, 306]). However, in practical calculations we sometimes produce simplicial elements which degenerate in various ways [65], see Fig. 4.1. Flat and narrow elements are also commonly used in covering thin slots, layers, gaps or strips of different materials (see Fig. 4.2) or to approximate functions that change more rapidly in one direction than in another direction [2]. Therefore, much effort has been spent in finding suitable (and practical) concepts, which are weaker than the Zlámal-type conditions. The first attempt in this direction was done by John L. Synge [275], however, without any application to the convergence of the finite element method. Already in 1957, he

NARROW TETRAHEDRA

spire / needle

splinter

spindle

spear

spike

FLAT TETRAHEDRA

wedge

spade

cap

Fig. 4.1 Classification of degenerated tetrahedra according to [65, 85]

sliver

4.4 The Maximum Angle Conditions

33

Fig. 4.2 Cross-section of a part of transformer windings. Thin isolation of copper wires can be covered by flat triangles but the optimal interpolation order O(h) will be still preserved

proved that linear triangular elements yield the optimal order of the interpolation error in the maximum norm under the so-called maximum angle condition: There exists a constant γ0 such that for any triangulation Th ∈ F and any triangle T ∈ Th we have γT ≤ γ0 < π

(Synge’s condition),

(4.19)

where γT is the maximum angle of the triangle T . This condition is weaker than the minimum angle condition, because one angle in triangles may tend to zero (cf. Fig. 4.5). Later, Pierre Jamet [149] proved optimal interpolation properties of triangular finite elements of kth polynomial degree in the W pm -Sobolev norm provided 1 < p ≤ ∞ and k + 1 − m >

2 p

and edges do not degenerate to one line. However, we immediately observe that the important case p = 2 and k = m, used often in convergence analysis of the finite element method, is not covered by the above inequalities (see also [164, p. 493]). This gap was resolved by Kenta Kobayashi and Takuya Tsuchiya, see [164]. The maximum angle condition for triangular elements was also investigated in terms of various norms in [4, 9, 15, 39, 163, 165, 190, 230]. To obtain optimal interpolation properties of linear tetrahedral elements one may impose (see [191]) the maximum angle condition for all triangular faces as well as a similar second condition for all dihedral angles between faces. Namely, there exists a constant γ0 < π such that for any tetrahedron T ∈ Th and any Th ∈ F we have γT ≤ γ0

and

ϕT ≤ γ0 ,

(4.20)

34

4 Angle Conditions

where γT is the maximum angle of all triangular faces of the tetrahedron T and ϕT is the maximum dihedral angle between faces of T . Now we show that these two conditions are independent. For the sliver (and also cap) tetrahedron from Fig. 4.1, the maximum angle condition γT ≤ γ0 holds for any triangular face, but some of the dihedral angles between faces may converge to π (cf. Example 4.3). On the other hand, for the spike tetrahedron the condition γT ≤ γ0 is violated for some faces, but all dihedral angles are less than some positive constant C < π for a certain manner of degeneracy (see Example 4.4 for more details). Further, a natural analogue of the maximum angle condition for prismatic finite elements has been recently proposed in [160]. Finally note that the maximum angle condition is, in fact, not necessary for convergence of finite element approximations, as shown in Examples 4.6 and 4.7 (see [131] for more details). The approach from [131] was later used by Václav Kuˇcera [210–212], and Peter Oswald [242]. Remark 4.8 Consider the standard cube-corner tetrahedron with vertices A0 = (0, 0, 0), A1 = (1, 0, 0), A2 = (0, 1, 0), and A3 = (0, 0, 1). Its solid (spatial) angle = π2 . Its 3-sine is by (4.10) equal at A0 , i.e., the solid angle of the first octant, is 4π 8 to 32 (1/6)2 = 1. sin3 ( Aˆ 0 |A0 A1 A2 A3 ) = 2!(1/2)3 For any d > 1 and a general hypercube-corner d-simplex S with vertices A0 = (0, . . . , 0), A1 = (1, 0, . . . , 0), . . . , Ad = (0, . . . , 0, 1), we, obviously, have vold S = 1/d! and vold−1 F j = 1/(d − 1)! for j = 1, . . . , d. Substituting these values into (4.10), a simple calculation also leads to sind ( Aˆ 0 |A0 A1 . . . Ad ) = 1. Remark 4.9 Denoting by A0 , A1 , A2 , A3 the vertices of the cap or sliver tetrahedron from Fig. 4.1 in an arbitrary way, we find by (4.10) that sin3 ( Aˆ 0 |A0 A1 A2 A3 ) → 0. Due to Corollary 4.1 and (4.13) we may present the following definition, which is equivalent to Definition 4.2. Definition 4.3 A family F is called a regular family of partitions of a polytope into simplices if there exists a C > 0 such that for any Th ∈ F and any S = conv{A0 , . . . , Ad } ∈ Th we have sind ( Aˆ i |A0 A1 . . . Ad ) ≥ C > 0 where sind is defined in (4.10).

∀ i ∈ {0, 1, . . . , d},

(4.21)

4.4 The Maximum Angle Conditions

35

Remark 4.10 From the previous section we know that the above definition is a natural d-dimensional generalization (see [48]) of Zlámal’s minimum angle condition, which is formulated as follows: There exists a constant α0 such that for any triangulation Th ∈ F and any triangle T ∈ Th we have αT ≥ α0 > 0,

(4.22)

where αT is the minimum angle of T . Clearly, sin αT ≥ sin α0 > 0, since sin is an increasing function over the interval [0, π/2]. Now we shall introduce a condition which is weaker than the regularity of a family of partitions. Definition 4.4 A family F is called a semiregular family of partitions of a polytope into simplices if there exists a C > 0 such that for any Th ∈ F and any S = conv{A0 , . . . , Ad } ∈ Th we can always find d edges of S which, when considered as vectors, constitute a (higher-dimensional) angle whose d-sine is bounded from below by the constant C. Remark 4.11 A construction of a semiregular family of tetrahedral partitions is presented in [187]. It differs from the family which is defined in the proof of Theorem 3.2 (e.g. for the Sommerville tetrahedron). A close relationship between the maximum angle condition (4.20) and the semiregularity property is analysed in [134]. Lemma 4.2 Any regular family of partitions of a polytope into simplices is semiregular. Proof For any simplex from the regular family of partitions the required d edges in the definition of semiregularity are just d edges emanating from some of its vertices. Remark 4.12 We observe that for the case d = 2, Definition 4.4 is equivalent to the maximum angle condition of Synge (4.19). To see this, we denote by Aˆ 0 the largest angle of T and then apply formula (4.11). We observe that in the family of triangulations which satisfies the maximum angle condition of Synge, we can have, in principle, an infinite sequence of triangles with decaying sizes whose minimum angles approach zero, which is not possible for Zlámal’s condition, see e.g. [190] for some examples of this, or Fig. 4.5. In 1965, Kang Feng [100, p. 206] proved the convergence of a finite difference scheme for elliptic problems under condition (4.19). Remark 4.13 The Synge condition guarantees the optimal interpolation order for triangles in the · 1,∞ -norm, which is not true if the maximum angle tends to π, see (4.44) and [131, 269]. Remark 4.14 Each d-simplex has d+1 = (d + 1)d/2 edges, i.e. their number 2 grows like the triangular numbers with dimension d. The d edges mentioned in

36

4 Angle Conditions

Definition 4.4 do not necessarily emanate from the same vertex. An example is a path-simplex with its d orthogonal edges forming a path in the sense of graph theory, see Fig. 5.7 (right). The path-simplex can degenerate (e.g. the d-dimensional sine of some of its angles can be close to zero and, therefore, the minimum angle condition does not hold) but the d-sine made by the orthogonal d edges stays the same. Remark 4.15 Consider a simplex S with vertices A0 , A1 , . . . , Ad . Let Ai denote any of its vertices. Then similarly as in (4.16) we have the following relations between sines of angles of various dimensions based on Ai : sin2 ( Aˆ i |Ai0 Ai1 Ai2 ) ≥ sin3 ( Aˆ i |Ai0 Ai1 Ai2 Ai3 ) ≥ · · · ≥ sind ( Aˆ i |A0 . . . Ad ), (4.23) where i 0 , i 1 , i 2 , i 3 , . . . are distinct indices from the set {0, 1, . . . , d} and i ∈ {i 0 , i 1 , i 2 }. By Remark 4.8 we see that the inequalities in (4.23) may become equalities e.g. for the hypercube-corner simplex. Lemma 4.3 Let F be a semiregular family of partitions of a polytope into simplices. Then there exists a constant C > 0 (depending on d only) such that for any Th ∈ F and any S ∈ Th we have (4.24) | det M S | > C, where the entries of the matrix M S are the scalar products ti t j , i, j = 1, . . . , d, of the unit vectors t1 , t2 , . . . , td parallel to those d edges of S which are required in Definition 4.4. Proof It is well known that the volume of the parallelotope P generated by the vectors t1 , t2 , . . . , td is equal to | det M|, where M = (ti t j )i,d j=1 . For convenience, we can suppose that these d vectors originate from the same vertex of the parallelotope. Let us denote it as Ai , where i ∈ {0, 1, . . . , d}, and let the d-simplex formed by Ai and ˜ Then the endpoints of these d vectors be denoted by S. d! vold S˜ = | det M S˜ | = vold P = | det M S |.

(4.25)

Let us denote the vertices of the simplex S˜ by A0 , A1 , . . . , Ad . From (4.10), we have ˜ d−1 = (d − 1)! (vold S) d d−1

d

vold−1 F˜ j · sind ( Aˆ i |A0 A1 . . . Ad ),

(4.26)

j=0, j =i

where F˜ j are facets of S˜ forming a d-dimensional angle at the vertex Ai . Further, due to the property of semiregularity, sind ( Aˆ i |A0 A1 . . . Ad ) is bounded from below by a positive constant, i.e.

4.4 The Maximum Angle Conditions

37 d

˜ d−1 ≥ C (d − 1)! (vold S) d d−1

vold−1 F˜ j ,

(4.27)

j=0, j =i

where C is a constant from the definition of semiregularity. Now, we notice that in (4.27) each term vold−1 F˜ j can be computed (and further estimated from below via a product of areas of relevant (d − 2)-dimensional facets) in a way similar to (4.26) and (4.27) using the semiregularity property and relations between sines of various dimensions (4.16). Finally, in order to prove (4.24), we apply the above arguments recursively to all involved areas of facets of dimensions d − 2, d − 3, . . . , 2 and notice at the end that after a finite number of steps the areas of one-dimensional facets are just t j = 1. Lemma 4.3 thus follows from (4.25) and (4.27) by induction. The next theorem gives the optimal interpolation order in the following maximum norm over S ∂w(x) ∂w(x) w1,∞,S = max max w(x), max , . . . , max x∈S x∈S x∈S ∂x1 ∂xd

(4.28)

for any Lipschitz function w. In what follows, we will also use the following matrix norm A1 = max |ai j |. j

i

The regularity assumption on F in Theorem 3.5 will now be replaced by a weaker semiregularity assumption. Theorem 4.6 Let F be a semiregular family of partitions of a polytope into simplices. Then there exists a constant C > 0 such that for any Th ∈ F and any S ∈ Th we have ∀v ∈ C 2 (S), (4.29) v − π S v1,∞,S ≤ Ch S |v|2,∞,S where π S is the standard Lagrange linear interpolant and h S = diam S. Proof We will present the proof from [133]. Consider a simplex S ∈ Th ∈ F. Let v ∈ C 2 (S) be arbitrary and let e be an arbitrary edge of S. Set w = v − π S v on S.

(4.30)

Since w = 0 at all vertices of S, there exists by Rolle’s theorem [249, p. 387] a point Q ∈ e such that (4.31) t grad w(Q) = 0, where t is the unit vector parallel to e. Set z = t grad w

(4.32)

38

4 Angle Conditions

in S. Let P be an arbitrary fixed point in the interior of S and let u be the unit vector parallel to Q P. Then by (4.31) and (4.32) we have z(Q) = 0 and P z(P) =

P

u grad z ds = Q

u (hes w)t ds,

(4.33)

Q

where hes w is the matrix of the second derivatives (Hessian) of w. Thus, by (4.32), (4.33), and (4.30), |t grad w(P)| ≤ dh S |w|2,∞,S = dh S |v|2,∞,S ,

(4.34)

where the upper estimate does not depend on P. Consider now those d edges of S required in Definition 4.4 and d corresponding unit vectors t1 , t2 , . . . , td parallel to them. Writing grad w(P) =

d

c j (P)t j , c j ∈ R,

(4.35)

j=1

we see that the coefficients c j = c j (P) fulfill the following linear system of algebraic equations d ti t j c j (P) = ti grad w(P), i = 1, 2, . . . , d, (4.36) j=1

with the Gram matrix M S = (ti t j )i,d j=1 . We can concisely write the above system as M S c(P) = b(P). Using standard definitions of vector and matrix norms and applying the constants from relevant equivalence relations, we observe from (4.35) and (4.36) that −1 grad w(P)∞ ≤ c(P)1 ≤ M−1 S 1 b(P)1 ≤ dM S 1 b(P)∞

(4.37)

holds for any point P. (In the above we also used the fact that t j 2 = 1 for j = 1, 2, . . . , d.) Let MS be the adjunct matrix of the entries of M S . Then M−1 S =

1 M , det M S S

and a straightforward calculation leads to the following estimation of the · 1 -norm of M S : (4.38) MS 1 ≤ d · (d − 1)! = d!. Hence, from (4.37), (4.38), and (4.34) we get

4.4 The Maximum Angle Conditions

39

Fig. 4.3 Three types of degenerating tetrahedra (wedge, needle, splinter) which do not deteriorate the optimal interpolation order [191]. The length h 2 can also be replaced by h 1+ε for any ε > 0

grad w(P)∞ ≤ dM−1 S 1 b(P)∞ ≤

d 2 d! h S |v|2,∞,S , | det M S |

(4.39)

which holds for any point P ∈ S. Now, using (4.39) and Lemma 4.3, we get that for any i = 1, . . . , d ∂w(x) max (4.40) ≤ Ch S |v|2,∞,S . x∈S ∂xi By [68, pp. 118–120], we have the estimate max w(x) ≤ Ch 2S |v|2,∞,S x∈S

2 ∀v ∈ W∞ (S)

(4.41)

without any regularity assumption on the family F. The relations (4.28), (4.40), and (4.41) now imply (4.29). Example 4.1 The needle, splinter, and wedge elements from Fig. 4.1 satisfy Definition 4.4. They yield the optimal interpolation order of linear elements provided the lengths of their edges are decaying as indicated e.g. in Fig. 4.3. Example 4.2 A higher-dimensional example can be constructed by taking positive numbers a1 , a2 , . . . , ad and forming a simplex with vertices A0 , A1 , . . . , Ad in the following way. We fix some number k such that 0 ≤ k ≤ d. The first k + 1 vertices of the simplex are defined as follows. Let A0 = (0, 0, . . . , 0, . . . , 0). Further, let A1 = (a1 , 0, . . . , 0, . . . , 0), A2 = (0, a2 , . . . , 0, . . . , 0), . . . , Ak = (0, . . . , 0, ak , 0, . . . , 0). The remaining vertices are: Ak+1 = (0, . . . , 0, ak+1 , 0, 0, . . . , 0), Ak+2 = (0, . . . , 0, ak+1 , ak+2 , 0, . . . , 0), Ak+3 = (0, . . . , 0, ak+1 , ak+2 , ak+3 , 0, . . . , 0), . . . ,

40

4 Angle Conditions

Ad = (0, . . . , 0, ak+1 , ak+2 , ak+3 , . . . , ad ). Therefore, for k = 0, we get the path-simplex, and for k = d the hypercube-corner simplex. Allowing some of the ak ’s to approach zero with different rates, in general, we arrive at various degenerated simplices still satisfying the semiregularity property. Now we show that for some degenerated tetrahedra we may lose the optimal interpolation order. Example 4.3 Assume that A0 = (h, 0, 0), A1 = (0, h 3 , h), A2 (−h, 0, 0), and A3 = (0, h 3 , −h) are the vertices of the tetrahedron T for some h ∈ (0, 1). From Fig. 4.1 we observe that T is a sliver tetrahedron when h → 0, i.e., the condition (4.24) is violated for any triple of different edges. We show that the optimal interpolation order of the linear interpolant πT is not guaranteed (cf. Theorem 4.6). Setting v(x1 , x2 , x3 ) = x12 , we immediately find that v(A0 ) + v(A2 ) = 2h 2 , v(A1 ) = v(A3 ) = 0. Using the linearity of πT v, we obtain ∂(πT v) (πT v)((A0 + A2 )/2) − (πT v)((A1 + A3 )/2) ∂ (v − πT v) = − = ∂x2 ∂x2 h3 v(A0 ) + v(A2 ) − v(A1 ) − v(A3 ) = = h −1 → ∞ as h → 0. 2h 3 Hence, the estimate (4.29) does not hold, since ∂ v − πT v1,∞,T ≥ (v − πT v) . ∂x2 0,∞,T Example 4.4 Assume that A0 = (0, 0, 0), A1 = (h, 0, 0), A2 = (h, 0, h 3 ), and A3 = (−h, h 3 , 0) are the vertices of the tetrahedron T for some h ∈ (0, 1). From Fig. 4.1 we see that T is a spike tetrahedron when h → 0. We again show that the optimal interpolation order of πT is violated, since (4.24) is not true. Putting v1 (x1 , x2 , x3 ) = x12 , we observe that v(A0 ) = 0, v(A1 ) = v(A2 ) = v(A3 ) = h 2 . Since πT v is linear, we deduce that

4.4 The Maximum Angle Conditions

41

∂ ∂(πT v) (πT v)(A0 ) − (πT v)((A1 + A3 )/2) (v − πT v) = − = 1 3 ∂x2 ∂x2 h 2 =−

v(A1 ) + v(A3 ) 2h 2 = − → −∞ h3 h3

whenever h → 0. Consequently, the estimate (4.29) is not valid. Recently, in [161] another maximum angle condition was introduced, which presents a natural generalization of conditions (4.19) and (4.20). Definition 4.5 A family F of partitions of a polytope into simplices satisfies the ddimensional maximum angle condition if there exists a constant C7 < π such that for any partition Th ∈ F, any simplex S ∈ Th and any subsimplex S ⊂ S (2 ≤ dim S ≤ d) with vertex set contained in the vertex set of S, the maximum dihedral angle γ S in S satisfies (4.42) γ S ≤ C7 . The following result was proved in [161]. Theorem 4.7 Definitions 4.4 and 4.5 are equivalent in Rd for any d ≥ 2. Remark 4.16 In [161] it is also proved that the condition of Definition 4.5 is equivalent to the condition of Jamet for simplices from [149]. In [11], Natalia V. Baidakova also proposed an angle condition which is equivalent to the condition of Jamet. Another (stronger) angle condition for simplices was proposed by Yuri N. Subbotin in [274]. See also the doctoral dissertation of Baidakova [12] for a comprehensive review of various interpolation results and associated conditions on simplicial partitions until 2018.

4.5 The FEM on Highly Distorted Partitions In [9, p. 223], [269, p. 138], and [300, p. 365] there are examples showing that if the maximum angle condition (4.19) does not hold then the linear triangular finite elements lose their optimal interpolation order. Their main idea is illustrated in the following example. Example 4.5 Take an arbitrary ε > 0 and the triangle T = T (ε) with vertices A0 = (−1, 0), A1 = (1, 0), and A2 = (0, ε) (see Fig. 4.4). Hence, the largest angle γT → π when ε → 0. Consider the function v(x1 , x2 ) = x12 and its linear interpolant (πT v)(x1 , x2 ) = −

x2 + 1 on T, ε

i.e., (πT v)(Ai ) = v(Ai ), i = 0, 1, 2.

(4.43)

42

4 Angle Conditions

x2

Fig. 4.4 Degenerating triangle with the largest angle γT → π as ε → 0

A2 A0

A1 x1

Using the standard Sobolev space notation, (4.43), and the fact

∂v ∂x2

= 0, we find that

∂π v 2 1 1 T v − πT v21,T ≥ = 2 vol2 T = → ∞ as ε → 0. ∂x2 0,T ε ε

(4.44)

We conclude that one badly shaped triangle in any triangulation Th ∈ F can yield an arbitrary large interpolation error in the Sobolev H 1 -norm as h → 0, and also in 1 -norm, due to the imbedding L ∞ (T ) ⊂ L 2 (T ). the W∞ In the case of tetrahedral elements similar discussions are given in Examples 4.3 and 4.4. Namely, if the maximal angle between two faces or the maximal angle between edges (within triangular faces) tends to π, then the interpolation error may tend to ∞ like in (4.44). Calculations analogous to (4.43)–(4.44) caused numerical analysts to believe that large angles of triangular elements (i.e., when the maximum angle condition (4.19) is not satisfied) produce a large discretization error when solving second order elliptic problems by the finite element method. For instance, Ivo Babuška and Kadir Aziz [9] state that the maximum angle condition (4.19) is essential for convergence of the finite element method, whereas Eduardo D’Azevedo and Bruce Simpson [79, p. 1063] assert that (4.19) is necessary and sufficient for convergence. However, in this section we show that the finite element method may converge even when (4.19) is violated for a quite large number of elements in the used partitions. Let us emphasize that Céa’s Lemma 3.2 gives only an upper bound for the discretization error by means of the interpolation error. Note that the discretization error can be, in some cases, of the same order as the interpolation error (see e.g. the left graph in Fig. 4.7). This was proved, for example, for uniform triangulations that satisfy the minimum angle condition (4.22) for a second order elliptic equation with smooth variable coefficients (see [207]). But in principle, the discretization error can also be much smaller than the interpolation error, as we will demonstrate later (see the right of Fig. 4.7). The fact that the maximum angle condition is not necessary was discovered by Antti Hannukainen during numerical tests (see [131]). We will introduce his main idea. Keeping in mind the result (4.44), we now show that the discretization error can be very small, whereas the interpolation error is large. For simplicity, consider

4.5 The FEM on Highly Distorted Partitions

43

Fig. 4.5 Family F1 satisfying the maximum angle condition. This is the so-called blue refinement, see [168, p. 122]

Fig. 4.6 Family F2 that does not satisfy the maximum angle condition

the Poisson equation with the homogeneous Dirichlet boundary conditions in Ω = (0, 1) × (0, 1), − Δu = f in Ω, u = 0 on ∂Ω, (4.45) where f ∈ L 2 (Ω). Since Ω is convex, its weak solution is from the space H 2 (Ω) and thus continuous by the Sobolev imbedding theorem (1.1). Example 4.6 We will define two special families F1 and F2 of triangulations of Ω. To this end we first introduce uniform rectangular meshes of the given unit square consisting of congruent rectangles. Its horizontal sides will be divided into 2k equal parts and the vertical sides will be divided into 4k equal parts for k = 0, 1, 2, . . . To construct the family F1 we divide each rectangle by its diagonal with a positive slope (see Fig. 4.5), whereas for the family F2 we take both diagonals (see Fig. 4.6). We observe that the first family F1 satisfies the maximum angle condition (4.19) with γ0 = π/2 for all k, whereas for the second family F2 we observe that γT → π for half of the triangles T from any Th ∈ F2 . Let Vh and Wh be finite element spaces of continuous piecewise linear functions over triangulations from F1 and F2 , respectively, vanishing on ∂Ω. Obviously, Vh ⊂ Wh .

(4.46)

Denote by u h ∈ Wh the standard finite element approximation of the weak solution u ∈ H 2 (Ω) of (4.45) uniquely defined by the relation (grad uh , grad wh )0 = (f, wh )0

∀wh ∈ Wh .

44

4 Angle Conditions 0

1

10

10

FEM Interpolation

0

10 −1

10

−1

10

FEM Interpolation

−2

10

−2

10

−1

−2

0

10

10

10

−2

−1

10

10

0

10

Fig. 4.7 The practical convergence rates for the families F1 (left) and F2 (right) from Example 4.6. The horizontal axis corresponds to the discretization parameter and the vertical axis corresponds to the H 1 -norm of the discretization and interpolation errors. The difference between interpolation and discretization errors on the left figure is extremely small, which cannot be seen from the graph

Let πh u stand for the linear interpolant of u in Vh . Then by Céa’s Lemma 3.2 and (4.46) there exists a constant C > 0 such that u − u h 1 ≤ C inf u − wh 1 ≤ C inf u − vh 1 wh ∈Wh

vh ∈Vh

≤ Cu − πh u1 ≤ C h|u|2 as h → 0,

(4.47)

where the interpolation error in the last inequality is bounded due to the maximum angle condition (4.19) valid for F1 and the constant C > 0 is independent of the discretization parameter h (see [9, 149, 190]). This example shows that the discretization error tends to 0 at least linearly in the H 1 -norm even though the maximal angle of every second triangle from any Th ∈ F2 tends to π. In Fig. 4.7 we present the practical rates of convergence on F1 and F2 for problem (4.45) with the following right-hand side f (x1 , x2 ) = π 2 sin πx1 sin πx2 . Example 4.7 Another supportive example is illustrated by Figs. 4.8 and 4.9. In this case, the family F3 satisfies even the minimum angle condition (4.22) and the maximal angle of every third triangle from any Th ∈ F4 tends to π (see Fig. 4.9). We can define finite element spaces Vh and Wh over triangulations from F3 and F4 as in the previous example so that Vh ⊂ Wh , and derive (4.47) again. Some generalizations of the previous results to arbitrary space dimension can be found in [131]. We refer also to the following papers [210–212, 242] which deal with similar topics.

4.5 The FEM on Highly Distorted Partitions

45

Fig. 4.8 Family F3 satisfying the minimum angle condition

Fig. 4.9 Family F4 that does not satisfy the maximum angle condition

Václav Kuˇcera [210, p. 134] proves that a given triangle whose angles are not greater than 120◦ cannot be partitioned into degenerating triangles all having their maximum angles arbitrarily close to π. Hence, the construction from [131] cannot yield triangulations containing only degenerate triangles (see Figs. 4.6 and 4.9). Now we will generalize this surprising result into the three-dimensional space. To this end recall that (see [249, p. 83]) ε = α1 + α2 + α3 − π

(4.48)

is said to be the spherical excess of a spherical triangle with angles α1 , α2 , α3 . It is measured in steradians and is also called the solid (or trihedral) angle. Lemma 4.4 The sum of all four solid angles of an arbitrary tetrahedron is less than 2π. Proof Let α1 , . . . , α6 be dihedral angles of a given tetrahedron and let ε1 , . . . , ε4 be its solid angles. Then by formula (4.48) for the spherical excess we have 4 i=1

εi = 2

6

α j − 4π < 6π − 4π = 2π,

j=1

since the sum of all dihedral angles is less than 3π, see [111].

From Fig. 4.1 we see that the upper bound 2π given in Lemma 4.4 cannot be reduced for the cap tetrahedron. Theorem 4.8 An arbitrary tetrahedron cannot be partitioned into only degenerating tetrahedra, whose maximum solid angles all tend to 2π.

46

4 Angle Conditions

Proof Assume to the contrary that such a partition of a given tetrahedron T exists. Denote by t the number of tetrahedra in this partition. Then the sum of all solid angles around all vertices is equal to 4πv I + 2πv B + s < 2πt,

(4.49)

where v I is the number of vertices in the interior of T , v B is the number of vertices in the interior of faces of T , s is the sum of the other solid angles at vertices that lie on edges of T , and the sharp inequality follows from Lemma 4.4. Since each interior vertex can be the vertex of at most two large solid angles and each boundary vertex can be the vertex of at most one large solid angle, we have t ≤ 2v I + v B . However, from (4.49) we get the opposite inequality 2v I + v B < t.

4.6 Superconvergence and Post-processing In 1966, Ivo Babuška, Milan Práger, and Emil Vitásek (see [10, Sect. 4.3]) developed a special finite difference scheme for the equation −( pu ) + qu = f in (0, 1) with mixed boundary conditions and smooth data p, q and f . Using the Marchuk identities and sophisticated numerical quadrature rules, they obtained the accuracy O(h 6 ) as h → 0 at uniformly distributed nodal points. The associated system of linear algebraic equations has only a tridiagonal matrix like for linear finite elements with accuracy O(h 2 ). During the development of the finite element method it has been found that the rate of convergence of finite element approximations at some exceptional points of a domain exceeds the optimal global rate if finite element partitions have some regular geometric structure. This phenomenon has come to be known as superconvergence. For instance, let Ω be an equilateral triangle and let all its triangulations consist of identical equilateral triangles. Then for a sufficiently smooth solution of the problem (4.45) we obtain O(h 4 ) superconvergence at nodal points (see [27]), whereas the optimal global rate is only O(h 2 ), see [68]. Unfortunately, this strong result cannot be generalized to regular tetrahedra due to Theorem 5.4 (see also Fig. 5.3). Special points of exceptionally high accuracy of the derivatives of finite element approximations

4.6 Superconvergence and Post-processing

47

Fig. 4.10 Triangulations exhibiting superconvergence phenomena: uniform, piecewise uniform, quasiuniform, locally symmetric, locally periodic and self-similar

of linear elliptic boundary value problems have also been observed, e.g. in [51, 193, 200, 219, 309, 310]. For nonlinear problems, see e.g. [223, 294]. A systematic study of superconvergence phenomena seems to have its beginning in the seventies. Great effort has been focused on the superconvergence at nodal points, but also the Gauss–Legendre, Jacobi, and Lobatto points, see [201]. However, later the term superconvergence was used in a much broader sense than before. One can recover the finite element solution or its derivatives by means of various postprocessing techniques and produce an acceleration of convergence. This is also called superconvergence, if the post-processing is easily computable, see e.g. (4.53) and [50, 200]. At present superconvergence of the finite element method is a quickly and dynamically developing field of research. Since the seventies thousands of papers have been written on this subject, see e.g. [204]. Superconvergence phenomena arise when solving differential and integral equations, variational inequalities and other problems of mathematical physics by numerical methods; in particular, by Galerkin and finite element methods, finite difference methods, collocation methods, finite volume methods, boundary element methods, box methods, etc. Great progress has been made especially in producing superconvergence meshes (i.e. locally symmetric, see Fig. 4.11), in introducing many post-processing methods leading to superconvergence and the use of various superconvergence phenomena for a posteriori error estimation [205]. A key assumption in proving many superconvergence phenomena is a high regularity of the exact solution and also some regular structure of the partitions used – uniform, piecewise uniform, quasiuniform, locally periodic, locally point-symmetric, self-similar, etc., see Fig. 4.10. In what follows we shall be dealing only with superconvergence of finite element approximations of elliptic partial differential equations over simplical partitions. Assume that the finite element space Vh consists of continuous piecewise linear functions. In 1969, Leonard A. Oganesjan and Leonid A. Ruhovec [238] proved a

48

4 Angle Conditions

Fig. 4.11 Point-symmetric patches of tetrahedral elements around edges

remarkable approximation phenomenon for the problem (4.45) over regular families of uniform partitions (i.e., when any two adjacent triangles form a parallelogram) of a rectangle Ω, namely that |u h − πh u|1 ≤ Ch 2 |u|3 ,

(4.50)

where πh is the standard linear Lagrange interpolation operator. Later, this phenomenon was called supercloseness, since |u − u h |1 ≤ Ch|u|2 and |u − πh u|1 ≤ Ch|u|2 are the optimal error estimates, which cannot be improved, in general. It is well known that the above seminorm | · |1 is equivalent to the norm · 1 due to Friedrichs’ inequality v1 ≤ C|v|1 ∀v ∈ H01 (Ω). Note that Oganesjan and Ruhovec proved the convergence of the finite element method for isosceles triangles without using Céa’s Lemma 3.2 and (3.13). They used the triangle inequality u − u h 1 ≤ u − πh u1 + πh u − u h 1 , where the first term on the right-hand side was estimated by the Interpolation Theorem 3.5 and the second term by the supercloseness property (4.50). The estimate (4.50) is at the basis of many superconvergence phenomena. Now we will present its generalization to an arbitrary space dimension for the Poisson problem (4.45) solved by linear elements. The corresponding bilinear form a(·, ·) is then the scalar product and (3.12) is called the orthogonality condition, cf. Fig. 1.1. Hence, the difference u h − πh u can be studied as follows. Assuming that u is smooth enough for πh u to be well defined, we get

4.6 Superconvergence and Post-processing

49

grad(u h − πh u)20 = (grad u h − grad(πh u), grad(u h − πh u))0 = (grad u − grad(πh u), grad(u h − πh u))0 .

(4.51)

Moreover, we assume that each partition Th is uniform, which means that it satisfies the following uniformity conditions: (U1) There exists d linearly independent vectors χ1 , . . . , χd such that for each S ∈ Th and each j ∈ {1, . . . , d} the simplex S has an edge parallel to χ j . (U2) For each internal edge e ⊂ ∂Ω in one of the directions χ1 , . . . , χd , the patch Pe of simplices sharing e is point symmetric with respect to the midpoint M of e, by which we mean that x ∈ Pe ⇔ 2M − x ∈ Pe for all x ∈ Pe (see Fig. 4.11). One can show after some puzzling that (U2)⇒(U1) for d ≤ 3. It is unclear, though not unlikely, that the same implication holds also for d > 3. The converse implication (U1)⇒(U2) only holds in the trivial case d = 1. Theorem 4.9 For a regular family of uniform simplical partitions we have |(grad u − grad(πh u), grad vh )0 | ≤ Ch 2 |u|3 |grad vh |0 ∀vh ∈ Vh ,

3/4

1/2

3/2

0 1/4

1/4

0

0

1/2

1/2

1/2

1/4

−1/4 0

3/4

−1/2

−1/2 1/2

−1/4

0

−1/2

(4.52)

1/2

1/6

1/4 1/6

1/6

1/6

1/6 1/6

Fig. 4.12 Weights of the averaged gradient of linear triangular elements at boundary and interior nodes

50

4 Angle Conditions

where u ∈ H s (Ω) ∩ H01 (Ω) with s = 3 if d ≤ 5 and s > d/2 if d ≥ 6. A proof is given in [50]. According to the Sobolev imbedding (1.1), the condition s > d/2 assures that u has a representation as a continuous function, and thus πh u is well defined. The condition s ≥ 3 is sufficient to get the factor h 2 in (4.52). Setting vh = u h − πh u in (4.52), we find by (4.51) that (cf. (4.50)) |u h − πh u|1 = grad(u h − πh u)0 ≤ Ch 2 |u|3 . This inequality is a key tool for proving many superconvergence phenomena. The main trick lies in the following triangle inequality grad u − G h (u h )0 ≤ grad u − G h (πh u)0 + G h (πh u − u h )0 , where G h is a suitable post-processing operator that makes both terms on the righthand side of order O(h 2 ). Functions vh ∈ Vh are piecewise linear, so their gradient is piecewise constant. This enables us to define a continuous piecewise linear vector function G h (vh ), called the averaged gradient, such that (G h (vh ))(x) =

1 grad vh | S , (d + 1)! S∈T

(4.53)

h S∩x =∅

where x is any interior node of Th . For the definition of G h (vh ) at boundary nodes we refer, e.g., to [139], cf. also Fig. 4.12. Various averaging techniques producing superconvergence at midpoints of edges can be found in [205], cf. also Fig. 4.11.

Chapter 5

Nonobtuse Simplicial Partitions

5.1 Preliminaries Nonobtuse simplicial partitions play an important role in many areas of science and engineering, as we shall see in Sects. 5.4 and 5.5. For instance, when solving the Laplace or Poisson equation by the finite element method, they yield irreducible and diagonally dominant stiffness matrices when the discretization parameter is small enough and guarantee the validity of the discrete maximum principle (see Sects. 5.4 and 9.3). Consider now a simplex in the Euclidean space Rd , d ∈ {1, 2, 3, . . . }. Opposite each vertex lies a (d − 1)-dimensional facet. For d = 1 facets are just points. For d ≥ 1 the dihedral angle α between two facets is defined by means of the inner product of their outward unit normals n 1 and n 2 , cos α = −n 1 · n 2 .

(5.1)

◦ If d = 1 these normals d+1 necessarily make an angle of 180 and thus α = 0. Each d simplex in R has 2 dihedral angles. If all dihedral angles of a given simplex are less than 90◦ (less than or equal to 90◦ ) we say that the simplex is acute (nonobtuse). Otherwise we say that the simplex is nonacute (obtuse). Let Ω ⊂ Rd be a domain. If the boundary of the closure ∂Ω of Ω is contained in a finite number of (d − 1)-dimensional hyperplanes, we say that Ω is polytopic. If, in addition, Ω is bounded, it is called a polytope; in particular, Ω is called a polygon for d = 2 and a polyhedron for d = 3.

Theorem 5.1 Convex compact sets that tile the space Rd are convex polytopes. For the proof see [255, p. 902]. The simplest convex polytope is a simplex. Now we generalize Definition 3.1 to unbounded polytopic domains.

© Springer Nature Switzerland AG 2020 J. Brandts et al., Simplicial Partitions with Applications to the Finite Element Method, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-55677-8_5

51

52

5 Nonobtuse Simplicial Partitions

Definition 5.1 By a partition (or triangulation) of a closed polytopic domain Ω we mean a set of simplices whose union is Ω, any two simplices have disjoint interiors, and each facet of a simplex is a facet of another simplex from the partition, or belongs to the boundary ∂Ω. Moreover, we assume that the set of vertices of all simplices does not have an accumulation point in Rd . Hence, for a bounded domain the number of simplices in a partition is finite. For an unbounded domain it is, of course, infinite. A partition is called acute (nonobtuse) if all its simplices are acute (nonobtuse). In contrast to Definition 3.1, the above definition allows us to consider simplicial partitions (face-to-face tilings) of the whole space Rd as well. Theorem 5.2 Let m be a positive integer. There does not exist a partition of R3 such that each edge is surrounded by exactly m tetrahedra. For the proof, see [193, p. 49].

5.2 Acute Partitions First we present a basic property of acute simplices discovered by Miroslav Fiedler (see e.g. [103], [105, p. 110], [43]): Theorem 5.3 Let d > 2. If a d-simplex is acute, then each of its facets is an acute (d − 1)-simplex. Remark 5.1 The converse implication does not hold. Indeed, the tetrahedron with vertices A = (−1, 0, 0),

B = (1, 0, 0), C = (0, −1, 1/2),

D = (0, 1, 1/2)

has congruent acute triangular faces, but the dihedral angles at the edges AB and C D are obtuse. Choosing C = (0, −1, 1) and D = (0, 1, 1), we get the so-called Sommerville tetrahedron (no. 1 according to the classification in [117]) with acute triangular faces and right dihedral angles at the edges AB and C D. The other dihedral angles are 60◦, see Fig. 3.5. Figure 5.1 presents a triangulation of an obtuse triangle and a square into seven and eight acute triangles, respectively, see the paper by Martin Gardner [113]. It was shown in [221, 229] that these numbers are minimal. Later, Charles Cassidy and Graham Lord [62] proved that for any n ≥ 10 there exists a triangulation of a square into n acute triangles. They also showed why for n = 9 such a triangulation does not exist. Each triangle and also each (possibly non-convex) quadrangle is a plane-filler. Using this fact we can easily construct acute “periodic” triangulations of the plane. Dividing the Penrose rhombic tiles into two isosceles triangles (see Roger Penrose [243]), we can generate acute “aperiodic” triangulations with maximal angle 72◦ , i.e.,

5.2 Acute Partitions

53

Fig. 5.1 Partition of an obtuse triangle and square into acute triangles

no such tiling is periodic [122]. Yuri D. Burago and Victor A. Zalgaller in [56] give an algorithm which constructs an acute triangulation for an arbitrary polygon. Joseph L. Gerver [115] presents an algorithm that enables us to decompose special polygons into “almost equilateral triangles” with maximal angle 72◦ . Hiroshi Maehara [228] proves that every n-gon can be triangulated into O(n) acute triangles (see also [298]). More details are provided in Tudor Zamfirescu’s short survey on acute triangulations [299]. It is obvious that in acute (nonobtuse) triangulations of planar domains, each inner vertex is surrounded by at least five (four) triangles (cf. Fig. 5.1). The ratio between the corresponding numbers in R3 is much bigger, namely 20:8. In [208, p. 165] it was proved that (see also Theorem 6.1): Theorem 5.4 In any acute (nonobtuse) partition of a polyhedral domain each inner vertex is surrounded by at least twenty (eight) tetrahedra. To show that these numbers are attainable, consider the classical Platonic bodies (being the regular tetrahedron, the cube, the regular octahedron, the regular dodecahedron, and the regular icosahedron). The regular icosahedron and the regular octahedron (depicted in Fig. 5.2) can be partitioned into twenty acute (eight nonobtuse) tetrahedra by taking the twenty (eight) convex hulls of the center of gravity with each of the triangular faces. Remark 5.2 For d = 4, the above ratio seems to be 600:16 (see Conjecture 5.1). In fact, around 1852, Ludwig Schläfli [254] studied regular polytopes in R4 , in particular the regular 600-cell and 16-cell (also called 4-orthoplex). Their three-dimensional surfaces are formed by congruent regular tetrahedra (see for instance [260, 266]) whose convex hulls with the center of gravity G of the regular polytope form 600 acute and 16 nonobtuse 4-simplices surrounding G, respectively. It is easy to verify that a vertex in R4 cannot be surrounded by less than 16 nonobtuse simplices, but the situation with acute simplices is much more difficult (cf. Theorem 5.4). Conjecture 5.1 Each vertex in an acute partition of R4 is surrounded by at least 600 simplices.

54

5 Nonobtuse Simplicial Partitions

Fig. 5.2 The regular octahedron and icosahedron

For d ≥ 5 the situation is very different, see Theorem 5.11 further on. Generating acute partitions in R3 is much harder than in R2 , as can be understood from the following open problem. Conjecture 5.2 Each tetrahedron can be partitioned into acute tetrahedra. Aristotle in his treatise On the Heavens (350 BC) incorrectly conjectured that the regular tetrahedron is a space-filler [5, Vol. 3, Chapt. 8]. This would require the dihedral angle between its faces to be equal to 72◦ . Since Aristotle was a recognized person, nobody doubted his statement. Only in the Middle Ages was it realized that he was mistaken (see Fig. 5.3). All dihedral angles of the regular tetrahedron are equal to arccos 13 which, rounded to entire degrees, gives 71◦ . Also, Averroës (1126– 1198) calculated [271, p. 127] that the length of each edge of the regular icosahedron, inscribed in the unit ball, is √ 1 10(5 − 5) ≈ 1.05, 5 which does not equal one, as would follow from Aristotle’s conjecture. Theorem 5.5 The regular tetrahedron does not tile R3 . An algorithm for partitioning R3 into acute tetrahedra was given in 2001, when Alper Üngör [283] published the following result. Theorem 5.6 There exists an acute partition of R3 . The proof of this theorem is constructive. It is based on the partition of the regular icosahedron (see Fig. 5.2) into twenty acute tetrahedra, as discussed already above. The projection of the regular icosahedron onto the plane on which it stands (on one of its triangular faces), is a regular hexagon, which is a plane-filler. Congruent regular

5.2 Acute Partitions

55

Fig. 5.3 The regular tetrahedron is not a space-filler. To a given face of the regular tetrahedron we may join face-to-face another regular tetrahedron in a unique way. Repeating this process, five regular tetrahedra may surround a common edge, but a small gap will appear, since all dihedral angles are approximately 71◦ only

icosahedra may thus be placed into a spatial regular lattice such that two neighboring icosahedra have a common edge, but not a common face. The remaining gaps can be partitioned by four different types of acute tetrahedra (see [283]). Remark 5.3 David Eppstein, John M. Sullivan, and Alper Üngör [88] introduced four more algorithms for generating acute partitions of R3 which use information about the position of atoms in crystals of zeolites. The volume of these minerals increases with pressure due to a special dense arrangement of atoms, such as in silicon dioxide. Their chemical structure was studied already in 1958 (see [108]). Half a century later they inspired mathematicians to construct acute partitions of R3 . The main idea is the following: Denote the centers of the particular atoms by A1 , A2 , . . . For each i ∈ {1, 2, . . . } define the corresponding Voronoï cells (also called Dirichlet regions [256]) in Rd . Properties of these convex polytopes (see [49]) were studied by Georgy Voronoï (1868–1908) in [293], though they were already defined much earlier by René Descartes in 1644 and also by Peter Gustav Lejeune Dirichlet (1805–1859) as Vi = {x ∈ Rd | x − Ai ≤ x − A j for all j = 1, 2, . . . }, where · denotes the Euclidean norm (see Fig. 5.4). The set Vi thus contains all points x ∈ Rd whose distance from Ai is less than or equal to the distances to each of the other points A j . It is possible to associate a so-called Delaunay triangulation (see Definition 5.2 below) to the face-to-face division into Voronoï cells, which is the required acute triangulation. Its vertices are {A1 , A2 , . . . } and each edge is surrounded by either five or six tetrahedra when d = 3.

56

5 Nonobtuse Simplicial Partitions

A9 A10

A8 A3

A11 A4

A17

A2 A7

A1 A12 A5 A13

A6

A14

A16

A15

Fig. 5.4 Voronoï cells in R2 are indicated by broken lines. Notice that the vertices of the Delaunay triangulation are not contained in the interiors of the circumscribed balls about the particular triangles

Theorem 5.7 Any Platonic body can be partitioned into acute tetrahedra. For the proof, see [166]. No more than 1370 acute tetrahedra are needed to partition a cube (see [286]). However, it is not known whether this number can be reduced. On the other hand, acute simplicial partitions of the d-cube do not exist for d ≥ 4. For the proof see [166]. Definition 5.2 Let D ⊂ Rd be the set of all vertices of all simplices from a given triangulation. If the interiors of the circumscribed balls about any simplex from the triangulation do not contain points from D (see Fig. 5.4), then the triangulation is said to be Delaunay. In the pioneering paper Sur la sphère vide (see [82]) by Boris Delone1 it was shown that for any finite number of points that do not all belong to the same hyperplane, such a triangulation exists. Moreover, if no d + 2 points from D lie on the surface of a ddimensional ball, the Delaunay triangulation is determined uniquely (cf. Remark 5.4). There also exist other constructive definitions of Delaunay triangulations in [83, 233, 239, 267]. Some authors say that the Delaunay triangulation is dual to a division into Voronoï cells (see Fig. 5.4). Remark 5.4 The four vertices of a square lie on a circle. They form two different Delaunay triangulations. 1 Usually

spelt (French) phonetically as Delaunay.

5.2 Acute Partitions

57

A4

A4

A3

A3

A1

A2

A1

A2

A5

A5

Fig. 5.5 Two adjacent acute tetrahedra A1 A2 A3 A4 and A1 A2 A3 A5 on the left do not form a Delaunay triangulations, whereas the three obtuse tetrahedra A1 A2 A4 A5 , A1 A3 A4 A5 , and A2 A3 A4 A5 on the right do form a Delaunay triangulation

Theorem 5.8 For d = 2, the Delaunay triangulation maximizes the minimal angle among all triangulations having the same finite set of vertices. For the proof, see [85, p. 145]. Remark 5.5 The above theorem cannot be generalized for d ≥ 3. Take, for instance (see Fig. 5.5), A1 = (0, 0, 0), A2 = (1, 0, 0), A3 = (1/2, A4 = (1/2,

√

3/6, 1/2), A5 = (1/2,

√

√

3/2, 0),

3/6, −1/2).

There are only two tetrahedral partitions of the hexahedron A1 A2 A3 A4 A5 being the convex hull of those vertices. The triangulation consisting of only two adjacent and congruent tetrahedra A1 A2 A3 A4 and A1 A2 A3 A5 is by Definition 5.2 not Delaunay even though all their dihedral angles (equal to 60◦ or 82.82◦ ) are acute. The reason is that√the point A5 lies inside the ball about A1 A2 A3 A4 , whose circumcenter O = (1/2, 3/6, −1/12) lies on the line A4 A5 , as |A1 O| = |A2 O| = |A3 O| = |A4 O| =

7 5 > |A5 O| = . 12 12

Consider now another tetrahedron A1 A2 A4 A5 for which |A1 A2 | = |A4 A5 | = 1. Its circumcenter is thus G = (1/2,

√

3/12, 0) and

58

5 Nonobtuse Simplicial Partitions

√ √ 39 5 3 |A1 G| = |A2 G| = |A4 G| = |A5 G| = < |A3 G| = . 12 12 Hence, the point A3 lies outside the circumscribed ball about the tetrahedron A1 A2 A4 A5 . Clearly, A1 A2 A3 is the equilateral triangle. By Definition 5.2, the Delaunay triangulation corresponding to vertices A1 , . . . , A5 ∈ D consists of three congruent obtuse tetrahedra A1 A2 A4 A5 , A1 A3 A4 A5 , and A2 A3 A4 A5 , whose dihedral angles are equal to 41.41◦ and 120◦ . We see that such a Delaunay triangulation for d = 3 does not maximize the minimum dihedral angle. Remark 5.6 Numerical experiments indicate (see [85, p. 204]) that sliver tetrahedra may appear right between other well-shaped tetrahedra inside a Delaunay triangulation. The sliver tetrahedron (see Fig. 4.1) is the only type of degenerated tetrahedron whose circumradius over the shortest edge length does not grow with decreasing volume. The following theorem is proved in [193]. Theorem 5.9 In any tetrahedral partition of R3 there exist an edge that is surrounded by at least six tetrahedra, and an edge that is surrounded by at most five tetrahedra. Corollary 5.1 In any tetrahedral partition of R3 there exist a dihedral angle less than or equal to 60◦ and a dihedral angle greater than or equal to 72◦ . From this we see again why Aristotle’s conjecture cannot hold. It should also be noted that one of the four above-mentioned algorithms in Remark 5.3 produces dihedral angles not greater than 74.2◦ (see [88]). It is not known whether there exists an acute partition of R3 with a smaller maximal dihedral angle. This angle cannot be less than 72◦ . This is due to the above Corollary 5.1. In [88] an algorithm is given which enables us to decompose an infinite slab of constant thickness into acute tetrahedra with maximal dihedral angle not greater than 87.7◦ . Shangyou Zhang [303] examines some problems arising during the successive refinement of tetrahedral partitions needed in numerical mathematics, see Sect. 8.3. In Fig. 5.6 we see one popular refinement technique (called red refinement) applied to an acute or nonobtuse tetrahedron. The four “exterior” corner subtetrahedra are similar to the original tetrahedron. Since the remaining four “interior” subtetrahedra share a common edge, the associated four dihedral angles are right or at least one of them is obtuse. Remark 5.7 It is not known how to refine acute partitions of a polyhedron Ω keeping all dihedral angles acute. The main obstacle here is the fact that a simple bisection of a half-space yields two dihedral angles that add up to 180◦ . In particular, edges of acute partitions that lie inside polygonal faces of Ω must be surrounded by at least three tetrahedra. Consequently, the acuteness condition seems considerably harder to fulfill (especially at the boundary) than the nonobtuseness condition which admits orthogonal bisections (cf. Figures 5.8–5.11, 8.16). On the other hand, in contrast

5.2 Acute Partitions

59

Fig. 5.6 Refinement of a tetrahedron by plane cuts through midlines of its faces

to nonobtuse partitions, allowing an acute partition is a stable property: any small enough perturbation of the original geometry still produces an acute partition. Theorem 5.10 The regular d-simplex does not tile Rd for any d ≥ 3. Proof The case d = 3 is treated in Theorem 5.5. The dihedral angle of the regular simplex in Rd is 1 α(d) = arccos < 90◦ . d For d ≥ 4 this angle is greater than 72◦ , since α(4) ≈ 76◦ and the sequence {α(d)}∞ d=2 is increasing. Consequently, 4α(d) < 360◦ < 5α(d) ∀d ≥ 4. Therefore, four regular d-simplices are not enough to surround a (d − 2)-dimensional hypersurface and five are already too much. Summarizing, this means that the regular simplex is not a space-filler of Rd for d ≥ 3. For d ≥ 5, the options are even more limited: Theorem 5.11 There is no acute partition of Rd for d ≥ 5. The proof [194] (see also Section 6.6) resembles Fermat’s method of infinite descent [199]. It uses some results from combinatorial topology, in particular the Euler–Poincaré formula [227, p. 147] which implies that in five-dimensional space a point cannot be surrounded by acute simplices. A general form of this formula states that

60

5 Nonobtuse Simplicial Partitions

f 0 − f 1 + f 2 − · · · + (−1)d−1 f d−1 = 1 + (−1)d−1 ,

(5.2)

where f m is the number of m-dimensional facets of a convex d-dimensional polytope. On the other hand, as already mentioned in Remark 5.2, a point in R4 can be surrounded by at least 600 acute simplices due to the existence of the 600-cell. In spite of that we believe that the following conjecture is true. Conjecture 5.3 There is no acute partition of R4 . Our belief is based on the following heuristics. If Conjecture 5.1 is true then each vertex is surrounded by at least 600 acute simplices. On the other hand, in an arbitrary partition this number should be “on average” much lower. For instance, in a uniform partition of R4 based on the Kuhn partition (see Theorem 5.20 below) each vertex is surrounded by (4 + 1)! = 120 simplices only. Moreover, we also have to take into account the fact that the sum Σ of all dihedral angles of each 4-simplex satisfies 720◦ < Σ < 1080◦ , whereas for acute simplices, the upper bound is only 810◦ (see Theorem 7.2 or Table 7.1 further on). The above ingredients seem to be mutually contradictory, but a formal proof has not been found yet. However, there is no acute partition R4 if all dihedral angles are less than some constant C < π, see [166]: Theorem 5.12 There is no periodic acute partition of the space R4 . In particular, there is no acute partition of the 4-cube. The construction and analysis of acute partitions is quite difficult. Therefore, in the following section we will also allow right dihedral angles to show up in the partition, which makes the analysis easier.

5.3 Nonobtuse Partitions and Path-Simplices Nonobtuse triangulations of polygons are very well studied, see for instance [13, 169, 173, 228, 298]. We start with a useful property of nonobtuse simplices in higher dimensions (see [43, 105]): Theorem 5.13 Let d > 2. If a d-simplex is nonobtuse, then each of its facets is a nonobtuse (d − 1)-simplex. The converse implication is false. To see that it is enough to consider the tetrahedron presented in Remark 5.1. Now we introduce two notions, which are not always consistently defined in the literature [254].

5.3 Nonobtuse Partitions and Path-Simplices

A

61

A

C

B

B

C

D D Fig. 5.7 Two types of ortho-simplices in R3 . The one on the left is a cube-corner tetrahedron. The one on the right is a path-tetrahedron whose three mutually orthogonal edges AB, BC, and C D form a path. All its faces are right triangles

Definition 5.3 An ortho-simplex in Rd , d ∈ {1, 2, . . . }, is a simplex having d mutually orthogonal edges. A path-simplex in Rd is an ortho-simplex whose d orthogonal edges form a path (in the sense of graph theory); in particular, for d = 3 we shall speak about a path-tetrahedron (or orthoscheme [78, 279]). A right triangle in R2 is an ortho-simplex and also a path-simplex. In Fig. 5.7 we see both types of ortho-simplices in R3 . The left one, often called a cube-corner tetrahedron, has three mutually orthogonal edges that share a common vertex B. Three of its faces are right triangles and the fourth face AC D is an acute triangle. The second type of ortho-simplex (see Fig. 5.7 on the right) only has four right triangular faces. We observe that its three orthogonal edges form a path. Therefore, it is a path-tetrahedron. The following three theorems can be found in the monograph by Miroslav Fiedler [105] (see also [22, 43, 169]) and Lemma 2.1 for the 3d case). Theorem 5.14 Each d-simplex has at least d acute dihedral angles. This theorem, originally published in [101, p. 315], is so nice that it was rediscovered and published fifty years later as [216], even though it can be found in Mathematical Reviews 0069507. From the next theorem it follows that there exist d-simplices that have exactly d acute dihedral angles. − d right dihedral angles. Theorem 5.15 Each ortho-simplex has d+1 2 Combining Theorems 5.14 and 5.15 shows that each ortho-simplex (and thus also each path-simplex) is nonobtuse.

62

5 Nonobtuse Simplicial Partitions

Theorem 5.16 Let d > 2. A d-simplex is a path-simplex if and only if each of its facets is a path (d − 1)-simplex. From this we inductively find that a d-simplex is path if and only if each of its two-dimensional faces is a right triangle (see [102, 105]). Let us present some further results. It is easy to see (compare with Fig. 5.9) that no cube-corner tetrahedron (see the left of Fig. 5.7) contains its circumcenter. (A formula for the circumradius of a simplex in Rd is given in (2.14)). On the other hand, the circumcenter of the path-tetrahedron on the right in Fig. 5.7 lies at the midpoint of the longest edge AD. This is also true in Rd (see [22, p. 194], [102, 105]): Theorem 5.17 An ortho-simplex contains its circumcenter if and only if it is a pathsimplex. Notice that the circumscribed ball about a path-simplex for d = 2 is, in fact, the Thales circle. In 1994 Vadakkedathu T. Rajan proved [247, p. 200] another assertion in this context. Theorem 5.18 If each simplex in a given triangulation in Rd contains its circumcenter, then the triangulation is Delaunay. In particular, each nonobtuse triangulation in the plane is Delaunay (see Fig. 5.4). The converse implication does not hold. Indeed, there exist Delaunay triangulations in R2 containing triangles with obtuse angles. For instance, when the set of vertices D ⊂ R2 consists of vertices of the regular pentagon. For d > 2, neither the nonobtuseness nor the acuteness of simplices implies that the triangulation is Delaunay. For instance, Alper Üngör in [283] (see also [88, p. 245]) presents a non-Delaunay partition consisting of two acute tetrahedra that do not contain their circumcenters. A combination of the previous two theorems results in: Theorem 5.19 A triangulation into path-simplices is Delaunay. In his paper [109] from 1942, Hans Freudenthal shows the following. Theorem 5.20 The unit cube [0, 1]d can be partitioned into d! path-simplices. Indeed, the path-simplices from the above theorem can be defined as: Sσ = {x = (x1 , . . . , xd ) ∈ Rd | 0 ≤ xσ(1) ≤ · · · ≤ xσ(d) ≤ 1}, where σ ranges over all permutations of the numbers 1, 2, . . . , d. See Fig. 5.8 for d = 2, 3. The Freudenthal triangulation is also called the Kuhn partition due to the paper [213]. Since each d-cube can be partitioned into m d congruent d-subcubes for any m ≥ 2, it can also be partitioned face-to-face into m d d! path-simplices. By Theorem 5.20, each d-dimensional cube can be partitioned into d! pathsimplices (see Fig. 5.8). However, the smallest number of simplices (without any

5.3 Nonobtuse Partitions and Path-Simplices

63

0 x1 x2 1

0 x2 x1 1 Fig. 5.8 Triangulation of a square (cube) into two (six) path-simplices Fig. 5.9 Partition of a cube into five tetrahedra having the same circumscribed ball

additional vertices) into which the d-dimensional cube can be partitioned without any additional vertices is given in [26, 146] (see also Table 11.3): 1, 2, 5, 16, 67, 308, 1493 for d ≤ 7 only. In Fig. 5.9 we see five tetrahedra that form a cube. One of them is the regular tetrahedron. The other four are cube-corner tetrahedra, hence ortho-simplices, but not path-tetrahedra. The center of the cube is the center of the circumscribed ball about each of the five tetrahedra. In [38], we studied partitions of the unit cube I d = [0, 1]d into simplices that have only nonobtuse dihedral angles and where the set of d + 1 vertices is a subset of vertices of I d . We proved that, surprisingly, for each d ≥ 3 there exist only two nonobtuse simplicial partitions of I d (cf. Figs. 5.8, 5.9, and Theorem 11.8). In 1893, Jacques Hadamard [127] stated the following conjecture:

64

5 Nonobtuse Simplicial Partitions e e e b 3a c

c b 3a

b

b

c

2a

a

e e

c

3a

b e

√ Fig. 5.10 Goldberg’s division of an infinite triangular prism into congruent tetrahedra. If 2b ≥ c, ◦ then no dihedral angle exceeds 90 (see [117, p. 353]). For instance, the choice a = 4, b = 11, and c = 13 yields integer lengths of edges

Conjecture 5.4 There exists a subset of d + 1 vertices of the d-cube whose convex hull is a regular d-simplex if d ≡ 3 mod 4 or d = 1. This conjecture is proved for d < 668, cf. Fig. 5.9 and [52]. We will return to this problem in Sect. 11.7. The space Rd can be first partitioned into d-cubes and then, by Theorem 5.20, also into path-simplices. Another idea was published by Michael Goldberg in [117], see also [262]. He divided R3 into congruent infinite prisms having an equilateral triangle as cross-section. Then each prism was partitioned into congruent tetrahedra (see Fig. 5.10), which can be chosen nonobtuse. If the partition of each triangular prism is a mirror image of an adjacent prism, we get a nonobtuse partition of the three-dimensional space in the sense of Definition 5.1, that is face-to-face. Theorem 5.21 There are infinitely many noncongruent nonobtuse tetrahedral spacefillers with integer lengths of edges. Proof Goldberg’s division (see Fig. 5.10) is constructed so that b2 = e2 + a 2 , c2 = e2 + 4a 2 . Eliminating e2 , we get the Diophantine equation

5.3 Nonobtuse Partitions and Path-Simplices

65

3a 2 + b2 = c2 . By inspection we can verify that its solutions are of the form a = 2mn, b = im 2 − jn 2 , c = im 2 + jn 2 , where i, j, m, n are positive integers such that im 2 > jn 2 and i j = 3.

(5.3)

√ If 2b ≥ c, then the tetrahedral space-filler is nonobtuse. This leads to the inequality

√ √ ( 2 − 1)im 2 ≥ ( 2 + 1) jn 2 ,

which holds for infinitely many integers m and n satisfying (5.3).

Remark 5.8 A special tetrahedral space-filler was found in 1923 by D. M. Y. Sommerville (see [117, 261, 263]). The lengths √ of its two longest opposite edges equal 2, while the other four edges have length 3, see Fig. 3.5. Dihedral angles at the two longest edges are right, whereas the other four dihedral angles equal 60◦ . This Sommerville tetrahedron can be partitioned into eight congruent tetrahedra that are all similar to the original one [189] (cf. Fig. 5.6 and Theorem 8.4). It can also be partitioned into four path-tetrahedra or into two congruent cube-corner tetrahedra [262]. As mentioned in [86, p. 288], there does not exist an acute tetrahedral space-filler. Partitions of Rd with congruent nonobtuse simplices are described in [80], [141], and [143]. Any right triangle can be bisected by the altitude onto the hypotenuse into two smaller right triangles. A partition of a path-tetrahedron into three smaller pathtetrahedra (see Figure 5.10) was described by Harold S. M. Coxeter [78] in 1989. The same construction was, in fact, implicitly used by Hans-Christof Lenhard already in 1960 to trisect tetrahedra of the so-called class T 1c (see [217]) to which pathsimplices belong. The result was generalized by us to any space dimension in [43]: Theorem 5.22 Each path-simplex in Rd can be partitioned into d (and also d + 1) smaller path-simplices. Induction is used to prove the statement for d + 1, after which the result for d follows as a degenerate case. Geometrically, part of the proof resembles the Gram– Schmidt orthogonalization process. In 1957, Hugo Hadwiger [128] stated the following conjecture: Conjecture 5.5 Each simplex in Rd can be dissected (not necessarily face-to-face) into a finite number of path-simplices. The correctness of this statement has so far been proved only for d ≤ 5. For example, each triangle can be bisected into two right triangles by means of the altitude onto the longest edge.

66 Fig. 5.11 Coxeter’s trisection of a path-tetrahedron ABC D into smaller path-tetrahedra AQ P B, B P Q D, and B PC D. The points P and Q are the orthogonal projections of the points B and P on the segment AC and AD, respectively. This technique is generalized by us to an arbitrary space dimension in [43]

5 Nonobtuse Simplicial Partitions

A P Q

B

C

D

In 1960 Hans-Christof Lenhard [217] described a method to decompose an arbitrary tetrahedron into at most twelve path-tetrahedra, though possibly not face-toface. Later, Johannes Böhm [29] showed that this number cannot be reduced. See also our result [172] for polyhedra. In 1977, Alexander B. Harazišvili [137] showed that an arbitrary 4-simplex can be decomposed into at most 730 path-simplices. This number was reduced in 1986 by Heiner Kaiser [152] to 610 and in 1993 by Katrin Tschirpke [279] to 500. The minimal number is still unknown. Tschirpke also investigated the case d = 5. In [281], based on the dissertation [280], it was shown that it is sufficient to use 12 598 800 path-simplices. Remark 5.9 Each nonconvex polytope can be cut into convex polytopes by a finite number of bisecting planes whose union contains ∂Ω (cf. (3.1) for d = 3). Each convex polytope can be easily decomposed into simplices. If the Hadwiger conjecture is valid, then each polytope can be decomposed into a finite number of path-simplices. Path-simplices in the geometry of polytopes would thus be basic building blocks like atoms in nature, even more elementary than general simplices themselves.

5.4 Applications in Numerical Mathematics Nonobtuse partitions are frequently employed in numerical mathematics. For example, nonobtuseness of simplicial partitions is sufficient for the Lagrange or Hermite interpolation to have optimal approximation properties (see Chap. 4 and [149, 161, 190, 191]). Also, the standard hypercube-corner reference simplex commonly used in the approximation theory of the finite element method is nonobtuse. The finite element method uses piecewise polynomials to approximate solutions of partial differential equations, see e.g. (4.45). If these solutions satisfy certain

5.4 Applications in Numerical Mathematics

67

Fig. 5.12 Illustration of the above formula (5.4) for d=3

Bi

Fj αi j Fi

Bj maximum principles [214, 246, 264], it is desirable that their finite element approximations satisfy certain discrete analogues of these principles. Nonobtuse and acute partitions indeed yield finite element approximations that satisfy so-called discrete maximum principles when solving (possibly nonlinear) elliptic [155, 157, 180, 196, 290] and parabolic problems [41, 91, 110, 147, 252], semiconductor equations [311], and convection-diffusion problems [1] by means of globally continuous piecewise linear functions. We will return to this topic in Chap. 9. A key observation in this context is that the gradient of a non-zero linear function on a simplex S that vanishes on a facet F j of S is a constant non-zero normal to F j . Hence, the sign of inner products between gradients of such functions is in one-to-one correspondence with the type of angle between the facets of S. To be more explicit, for d ≥ 1 we have the following expression, which was derived in [45] directly from [43] and [196], grad v i · grad v j = −

cos αi j , i, j = 0, . . . , d, i = j, hi h j

(5.4)

where αi j is the dihedral angle between faces Fi and F j of S, and h i is the height in S above Fi , see (2.11). Finally, v i is the linear function that vanishes on Fi and has value one at the vertex Bi opposite Fi (see Fig. 5.12 and formulae (3.8) and (3.10)). If d = 1, then the outward unit normals to an interval form an angle of 180◦ and thus by (5.1), αi j = 0, implying that cos αi j = 1. A similar expression to (5.4) was also introduced by Jinchao Xu and Ludmil Zikatanov [297]. Basically, the discrete Laplacian that results from the standard finite element method has a non-negative inverse if each of the above inner products in the partition is non-positive for distinct i and j, which is the case for nonobtuse partitions. If the partition is in fact acute, the discrete Laplacian has a positive inverse and then both reaction and convection terms of small enough size can be handled using perturbation arguments. See for instance the paper [70] by Philippe G. Ciarlet and Pierre-Arnaud Raviart, where the presence of a reaction term in a differential equation led to the condition that the partition should be acute and the diameters of the simplices small enough. In that paper, the discrete maximum principle was, in turn, used to derive

68

5 Nonobtuse Simplicial Partitions

Fig. 5.13 Triangulation with a single obtuse triangle ● ●

●

the uniform error bounds for the finite element method. The discrete maximum principle is also of interest when avoiding negative numerical values of typical positive physical quantities like concentration, temperature (in Kelvin), and density, see the next example. Example 5.1 A single obtuse simplex in the partition can already destroy the discrete maximum principle, see [253] (cf. also [84]). To show this consider the Poisson equation −Δu = f with homogeneous Dirichlet boundary conditions on the domain (0, 4) × (0, 2), triangulated as in Fig. 5.13 below. The space of continuous piecewise linear functions relative to this triangulation and satisfying the boundary condition has dimension three: values at the vertices indicated with bullets are the degrees of freedom. Assume that they have coordinates v1 = (1, 1), v2 = (3, 1), and v3 = (2, 1 + p). For all p ∈ (0, 1) they are the vertices of the only obtuse triangle in the partition, and the discrete Laplacian does not have a non-negative inverse. For example, for p = 21 this inverse equals ⎛

⎞ 63 1 1 − ⎜ 248 248 16 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 63 1 ⎟ ⎜− 1 ⎟ ⎜ 248 248 16 ⎟ . ⎜ ⎟ ⎜ ⎟ ⎝ 1 37 ⎠ 1 16

16 160

It can easily be shown that each non-positive continuous function f = 0 whose support does not intersect the supports of the finite element functions that vanish at v1 , gives rise to an approximation u h of u that is positive at v2 , hence violating the discrete maximum principle. It should be mentioned that an alternative approach to enforce the validity of discrete maximum principles is to modify the finite element method in such a way

5.4 Applications in Numerical Mathematics

69

that obtuse angles do not stand in their way anymore. This approach was followed by Erik Burman and Alexandre Ern in [58, 59]. The modified method however also needs a modified analysis and implementation. Remark 5.10 In general, multilinear block finite element approximations with respect to partitions into block-type elements may not satisfy the discrete maximum principle for d ≥ 2. In fact, for d ≥ 4, the finite element stiffness matrix for the Laplace operator is never diagonally dominant for block-type elements [157]. To avoid this drawback we can apply linear simplicial elements. In Theorem 11.8 we state that there are just two qualitatively different nonobtuse partitions of the unit d-cube, i.e., the discrete maximum principle can be preserved. Problems with d ≥ 4 are encountered in financial mathematics and theoretical physics, see the references in [50]. In [173] the Coxeter trisection from Fig. 5.11 is used recursively to construct local nonobtuse refinements of partitions in a neighborhood of vertices of a polyhedron by means of path-tetrahedra. For instance, in Fig. 8.16 we see such a refinement near a reentrant corner, where singularities of the solution are expected. Kuhn’s triangulation is employed for preconditioning of large problems solved by multigrid methods (see [7, 24]). In [50] it is used to prove gradient superconvergence of linear finite elements in Rd . Allain Bossavit in [31] uses partitions formed by the Sommerville nonobtuse tetrahedra to approximate solutions of electric network problems. Again, one poorly shaped triangle or tetrahedron may already increase the condition number of the matrix associated with the discretization (see e.g. [253, Sect. 5] and [282]). Acute type partitions are required in the tent-pitcher algorithm (see [284] and [88, p. 238]). Nonobtuse partitions are also employed in the finite volume method [1]. For instance, in [90] this method requires a strict Delaunay triangulation, which means that the circumscribed ball about each simplex from the partition does not fully contain another simplex from the partition. In the box method, integration over elements from the Delaunay triangulation is replaced by the integration over Voronoï cells (boxes) or similar regions centered at nodal points. If the triangulation is nonobtuse, then each cell (box) is entirely included in the patch of triangles surrounding a given inner nodal point, which is necessary for patches close to the boundary of the domain. As mentioned in [57, p. 481] even one obtuse triangle can lead to a large spike in the numerical solution (cf. Example 5.1).

5.5 Further Applications Acute and nonobtuse simplices also show up in other areas of mathematics. In algebra one considers groups of symmetries of the Platonic bodies and their generators. For instance, there are four generators for the cube. The corresponding four planes of symmetries bound a path-tetrahedron which is called the fundamental domain. All planes of symmetry divide the Platonic bodies into path-tetrahedra (see [49]).

70

5 Nonobtuse Simplicial Partitions

Ortho-simplices served as a tool for the evaluation of an integral. Using the trisection of a path-tetrahedron (see Fig. 5.11) into three smaller path-tetrahedra, Coxeter proved [78] that 1

6

sec−1 x √ (x + 2) x + 1

√

1 x +3

+ 2 dx =

2 2 π . 15

A tetrahedron is called isosceles if the lengths of each pair of opposite edges are equal. We can easily show that all faces of an isosceles tetrahedron are congruent acute triangles and that the tetrahedron itself is acute. In [54] we prove that all spatial angles of a tetrahedron T have the same size if and only if T is isosceles. Moreover, we have that all four faces of a tetrahedron have the same perimeter (resp. the same area) if and only if a tetrahedron is isosceles. In this special case only two dihedral angles can be obtuse due to Theorem 5.13. They must have the same size. A large number of (computational) geometric applications of nonobtuse simplices (in particular, path-tetrahedra) are given in the Geometry Junkyard [312]. Using graph theory it is shown in [105, Chapt. 14] how to use path-simplices to establish the structure of electric networks. The main theorem (based on the paper [104]) finds the possible networks composed only of resistors inside a “black box” having n ∈ {2, 3, . . . } outlets. It is also convenient to use acute triangulations in geodesy and cartography. The closer the triangle is to an equilateral triangle, the more accurately we can establish the coordinates of particular triangulation points by means of measurement of lengths of edges (and angles). Finally, nonobtuse simplices are also used in computer graphics [88], mathematical genetics [224], crystallography [108, 256], the finite difference method [159], and in the Monte Carlo method for solving partial differential equations [304, p. 210]. In [42, 44], we use nonobtuse partitions to prove the strengthened Cauchy–Bunyakowski–Schwarz inequality. Surely, there are numerous applications and occurrences of nonobtuse and acute simplices that are not listed in this section.

Chapter 6

Nonexistence of Acute Simplicial Partitions in R5

6.1 Preliminaries Acute simplicial partitions are very useful in numerical analysis, since they yield inverse nonnegative stiffness matrices, when solving the equation −Δu + bu = f by standard linear conforming finite elements in a bounded polytopic domain in Rd with some boundary conditions and b ≥ 0 small enough. In this case the discrete maximum principle applies, see [45, 169] (and also [155, 157] for nonlinear problems). The necessity of solving partial differential equations for dimensions d > 3 arises in statistical physics, financial mathematics, general relativity, particle physics, etc. Therefore, it would be useful to have an algorithm that produces acute partitions in higher-dimensional spaces. However, in this chapter we prove that a point in the Euclidean space Rd cannot be surrounded by a finite number of acute simplices for d ≥ 5, i.e., even locally there are no acute partitions (see Theorems 5.11 and 6.3). Our proof resembles Fermat’s method of infinite descent [199]. For d = 4 the existence of acute partitions is an open problem. In two-dimensional acute partitions each vertex is obviously surrounded by at least 5 triangles. In Theorem 6.1 we prove that each vertex in acute partitions of R3 has to be surrounded by at least 20 tetrahedra (and this number is attainable, since the regular icosahedron can be partitioned into 20 acute tetrahedra). In Theorem 6.2 we prove that there is no partition of R5 into acute simplices. We also present some extensions for d ≥ 6. Recall that for d > 1 the dihedral angle between two facets is defined by formula (5.1). A d-simplex is said to be acute if all its dihedral angles are less than π/2.

© Springer Nature Switzerland AG 2020 J. Brandts et al., Simplicial Partitions with Applications to the Finite Element Method, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-55677-8_6

71

72

6 Nonexistence of Acute Simplicial Partitions in R5

6.2 Auxiliary Lemmas By Definition 5.1 each partition of a polytope is formed only by a finite number of simplices, since it is bounded. , . . . , Sk surround a point A if A is a vertex Definition 6.1 We say that simplices S1 of each Si , A lies in the interior of Ω = i Si , and S1 , . . . , Sk form a partition of Ω. Lemma 6.1 The sum of all six dihedral angles in a tetrahedron is greater than 2π. Proof Let A0 A1 A2 A3 be a tetrahedron and let Fi be faces opposite to Ai for i = 0, 1, 2, 3. Let αi j be the dihedral angle between faces Fi and F j (see Fig. 6.1). Consider 4 spherical triangles that arise by intersecting the tetrahedron A0 A1 A2 A3 with 4 sufficiently small spheres centered at the vertices Ai . Then from spherical (Riemannian) geometry we get (see Fig. 6.2)

Fig. 6.1 The dihedral angle between faces Fi and F j is denoted by αi j and ϕ is the angle between two edges A0 A2 and A0 A3

A1

α

03

α

F3

02

A0 ϕ

α 13

α

23

F1

F

α 12

2

α01

A2 Fig. 6.2 Dihedral angles α12 , α13 , and α23 around the vertex A0

A3

α 23

α 13

A0

α12

6.2 Auxiliary Lemmas

73

α12 + α13 + α23 > π, α02 + α03 + α23 > π, α01 + α03 + α13 > π, α01 + α02 + α12 > π. Summing these inequalities, we obtain the desired result

αi j > 2π.

0≤i< j≤3

Note that the lower bound 2π in Lemma 6.1 cannot be enlarged for acute tetrahedra, since the sum of all dihedral angles of the acute tetrahedron with vertices A0 = (−1, 0, −ε), A1 = (−1, 0, ε), A2 = (1, −ε, 0), and A3 = (1, ε, 0) tends to 2π for ε → 0, see also the slivers from Fig. 4.1. Lemma 6.2 The sum of all 10 dihedral angles in a 4-simplex is greater than 4π. The proof can be done in a similar way as that of Lemma 6.1. It also immediately follows from a more general result [112, p. 96] proved for a general d-simplex, which states d(d − 1) 1 2 αi j < π π d < , 4 2 0≤i< j≤d where d ≥ 3 and r stands for the integer part of a real number r , see also Theorem 7.1. Now let d ≥ 3 and let F1 , F2 , and F3 be facets of a d-simplex S. Since F1 is a (d − 1)-simplex, its inner angle ϕ between its (d − 2)-dimensional faces F1 ∩ F2 and F1 ∩ F3 is defined similarly to (5.1), but in the hyperplane containing F1 . The intersection I = F1 ∩ F2 ∩ F3 has dimension d − 3. Let L be a threedimensional space orthogonal to I (for d = 3, let L = R3 ). Then S ∩ L is a tetrahedron. Applying the Cosine theorem from spherical trigonometry to a sufficiently small sphere centered at one of its vertices contained in I , we get (see [103, p. 465] for details) cos α = − cos β cos γ + sin β sin γ cos ϕ, where α = ∠F2 F3 , β = ∠F1 F3 , and γ = ∠F1 F2 are defined as in (5.1). In Lemma 6.3 below we prove that the angle ϕ of an acute simplex is less than the associated dihedral angle α (in terms of Fig. 6.1, α = α23 ). Lemma 6.3 Let d ≥ 3. If a d-simplex is acute, then in the above notation we have ϕ < α.

6 Nonexistence of Acute Simplicial Partitions in R5

74

Proof Since the dihedral angles α, β, and γ are less than π/2, we find by the above Cosine theorem that cos ϕ =

cos α + cos β cos γ > cos α + cos β cos γ > cos α. sin β sin γ

6.3 The Proposed Proof Technique for d = 3 The proof technique used for five-dimensional space will first be illustrated on two lower-dimensional cases for d = 3 and d = 4. First we prove Theorem 5.4 from the previous chapter. Theorem 6.1 Each vertex in an acute partition of R3 is surrounded by at least 20 tetrahedra. Proof Let S1 , . . . , St be all tetrahedra that surround an arbitrary vertex A in an acute partition of R3 . The existence of such a partition is guaranteed by Theorem 5.6. Then P=

t

Si

i=1

is a convex polyhedron with t triangles on its boundary ∂ P. By the classical Euler formula v + t = e + 2, (6.1) where v is the number of vertices and e is the number of edges on ∂ P. Clearly, 2e = 3t.

(6.2)

Since the Si are acute, each interior edge containing the point A is shared by at least 5 tetrahedra. This means that any vertex on ∂ P is the intersection of at least 5 edges from ∂ P. Since each edge has 2 vertices, we obtain 5v ≤ 2e.

(6.3)

From (6.2), (6.3), and (6.1) we find that t = 4e − 5t ≥ 10v + 10t − 10e = 20.

Notice that the number of triangular faces of the polyhedron P is always an even number due to (6.2).

6.3 The Proposed Proof Technique for d = 3

75

Denote by α0T , α1T , α2T the angles of a given triangle T ⊂ ∂ P and by α1V , . . . αnVV the angles about a vertex V ∈ ∂ P. Then πt =

2 T

αiT

=

i=0

nV V

αVj < 2πv,

j=1

where the last inequality follows from Lemma 6.3 and the sums T and V are taken over all triangles T and vertices V from ∂ P, respectively. Consequently, from (6.3) and (6.2) we find that the number of edges and triangular faces is quite limited by the number of vertices for acute tetrahedra, namely 5v ≤ 2e = 3t < 6v.

6.4 The Proposed Proof Technique for d = 4 For the time being we do not know whether there exists an acute simplicial partition of R4 . It is not clear whether the constructive proof of Theorem 5.6 (see [88]) can be generalized to four-dimensional space. Anyway, a given point A can be surrounded by acute simplices of R4 . By [254] (see also [76]) there exists a regular polytope in R4 , called the 600-cell, whose three-dimensional surface is formed by 600 regular tetrahedra. It has 120 vertices (its dual is another regular polytope called the 120-cell). So let c Si , P= i=1

where S1 , . . . , Sc are all acute 4-simplices containing a given vertex A. We see that P is a convex polytope, since it can be represented as the intersection P=

c

Hi ,

i=1

where Hi are closed half-spaces such that Si ⊂ Hi and ∂ Hi contains that facet of Si , which is opposite A. In this case the famous Euler–Poincaré formula (5.2) has the form v + t = e + c, (6.4) where v, e, t, and c, respectively, are the number of vertices, edges, triangles, and tetrahedra on the surface ∂ P. Since each facet is a tetrahedron, it has 4 triangular faces, and since each triangular face is shared by exactly 2 adjacent tetrahedra, we get the equality 2t = 4c (6.5)

6 Nonexistence of Acute Simplicial Partitions in R5

76

Table 6.1 Relations valid on the surface ∂ P d Euler–Poincaré formula 2 3 4 5

v=e v+t =e+2 v+t =e+c v+t + f =e+c+2

Simplicial equality

Acuteness inequality

2v = 2e 2e = 3t 2t = 4c 2c = 5 f

5≤v 5v ≤ 2e 5e ≤ 3t 5t ≤ 4c

(see Table 6.1, where we intentionally do not reduce by 2 to indicate that two simplices always share a common (d − 2)-dimensional face). This means that the regular 600-cell has 1200 triangular faces and from (6.4) we find that it has 720 edges. Since each tetrahedron has 6 dihedral angles of value α, the total sum of all dihedral angles is 600 · 6α. Therefore, the sum Σ of all dihedral angles of regular tetrahedra sharing a given edge is Σ=

3600α = 5α ≈ 1.959π < 2π. 720

The fact that Σ is less than 2π is a consequence of Lemma 6.3. Every edge is shared by exactly 5 tetrahedra and the small gap from Fig. 5.3 does not appear for the 600-cell, since each tetrahedral cell is in a different hyperplane. From the above we observe that in four-dimensional space a point can be surrounded by 600 acute 4-simplices. Each of them can be defined as the convex hull of the center of gravity of the 600-cell and a particular regular tetrahedron from its boundary. Equalities (6.4) and (6.5) hold for more general clusters of 4-simplices, in particular, for any convex polytope in R4 whose three-dimensional surface is formed by tetrahedra. Each interior triangle has to be surrounded by at least 5 simplices, because each Si is acute. Therefore, for an acute partition of such a polytope we get the acuteness inequality 5e ≤ 3t, (6.6) since each edge from ∂ P has to be shared by at least 5 triangular faces (each having 3 edges). Table 6.1 surveys relations (6.1), (6.2), (6.3), (6.4), (6.5), and (6.6), and their analogues, for d = 2, 3, 4, 5. Denote by α1C , …, α6C all dihedral angles of a given tetrahedron C ⊂ ∂ P. Using the lower bound stated in Lemma 6.1, we have 2π

90◦ , α + β + γ + δ > 180◦ , and α + β + γ + δ + ε > 270◦ . Proof Assume to the contrary that α + β + γ ≤ 90◦ . By Lemma 6.1 (see also Table 7.1) we know that the sum of all the dihedral angles in an arbitrary tetrahedron is greater than 360◦ . Consequently, δ + ε + ϕ ≥ 270◦ , and thus at least one of these dihedral angles has to be greater than or equal to 90◦ , which contradicts the acuteness. The other cases can be investigated analogously.

Chapter 8

Refinement Techniques

8.1 Refinements of Unstructured Partitions Suitable refinements (adaptivity) of finite element partitions can often increase the accuracy of finite element approximations if they are done near those points, where singularities or oscillations of the solution (and its derivatives) of problems of mathematical physics occur, see Figs. 8.1 and 8.2. The theory of adaptive procedures in the finite element method for solving differential equations began in 1976 with Babuška’s pioneering paper [8]. Various error estimators are now available, which give us information about the regions where the refinements are actually needed to increase accuracy and minimize computational cost. Such an approach can be then used to construct multilevel sequences of finite element spaces, multilevel preconditioning, adaptive subspace selection, etc., see e.g. [30, 241, 302] and the references therein. Local refinements of unstructured (and also structured) triangulations of a polygon are often done with the help of midlines and medians of triangles, see the broken lines in Fig. 8.3 (a different approach is presented in [184, 185]). In the next section we discuss generalizations of this technique for tetrahedral partitions. However, we notice that, in contrast to the two-dimensional case, not any tetrahedron can be subdivided, in general, into smaller congruent tetrahedra that are similar to the original tetrahedron (see Theorem 8.4). This considerably complicates the analysis of properties of produced simplicial partitions in higher dimensions. In [209] an algorithm that performs a local refinement of tetrahedral partitions is presented. The main theorem states that the refined tetrahedra satisfy the regularity ball condition from Definition 8.1, i.e., they do not degenerate as the discretization parameter tends to zero. In particular, there exists a constant κ > 0 such that any tetrahedron T from any refinement level contains a ball with radius κ diam T . The proof is based on affine mapping from a special reference tetrahedron T˜ onto T . There are several similar articles on this subject (see e.g. [14, 106, 118, 240]).

© Springer Nature Switzerland AG 2020 J. Brandts et al., Simplicial Partitions with Applications to the Finite Element Method, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-55677-8_8

87

88

8 Refinement Techniques 1

Iteration1000

0

−1 −1

0

1

Fig. 8.1 Examples of planar face-to-face refined triangulations

a)

b)

Fig. 8.2 The refinement technique towards a vertex due to Benji Guo [124]

Fig. 8.3 Red and green refinements

8.1 Refinements of Unstructured Partitions

89

In [129, 130] we introduce a mesh density function which prescribes where and how much to refine a given simplicial partition (see e.g. Fig. 3.1).

8.2 Red and Green Refinements of Tetrahedra The following definition is, in fact, a particular case of Definition 4.2. Definition 8.1 A family F = {Th }h→0 of partitions into tetrahedra is said to be regular if there exists a constant κ > 0 such that for any Th ∈ F and any T ∈ Th we have (8.1) κh T ≤ r T , where r T is the radius of the inscribed ball to T and h T = diam T. Remark 8.1 Let P be the area of each face of the regular tetrahedron Tˆ with h Tˆ = 1. √ The length of each spatial altitude of Tˆ is by (2.12) obviously equal to v = 6/3. By (2.6) we find that √ 1 Pv 6 3 = . r Tˆ = 3 4P 12 Hence, for the constant κ from Definition 8.1 we have the following estimate κ ≤

√ 6 . 12

Definition 8.2 Let Th and Th be two partitions of Ω into tetrahedra. Then the set Th is called a nested partition or subpartition (subdivision) of Th if T ∈Th

∂T ⊂

∂ T.

T ∈Th

First we show how to naturally generalize the red and green refinements of triangles from Fig. 8.3 to tetrahedra (see also [119] for any space dimension). In Fig. 8.4 we see a generalization of the green refinement to three-dimensional space and the so-called red-green refinement into four subtetrahedra. Let ABC D be an arbitrary tetrahedron. Denote by M1 , M2 , M3 , M4 , M5 , M6 the midpoints of its edges AB, BC, AC, C D, AD, and B D. In Fig. 8.5 (see also Figs. 5.6 and 8.7) we observe the red refinement of a tetrahedron into eight subtetrahedra (see [189, 240]): AM1 M3 M5 , M1 B M2 M6 , M3 M2 C M4 , M5 M6 M4 D, M6 M1 M3 M5 , M1 M3 M2 M6 , M3 M2 M6 M4 , M5 M6 M4 M3 .

90

8 Refinement Techniques

Fig. 8.4 Green and red-green refinements of a tetrahedron into two and four tetrahedra, respectively Fig. 8.5 Red refinement of a tetrahedron

D

M4 M5

M6

C M2

M3 A

M1

B

Note that the octahedron M1 M2 M3 M4 M5 M6 can be subdivided into 4 tetrahedra (=2 pyramids), whose common edge can be M1 M4 or M2 M5 or M3 M6 (see Fig. 8.5 for the last case). Thus we have three different ways to subdivide the tetrahedron ABC D into 8 tetrahedra so that all faces of ABC D are divided by midlines. Lemma 8.1 Let ABC D be an arbitrary tetrahedron and let Mi be midpoints of its edges as sketched in Fig. 8.5. Then the distance between midpoints of two opposite edges, e.g., M1 ∈ AB and M4 ∈ C D is given by |M1 M4 | =

1 |AC|2 + |AD|2 + |BC|2 + |B D|2 − |AB|2 − |C D|2 . 2

(8.2)

Proof Let M be the midpoint of the side AB of a triangle ABC. First, we shall employ the Cosine theorem for the triangles AMC and B MC. Thus, we find that

8.2 Red and Green Refinements of Tetrahedra

91

|AC|2 = |AM|2 + |C M|2 − 2|AM||C M| cos μ, |BC|2 = |B M|2 + |C M|2 + 2|B M||C M| cos μ, where μ = ∠AMC. Since |AM| = |B M|, we can eliminate cos μ and after some manipulations we find the Stewart formula for the length of the median |C M| (see [259, p. 71]) 1 (8.3) |C M|2 = (2|AC|2 + 2|BC|2 − |AB|). 4 Now applying formula (8.3) three times successively for the triangles (see Fig. 8.5) AB M4 with M1 being the midpoint of AB, AC D with M4 being the midpoint of C D, BC D with M4 being the midpoint of C D, we obtain 4|M1 M4 |2 = 2|AM4 |2 + 2|B M4 |2 − |AB|2 1 = [(2|AC|2 + 2|AD|2 − |C D|2 ) + (2|BC|2 + 2|B D|2 − |C D|2 ) − 2|AB|2 ] 2 = |AC|2 + |AD|2 + |BC|2 + |B D|2 − |AB|2 − |C D|2 . Example 8.1 For the regular tetrahedron ABC D with |AB| √ = 1, the distance between two midpoints of opposite edges is by (8.2) equal to 2/2. Therefore, after dividing ABC √ D into eight tetrahedra, the longest edge of each interior tetrahedron has length 2/2 and the other edges have length 1/2. Now we similarly divide each interior tetrahedron into 8 smaller subtetrahedra by connecting the midpoint of the longest edge and its opposite edge of length 1/2. Their distance is by (8.2) equal to 1 2

√ 1 1 (1 + 1 + 1 + 1 − 1 − ( 2)2 = . 22 4

In this way the interior suboctahedron is divided into two regular subtetrahedra √ whose edges have length 1/4 and two subtetrahedra whose one edge has length 2/4 and the other have length 1/4. Now we can repeat this process recursively and obtain only two types of nonobtuse tetrahedra. This partition is well-known in crystallography, since three-dimensional space can be tessellated by regular tetrahedra and regular octahedra, forming a diamond lattice. For a general tetrahedron, if the subdivision is done properly, we obtain by [303] at most six different shapes of subtetrahedra. Hence, the associated family of partitions is strongly regular (cf. Theorem 3.2).

92

8 Refinement Techniques

Fig. 8.6 Gluing red, red-green, and green refinements of tetrahedra

Remark 8.2 The red refinement algorithm has a lot of applications in multigrid methods. Nested refinements of tetrahedra are necessary for hierarchical-basis methods and domain decomposition methods, as well. In practical calculations, the shortest spatial diagonal of M1 M4 , M2 M5 , and M3 M6 is usually chosen, since then the tetrahedra do not degenerate when h → 0 in the sense of Definition 8.1. This was proved by Zhang [303] by means of Lemma 8.1. He also gives an example showing that tetrahedra may degenerate (see Fig. 4.1) if we do not take the shortest diagonal. Figure 8.6 shows how to make post-refinements around the red refinement to get a face-to-face partition. The leftmost tetrahedron is subdivided into 8 tetrahedra (red refinement), its neighbor into 4 tetrahedra (red-green refinement), and another neighbor into 2 tetrahedra (green refinement). Repeating this refinement process, one can prove (see [209, p. 209]) that the corresponding family of partitions is regular. Remark 8.3 The bigger κ is in (8.1), the higher the quality of partition obtained. There are several ways to measure the quality of subdivisions of individual tetrahedra. For instance, the quality factor Q T ∈ (0, 1] of a tetrahedron T (often used in the meshing community) can be defined as follows (see [118, p. 794]): QT = 3

rT , RT

(8.4)

where r T is the radius of the inscribed ball of T and RT is the associated circumradius (see (2.6) and (2.7)). For the regular tetrahedron we have Q T = 1, since the ratio at which all spatial altitudes are divided is 1 : 3 = r T /RT . Consequently, tetrahedra which are almost regular have a quality factor near 1, while for flat tetrahedra this factor is near 0, cf. Fig. 4.1. More on this topic can be found in [222].

8.2 Red and Green Refinements of Tetrahedra

93

Theorem 8.1 Let F be a regular family of tetrahedral partitions. Then Q T ≥ 100κ 4 ∀T ∈ Th ∀Th ∈ F. Proof Let ABC D be an arbitrary tetrahedron T from Th ∈ F. Set l1 = A − B, l4 = C − D, etc. Then by (8.4), (2.7), (2.8), and (8.1) we find that QT = 3

r T 4 πr 3 rT r T vol3 T πr 4 = 72 √ > 72 √3 T ≥ 96 T > 100κ 4 . RT ZT ZT 6h 8 T

8.3 Properties of Red Refinement Techniques The red refinement of an arbitrary triangle produces only congruent subtriangles (see Fig. 8.3). However, this is not true for an arbitrary tetrahedron. In this section we show that large dihedral angles of tetrahedra may have a bad influence on the convergence of the finite element method. Consider the standard 3d red refinement algorithm of a given tetrahedron into eight subtetrahedra (see Fig. 8.5). Using this algorithm recursively, we obtain a family F = {Th }h→0 of tetrahedral partitions. Theorem 8.2 The maximum (minimum) dihedral angles of all tetrahedra K ∈ Th ∈ F form a nondecreasing (nonincreasing) sequence as h → 0. Proof The proposition follows immediately from the fact that the four “exterior” subtetrahedra arising from the red refinement algorithm are similar to the original tetrahedron (see Fig. 5.6). Analogously we can prove: Theorem 8.3 The maximum (minimum) angles between edges in all faces of all tetrahedra K ∈ Th ∈ F form a nondecreasing (nonincreasing) sequence as h → 0. Similar statements can be proved for any dimension d > 1. Further, we will use two different basic strategies, the longest-diagonal and shortest-diagonal refinements, for dividing the interior octahedron (see Fig. 5.6). Let Ω = (0, 1)3 and consider the problem − Δu = sin π x1 sin π x2 sin π x3

in

Ω, u = 0 on ∂Ω.

(8.5)

Let the initial partition of Ω be the Kuhn division of the cube into six nonobtuse tetrahedra (see Fig. 5.8). In the longest-diagonal refinement, the interior octahedron is divided by taking the longest diagonal as a common edge for all four resulting subtetrahedra. Shangyou Zhang [303] found that the inscribed ball condition does not hold for the longestdiagonal red refinement algorithm. Our numerical tests, moreover, show that even the

94

8 Refinement Techniques

maximum angle condition may not hold. The maximal dihedral angle of subsequent partitions is 90◦ , 135◦ , 144.74◦ , 161.57◦ ,. . . , i.e. some subtetrahedra degenerate quite fast (see Theorem 8.2). On the other hand, in the shortest-diagonal red refinement (see Fig. 8.5), the shortest edge is chosen as the common edge and the resulting subtetrahedra do not degenerate, i.e., all dihedral angles and also angles between edges are bounded from below by a positive constant. The longest-diagonal refinement will lead to considerably slower decay of h in comparison to the shortest-diagonal refinement. In practice, this means that a certain value of h is obtained for a smaller number of degrees of freedom for the shortest-diagonal refinement, compared to the longest-diagonal refinement. Therefore, the usage of h as a measure for convergence is not quite correct in this example. The optimal convergence rate O(h) could be theoretically and also practically obtained only for the shortest-diagonal red refinement algorithm. Theorem 8.4 There exists only one type of a tetrahedron T (up to similarity) whose red refinement produces eight congruent subtetrahedra similar to T . It is the Sommerville tetrahedron T1 . Proof Let us consider a tetrahedron T such that its red refinement produces eight congruent subtetrahedra similar to T . Its faces are obviously partitioned into four congruent subtriangles. The four exterior subtetrahedra and the four interior subtetrahedra obtained by plane cuts passing through the midlines of its faces are shown in Fig. 8.7. We show that T is similar to the Sommerville tetrahedron T1 . Let o be the operator that assigns to a given edge of any tetrahedron its opposite edge and let us denote by a, b, c, d, e, f the edges of the front exterior subtetrahedron such that (see the lower part of Fig. 8.7) o(a) = b, o(c) = d, o(e) = f. Parallel edges of the same length are denoted, for simplicity, by the same letters. The exterior corner subtetrahedra are obviously similar to the original tetrahedron T . Denote by g the inner edge that is surrounded by all four interior subtetrahedra. Consider the right interior and right exterior subtetrahedra. Their five edges are a, b, c, d, e. Since these two subtetrahedra are congruent, the remaining sixth edges must have the same length, i.e., | f | = |g|. Similarly, for the lower interior and lower exterior subtetrahedra we find that |e| = | f |. Since the regular tetrahedron cannot be divided into eight congruent subtetrahedra, at least two edges of T have a different length. Without loss of generality, we may assume that |a| = |e|, since e, f , and g are in all cases opposite edges (otherwise we rename the edges a, b, c, and d). Now consider four cases: 1. Let |a| ∈ / {|b|, |c|, |d|}. From the right exterior, right interior, and the lower interior subtetrahedron we see that o(a) = b, o(a) = c, and o(a) = d. Hence, |b| = |c| = |d|, since a is obviously mapped only onto a during “congruence mapping”. Consider the right interior subtetrahedron. If |b| = |d| = |e| = |g|, then the four

8.3 Properties of Red Refinement Techniques

95

b

d e a

c a

a d

g e

d

c

g

d g

b

d e a

c

e d

b e

g

d c

c

b

f

a a

a

c

f d

b e f

a

c

Fig. 8.7 Red refinement of a tetrahedron T by plane cuts through midlines of its faces (left) and its exploded version (right). The lower interior and exterior subtetrahedra have the same volume, because they have equal bases formed by the edges a, c, f and they have the same height as they are adjacent. The same is true for the other three pairs of exterior and interior adjacent subtetrahedra

dihedral angles at these edges have the same size. They cannot be nonacute, since any tetrahedron has at least three acute dihedral angles, see [169, p. 727]. Similarly we find that dihedral angles at g are acute for all four interior subtetrahedra, which is a contradiction. Thus |b| = |c| = |d| = |e| = | f | = |g|, but then the right interior and right exterior subtetrahedra are not congruent (they are only indirectly congruent up to mirroring), which is a contradiction. 2. So let |a| = |b|. Then we easily find that |b| = |c| = |d|. The cases 3. |a| = |c| and 4. |a| = |d| can be treated similarly. Therefore, altogether we obtain |a| = |b| = |c| = |d| = |e| = | f | = |g|.

(8.6)

Due to the mirror image symmetry of T and its eight subtetrahedra, the edge e is perpendicular to the plane passing through the edges f and g. Similarly, the edge f is perpendicular to the plane of symmetry containing e and g. Hence, we find that (see Fig. 8.7) e ⊥ g ⊥ f ⊥ e.

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8 Refinement Techniques

Now applying the Parseval equality, we come to (2|a|)2 = |e|2 + |g|2 + | f |2 and thus, (8.6) implies that 2|a| =

√

3|e|.

(8.7)

From this we see that T is the Sommerville tetrahedron T1 up to similarity and there is no other type of tetrahedron that can be partitioned into eight congruent subtetrahedra that are similar to the original one. Red refinement of a tetrahedron that produces congruent subtetrahedra is also treated in [235]. Some authors allow mirroring of congruent tetrahedra. Zhang in [303] presents a different proof of Theorem 8.4. Consider now a red refinement of a 4-simplex S, i.e., it is partitioned into 16 subsimplices. Then we get a situation which is a bit difficult to imagine. Namely, we first cut off 5 congruent corner subsimplices that are similar to S. The remaining polytopic domain then has 10 three-dimensional facets and it is partitioned into 16 − 5 = 11 subsimplices. Theorem 8.5 There is no 4-simplex whose three-dimensional facets are all Sommerville tetrahedra. Proof From the well-known Euler–Poincaré formula (6.4) we find that a 4-simplex has 5 vertices, 10 edges, 10 triangular faces, and there are 5 tetrahedral threedimensional facets. Now we show that there is no 4-simplex whose five facets are all the Sommerville tetrahedra T1 . Suppose to the contrary that such a 4-simplex S exists. Denote its 10 edges by a, b, c, d, e, f, g, h, i, j as indicated in Fig. 8.8. Let one of its facets be the Sommerville tetrahedron T1 . Without √ loss of generality we may assume that its edges satisfy |a| = |b| = |c| = |d| = 3 and |e| = | f | = 2. Since e is opposite to h and i; and f is opposite to g and j, we get |g| = |h| = |i| = | j| = 2. However, this relation does not allow all five facets to be the Sommerville tetrahedra T1 , since the edges g, h, i, j contain a common point and thus their pairs are not opposite. This is a contradiction. Theorem 8.6 The red refinement of an acute simplex for d > 2 never yields subsimplices that are mutually congruent. Proof Assume to the contrary that there exists an acute simplex whose red refinement produces mutually congruent subsimplices, which should then obviously be acute as the exterior subsimplices are always similar to the father simplex. As the red

8.3 Properties of Red Refinement Techniques

97

Fig. 8.8 Schematic illustration of a 4-simplex and notation of its edges

b

d

j

e i

h f

a

g c

refinement of the simplex implies by induction the red refinement of all its lowerdimensional facets, any of its three-dimensional facets would be partitioned as in Fig. 8.7. But then some nonacute angles between lower-dimensional faces appear, since the inner edge g is surrounded by four tetrahedra. By Theorem 5.3 this contradicts the assumption that all subsimplices are acute. Remark 8.4 In fact, from the above proof we observe an even stronger result than the one stated in Theorem 8.6. The red refinement of a d-simplex never produces only acute subsimplices for d > 2. Now we define the “red refinement” of a simplex in higher dimension by a technique due to Hans Freudenthal [109]. The unit hypercube I d = [0, 1]d can be partitioned into d! simplices of dimension d defined as Sσ = {x ∈ Rd | 0 ≤ xσ (1) ≤ · · · ≤ xσ (d) ≤ 1},

(8.8)

where σ ranges over all d! permutations of the numbers 1 to d. Obviously, all simplices have the same volume 1/d!. The unit hypercube I d can also be trivially partitioned into 2d congruent subhypercubes. Each of the sub-hypercubes can thus be partitioned into d! simplices as in (8.8). This will result in a face-to-face partition of I d into d!2d subsimplices. All the subsimplices that are contained in the reference simplex Sˆ = {x ∈ Rd | 0 ≤ x1 ≤ · · · ≤ xd ≤ 1} ˆ In form a face-to-face partition which will be said to form the red refinement of S. this case the permutation σ is the identity. The partition contains 2d subsimplices (see Fig. 8.5 for d = 3).

98

8 Refinement Techniques

8.4 Refinements into Path-Tetrahedra First, we will present the following elementary property. Lemma 8.2 Let O be the circumcenter of a tetrahedron T . Let P be the circumcenter of a face F of T . If P = O then the line segment P O is perpendicular to the plane containing F. Proof The set of all points which have the same distance to all three vertices of F is a straight line perpendicular to the plane containing F. Obviously, both O and P lie on this line. In [169], we present sufficient conditions for the existence of partitions into pathtetrahedra with an arbitrarily small mesh size, as formulated in the following theorem. Theorem 8.7 Let each tetrahedron in a nonobtuse partition contain its circumcenter. Then there exists a family of refined partitions into path-tetrahedra. Its proof is constructive. Each face is nonobtuse by Theorem 5.13. It is first partitioned into four or six right triangles whose common vertex is its circumcenter. Then each tetrahedron from the initial partition is partitioned into 8, 12, 16, 20, or 24 path-tetrahedra, by taking the convex hulls of the right triangles on its surface with its circumcenter (see Fig. 8.10). Such a refinement technique is called yellow (see [169]). In this case, common faces of adjacent tetrahedra from the initial partition are partitioned in the same manner. The proof then proceeds by induction, since by Theorems 5.14, 5.15, 5.17, the assumptions of Theorem 8.7 are satisfied for any refined partition. In [206] the nonobtuseness assumption in Theorem 8.7 is replaced by a weaker condition that requires that only faces are nonobtuse. Theorem 8.8 Let each tetrahedron in a given partition contain its circumcenter and let all its faces be nonobtuse. Then there exists a family of refined partitions into path-tetrahedra. We will outline the proof from [206]. The assumptions are satisfied, for example, for the sliver tetrahedron from Fig. 4.1, which is obtuse, but its faces are acute. Since all faces are nonobtuse, any triangular face F of T always contains its circumcenter P. So we have two possibilities: (a) If P lies in the interior of F, then F is an acute triangle and we divide it into 6 = 3 × 2 right triangles whose common vertex is P, and other vertices are the vertices of F and midpoints of sides of F, see Fig. 8.9 (left). (b) If P lies on the edge of F, then F is, obviously, a right triangle and similarly to (a) we divide F into 4 = 2 × 2 right triangles whose common vertex is P, see Fig. 8.9 (right). The interior of T will be decomposed as follows. We denote by O the circumcenter of T . Consider again two cases.

8.4 Refinements into Path-Tetrahedra

99

P P

Fig. 8.9 Two different partitions of nonobtuse triangular faces Fig. 8.10 Yellow partition into 24 path-tetrahedra of an acute tetrahedron containing the circumcenter in its interior

(1) If O lies in the interior of T then from Thales’ theorem it follows that each face is an acute triangle which will be subdivided as described in (a) and thus T will be decomposed into 24 = 4 × 6 subtetrahedra whose common vertex is O, see Fig. 8.10. (2) If O lies on the boundary of T , then T will be decomposed into 18, 16, 14, 12, 10 or 8 subtetrahedra. The number of subtetrahedra is equal to the number of all subtriangles on those faces which do not contain the center O. In both cases (1)–(2), subtetrahedra are defined as the convex hull of O and subtriangles which lie on faces which do not contain O. Now we prove that all subtetrahedra are nonobtuse. Let AB be an arbitrary edge of a given face F which does not contain O and let M be the midpoint of AB. Then the line segment O P is perpendicular to F by Lemma 8.2 and, thus O P ⊥ M P and also O P ⊥ AM. Moreover, obviously AM ⊥ M P. Hence, the subtetrahedron AM O P is a path-tetrahedron and thus nonobtuse. Each subtetrahedron satisfies the assumptions of Theorem 8.8. So it can be further decomposed into 8 subtetrahedra which are again nonobtuse, etc., see Fig. 8.11. The main trick is that there are four subtetrahedra around the interior edge which connects midpoints of opposite edges, and all interior angles around this edge are right. Common faces of adjacent tetrahedra in the refined partition are obviously divided in the same fashion, which is necessary to obtain face-to-face partitions.

100

8 Refinement Techniques

D

O C B A Fig. 8.11 Yellow partition of a path-tetrahedron into eight path-subtetrahedra

8.5 Partitions into Well-Centered Simplices Definition 8.3 A d-simplex S is said to be well-centered if its circumcenter lies in the interior of S. Obviously , a triangle is acute if and only if it is well-centered. However, for d > 2 the relation between these two types is more complicated. Remark 8.5 The fact that a tetrahedron is well-centered does not imply that its faces are acute. To show this, we set A = (5, 0, −5),

B = (3, 4, −5), C = (3, −4, −5),

D = (−4, 0,

√

34).

Then ABC is an obtuse triangle (in the plane z = −5) and the circumcenter O = (0, 0, 0) lies in the interior of the nonacute tetrahedron ABC D (which is easy to see in the plane y = 0). Obviously, |AO| = |B O| = |C O| = |D O| =

√

50.

Remark 8.6 The fact that a tetrahedron is acute does not imply that it is wellcentered. To see this, take ε ≥ 0 and consider the tetrahedron with vertices A = (−ε, −ε, −ε),

B = (1, 0, 0), C = (0, 1, 0),

D = (0, 0, 1).

If ε = 0 then ABC D is the standard nonobtuse cube-corner tetrahedron whose circumcenter O = ( 21 , 21 , 21 ) lies outside ABC D. Therefore, if ε > 0 is small enough, the circumcenter still lies outside ABC D and this tetrahedron is acute.

8.5 Partitions into Well-Centered Simplices

101

Radim Hošek presents in [141] an algorithm which fills R3 by congruent wellcentered tetrahedra. The paper [143] generalizes a tessellation of R3 by the Sommerville tetrahedron to an arbitrary space dimension d > 3. In [144], Hošek proves the following statement: Theorem 8.9 There exists a one-parametric family of partitions of R4 by wellcentered congruent simplices. Its proof is constructive. The cross section of each proposed partition by the three-dimensional hyperplane x4 = 0 is tiled by the Sommerville tetrahedra. Thus, Theorem 8.8 and the yellow refinement can be generalized to four-dimensional simplices. In contrast to Theorem 6.2, Hošek in [144] proves the following theorem. Theorem 8.10 There exists a partition of R5 by well-centered simplices. It is an open problem for which d > 5 there still exists a partition of Rd by well-centered simplices (compare Theorems 8.10 and 8.16). Now we shall present several further properties of well-centered simplices. Theorem 8.11 A d-simplex is well-centered if and only if its vertices do not lie on a hemisphere of the circumscribed sphere. For the proof see [53]. The following two theorems are also proved there. Theorem 8.12 The set of unit outward normals to the facets of any d-simplex S is the set of vertices of a well-centered simplex S˜ with circumsphere Sd−1 . Theorem 8.13 Endow Sd−1 with the uniform probability measure. Then the probability that n 0 , . . . , n d ∈ Sd−1 are outward normals to a simplex S equals 2−d . Let S be an arbitrary d-simplex. Then the unique d-simplex, whose vertices are the points of contact of the inscribed sphere to S, will be denoted by S and called the dual d-simplex of S. In [54] we prove the following two statements. Theorem 8.14 Every dual d-simplex is well-centered. Theorem 8.15 Every well-centered simplex is the dual d-simplex S of some dsimplex S. Jon Eivind Vatne [289] proved that simplices rarely contain their circumcenter in high dimensions, which can be expressed as follows. Theorem 8.16 The probability that d + 1 vertices A0 , . . . , Ad , selected uniformly randomly from the sphere Sd−1 , are vertices of a well-centered simplex equals 2−d . In [53] we give an alternative proof of this statement which is based on Theorems 8.12 and 8.13.

102

8 Refinement Techniques

8.6 Local Nonobtuse Refinements Now we will show how to generalize the local nonobtuse refinement towards a vertex from Fig. 8.12 to the three-dimensional case. Consider the cube ABC D E F G H as sketched in Fig. 8.13. Local refinements will be generated near the vertex A. The algorithm presented in [170] consists of five steps: (1) First we decompose the cube into 5 nonobtuse tetrahedra as illustrated in Fig. 8.13 (left). (2) Each of the tetrahedra C B DG, F B E G, and H D E G is bisected into 2 pathtetrahedra as indicated in Fig. 8.13 (right). For instance, the tetrahedron C B DG is divided into C L BG and C L DG, where L is the midpoint of the diagonal B D.

Fig. 8.12 A simple local nonobtuse refinement towards a vertex E

E

F

A

F M

A B

B K

L H

H

G

D

C a)

Fig. 8.13 Initial division steps

G

D

C b)

8.6 Local Nonobtuse Refinements

103

E

E M

A

M

U B

R

K

K

S

B

T

A

O

P

O Q L

L G

D

D

a)

b)

Fig. 8.14 Further division steps

(3) Denote by O the center of the equilateral triangle B D E. The interior regular tetrahedron B D E G is decomposed into 6 path-tetrahedra G O L B, G O L D, G O K D, G O K E, G O M E, G O M B, whose common edge is G O (see Fig. 8.14 (left)). (4) It remains to decompose the last tetrahedron AB D E. Let the points P, Q, and R be the orthogonal projections of the point O onto the faces AB E, AB D, and AD E, respectively. This enables us to define the cube AT Q SU P O R as can be seen in Fig. 8.14 (right). The remaining part can be decomposed into 3 congruent tetrahedra B D Q O, B E P O, D E R O, and 3 congruent pyramids O QT P B, O PU R E, O Q S R D with square faces. In Fig. 8.15 we observe how to bisect each of the above 6 elements into 2 path-tetrahedra. (5) The cube AT Q SU P O R can further be decomposed into 5 nonobtuse tetrahedra in a way similar to Fig. 8.13a. For local nonobtuse tetrahedral refinements towards a concave corner (see Fig. 8.16), we refer to [19]. If several congruent cubes meet at a singularity point, we can apply the above refinement algorithm to each of them so that the whole partition remains face-to-face (see Fig. 8.16). Local nonobtuse local tetrahedral refinements towards a polygonal face/interface are described in [174] and for local nonobtuse tetrahedral refinements around an edge, see [175].

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8 Refinement Techniques

B O

P B

Q T L

O Q

D

a)

b)

Fig. 8.15 Partition into nonobtuse tetrahedra Fig. 8.16 A locally refined partition of the Fichera domain into path-tetrahedra

Now we will present another technique for generating local nonobtuse tetrahedral partitions, which differs from that of Fig. 8.14. Its main idea is exposed in the following theorem, whose proof is constructive. It is based on Coxeter’s result that each path-tetrahedron can be subdivided into three path-tetrahedra (see Fig. 5.11 or [78]). Theorem 8.17 Let ABC D be a path-tetrahedron whose edges AB, BC, and C D are mutually perpendicular. Then there exists an infinite regular set of nonobtuse partitions of this tetrahedron into path-tetrahedra that refine ABC D locally in a neighborhood of the vertex A.

8.6 Local Nonobtuse Refinements Fig. 8.17 A partition of the path-tetrahedron AB Q P into three path-tetrahedra

105

A U

V

P Q

B Proof Let P be the orthogonal projection of the point B onto the line AC. Obviously, P lies in the interior of the line segment AC, since ABC is a right triangle. Further, let Q be the orthogonal projection of the point P onto the line AD. Since AC D is a right triangle, A P D has an obtuse angle at P, and thus the point Q lies in the interior of the line segment AD. We observe that the line segment B P is perpendicular to the face AC D. Therefore, B P is perpendicular to any line that is contained in the plane AC D. From this property we easily find that the original tetrahedron ABC D can be decomposed into the following three path-tetrahedra (see Figs. 5.11 and 8.18): B PC D with B P Q D with AQ P B with

B P ⊥ PC ⊥ C D ⊥ B P, B P ⊥ P Q ⊥ Q D ⊥ B P, AQ ⊥ Q P ⊥ P B ⊥ AQ.

Now we decompose the last path-subtetrahedron AQ P B again into three pathsubtetrahedra following the same rules as above. In this way we obtain a tetrahedron which is similar to the original tetrahedron ABC D, which will later help us to prove the regularity of the set of partitions. So, let U be the orthogonal projection of the point Q onto the line A P, and let V be the orthogonal projection of the point U onto the line AB. Then the pathtetrahedron AQ P B can be decomposed into the following three path-subtetrahedra (see Fig. 8.17): QU P B QU V B AV U Q

with with with

QU ⊥ U P ⊥ P B ⊥ QU , QU ⊥ U V ⊥ V B ⊥ QU , AV ⊥ V U ⊥ U Q ⊥ AV .

Thus, the five path-subtetrahedra B PC D, B P Q D, QU P B, QU V B, and AV U Q form a face-to-face partition of the original path-tetrahedron ABC D (see Fig. 8.18): Since U is the orthogonal projection of Q onto the line AC, the line segments QU and DC are parallel. Similarly we find that V U and BC are parallel, since V is the orthogonal projection of U onto the line AB. From here we conclude that the face V U Q is parallel to BC D, and thus, the path-subtetrahedron AV U Q is similar to the original tetrahedron ABC D.

106

8 Refinement Techniques

A U

V Q

B

P

C

D Fig. 8.18 A partition of a path-tetrahedron ABC D into five path-tetrahedra

The subtetrahedron AV U Q can be now decomposed into 5 subtetrahedra in a similar way (as ABC D), and thus we can get a further refinement near the vertex A. By this recursive procedure, we obtain the required infinite set of face-to-face tetrahedral partitions with regularity of the generated set preserved as any tetrahedron is similar to one of the five from Fig. 8.18. Since any path-simplex is nonobtuse, each partition from this set is nonobtuse as well.

8.7 The Longest-Edge Bisection Algorithm The longest-edge bisection algorithm is one of the most popular refinement algorithms in the finite element method, since it is very simple and can be easily applied in higher dimensions, see [250, 251]. It is also used in visualization techniques and geometrical modeling. Martin Stynes in his papers [272, 273] examined the regularity of a family of triangulations obtained by a recursive use of the longest-edge bisection algorithm. This algorithm bisects simultaneously all triangles by medians to the longest edge of each triangle in a given triangulation (see Fig. 8.19). In this way, a family of nested triangulations is obtained.

Fig. 8.19 The classical longest-edge bisection algorithm may produce triangulations that are not face-to-face. Broken lines indicate new bisections

8.7 The Longest-Edge Bisection Algorithm

107

Fig. 8.20 A modified longest-edge bisection algorithm that always produces face-to-face triangulations Fig. 8.21 Bisection of tetrahedra sharing the longest edge

M. Stynes proved that during the infinite bisection process only a finite number of similarity-distinct subtriangles are generated, which, obviously, provides the regularity of the resulting family of triangulations, since Zlámal’s minimum angle condition (4.22) is satisfied. However, this refinement may lead, in general, to so-called hanging nodes and thus refined triangulations may not all be face-to-face, in general (see Fig. 8.19). Therefore, in [178] a modified version of the bisection algorithm by medians was introduced, where all simplices sharing the longest edge of the whole triangulation are bisected at each step (see Fig. 8.20). In this way, all produced triangulations are face-to-face and moreover, the corresponding family of triangulations satisfies the minimum angle condition (4.22) with computable and known α0 , see [178]. To avoid the generation of nonconforming partitions in 3d, it is essential to divide all tetrahedra surrounding the longest edge, see Fig. 8.21. In [130] we modified the above bisection algorithm further so that it may also generate locally refined simplicial faceto-face partitions. This goal was achieved by introducing a Lipschitz continuous mesh density function which tell us which simplices have to be chosen for bisections. We proved that this generalized algorithm also produces a family of nested face-to-face partitions. Bisection-type algorithms are very popular in the finite element community, since they enable us to perform local mesh refinements quite easily (see [265]). In [30, 209] local mesh refinements are generated by means of the famous red refinement.

108

8 Refinement Techniques

1 3

2 2

1

1

3 2

3 2

2

3 2

3 2 1

1

2 2

3 1 2 2 2 2

1

Fig. 8.22 The first three steps of the longest-edge algorithm applied to the reference pathtetrahedron

This process is quite complicated from a combinatorial and programming point of view, although the corresponding family of partitions is regular (see [209] for the proof). One step of the face-to-face longest-edge bisection algorithm is defined as follows: (I) choose the longest edge in a given face-to-face simplicial partition; (II) bisect each simplex S sharing by a hyperplane passing through the midpoint of and the vertices of S that do not belong to (see Figs. 8.21 and 8.22). From (II) we observe that all partitions generated by the longest-edge bisection algorithm remain face-to-face. In fact, this algorithm performs the so-called green refinements. Lemma 8.3 Let F = {Th }h→0 be a regular family of simplicial partitions of a bounded polytope Ω and let S be an arbitrary d-simplex from an arbitrary partition Th ∈ F. Denote by a the length of the shortest edge of S and by e the length of its longest edge. Then a ≥ κe, where κ > 0 is a fixed constant independent of h. Proof Let A1 , A2 , . . . , Ad+1 be vertices of S, let Fi be the facet of S opposite to the vertex Ai , and let vi be the altitude from the vertex Ai to the facet Fi . Without loss of generality we may assume that the shortest edge is A1 A2 . Then clearly a ≥ v1 and from vold S ≥ C(diam S)d we obtain Ced = C(diam S)d ≤ vold S =

1 a a v1 vold−1 F1 ≤ vold−1 F1 ≤ ed−1 , d d d

where the last inequality is due to the fact that the (d − 1)-simplex F1 is contained in a (d − 1)-dimensional hypercube whose edge has length e. Dividing the left-hand and right-hand sides of the above inequality by ed−1 we complete the proof. Definition 8.4 A family F = {Th }h→0 of simplicial partitions of a bounded polytope Ω is called strongly regular if there exists a constant C > 0 such that for all partitions Th ∈ F and for all simplices S ∈ Th we have

8.7 The Longest-Edge Bisection Algorithm

vold S ≥ Ch d .

109

(8.9)

Theorem 8.18 Let d be an arbitrary space dimension and let F = {Th }h→0 be a family of simplicial partitions of Ω generated by the face-to-face longest-edge bisection algorithm. Then F is regular if and only if it is strongly regular. Proof Every strongly regular family is clearly regular. To prove the converse implication assume that all edges from the initial partition were already halved at least once, which occurs after a finite number of steps. It is enough to analyze only partitions generated after these initial refinement steps. Denote such partitions by Th , where h is the length of the longest edge. Let S be any d-simplex (with the shortest edge of length a) from some partition Th . Since all edges from the initial partition were already halved, there exists exactly one mother d-simplex S from some previous partition such that the bisection of S in the next step yielded S. Let the diameter of S be h , i.e. S ∈ Th . Denoting by e the longest edge of S, we find that (8.10) 2e ≥ h ≥ h. From Lemma 8.3 and (8.10) we get vold S ≥ C(diam S)d ≥ Ca d ≥ Cκ d ed ≥ Cκ d 2−d h d . Therefore, (8.9) holds.

Tetrahedra have more angles than triangles, which obviously complicates the analysis and numerical implementation in 3d. In addition, it is often not even possible to formulate or observe the properties of bisected tetrahedra, which could provide natural analogues of the performance of the bisections in 2d. To illustrate the latter fact, we give several examples. Set A = (−1, 0, 0), B = (1, 0, 0), C = (−2, 2, −1), D = (2, 2, 1). Then the dihedral angles at the edges are: 25.21◦ at |C D| = 4.47, 28.56◦ at |B D| = |AC| = 2.45, 53.13◦ at |AB| = 2, 133.09◦ at |BC| = |AD| = 3.75. Observation 1 The largest (dihedral) angle is not opposite to the longest edge C D. Observation 2 The tetrahedron is obtuse. However, after the first bisection of the longest edge C D the largest (dihedral) angle increases to 150.79◦ . Observation 3 If we bisect the edge AD opposite the biggest (dihedral) angle 133.09◦ , then one of the new angles is bigger, namely 140.77◦ . The numerical tests from [132] indicate that the face-to-face longest-edge bisection algorithm seems to produce regular (and, therefore, by Theorem 8.18, strongly regular) families of tetrahedral partitions. However, a rigorous proof of this statement is an open mathematical problem and it would not be easy to obtain due to the non-uniqueness phenomena in the bisection process as illustrated in Fig. 8.23.

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8 Refinement Techniques

Fig. 8.23 The bisection process of the regular tetrahedron does not produce uniquely determined shapes of subtetrahedra after the second bisection step. The first bisection midpoint E can be chosen arbitrarily. However, the next bisection midpoint F can be chosen on the opposite edge or an adjacent edge, which yields different shapes

In practical calculations, when two or more longest edges have the same length, then the newest longest edge is usually proposed for bisection. More material on bisection-type refinement techniques can be found in the review paper [183].

8.8 Refinements of Triangular Prisms By a prism (or more precisely triangular prism) we shall mean a prism with two parallel triangular faces and three rectangular faces. In what follows we consider only bounded polyhedra Ω ⊂ R3 which can be decomposed into prisms. Let Ph be a face-to-face partition of one such polyhedron into prisms (see Fig. 8.24). Here h stands for the usual discretization parameter, i.e., the maximum diameter of all prisms from Ph . It is evident from the definition of a prism that any partition Ph into prisms consists of parallel layers of prisms. In the following we may assume these layers to be horizontally oriented and we call the bottom plane of a layer its base and the triangular face of a prism the base triangle of the prism. We shall consider tetrahedralizations of Ph such that the triangular faces of the prisms are not cut. Hence, the different layers of prisms can be subdivided independently into tetrahedra and these altogether provide us with a conforming tetrahedral mesh over Ω. We shall subdivide each prism into three tetrahedra as marked in Fig. 8.25 (left). We see that its rectangular faces are divided by diagonals into triangles and these diagonals determine three tetrahedra in the subdivision. However, these diagonals cannot be chosen arbitrarily. In Fig. 8.25 (right) we observe a division of three rect-

8.8 Refinements of Triangular Prisms

111

Fig. 8.24 Partition into prisms

Fig. 8.25 Two subdivisions of rectangular faces of a prism. The subdivision on the right does not correspond to any partition of the prism into tetrahedra, see [176]

angular faces of a prism which does not correspond to any partition of the prism into tetrahedra. Therefore, we have to divide rectangular faces in the whole partition carefully. In the next theorem we show how to practically construct from a given prismatic partition Ph a face-to-face tetrahedralization, thus avoiding the situation illustrated in Fig. 8.25 (right, or its mirror image) when dividing rectangular faces by diagonals. Theorem 8.19 For any conforming partition into prisms there exists a face-to-face subdivision into tetrahedra. Proof From the beginning of this section we know that any partition of Ph into prisms consists of parallel layers which can be tetrahedralized independently (see Fig. 8.24). Consider one such layer and let Th be the triangulation of its base corresponding to the partition Ph . Take an arbitrary vector v = 0 in the plane containing the triangulation Th (for instance v = (1, 0, 0)). Now we define the orientation e of each edge e of the triangulation Th such that ( v , e) ≥ 0. (8.11) If an edge e is perpendicular to v, we may take an arbitrary orientation of e. In this way we get the planar digraph G h = (N , E), where N is the set of nodes and E is the set of directed edges.

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8 Refinement Techniques

Fig. 8.26 Non-allowed edge orientations

Fig. 8.27 A possible orientation of edges of a triangular base that leads to a partition into three tetrahedra

C

A B

C

A B

It is clear that G h does not contain a directed circle whose edges form a triangle of Taking Th . Indeed, if, on the contrary, e1 , e2 , e3 form a circle then e1 + e2 + e3 = 0. the scalar product of both sides by v and using (8.11), we get that v is perpendicular to the triangle with side vectors e1 , e2 , e3 , which is a contradiction. The algorithm can be summarized as follows: 1. Orient all edges in the triangulation of the base according to formula (8.11). 2. Subdivide all vertical rectangular faces in the direction defined by the orientation of edges of the base as indicated in Fig. 8.27. Remark 8.7 Non-allowed edge orientations are sketched in Fig. 8.26. In Fig. 8.27 we see a partition of the prism ABC A B C into three tetrahedra different from that in the left part of Fig. 8.25. We demonstrate now that the tetrahedral mesh generated as above is regular whenever the original prismatic mesh is regular as well. Definition 8.5 A family of partitions F = {Ph }h→0 of a polyhedron Ω into triangular prisms is said to be regular (strongly regular) if there exists a constant κ > 0 such that for any partition Ph ∈ F and any element P ∈ Ph we have

8.8 Refinements of Triangular Prisms

κh 3P ≤ vol3 P

113

(κh 3 ≤ vol3 P),

(8.12)

where h P = diam P. Theorem 8.20 If a family of partitions {Ph }h→0 of a polyhedron Ω into prisms is regular (strongly regular), then the associated family of partitions {Th }h→0 into tetrahedra is also regular (strongly regular). Proof It is evident that the volume of each of the three tetrahedra from Fig. 8.25 (left) is equal to one third of the volume of the prism. Therefore, if inequalities (8.12) hold for prisms, then the same relations also hold for tetrahedra with another constant κ = κ/3, see (4.2). Remark 8.8 Assume that a family of partitions of Ω into prisms is regular and that Ω has Lipschitz boundary. Then by Theorem 8.20 the optimal interpolation properties of tetrahedral elements in Sobolev space norms are satisfied. In practice we sometimes meet polyhedral domains which do not have Lipschitz boundary in the sense of Neˇcas [237]. Let us point out that domains with Lipschitz boundaries often stand as an important assumption in a number of useful theorems, such as various imbedding and density theorems, the trace theorem, the Poincaré–Friedrichs theorem, etc. Many authors then apply these theorems for polyhedral domains assuming (incorrectly) that any polyhedral domain has Lipschitz boundary. In the left part of Fig. 8.28 we observe a polyhedral domain whose boundary is not Lipschitz (see [189, p. 48]) near points marked by black dots. In any neighborhood of these points the boundary is not a graph of a function. It can be expressed only by means of a multivalued function in any coordinate system (whereas any Lipschitz function is one-valued). Note that the polyhedral domain of Fig. 8.28 satisfies the classical external (and also internal) cone condition.

Fig. 8.28 A polyhedral domain whose boundary is not Lipschitz (left) and its partition into triangular prisms (right)

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8 Refinement Techniques

The right part of Fig. 8.28 shows its partition into rectangular blocks and their subdivision into triangular prisms. We see that this partition is somewhat different from that sketched in Fig. 8.24 and thus the face-to-face subdivision into tetrahedra has to be done carefully on the intersection of the two bars (see Theorem 8.19) but it is, in principle, possible.

Chapter 9

The Discrete Maximum Principle

9.1 The Maximum Principle for an Elliptic Problem The maximum principle is an important feature of scalar second-order elliptic equations that distinguishes them from higher order elliptic equations and systems of elliptic equations (cf. [116, 197, 198, 214, 296]). This principle, in its simplest form, was first discovered for harmonic functions: Any nonconstant harmonic function u (i.e. Δu = 0) takes its minimum and maximum values only on the boundary ∂Ω of any domainΩ in which u ∈ C(Ω), min u(s) < u(x) < max u(s)

s∈∂Ω

s∈∂Ω

∀x ∈ Ω.

(9.1)

The relation (9.1) gives, in fact, an a priori estimate for u(x) in Ω via its values on ∂Ω. This nicely illustrates, for instance, the temperature of a steady-state heat conduction problem with zero heat sources in homogeneous and anisotropic media. Later, maximum principles were formulated for various second order boundary value problems (see, e.g., [225, 246, 264]). We consider the following model problem: Find a function u such that − div(A grad u) + cu = f in Ω, u = g on ∂Ω,

(9.2)

where Ω ⊂ Rd is a bounded polytopic domain with Lipschitz boundary ∂Ω and g is a given continuous function. The diffusive tensor A = A(x) is assumed to be a symmetric and uniformly positive definite matrix, i.e. there exists a constant C > 0 such that ζ A(x)ζ ≥ Cζ2 ∀ζ ∈ Rd ∀x ∈ Ω. The reactive coefficient c ≥ 0 and the source function f are assumed to be nonnegative in Ω. All these quantities will be specified more precisely in what follows. © Springer Nature Switzerland AG 2020 J. Brandts et al., Simplicial Partitions with Applications to the Finite Element Method, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-55677-8_9

115

116

9 The Discrete Maximum Principle

The classical solutions of elliptic problems of the second order are known to satisfy the so-called maximum principles (MPs), see e.g. [155, 214]. For our model problem the corresponding MP is the following implication: f ≤0

max u(x) ≤ max{0 , max g(s)}.

=⇒

(9.3)

s∈∂Ω

x∈Ω

For MPs for the other types of elliptic problems (nonlinear ones, with mixed boundary conditions), see e.g. [94, 95, 155, 157] and the references therein. The first reasonable discrete maximum principle (DMP) and conditions providing its validity were formulated in 1966 by Richard Varga [288] for the finite difference method. Later, in 1970 in [67] (and [70]), Philippe G. Ciarlet presented a more general form of DMP suitable for both finite element (FE) and finite difference types of discretizations. He also proposed a practical set of sufficient conditions on the matrices involved, proving the validity of his DMP. Since that time these conditions have become popular in the numerical community, see e.g. [155, 157, 180, 196] and the references therein, for proving various DMPs for problems of elliptic type. In this chapter we consider the issue of weakening the conditions proposed by Ciarlet and its relation to the geometric properties of the partitions used.

9.2 Finite Element Discretization Standard linear schemes for constructing FE approximations for the (unknown) solution u of (9.2) are based on the so-called weak formulation: Find u ∈ g + H01 (Ω) such that a(u, v) = F(v) ∀v ∈ H01 (Ω), where a(u, v) =

Ω

A grad u · grad v dx +

cuv dx,

Ω

F(v) =

Ω

f v dx.

The matrix A is assumed to be in (L ∞ (Ω))d×d , c ∈ L ∞ (Ω), g ∈ H 1 (Ω) ∩ C(Ω), and f ∈ L 2 (Ω). The existence and uniqueness of the weak solution u is provided by the Lax–Milgram Lemma 3.1, see also [68]. Let Th be a partition of Ω with interior nodes B1 , . . . , B N lying in Ω and boundary nodes B N +1 , . . . , B N +N ∂ lying on ∂Ω. Further, let Vh be a finite-dimensional subspace of H 1 (Ω), associated with Th and its nodes, being spanned by the basis functions φ1 , φ2 , . . . , φ N +N ∂ with the following properties: φi ≥ 0 in Ω, i = 1, . . . , N + N ∂ , and N +N ∂ φi ≡ 1 in Ω. (9.4) i=1

9.2 Finite Element Discretization

117

We also assume that the basis functions φ1 , φ2 , . . . , φ N (associated with the interior nodes) vanish on the boundary ∂Ω. Thus, they span a finite-dimensional subspace Vh0 of H01 (Ω). Let, in addition, gh =

N +N ∂

gi φi ∈ Vh

i=N +1

be a suitable approximation of the function g via the basis functions associated with the boundary nodes. The FE approximation is defined as a function u h ∈ gh + Vh0 such that a(u h , vh ) = F(vh ) ∀vh ∈ Vh0 , whose existence and uniqueness are also provided by the Lax–Milgram lemma. N +N ∂ Algorithmically, u h = i=1 yi φi , where the coefficients yi are the entries of the solution y¯ = (y1 , . . . , y N +N ∂ ) of the following square system of N + N ∂ linear algebraic equations ¯ y = F, ¯ A¯ (9.5) where

−1 F A −A−1 A∂ A A∂ −1 ¯ ¯ ¯ , F = ∂ , and A = . A= F 0 I 0 I

(9.6)

In the above, the blocks A and A@ are matrices of size N × N and N × N ∂ , respectively, I stands for the unit matrix, and 0 for the zero matrix. The entries of ¯ are denoted by ai j = a(φ j , φi ), i = 1, . . . , N , j = 1, . . . , N + N ∂ . The vector A F consists of entries f i = F(φi ), i = 1, . . . , N , and the vector F@ has entries f i = gi , i = N + 1, . . . , N + N ∂ , given by the boundary data. For later reference we ¯ −1 in (9.6). Notice that A ¯ is nonsingular if and only if A included the formula for A is nonsingular.

9.3 Sufficient Algebraic Conditions We will distinguish two essentially different types of DMPs. Algebraic DMP: A natural algebraic analogue of (9.3) is as follows (cf. (9.5)): F≤0

=⇒

max

i=1,...,N +N ∂

yi ≤ max{0,

max

j=N +1,...,N +N ∂

y j }.

Functional DMP: A natural functional imitation of (9.3) is as follows: f ≤0

=⇒

max u h ≤ max{0, max u h }. Ω

∂Ω

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9 The Discrete Maximum Principle

Remark 9.1 It is easy to see that the above types of DMPs are equivalent in the case of linear and multilinear finite elements. However, these DMPs are not equivalent, in general, for higher-order finite elements, see [292]. In [67], Ciarlet formulated and proved the following theorem: Theorem 9.1 The algebraic DMP is satisfied if and only if −1 (A) A¯ is monotone (i.e., A¯ nonsingular and A¯ ≥ 0), (B) ξ + A−1 A∂ ξ ∂ ≥ 0, where ξ and ξ ∂ are vectors of all ones of sizes N and N ∂ , respectively.

Since conditions (A) and (B) are difficult to verify, Ciarlet suggested in [67] the following set of sufficient conditions, which is more practical. Theorem 9.2 The algebraic DMP is valid provided the matrix A¯ satisfies (a) aii > 0, i = 1, . . . , N , (b) ai j ≤ 0, i = j, i = 1, . . . , N , j = 1, . . . , N + N ∂ , N +N ∂ (c) ai j ≥ 0, i = 1, . . . , N , j=1

(d) A is irreducibly diagonally dominant. Ciarlet essentially proposed the above conditions in order to utilize the following result of Varga [287, p. 85]: Lemma 9.1 If A ∈ R N ×N is an irreducibly diagonally dominant matrix with strictly positive diagonal and nonpositive off-diagonal entries then A−1 > 0. Now, we can easily demonstrate the proof of Theorem 9.2. We follow the steps of [67]. Conditions (a), (b), and (d) together with Lemma 9.1 imply A−1 ≥ 0 and, hence, condition (A). Further, condition (c) is equivalent to Aξ + A∂ ξ ∂ ≥ 0 and since A−1 ≥ 0, we conclude that (B) is valid as well. Theorem 9.1 then guarantees the algebraic DMP. A local version of DMP was investigated in [16]. Remark 9.2 In the case of homogeneous Dirichlet boundary conditions system (9.5) reduces to a simpler form Ay = F (cf. [196]). Then the algebraic DMP holds if and only if A−1 ≥ 0, i.e., if and only if A is monotone.

9.4 Associated Geometrical Conditions ¯ can be computed explicitly, For some types of finite elements, the entries of A therefore the validity of (b) can often be guaranteed a priori by imposing suitable geometrical requirements on the shape (and size) of partitions. Thus, if A in (9.2) is a diagonal matrix with the same entries then there exist the following popular geometrical conditions guaranteeing (b):

9.4 Associated Geometrical Conditions

119

(i) For simplicial finite elements (d ≥ 2)—all dihedral angles between facets of simplices have to be nonobtuse or acute [45, 70, 155, 157, 171, 182, 196], depending on the magnitude of c. √ (ii) For bilinear √ elements—all rectangular elements have to be nonnarrow (i.e. 2/2 ≤ b1 /b2 ≤ 2, where b1 , b2 are the edges of the rectangle) and trilinear elements have to be cubes, see e.g. [157]. (iii) For 3d partitions consisting of right triangular prisms the altitudes of prisms are limited from both sides by certain quantities dependent on the area and angles of the triangular base and on the magnitude of the reaction coefficient c, see [136, 291]. The case of a nondiagonal matrix A in (9.2) is investigated in [181].

9.5 Typical Problems with Standard Conditions Not only condition (b) but also the other conditions (a), (c), and (d) in Theorem 9.2 have to be addressed too. The positivity of diagonal entries (a) is trivially satisfied for standard elliptic problems. Also the row sums (c) are nonnegative automatically for problem (9.2), because in many cases the basis functions form a partition of unity (9.4): N N +N ∂ +N ∂ ai j = a φ j , φi = a(1, φi ) j=1

j=1

=

Ω

cφi dx ≥ 0, i = 1, . . . , N .

(9.7)

On the other hand, the irreducibility of A required in (d) is not always obvious. We present three simple examples of triangulations which lead to reducible FEM matrices in Fig. 9.1 (cf. [135]). It might be a difficult task to satisfy all conditions (a)–(d), especially in 3d. A serious problem in 3d is to keep the desired geometrical limitations on the elements during global and local refinements of simplicial partitions. Condition (b) leads to a

Fig. 9.1 Examples of partitions leading to a reducible matrix A for the Poisson problem with zero Dirichlet boundary conditions

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9 The Discrete Maximum Principle

severe limitation in the case of 3d rectangular blocks (only cubes are allowed), see the point (ii) above. Moreover, if the diffusive tensor A is not diagonal, then proving the irreducibility (d) could be a nontrivial task.

9.6 Less Severe Conditions Based on Stieltjes Matrices Conditions (a)–(d) from Theorem 9.2 can be weakened using the concept of Mmatrices and Stieltjes matrices [287]. A real square matrix A is an M-matrix if its off-diagonal entries are all nonpositive and if it is nonsingular and A−1 ≥ 0. A real square matrix A is a Stieltjes matrix if its off-diagonal entries are all nonpositive and if it is symmetric and positive definite. The following result [287, p. 85] enables us to eliminate conditions (a), (c), and (d) from the standard set as we state in Theorem 9.3 below. Lemma 9.2 If A is a Stieltjes matrix, then it is also an M-matrix. ¯ associated to (9.2), satisfies condition Theorem 9.3 If the finite element matrix A, (b) from Theorem 9.2, then the algebraic DMP is valid. Proof We verify conditions (A) and (B) of Theorem 9.1. (A) For problem (9.2), the FE matrix A is always symmetric and positive definite and hence if condition (b) is satisfied then A−1 ≥ 0 by Lemma 9.2. Further, since ¯ −1 ≥ 0, see (9.6). A∂ ≤ 0 by (b), we obtain −A−1 A∂ ≥ 0 and thus A (B) Condition (c) is satisfied for problem (9.2) due to (9.7) and, as we already mentioned, (c) implies (B). Remark 9.3 In [180] we prove that the discrete maximum principle for continuous piecewise linear tetrahedral finite element approximations for the Poisson problem (3.6)–(3.7) is satisfied if some dihedral angles are slightly greater than 90◦ . In this case the associated stiffness matrix is a Stieltjes matrix and thus also an M-matrix by Lemma 9.2. The nonobtuseness of tetrahedra is therefore not a necessary condition to satisfy the discrete maximum principle. Remark 9.4 Nonobtuse and acute simplicial partitions are needed when continuous and discrete maximum principles are investigated for systems of elliptic equations [156], for parabolic (linear and nonlinear) equations and systems of parabolic equations, see for details e.g. [92, 93].

Chapter 10

Variational Crimes

10.1 What Are Variational Crimes? The solution of elliptic boundary value problems is usually transformed to the minimization of a convex coercive functional or solving a variational problem. This is their variational formulation. However, in practical implementations of the finite element method in order to solve elliptic boundary value problems numerically, so-called variational crimes are usually committed, see [268]. They include the following situations: (1) Integrals are evaluated numerically with the use of quadrature or cubature formulae (cf. (10.1) and (10.26)). (2) The given domain with a piecewise curved boundary is approximated by polygonal (in 2d ) or polyhedral (in 3d ) domains (cf. (10.30) and Fig. 10.1). (3) Boundary conditions are approximated (interpolated) by finite element functions (cf. (10.23)). In each of the above three cases we observe some violations of the variational formulation. A detailed analysis of variational crimes for linear boundary value problems is given, e.g., in [68, 69]. Its extension to a class of nonlinear elliptic problems of monotone type in the plane was first done in [98]. This analysis was later generalized in several directions: to pseudomonotone operators [96, 99], to problems with discontinuous coefficients [97, 301], to polyhedral domains [196], etc. In the next section we introduce some useful numerical integration formulae. In Sect. 10.3, we apply isoparametric quadratic triangular elements to curved boundaries in 2d . Furthermore, we show how to treat curved boundaries in 3d . In particular, in Sect. 10.4, we shall deal with a typical three-dimensional nonlinear elliptic problem. Then we approximate a smooth boundary by a polyhedral one, we use appropriate numerical quadrature formulae (see [87]) to evaluate all integrals, and also interpolate

© Springer Nature Switzerland AG 2020 J. Brandts et al., Simplicial Partitions with Applications to the Finite Element Method, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-55677-8_10

121

122

10 Variational Crimes

Γh Γ

Fig. 10.1 Approximation of a smooth boundary Γ = ∂Ω by a polygonal boundary Γh = ∂Ωh

boundary conditions. Doing such approximations, we therefore commit variational crimes, in virtue of which the used finite element approximations become nonconforming. We employ linear tetrahedral finite elements and sketch the proof of the convergence of approximate solutions on polyhedral domains in the H 1 -norm to the true solution without any additional regularity assumptions. For simplicity, we shall consider a bounded domain Ω ⊂ R3 with smooth boundary. There are several approaches to treat the curved boundary ∂Ω. The first one is to employ isoparametric elements. However, they do not have a simple form in R3 (see [23, 68]). The associated shape ansatz functions are nonpolynomial, in favorite cases rational, but, in general, they are quite complicated. Furthermore, note that for a small discretization parameter h it is not possible to decompose Ω into tetrahe-

Fig. 10.2 A partition of a ball into one-face curved tetrahedra. However, in carrying out refinements with h → 0, difficulties illustrated in Fig. 10.6 will necessarily appear

10.1 What Are Variational Crimes?

123

dral elements having at most one face curved (cf. Fig. 10.2 and [202, p. 77]). Thus, isoparametric elements with at least two curved faces are used (see [218]). Another approach is to approximate Ω by a polyhedron Ωh and then decompose Ω h into standard (non-curved) elements. Since we usually have no a priori information about the regularity of the true solution of a nonlinear problem, lower order finite elements on Ωh are usually applied in this situation. This approach is often used in practical calculations, but for a complete finite element analysis the entire shape of the domain Ω should be taken into account. Here we present a different approach. We shall assume for simplicity that Ω is convex, smooth, and Ωh ⊂ Ω. The set Ω \ Ωh is decomposed into two kinds of special elements—hat and slice elements to which approximations constructed on Ωh are extended, see Figs. 10.7 and 10.8. These special elements are not applied for computer implementation, but only to prove the convergence. Some problems which we meet, when ∂Ω is not smooth, are outlined in Remarks 10.5 and 10.6. The mixed boundary conditions are not considered, since the set of points, where one boundary condition changes to another, is technically difficult to treat.

10.2 Efficient Quadrature Formulae on Simplices Consider a general numerical quadrature formula v(x) dx ≈ vold S S

Q

cq v(Bq ),

(10.1)

q=1

where S is a simplex, the numbers cq ∈ R are called coefficients (or weights), c1 + · · · + cd = 1, and Bq ∈ S are called nodal points (or just nodes). If (10.1) is exact for all polynomials p ∈ Pk (S) of degree k > d − 1, then due to the Sobolev imbedding W1k+1 (S) ⊂ C(S) (cf. (1.1)) the following error estimate holds Q k+1 v(x) dx − vold S cq v(Bq ) ≤ Chk+1 (S), S |v|k+1,1,S ∀v ∈ W1 S

(10.2)

q=1

where hS = diam S. In this section we briefly introduce several useful numerical integration formulae which have all nodes in the interior of an element S. The choice of such formulae is advisable especially when the integrated function v has a jump on the common side (face) of two adjacent elements, as then we need not evaluate v on the boundary of S. This situation occurs, e.g., in problems with composite materials. Moreover, such formulae have, in general, a higher approximation order than numerical quadrature formulae whose nodes lie on the boundary ∂S. For example, the 7-point formula

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10 Variational Crimes

Fig. 10.3 Integration points of quadrature formula (10.5)

C

D E G F

B

A

from Fig. 10.3 is, by (10.2), of order h6S . On the other hand, the 7-point integration formula from [68, p. 183], whose nodes lie at vertices, midpoints and the center of gravity of a triangle, is only of order h4S . For a rough approximation of integrals in Rd , d = 1, 2, . . . , we can always apply the well-known centroid rule v(x) dx ≈ vold S · v(G), (10.3) S

where G is the center of gravity of the element S. It is easy to see that (10.3) is exact for all v ∈ P1 (S), if S is a simplex. Further we introduce some higher order quadrature formulae in two- and threedimensional space. Example 10.1 The numerical integration formula (see [305, p. 156])

v(x) dx ≈ vol2 T c1 (v(A) + v(B) + v(C)) + c2 v(G)

(10.4)

T

is exact for all v ∈ P3 (T ) (and thus by (10.2) of the order O(h4T )) if we choose the triangular coordinates (a special case of barycentric coordinates) of the nodes as follows A = (a, b, b), B = (b, a, b), C = (b, b, a), G = 13 , 13 , 13 , and coefficients c1 = 25/48, c2 = −9/16, where a = p. 156].)

3 5

and b = 15 . (Note that the values of a and b are incorrectly cited in [305,

10.2 Efficient Quadrature Formulae on Simplices

125

The negative coefficient c2 may cause us to subtract two numbers of almost the same size, which may lead to a significant loss of accuracy (see [55]). Example 10.2 The following formula (see [202, p. 58])

v(x) dx ≈ vol2 T c1 (v(A) + v(B) + v(C))

T

+ c2 (v(D) + v(E) + v(F)) + c3 v(G)

(10.5)

is of the order O(h6T ) if we choose seven integration points A, B, . . . , G lying on medians as shown in Fig. 10.3: A = (a1 , b1 , b1 ), B = (b1 , a1 , b1 ), C = (b1 , b1 , a1 ), D = (a2 , b2 , b2 ), E = (b2 , a2 , b2 ), F = (b2 , b2 , a2 ), G =

1 3

, 13 , 13 ,

and coefficients c1 = (155 − where1

√ √ 15)/1200, c2 = (155 + 15)/1200, c3 = 9/40,

√ √ a1 = (9 + 2 15)/21, a2 = (9 − 2 15)/21, b1 = (6 −

√

15)/21,

b2 = (6 +

√

15)/21.

In [305] these irrational numbers are rounded to rational numbers with only several significant digits. The formula (10.5) is exact for all quintic polynomials v ∈ P5 (T ). This means that the first 1 + 2 + · · · + 6 = 21 terms (!) in the Taylor expansion of an integrated function are computed exactly. In this case the error of numerical integration is often smaller than round-off errors. Integration formulae on a triangle up to order of accuracy 8 are surveyed in [232]. Example 10.3 Let T be an arbitrary tetrahedron with vertices A1 , A2 , A3 , A4 (see Fig. 10.4). Set Ar (10.6) Bq = aAq + b r =q

for every q = 1, 2, 3, 4, where √ √ a = (5 + 3 5)/20, b = (5 − 5)/20. Note that the tetrahedral (barycentric) coordinates of the points B1 , . . . , B4 are (a, b, b, b), . . . , (b, b, b, a), respectively. Each node Bq lies on the spatial median 1 The

value of a1 is incorrectly written in [305, p. 156].

126

10 Variational Crimes

Fig. 10.4 Integration points of cubature formula (10.6)

A4 B4

B1 A1

B3

A3

B2 A2

passing through the vertex Aq . Now a more accurate cubature formula than (10.3) in R3 reads 4 1 v(x) dx ≈ vol3 T v(Bq ) . 4 T q=1 It is exact for all v ∈ P2 (T ). Example 10.4 Let A1 , A2 , A3 , and A4 be vertices of a tetrahedron T . Another cubature formula which is exact for all v ∈ P3 (T ) reads v(x) dx ≈ vol3 T T

where2 Cq = aAq + b

4 9 4 v(Cq ) − v(G) , 20 q=1 5

(10.7)

Ar , q = 1, 2, 3, 4,

r =q

are in the interior of T , a = 21 , b = 16 , and the tetrahedral coordinates of the center of gravity G are ( 41 , 41 , 41 , 41 ), i.e., 1 Aq . 4 q=1 4

G=

For higher order cubature formulae with 11 and 24 integration points inside tetrahedra we refer to [158]. They are exact for all polynomials of the fourth and sixth degree, respectively.

2 The

value of a is incorrectly written in [305, p. 156].

10.2 Efficient Quadrature Formulae on Simplices

127

Example 10.5 Consider now a 4-simplex S with vertices A1 , A2 , A3 , A4 , and A5 . The following quadrature formula is exact for all cubic polynomials v ∈ P3 (S),

5 49 25 v(x) dx ≈ vol4 S v(Dq ) − v(G) , 120 q=1 24 S

where Dq = aAq + b

(10.8)

Ar , q = 1, 2, 3, 4, 5,

r =q

are in the interior of S, a = 37 , b = 17 , and the simplicial (barycentric) coordinates of the center of gravity G are ( 15 , 15 , 15 , 15 , 15 ). Further numerical integration formulae for d -simplices can be found in [81] and [123]. Remark 10.1 Suppose that A0 = (0, 0, . . . , 0), A1 = (1, 0, . . . , 0),. . . , Ad = (0, 0, . . . , d ) are the vertices of the generalized cube-corner d -simplex Sd . According to (2.10), its volume is 1/d !. Let x = (x, y, z, . . . ) ∈ Sd and let e ∈ {1, 2, 3, . . . } be an arbitrary exponent. Now we will show how to evaluate integrals of the monomial function xe over Sd . By the Fubini theorem (or formulae (10.3), (10.4), (10.7), (10.8), . . . ) we find that ⎡

∞ d! xe dx Sd

d ,e=1

1 ⎢2 ⎢ ⎢ ⎢1 ⎢ ⎢3 ⎢ ⎢ ⎢1 ⎢ ⎢ ⎢4 =⎢ ⎢ ⎢1 ⎢ ⎢5 ⎢ ⎢ ⎢1 ⎢ ⎢ ⎢6 ⎢ ⎣ .. .

1 3

1 4

1 5

1 1 1 6 10 15 1 1 1 10 20 35 1 1 1 15 35 70 1 1 1 21 56 126 .. .

.. .

.. .

⎤ 1 ... ⎥ 6 ⎥ ⎥ ⎥ 1 . . .⎥ ⎥ 21 ⎥ ⎥ ⎥ 1 ⎥ . . .⎥ ⎥ 56 ⎥. ⎥ ⎥ 1 . . .⎥ ⎥ 126 ⎥ ⎥ ⎥ 1 ⎥ . . .⎥ ⎥ 252 ⎥ ⎦ .. . . . .

We observe that the second row contains the well-known triangular numbers (d + 1)(d + 2)/2 in the denominators, the third row contains the tetrahedral numbers (d + 1)(d + 2)(d + 3)/6 in the denominators, etc. Moreover, this infinite matrix is symmetric, i.e., d!

xe dx = e! Sd

xd dx. Se

(10.9)

128

10 Variational Crimes

Formula (10.9) has a useful application. It enables us to evaluate easily integrals of lower degree monomials over higher-dimensional simplices. For instance, if d = 100 and e = 1 then we can integrate a higher degree monomial over the interval [0, 1] instead of integrating x over the 100-dimensional cube-corner simplex, that is,

1 x dx = 100! S100

1

x100 dx =

0

1 . 101!

Similarly, for d = 5 and e = 2 we get by (10.9) that x2 dx = S5

2! 5!

x5 dxdy = T

1 60

1

0

1−y

x5 dxdy =

0

1 360

1

(1 − y)6 dy =

0

1 , 2520

where T is the reference triangle with vertices (0, 0), (1, 0), and (0, 1). Integration formulae on rectangular domains are usually derived from one-dimensional Gaussian formulae (see [202]). More details on numerical quadrature can be found in [72, 73, 87, 150, 232, 270, 305], for example.

10.3 Isoparametric Quadratic Elements Let Tˆ be the reference triangle with vertices Aˆ 0 = (0, 0), Aˆ 1 = (1, 0), Aˆ 2 = (0, 1) and the midpoints Aˆ 3 = (1/2, 1/2), Aˆ 4 = (1/2, 0), and Aˆ 5 = (0, 1/2). Assume that Ai , i = 0, . . . , 5, are such that A0 , A1 and A2 do not lie on the same straight line, A4 and A5 are the midpoints of the straight line segment A0 A1 and A0 A2 , respectively. If A3 lies in the shadow area of Fig. 10.5, then the transformation x = F(ˆx) =

5

pˆ i (ˆx)Ai , xˆ ∈ Tˆ ,

(10.10)

i=0

where pˆ i , i = 0, . . . , 5, are the standard quadratic basis functions on the reference element (ˆpi (Aˆ j ) = δij ), is a one-to-one mapping (see [150, p. 242]) from the reference triangle onto a curved triangle T = F(Tˆ ) = {x ∈ R2 | x = F(ˆx), xˆ ∈ Tˆ }.

(10.11)

Hence, F(Aˆ i ) = Ai for i = 0, 1, . . . , 5. The curved side A1 A2 of T is a parabolic arc. The function space PK = {p | p(x) = pˆ (F −1 (x)), x ∈ T , pˆ ∈ P2 (Tˆ )} contains pull-back quadratic polynomials.

10.3 Isoparametric Quadratic Elements

129

Fig. 10.5 The curved triangle T with vertices A0 , A1 , A2 , and A3 . The point A3 = FT (Aˆ 3 ) should be in the shaded area, which is uniquely defined by the distances d and e

A2

e

4

e

A3

d

T

d

4

A0

A1

Let M = 21 (A1 + A2 ). Setting s = A3 − M and w = 4s, transformation (10.10) can be rewritten as the bilinear function x = F(ˆx) = (A1 − A0 , A2 − A0 )ˆx + A0 + xˆ 1 xˆ 2 w,

(10.12)

where (A1 − A0 , A2 − A0 ) is a nonsingular matrix and xˆ = (ˆx1 , xˆ 2 ). The Jacobian matrix of the bilinear transformation (10.12) is JT (ˆx) =

∂F ∂F ∂F = = (A1 − A0 + xˆ 2 w, A2 − A0 + xˆ 1 w), , ∂ xˆ ∂ xˆ 1 ∂ xˆ 2

and its Jacobian det JT (ˆx) = det(A1 − A0 , A2 − A0 ) + xˆ 1 det(A1 − A0 , w) + xˆ 2 det(w, A2 − A0 ) is a linear function in xˆ 1 and xˆ 2 . Therefore, the following statement holds. Theorem 10.1 Under the above assumptions det JT (ˆx) = 0 ∀ˆx ∈ Tˆ if and only if its values at the vertices of Tˆ : det JT (Aˆ 0 ), det JT (Aˆ 1 ), det JT (Aˆ 2 ) are simultaneously all positive or all negative. By the substitution theorem we have

v dx = T

Tˆ

v| ˆ det JT | dˆx.

This formula is employed to calculate the element mass matrix or the element load vector corresponding to the curved element T on the straight reference element Tˆ .

130

10 Variational Crimes

Let us show how to calculate the stiffness matrix, for instance, in the case of the Laplace operator. The entries of the corresponding element stiffness matrix are aij = (grad v i , grad v j )0,T , where v i and v j are pull-back polynomial finite element basis functions. This scalar product is again computed by the substitution theorem on the reference triangle Tˆ as follows (10.13) aij = ((JT−1 ) grad vˆ i , (JT−1 ) grad vˆ j | det JT |)0,Tˆ , where grad v i = (JT−1 ) grad vˆ i . Inserting JT−1 =

1 J ∗, det JT T

where JT∗ is the adjunct matrix, into (10.13), we see that we have to integrate numerically a rational function over Tˆ (see the previous section for efficient quadrature rules in this respect). The corresponding error estimates can be found in [68, pp. 251–259]. Remark 10.2 To avoid the calculation of the ratio 00 at integration points of some quadrature formula, one should employ another numerical quadrature formula. If A4 and A5 are not the midpoints of sides of the straight line segment A0 A1 and A0 A2 , respectively, then the transformation x = F(ˆx) =

5

pˆ i (ˆx)Ai , xˆ ∈ Tˆ ,

i=0

is not bilinear but quadratic, i.e., F ∈ (P2 (Tˆ ))2 . In this case the element T = F(Tˆ ) has, in general, three curved sides (cf. [307, 308]). Example 10.6 Let Tˆ be the triangle with vertices Aˆ 0 = (1, 0), Aˆ 1 = (2, 0), and Aˆ 0 = (2, 2π). Consider the mapping F: Tˆ → T such that F(ˆx1 , xˆ 2 ) = (ˆx1 cos xˆ 2 , xˆ 1 sin xˆ 2 ) and T = F(Tˆ ). Then the Jacobian det

∂F = (cos xˆ 2 )(ˆx1 cos xˆ 2 ) + (sin xˆ 2 )(ˆx1 sin xˆ 2 ) = xˆ 1 ∂ xˆ

is continuous and non-zero over Tˆ . However, we observe that the mapping F is not one-to-one, since F(Aˆ 1 ) = F(Aˆ 2 ). This example shows that the condition that the Jacobian is non-zero is not sufficient to guarantee that a mapping is one-to-one.

10.4 Setting the Problem

131

10.4 Setting the Problem Now we will show how to handle variational crimes in the context of a nonlinear stationary heat conduction problem which has wide applications in practice, see e.g. [203]. It is described by the following quasilinear elliptic problem whose classical formulation reads [140]: Find u ∈ C(Ω) such that u|Ω ∈ C 2 (Ω) and − div(A(· , u)grad u) = f

in Ω,

u = u on ∂Ω,

(10.14) (10.15)

where Ω is a bounded convex domain in R3 with a C 2 -smooth boundary ∂Ω. Since quadrature formulae will be employed later on, we need stronger smoothness assumptions on input data. Thus, let A = (aij )3i,j=1 , aij = aij (x, ξ), x ∈ Ω, ξ ∈ R, and let aij , ∂aij /∂xk , ∂aij /∂ξ be continuous and bounded in Ω × R for all i, j, k = 1, 2, 3. We assume that there exists a positive constant C such that ∂aij ∂ξ (x, ξ) ≤ C ∀x ∈ Ω ∀ξ ∈ R. (10.16) The boundedness of the derivatives ∂aij /∂ξ obviously implies the Lipschitz-continuity of aij with respect to ξ, i.e. there exists a constant CL > 0 such that for all ζ, ξ ∈ R3 and almost all x ∈ Ω we have ∂aij (x, ξ) ≤ C, |aij (x, ξ)| ≤ C, ∂xk

|aij (x, ζ) − aij (x, ξ)| ≤ CL |ζ − ξ|, i, j = 1, 2, 3.

(10.17)

Moreover, let there exist a C0 > 0 such that for almost all x ∈ Ω C0 η η ≤ η A(x, ξ)η ∀ξ ∈ R ∀η ∈ R3 .

(10.18)

1 (Ω), u ∈ Wp1 (Ω) with p > 3 fixed and Finally, let f ∈ W∞

V = H01 (Ω).

(10.19)

For simplicity, a possible dependence of A on x will usually not be explicitly indicated in what follows. Set a(y; w, v) = (A(y)grad w, grad v)0,Ω , y, w, v ∈ H 1 (Ω), F(v) = (f , v)0,Ω , v ∈ H 1 (Ω).

(10.20) (10.21)

Since the entries of A are bounded (cf. (10.16)), we observe that the right-hand side of (10.20) is finite, i.e., a(· ; · , ·) is well-defined.

132

10 Variational Crimes

Definition 10.1 A function u ∈ H 1 (Ω) is said to be a weak solution of the problem (10.14)–(10.15) if u − u ∈ V and a(u; u, v) = F(v) ∀v ∈ V . According to [140], there exists precisely one weak solution u ∈ H 1 (Ω). From the properties (10.16) of the matrix A it follows that |a(y; w, v)| ≤ C w 1,Ω v 1,Ω ∀y, w, v ∈ H 1 (Ω). Moreover, from the Lipschitz continuity (10.17) we have |a(y; w, v) − a(z; w, v)| ≤ |((A(· , y) − A(· , z))grad w, grad v)0,Ω | ≤ A(· , y) − A(· , z) 0,Ω grad w 0,Ω grad v 0,∞,Ω 1 ≤ d 2 CL y − z 0,Ω w 1,Ω v 1,∞,Ω ∀y, z, w ∈ H 1 (Ω) ∀v ∈ W∞ (Ω).

Further we introduce a polyhedral approximation Ωh of Ω. For a given discretization parameter h ∈ (0, h0 ) let Th consist of closed tetrahedra T such that (1) (2) (3) (4) (5)

hT = diam T ≤ Ch for all T ∈ Th , Ω h = T ∈Th T ⊂ Ω, Ωh is convex, all vertices of Ω h belong to ∂Ω, any face of any tetrahedron T ∈ Th is a face of another tetrahedron from Th or a part of the boundary ∂Ωh .

Assumption (3) is used later to define slice and hat elements, see Fig. 10.6. Note that it does not follow from the convexity of Ω, in general. To see this we can easily construct a nonconvex polyhedron which consists only of two tetrahedra and is inscribed in a ball. The set Th will be called a partition of Ω h into tetrahedra. Consider families {Ωh }, h ∈ (0, h0 ), of polyhedral approximations of Ω and {Th }, h ∈ (0, h0 ), of partitions of Ω h into tetrahedra (with h0 > 0 sufficiently small). Let the family {Th }, h ∈ (0, h0 ), be strongly regular, i.e., there exists a C > 0 such that Ch3 ≤ vol3 T for all T ∈ Th and all h ∈ (0, h0 ). A constructive proof of the existence of such a family is given in [142] (for the case d = 2 we refer to [179, 188]). It is clear that at most three vertices of any T ∈ Th belong to ∂Ω provided h is sufficiently small. For any h ∈ (0, h0 ) we set Xh = {vh ∈ C(Ω h ) | vh |T ∈ P1 (T ) ∀T ∈ Th }, Vh = {vh ∈ Xh | vh |∂Ωh = 0},

10.4 Setting the Problem

133

a˜ h (y; w, v) = (A(y) grad w, grad v)0,Ωh , y, w, v ∈ H 1 (Ωh ),

(10.22)

F˜ h (v) = (f , v)0,Ωh , v ∈ H 1 (Ωh ), where P1 (T ) is the space of linear polynomials over the tetrahedron T . In view of the Sobolev imbedding theorem we have u ∈ C(Ω) and, thus, it makes sense to define the Lagrange interpolant πh u ∈ Xh . Recall that πh u(P) = u(P) for every vertex P of T ∈ Th . We set uh = πh u.

(10.23)

Lemma 10.1 The Lagrange interpolant uh has the following properties lim u − uh 1,Ωh = 0,

(10.24)

uh 1,Ωh ≤ C ∀h ∈ (0, h0 ).

(10.25)

h→0

The proof can be done as in [97, Lemma 3.1.3].

10.5 Approximate Solution In solving nonlinear problems, one usually has to employ numerical integration, since integrals over elements cannot be evaluated analytically, in general. We will use the following quadrature formula over an element T ∈ Th , v(x) dx ≈ vol3 T T

Q

cq v(Bq ),

(10.26)

q=1

where v is a continuous function, the weights cq ∈ R are such that cq > 0 and

Q

cq = 1,

q=1

and the nodes Bq ∈ T for q = 1, . . . , Q, see e.g. (10.6). For any vh ∈ Xh and any T ∈ Th we set vT = vh |T .

(10.27)

134

10 Variational Crimes

Definition 10.2 A function uh ∈ Xh is said to be an approximate solution of the problem (10.14)–(10.15) if uh − uh ∈ Vh , ah (uh ; uh , vh ) = Fh (vh ) ∀vh ∈ Vh , where ah (yh ; wh , vh ) =

vol3 T

T ∈Th

Q

cq (A(Bq , yK (Bq ))grad wT ) grad vT

(10.28)

q=1

for yh , wh , vh ∈ Xh and Fh (vh ) =

vol3 T

T ∈Th

Q

cq f (Bq )vK (Bq ), vh ∈ Xh .

q=1

To state the main convergence theorem, we introduce several auxiliary results. Lemma 10.2 The seminorm | · |1,Ωh (in H 1 (Ωh )) is a norm on Vh uniformly equivalent to the norm · 1,Ωh , i.e., there exists a constant C > 0 such that C vh 1,Ωh ≤ |vh |1,Ωh ≤ vh 1,Ωh ∀vh ∈ Vh ∀h ∈ (0, h0 ).

(10.29)

Proof It is well-known that | · |1,Ω and · 1,Ω are equivalent on V = H01 (Ω), i.e., there exists a constant C > 0 independent of v such that C v 1,Ω ≤ |v|1,Ω ≤ v 1,Ω ∀v ∈ V . Any function vh ∈ Vh may be extended by zero on Ω \ Ωh and, thus, (10.29) holds with the same constant. Lemma 10.3 Let (10.27) hold. Then there exist positive constants C1 , C2 , C3 such that ah (yh ; vh , vh ) ≥ C1 vh 21,Ωh ∀yh ∈ Xh ∀vh ∈ Vh , |ah (yh ; wh , vh )| ≤ C2 wh 1,Ωh vh 1,Ωh ∀yh , wh , vh ∈ Xh , |Fh (vh )| ≤ C3 vh 1,Ωh ∀vh ∈ Xh .

The proof can be found in [167].

10.5 Approximate Solution

135

Theorem 10.2 Let the assumptions of Lemma 10.3 and assumptions (10.16) on aij be satisfied. Then, for any h ∈ (0, h0 ) there exists an approximate solution uh ∈ Xh from Definition 10.2. Moreover, there exists a constant C > 0 such that

uh 1,Ωh ≤ C ∀h ∈ (0, h0 ).

Proof The proof is based on Lemma 10.3 and the Brouwer fixed-point theorem. It can be done as in [203, p. 182]. Lemma 10.4 Let (10.27) hold and let the assumptions (10.16) on aij be satisfied. Then there exists a constant C independent of h such that |˜ah (yh ; wh , vh ) − ah (yh ; wh , vh )| 3

≤ Ch|wh |1,Ωh |vh |1,Ωh + Ch1− p |yh |1,Ωh |wh |1,α,Ωh |vh |1,β,Ωh ∀yh , wh , vh ∈ Xh , where p > 3 and either α = p, β = 2 or α = 2, β = p, and a˜ h (·; ·, ·) and ah (·; ·, ·) are defined in (10.4) and (10.28), respectively. Proof See [203, p. 183] with Ω replaced by Ωh . The proof is based on the inverse inequality |vh |1 ≤ Ch−1 |vh |0 with C > 0 (see [68, p. 142]), which holds for any strongly regular family of partitions.

10.6 Slice and Hat Elements Let ωh = Ω \ Ω h . Then,

Ω = Ω h ∪ ωh .

We will decompose the set ω h into special elements of two kinds. Let T1 , . . . , TI¯ be those tetrahedra from Th which have three vertices on ∂Ω. Let Fi = Ti ∩ ∂Ωh , i = 1, . . . , I¯ , i.e., ∂Ωh =

I¯ i=1

Fi .

136

10 Variational Crimes

Fig. 10.6 Partition by hat and slice elements at a smooth curved boundary Fig. 10.7 Hat element

Let F1 , . . . , FI , I ≤ I¯ , be only those faces such that Fi ⊂ ∂Ω, i = 1, . . . , I . If Ω is strictly convex, then I = I¯ . However, if ∂Ω contains a part of a plane, then we can get I < I¯ . Let Ti⊥ be a tetrahedron symmetric to Ti with respect to the plane containing the face Fi , i.e., Fi = Ti ∩ Ti⊥ . (Note that Ti⊥ ∈ / Th .) Define Hi = Ti⊥ ∩ Ω for i = 1, . . . , I . Such a set will be called a hat element, because it looks like a hat on the element Ti (see Figs. 10.6 and 10.7). Since Ωh is convex, the interiors of all hat elements are mutually disjoint. Let S1 , . . . , SJ be the closures of components of the set ωh \ Ii=1 Hi . Any such set Sj , j = 1, . . . , J , will be called a slice element, because it looks like an orange slice (see Figs. 10.6 and 10.8). Thus, we can decompose ω h into hat and slice elements, i.e., ωh =

I i=1

J Hi ∪ Sj . j=1

10.6 Slice and Hat Elements

137

Fig. 10.8 Slice element

Fig. 10.9 The slice elements need not appear between two hat elements

c

F2

F1

F3

F4

Remark 10.3 We introduce a simple example which shows that the slice element need not occur between two neighboring Ti⊥ and Tj⊥ even if Ω is a strictly convex domain. Let Ω be the unit ball and let c be a small circle on its surface ∂Ω. Figure 10.9 shows the position of four elements T1 , T2 , T3 , T4 , each of which has three vertices on c and the fourth vertex is common to all these four elements. For simplicity we have only marked faces F1 , . . . , F4 in the figure. It is easy to see that the hat element H1 is surrounded by three other hat elements H2 , H3 , H4 in such a way that there are no slice elements between H1 and Hi , i = 2, 3, 4. Remark 10.4 The reason why we define two kinds of elements for decomposing ω h is that they help us to extend continuous piecewise linear functions from Ωh to the whole domain Ω so that the extended functions remain continuous and piecewise linear. If we had only considered one kind of element (say, “hat” elements), than

138

10 Variational Crimes

we would not be able to get the continuity of extended functions between adjacent “hat” elements, since any linear function in R3 is uniquely determined by values only at four points, which do not belong to one plane. Three of these values are given at vertices of Fi , i ∈ {1, . . . , I }. The fourth value in each “hat” element cannot guarantee the required continuity over the whole domain Ω. Lemma 10.5 There exists a positive constant C independent of h such that vol3 ωh ≤ Ch2 , h ∈ (0, h0 ). Proof We will present only a short sketch of the proof (for details see [142]). There exists a finite number of overlapping parts of ∂Ω such that each part is a graph of a C 2 -function in some coordinate system. In the same system, a part of ∂Ωh represents a graph of a continuous piecewise linear function which interpolates the part of ∂Ω. Using interpolation properties of linear elements in the C-norm, we come to max dist (x, ∂Ωh ) ≤ Ch2 ,

x∈∂Ω

which can be used to prove the lemma.

(10.30)

Remark 10.5 Even an arbitrarily short part of an edge on the boundary of a convex domain need not lie in one plane, see, e.g., the intersection of two cylinders in Fig. 10.10. Thus, it is not always possible to decompose a neighborhood of such an edge into elements (hat elements, slice elements, tetrahedral elements, …) which have at most one face curved as it is obvious that any edge of one-face curved elements belong to some plane. This is the reason why we do not consider Ω with edges here. Remark 10.6 There are problems also with isolated vertices. Consider, for example, a domain which looks like a drop. Its boundary is, of course, non-differentiable at one point (vertex) only. It is easy to show that the second derivatives of the associated graph are unbounded near this point. To see this assume that the singular point of the drop is located at the origin and let its neighborhood be described by the function f (x1 , x2 ) = − x12 + x22 . Then the second mixed derivative ∂2f x1 x2 (x1 , x2 ) = 2 ∂x1 ∂x2 (x1 + x22 )3/2 is unbounded near the origin. Hence, we are not able to prove Lemma 10.5 for a drop domain by the presented technique.

10.7 A Convergence Result

139

Fig. 10.10 The intersection of two cylinders cannot be partitioned into tetrahedra with at most one curved face

10.7 A Convergence Result Before we state the main convergence theorem, we have to extend functions from Xh to Ω. Using the terminology of [98], we first define an analogue of an “associated function” under our assumption on the given three-dimensional convex domain Ω: if vh ∈ Vh then we extend it by zero in Ω \ Ω h and the resulting function (defined on Ω) is denoted by vˆh and called the associated function with vh . Finding an analogue of a “natural extension” (see also [98] for terminology) is much more complicated in the three-dimensional case. We have to extend functions from Xh to hat and slice elements, both being parts of ω h . Note that our extended functions need to be continuous. This is “automatic” in the two-dimensional case, but in the three-dimensional case we meet several obstacles to guarantee the continuity over Ω. The natural extension vh∗ of vh ∈ Xh will be a piecewise linear function which is continuous on Ω such that vh∗ |Ωh = vh . If vh ∈ Xh then vh∗ on each hat element Hi is defined by vh |Ti using the symmetry with respect to the face Fi , i = 1, . . . , I , i.e., for any x⊥ ∈ Hi we set vh∗ (x⊥ ) = vh (x), where x is the mirror image of x⊥ with respect to Fi .

140

10 Variational Crimes

For the definition of vh∗ on slice elements, we refer to [167]. Now for each h ∈ (0, h0 ) we define a function uh ∈ H 1 (Ω) corresponding to an approximate solution uh as follows: according to Definition 10.2 we express uh in the following form: uh = uh + zh with zh ∈ Vh . Then

uh = u∗h + zˆh ,

where u∗h is the natural extension of uh and zˆh is the associated function. The main convergence result reads as follows. Theorem 10.3 We have uh → u in H 1 (Ω) as h → 0 and lim u − uh 1,Ωh = 0.

h→0

For the proof, see [167].

Chapter 11

0/1-Simplices and 0/1-Triangulations

Write Bd = {0, 1}d for the set of vertices of the unit d-cube I d = [0, 1]d . The convex hull of k + 1 elements of Bd is called a 0/1-simplex whenever it has dimension k. As such it is an example of a 0/1-polytope [154], which is the convex hull of any number of elements of Bd . Questions involving 0/1-simplices that will be reviewed here are related to: • • • •

counting and enumeration, representation as 0/1-matrices, 0/1-simplices that are nonobtuse, orthogonal, or acute, 0/1-matrices and the Hadamard determinant conjecture, triangulations of I d using 0/1-simplices.

In many of these questions it is useful to study 0/1-simplices modulo d-cube symmetries. The motivation to study 0/1-simplices and triangulations varies from attempting to tackle the still unsolved 1893 Hadamard Determinant Conjecture, to trying to find the minimal triangulation of the unit d-cube (solved only up to d = 6) as well as to uncover new algebraic and combinatorial structures that are associated with the rich topic of 0/1-matrices, in general.

11.1 Congruence Versus 0/1-Equivalence Two 0/1-simplices S1 , S2 are called 0/1-equivalent if there exists an element h : I d → I d of the hyperoctahedral group of symmetries of I d such that h(S1 ) = S2 . Since any symmetry is a linear isometric bijection, 0/1-equivalence implies congruence, but unless d ≤ 3, not every congruence of 0/1-simplices is a 0/1-equivalence.

© Springer Nature Switzerland AG 2020 J. Brandts et al., Simplicial Partitions with Applications to the Finite Element Method, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-55677-8_11

141

142

11 0/1-Simplices and 0/1-Triangulations

Fig. 11.1 The two congruent but not 0/1-equivalent 0/1-tetrahedra T1 (left) and T2 (right)

Example 11.1 Consider the matrices ⎡

⎤ 111 ⎢1 0 0⎥ ⎥ P1 = ⎢ ⎣0 1 0⎦ 001

⎡

11 ⎢1 0 and P2 = ⎢ ⎣0 1 00

⎤ 0 1⎥ ⎥. 1⎦ 0

(11.1)

The origin in R4 together with the columns of P2 form the four vertices of a 0/1tetrahedron T2 that is contained in the 3-dimensional facet x4 = 0 of I 4 , whereas the origin together with the columns of P1 form a 0/1-tetrahedron T1 that is not contained in any 3-facet of I 4 . See Fig. 11.1. Thus, T1 and T2 are not 0/1-equivalent, because for any d, each d-cube symmetry d maps any -facet of I d to an -facet of √ I . However, S1 and S2 are both regular tetrahedra with all their edges equal to 2 and hence congruent.

11.2 Representation as 0/1-Matrices In order to study 0/1-simplices modulo 0/1-equivalence, we may without loss of generality assume that one of its vertices is located at the origin. The remaining k vertices can then be represented as a 0/1-matrix of size d × k, similarly as in (11.1). Such a 0/1-matrix is then called a matrix representation of the 0/1-simplex. Matrix representations are far from unique. To see this, let P be a 0/1 matrix representation of a 0/1-simplex S. Swapping columns i and j of P merely changes the order in which the vertices of S are listed in P and thus the resulting matrix represents the same 0/1-simplex S. Furthermore, swapping rows i and j of P corresponds to the cube symmetry of reflection in the hyperplane xi = x j , and thus, modulo 0/1-equivalence,

11.2 Representation as 0/1-Matrices

143

the resulting matrix represents the same 0/1-simplex S. Finally, selecting the jth column c = (c1 , . . . , cd ) of P and then replacing each remaining column d of P by (c + d) mod 2 corresponds to consecutive reflections in those hyperplanes 2xi = 1 for which ci = 1. This product of reflections maps the vertex of S represented by the jth column of P to the origin. We will call the corresponding matrix operation a row complementation induced by a given column. Modulo 0/1-equivalence, the resulting matrix represents the same 0/1-simplex S. Observe that there is a potentially huge number d!k!(k + 1) = d!(k + 1)! of distinct matrix representations of a 0/1-simplex of dimension k in I d from the same 0/1-equivalence class. Example 11.2 The following are matrix representations of 0/1-equivalent tetrahedra in I 4 , ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 101 011 100 100 ⎢1 1 1⎥ ⎢1 1 1⎥ ⎢1 1 1⎥ ⎢0 1 0⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣1 0 0⎦ ∼ ⎣0 1 0⎦ ∼ ⎣0 1 0⎦ ∼ ⎣1 1 1⎦. 010 100 011 110 Swapping columns 1 and 2 gives the second matrix. Swapping rows 1 and 4 yields the third. Finally, add the second column of the third matrix to its first and third column modulo 2. Minimal matrix representation. For any d × k matrix representation P of a 0/1-simplex S, let ν(P) = (20 , 21 , . . . , 2d−1 )PΠ ∈ Nk , where Π is the unique k × k permutation matrix for which the integer entries of the resulting k-vector ν(P) are in increasing order. The matrix P ∗ with ν(P) lexicographically minimal over all matrixrepresentations P of the same 0/1-simplex S modulo 0/1-equivalence is unique and will be called the minimal matrix representation of S. The simplex defined by the origin and the columns of P ∗ we will call the minimal representative of its 0/1-equivalence class. See Fig. 11.2 in which we display four 0/1-equivalent path-tetrahedra together with their matrix representation and vectors ν(P). The matrix P3 is the minimal matrix representation.

11.3 Counting and Enumeration of 0/1-Equivalence Classes The simplest non-trivial 0/1-simplices are the 0/1-triangles, which are the convex hulls of all triples of distinct elements from Bd . Using the concept of minimal matrix representations explained in the previous section, the following results on counting were proved in [34]:

144

11 0/1-Simplices and 0/1-Triangulations

Fig. 11.2 Four 0/1-equivalent 0/1-tetrahedra with corresponding matrix representations

• 0/1-triangles are congruent if and only if they are 0/1-equivalent, • the number of 0/1-equivalence classes of right 0/1-triangles in I d equals d d , Rd = 2 2 • the number of 0/1-equivalence classes of acute 0/1-triangles in I d equals Ad =

2d 3 + 3d 2 − 6d + 9 . 72

By an acute triangle we mean a triangle with three acute angles. Since no 0/1triangle has an obtuse angle, the total number of equivalence classes of 0/1-triangles equals 3 d 2d + 3d 2 − 6d + 9 d + . Δd = Rd + Ad = 2 2 72 In [34] it is also shown how these right and acute 0/1-triangles can be efficiently enumerated. We now generalize the concept “right triangle” to “orthogonal simplex” in higher dimensions.

11.4 Orthogonal 0/1-Simplices Fiedler proved in [103] that any d-simplex has at least d acute dihedral angles, see − d dihedral angles are all also Theorem 5.14. A simplex whose remaining d+1 2 right is called orthogonal or ortho-simplex. The subclass of orthogonal 0/1-simplices

11.4 Orthogonal 0/1-Simplices

145

Fig. 11.3 There are d! upper triangular matrix representations of orthogonal 0/1-simplices

can be characterized recursively as follows [38]: a d-dimensional 0/1-simplex S in I d is orthogonal if and only if it has a (d − 1)-dimensional facet F such that: • F is an orthogonal (d − 1)-simplex contained in a (d − 1)-facet of I d , • the vertex v of S opposite F orthogonally projects onto a vertex of F. At the basis of this recursive characterization lies the single cube edge, or in other words I 1 . A consequence of this recursion is the following result. Theorem 11.1 There exist d! distinct upper triangular 0/1-matrices that represent orthogonal 0/1-simplices in I d . Some of these matrices are enumerated in Fig. 11.3, in “white-black” instead of 0/1-format. The essential feature of the enumeration pattern is that the last column of any k × k matrix equals one of the previous columns (or the zero column) plus the last kth standard basis vector ek , such that the vertex represented by this column orthogonally projects onto a vertex of the opposite (orthogonal) facet. This also immediately proves the proposition. Remark 11.1 Modulo 0/1-equivalence, there remain as many as the number of unlabeled trees on n + 1 vertices. Indeed, it is not hard to verify that two matrices representing orthogonal 0/1-simplices can be transformed into one another using row and column permutations and row complementations if and only if the spanning trees of orthogonal edges of the simplices corresponding to these matrices are isomorphic as graphs. The isomorphism problem for unlabeled trees is thus rephrased as a problem of 0/1-equivalence of 0/1-simplices. In I 3 there are modulo 0/1-equivalence only two orthogonal tetrahedra, depicted in the left half in Fig. 11.4, and only four tetrahedra altogether.

146

11 0/1-Simplices and 0/1-Triangulations

Fig. 11.4 The only four 0/1-tetrahedra in I 3 modulo 0/1-equivalence. The two on the left are orthogonal tetrahedra, usually called the cube-corner (far left) and the path-tetrahedron Table 11.1 The number a(d) of acute 0/1 d-simplices in I d related to their total number s(d), which includes all the degenerate ones. All cardinalities are modulo 0/1-equivalence d 1 2 3 4 5 6 7 8 9 10 11 a(d) 1 s(d) 1

0 1

1 6

1 27

2 6 472 19735

13 2773763

29 67 1245930065 1.8e12

162 8.7e15

392 1.3e20

Fig. 11.5 The upper Hessenberg matrix Hλ corresponding to the composition λ = λ1 , . . . , λk of d − 1, and its determinant as the numerator of the continued fraction [λ1 ; λ2 , . . . , λk ]

11.5 Acute 0/1-Simplices An acute 0/1-simplex is a 0/1-simplex whose dihedral angles are all acute. As it turns out, they are quite rare in comparison to the total number of 0/1-simplices, see Table 11.1. The computational enumeration of acute 0/1-simplices that led to Table 11.1 is a difficult task, which involves Pólya’s theory of counting. Details, including an overview of minimal matrix representations of all d-dimensional acute 0/1-simplices in I d for d ≤ 9 are given in [35]. A subset of the acute 0/1-simplices that can be fully characterized are those that have a matrix representation that is unreduced upper Hessenberg. To illustrate this, let d ≥ 3, and let the ordered tuple λ = λ1 , . . . , λk be a composition of d − 1 into k parts with λ1 ≥ 2 ≤ λk . Define Hλ ∈ Rd×d as is done in Fig. 11.5 for λ = 3, 1, 2, 2 . As illustrated in Fig. 11.5, the matrix Hλ has identity matrices I j of size λ j × λ j whose entries equal to one fill the lower co-diagonal. Further, I1 , . . . , Ik define a

11.5 Acute 0/1-Simplices

147

checkerboard pattern in Hλ above I1 , . . . , Ik , with blocks containing either ones or zeros. Each block bordering I j and I j+1 contains only ones. In [35] we proved the following results in this context. Theorem 11.2 Let H be a d × d unreduced upper Hessenberg 0/1-matrix whose columns and the origin are the d + 1 vertices of a 0/1-simplex S ⊂ I d with acute dihedral angles only. Then, possibly after exchanging its first two rows and/or last two columns, H is equal to the matrix Hλ for some composition λ = λ1 , . . . , λk of d − 1 with first and last parts larger than one. Moreover, | det(Hλ )| = f k ,

where

fk 1 = λ1 + gk λ2 + ..

1

,

gcd( f k , gk ) = 1.

(11.2)

. + λ1k

Conversely, each such matrix Hλ has the property that its Gramian is strictly ultrametric [236], which implies that its columns together with the origin are the vertices of an acute 0/1-simplex. As a corollary of this theorem, all attainable absolute values of the determinant function on the set of all unreduced d × d upper Hessenberg 0/1-matrices H for which (H H )−1 is a diagonally dominant Stieltjes matrix with negative off-diagonal entries can be explicitly read from a part of Kepler’s Tree of Fractions [162]. This part is depicted in Fig. 11.6. At the root with fraction 21 we have d = 3. The children p q and p+q . Transversing the tree level by level of a vertex qp with p, q ∈ N are p+q yields an enumeration of all the rationals Q ∩ (0, 1). The circled integers displayed below each vertex in Fig. 11.6 equal the sum of the numerator and denominator of the fraction belonging to that vertex. At level k these integers correspond to the absolute values of the determinants of each of the 2k matrices Hλ of size (k + 4) × (k + 4). The leftmost branch of this tree corresponds to the so-called [40] antipodal simplices in I d . These√are the convex hulls of a vertex v of I d , together with the d vertices of I d at distance d − 1 to v. The determinants of their matrix representations in dimension d equal d − 1. The determinants in the rightmost branch in the tree equal the Fibonacci numbers, which were proved in [66] to be the maximal values of the determinant function over all d × d Hessenberg 0/1-matrices. We can now conclude that this maximum is (also) attained by matrices representing acute simplices. More generally we see that any branch of the tree that, starting at a given vertex p/q corresponding to the value p + q of the determinant, extends only to the right yields a Fibonacci-type sequence F( j), j = 0, 1, 2, . . . , F( j + 2) = F( j) + F( j + 1) with F(−1) = p,

F(0) = q and F(1) = p + q,

(11.3)

148

11 0/1-Simplices and 0/1-Triangulations

Fig. 11.6 Part of Kepler’s Tree of Fractions and absolute determinants of the matrices Hλ

whereas any branch from a vertex p/q that extends only to the left yields a family of acute 0/1-simplices with determinants increasing linearly as L( j) = j p + q.

(11.4)

The corresponding matrices Hλ in this latter case have integer compositions of which the last part increases by one when the size of Hλ increases by one while all the other parts of λ remain the same. The existence of such families with linearly increasing determinants was first observed in [40]. They will also be discussed in Sect. 11.7.

11.6 Neighbor Theorems for Nonobtuse Simplices For each v ∈ Bd , write v for its antipodal in I d . In other words, v + v = (1, 1, . . . , 1) .

(11.5)

An interesting property [38] of any d-dimensional acute 0/1-simplex S is that, face-to-face to each of its (d − 1)-facets, it has at most one acute neighboring ddimensional 0/1-simplex. Theorem 11.3 Let S ⊂ I d be a d-dimensional acute 0/1-simplex and v ∈ Bd a vertex of S opposite a facet F of S of dimension d − 1. If the convex hull of w ∈ Bd and F is also a d-dimensional acute 0/1-simplex, then w = v. This theorem can be understood as follows. Assume that S is an acute 0/1-simplex and that the origin v = 0 is a vertex of S. Write F for the facet of S opposite v. Because

11.6 Neighbor Theorems for Nonobtuse Simplices

149

S is acute, v orthogonally projects into the interior of its opposite facet. Thus, the perpendicular from v onto F points into the interior of I d . In particular, the vector q that represents this perpendicular has positive entries only. It is now easy to verify that for each vertex w of I d other than the origin or its antipodal e, the point w + αq lies in I d only for α = 0, and thus w cannot orthogonally project into F. Finally, note that this argument does not only apply to the vertex of S that is located at the origin. See Fig. 11.7 for an illustration. Remark 11.2 Theorem 11.3 does not claim that the convex hull of v and F is an acute simplex, only that v is the only vertex other than v for which this may be the case. Indeed, the acute tetrahedron in Fig. 11.7 only has cube corners as face-to-face neighbors, which are not acute. Remark 11.3 Another geometric consequence is that if S is an acute 0/1-simplex, none of its k-dimensional facets is contained in a k-dimensional facet of I d for k ∈ {1, . . . , d − 1}. Indeed, given a k-facet F of I d , each vertex of I d orthogonally projects on a vertex of F and hence not in its interior. See Fig. 11.8 for an illustration. A linear algebraic consequence in terms of matrix representations is the following. Let P ∈ Bd×d be a matrix representation of an acute 0/1-simplex. Consider the matrix Q = P − for which obviously Q P = I. (11.6) Thus, the jth column q j of Q is orthogonal to all the columns pi of P with i = j whereas q j p j = 1, and hence q j is an inward pointing normal vector to the facet

Fig. 11.7 Illustration of the neighbor theorem for acute 0/1-simplices

Fig. 11.8 For each facet F of I d , each vertex of I d projects onto a vertex of F. Consequently, no acute 0/1-simplex has a facet contained in a facet of I d of the same dimension

150

11 0/1-Simplices and 0/1-Triangulations

F j of P opposite p j . In particular, p j − q j points into I d , which lies at the basis of the following theorem. Recall that a doubly stochastic matrix is a square matrix with nonnegative entries such that each row sum and each column sum is equal to one, and that a row substochastic matrix is a square matrix with nonnegative entries such that each row sum is at most one. Theorem 11.4 Let P ∈ Bd×d be a matrix representation of an acute 0/1-simplex S, and write Q = P − . Then qi j > 0 ⇔ pi j = 1 and qi j < 0 ⇔ pi j = 0.

(11.7)

Defining 0 ≤ C = (ci j ) and 0 ≤ D = (di j ) by C=

1 (|Q| − Q) 2

and D =

1 (|Q| + Q) , 2

(11.8)

where |Q| is the matrix whose entries are the moduli of the entries of Q, we have that Q = D − C, (11.9) where D is doubly stochastic and C row substochastic. For the proof, see [36]. Theorem 11.4 is illustrated in Fig. 11.9. The entries equal to one in P are boxed, as are the entries at the corresponding positions in P − . These form the doubly stochastic matrix D. For nonobtuse 0/1-simplices, similar neighbor results are much harder to prove. This is motivated by the simple example in Fig. 11.10. The path-tetrahedron in

Fig. 11.9 Illustration of Theorem 11.4

Fig. 11.10 Four vertices of I 3 can move in the direction of ±q and remain in I 3

11.6 Neighbor Theorems for Nonobtuse Simplices

151

Fig. 11.10 has a normal q to the indicated interior facet F that is parallel to two opposite facets of the cube. Hence, q has an entry equal to zero. This is very different than in the acute case. Thus, there are four vertices of I 3 (the white ones) that, when following the direction ±q, remain in I 3 . Remark 11.4 Generally, I d has 2 vertices that remain in I d when translating them in the direction ±q whenever q has entries equal to zero. Nevertheless, no example is yet known of a nonobtuse 0/1-simplex having more than one nonobtuse neighbor to any of its facets. The corresponding conjecture is called the one-neighbor conjecture for nonobtuse 0/1-simplices. Proofs of this conjecture for special subclasses of nonobtuse 0/1-simplices can be found in [36]. The following theorem is similar to Theorem 11.4 and is a consequence of the above observations. Its proof differs from the proof of Theorem 11.4 in the sense that p j − αq j is now merely an element of I d including its boundary, instead of the interior of I d . Theorem 11.5 Let P ∈ Bd×d be a matrix representation of a nonobtuse 0/1-simplex S, and write Q = P − . Then qi j > 0 ⇒ pi j = 1 and qi j < 0 ⇒ pi j = 0.

(11.10)

Defining 0 ≤ C = (ci j ) and 0 ≤ D = (di j ) by C=

1 (|Q| − Q) 2

and D =

1 (|Q| + Q) , 2

(11.11)

where |Q| is the matrix whose entries are the moduli of the entries of Q, we have that Q = D − C, (11.12) where D is doubly stochastic and C row substochastic.

11.7 The Hadamard Conjecture In 1893, Jacques Hadamard [127] proved that for any 0/1-matrix P ∈ Bd×d , √ d+1 d +1 | det(P)| ≤ Δd := 2 , 2

(11.13)

which is called the Hadamard inequality. It has many applications in number theory, combinatorics, the theory of error-correcting codes, group theory, and the theory of Fredholm integral equations, see [52] for several other applications.

152

11 0/1-Simplices and 0/1-Triangulations

As a consequence of (11.13), Δd /d! is an upper bound for the volume of a 0/1simplex in I d . For d = 3 this upper bound Δ3 1 = 3! 3 is attained by the regular 0/1-tetrahedra in the cube that result after slicing away four of its corners, see the far right cube in Fig. 11.4. The Hadamard maximum determinant conjecture states that for any dimension d whose remainder after division by 4 equals 3, there exists a P ∈ Bd×d for which the upper bound in (11.13) is attained. Hadamard already proved [127] that it cannot be attained for other values of d. Theorem 11.6 If d > 1 is not congruent to 3 modulo 4 then the bound (11.13) cannot be attained. Currently, the conjecture has been successfully verified for all d ≡ 3 mod 4 up to d = 667, as well as for a number of infinite parametrized families. It is not difficult to prove [120] that a 0/1-simplex whose volume attains the upper bound is regular, and thus in particular acute. Theorem 11.7 Any 0/1-simplex whose volume equals Δd /d! is regular. In fact, whenever P ∈ Bd×d is a matrix representation of such a regular simplex, then

d +1 Id + ed ed , (11.14) P P = 4 where

ed = (1, . . . , 1) ∈ Rd ,

and Id ∈ Rd×d is the identity matrix. Thus, Hadamard’s conjecture is equivalent to the existence of a regular 0/1-simplex S in I d for all d ≡ 3 mod 4.

(11.15)

A congruent copy Sˆ of S exists as a 0/1-simplex in I m with m = 41 (d + 1)2 > d. A simple proof of this observation is to define = 41 (d + 1) ∈ N and to write ⊗ for the Kronecker product. Then with

e K = e ⊗ d Id we have that K K =

d +1 Id + ed ed , 4

(11.16)

11.7 The Hadamard Conjecture

153

showing that K ∈ Bm×d is a matrix representation of Sˆ in I m . Therefore, the general question that remains is how to orthogonally transform Sˆ ⊂ I m into I d . Example 11.3 For d = 3 and = 1 and m = 4, the matrix K in (11.16) equals the matrix P1 from (11.1), and it can be orthogonally transformed into P2 from (11.1) as follows, ⎡

⎤⎡ 11 1 1 1 −1 ⎥⎢1 0 1⎢ 1 1 −1 1 ⎥⎢ QK = ⎢ 2 ⎣ 1 −1 1 1 ⎦ ⎣ 0 1 −1 1 1 1 00

⎤ ⎡ 11 1 ⎢1 0 0⎥ ⎥=⎢ 0⎦ ⎣0 1 00 1

⎤ 0 1⎥ ⎥= H . 1⎦ 0 0

(11.17)

Since the last coordinate of each column of Q K vanishes, the tetrahedron with the origin and the columns of Q K as vertices lies in a 3-dimensional facet of I 4 and hence we have found the required regular tetrahedron in I 3 using its congruent copy in I 4 as a starting point. In general, of course, the orthogonal transformation I m → I d of the embedding is not known. The above discussion may also raise the question in which dimensions acute 0/1-simplices have maximal volume, over all simplices. In Table 11.2 we display d! times the maximal volume Vd of any 0/1-simplex versus d! times the maximal volume Ad of acute 0/1-simplices. The numbers d!Vd can be found in the On-Line Encyclopedia of Integer Sequences as item A003432, whereas the numbers d!Ad were computed in [35]. Not before dimension d = 9, but also for d = 10 and d = 13 do the numbers differ, illustrating Günter Ziegler’s claim [154]: Low-dimensional intuition does not work! Generally, of course, Ad is a lower bound for Vd . In other words, the maximum volume among all 0/1-simplices is attained for an acute simplex if d = 3, 4, 5, 6, 7, 8. However, the fact that some property is satisfied for d = 3, 4, 5, 6, 7, 8 does not mean that it holds for all d > 8. This is expressed by the strong law of small numbers, see [125, 126]. Numerical experiments show that for d = 9 the 0/1-simplex with the maximum volume 144/(9!) is surprisingly not acute, see Table 11.2.

Table 11.2 Maximum determinants of all, versus of all acute 0/1-simplices. The columns corresponding to (11.15) are in bold d 2 3 4 5 6 7 8 9 10 11 12 13 d!Vd d!Ad

1 1

2 2

3 3

5 5

9 9

32 32

56 56

144 96

320 224

1458 1458

3645 3645

9477 7290

154

11 0/1-Simplices and 0/1-Triangulations

Fig. 11.11 Possible determinants of 0/1-matrices representing acute 0/1-simplices

The attainable volumes of all acute simplices, or equivalently, the determinants of all matrix representations of acute 0/1-simplices, are partly displayed in Fig. 11.11. Some of the families with linearly increasing determinant, that were already identified in Sect. 11.5, are highlighted, such as the left branch in Fig. 11.6 with determinants 3, 4, 5, 6, . . . starting for d = 4 at the fraction 21 (called the antipodal family), the left branch with determinants 5, 7, 9, 11, . . . starting for d = 5 at 23 , the left branch with determinants 7, 10, 13, . . . starting for d = 6 at 34 , and the left branch with determinants 8, 11, 14, . . . starting for d = 6 at 35 . All possible determinants of acute 0/1-simplices up to dimension 13 have been computed but are not included in Fig. 11.11 as the range on the horizontal axis would need to be 7290.

11.8 Triangulations of I d Using 0/1-Simplices The topic of triangulating the unit d-cube I d = [0, 1]d using 0/1-simplices started with the 1976 paper [231] by Patrick Scott Mara, who raised the question of the minimal number Md of 0/1-simplices needed to triangulate I d . It is relatively easy

11.8 Triangulations of I d Using 0/1-Simplices

155

to show that M3 = 5. The corresponding triangulation was already mentioned in Sect. 11.7 and consists of the regular tetrahedron and four cube corners. Its other triangulations consist of six path-tetrahedra. Note that Md ≥ 2Md−1 . Indeed, each triangulation of I d induces triangulations of its facets, and simplicial facets contained in opposite cube facets cannot belong to the same simplex. In [40], the following strengthening of the inequality Md ≥ 2Md−1 was proved, using the upper bound Δd from (11.13) on the volume of a binary 0/1-simplex, Md ≥ 2Md−1 +

(d − 1)!(d − 2) . Δn

(11.18)

Together with the trivial M2 = 2, this bound immediately implies that M3 ≥ 5 and moreover that M4 ≥ 14. This latter bound is however not attainable. For this, note that the total volume of the at least 2M3 = 10 simplices that have exterior facets in two opposite facets of I 4 equals 21 . The key observation, proved in [40], is now that the remaining volume of 21 can contain only one 0/1-simplex of maximum 3 volume 24 , because all such simplices intersect and thus at most one can figure in a 9 2 triangulation. To fill the remaining volume of 24 at least 4 simplices of volume 24 1 are required and one of volume 24 . This shows that M4 ≥ 16 without the need to use a computer, as was done in the original proof that M4 indeed equals 16 in [74]. Table 11.3 shows the currently known values of Md . The next line contains the smallest number Nd of nonobtuse 0/1-simplices that is needed to triangulate I d . This topic was studied in [38]. The topic of triangulating I d using only nonobtuse 0/1-simplices starts with I 3 . Modulo 0/1-equivalence, there are only four distinct 0/1-tetrahedra in I 3 , of which three are nonobtuse.

Table 11.3 The minimum number Md of 0/1-simplices together with the minimum number Nd of nonobtuse 0/1-simplices that are needed to triangulate I d d 1 2 3 4 5 6 7 8 9 Md Nd

1 1

2 2

5 5

16 18

67 87

308 518

1493 3621

28962

260651

Fig. 11.12 The 0/1-distinct 0/1-tetrahedra in I 3 . Only the one at the right is not nonobtuse

156

11 0/1-Simplices and 0/1-Triangulations

It is easy to verify that there are only two 0/1-distinct nonobtuse triangulations of I 3 . One consists of the regular tetrahedron (displayed at the left in Fig. 11.12) and four cube corners (displayed next to it), see Fig. 5.9. The other is the Kuhn triangulation consisting of six path-tetrahedra, one of which is to the right of the cube corner. In particular, the minimal triangulation of I 3 consists of nonobtuse simplices. This observation led to the question of whether this is more generally the case. The following is the main result in [38]. Theorem 11.8 Let d ≥ 3. Apart from the Kuhn triangulation of I d into d! path-simplices, there exists modulo the symmetries of I d only one other family of triangulations of I d consisting of nonobtuse 0/1-simplices only. It consists of Nd 0/1-simplices, where Nd = d(Nd−1 − 1) + 2 with N3 = 5.

(11.19)

With e = 2.718281 . . . this can also be expressed as d 1 Nd = 1 + d! = d!(e − 2) . k! k=2

(11.20)

Because d!(e − 2) < d! this triangulation is the minimal nonobtuse triangulation of I d . Some smaller values of Nd are displayed in Table 11.3. Obviously, already for d = 4, the minimal triangulation does not consist of nonobtuse 0/1-simplices only. The charm of Theorem 11.8 lies in the fact that it completely describes all nonobtuse triangulations of I d in any dimension. We will now describe the construction of the family of minimal nonobtuse triangulations from Theorem 11.8. It is a special case of what is called coning off to a vertex of a convex polytope Ω. In general, a triangulation of Ω can be constructed by choosing a vertex p and then triangulating all (d − 1)-dimensional facets of Ω that do not have p as a vertex. The convex hulls of p with each of the (d − 1)-dimensional simplices just created then results in a triangulation of Ω itself. Applying this procedure to I d means to triangulate the d facets of I d that do not contain p and to take convex hulls with p. This is illustrated for d = 3 in Fig. 11.13. In the first two pictures in Fig. 11.13, we look into the open box formed by three facets of I 3 . On the left, √ these facets are triangulated so that all six triangles have a vertex at distance 2 from the origin (the white bullet). Thus, the orthogonal projection of p = (1, 1, 1) onto such a facet lands on a vertex of that facet, which is a necessary and in this case also sufficient condition that coning off to p results in six nonobtuse (in this case path-)tetrahedra. In the middle picture, coning off to p results in three path-tetrahedra and three obtuse tetrahedra of the type depicted in the right of Fig. 11.12. Slicing off a cube corner from I 3 and then coning off to p, as in the right picture in Fig. 11.12, again leads to a nonobtuse triangulation.

11.8 Triangulations of I d Using 0/1-Simplices

157

Fig. 11.13 Coning off to a vertex p of I 3 , and of I 3 with a cube corner sliced off

Now, consider the four-cube I 4 . Slice away the 4-cube corner at the origin. This corresponds to slicing away the 3-cube corners of all four three-dimensional facets of I 4 at the origin. Each of these facets can be triangulated using one regular tetrahedron √ and three 3-cube corners. Each of these four tetrahedra has a vertex at distance 3 from the origin, as is visible in the right picture in Fig. 11.13. Thus, coning off to p = (1, 1, 1, 1) then leads to a nonobtuse triangulation of I 4 with the cube corner removed. Gluing back the cube corner then results in a triangulation of I 4 into 18 nonobtuse 0/1-simplices: 4 × 4 from the convex hulls with the tetrahedra in the facets of I 4 , one 4-cube corner, and its antipodal. Note that 18 = 4 × 4 + 1 + 1 equals d(Nd−1 − 1) + 2 for d = 4, see the recursion (11.19). An inductive construction now leads to the nonobtuse family from Theorem 11.8. A similar inductive construction can also be used for the standard triangulation into d! path-simplices. An important ingredient in the proof of uniqueness is that any nonobtuse simplex has nonobtuse facets [103]. Thus, any nonobtuse triangulation of I d induces a nonobtuse triangulation of each of its 2d facets of dimension d − 1. This imposes severe restrictions on triangulations of I d for them to consist of nonobtuse 0/1-simplices. Ultimately, any 3-dimensional facet of I d needs to be triangulated by one of its two nonobtuse triangulations. Using neighbor-theorems like in Sect. 11.6 it can be shown that if a nonobtuse triangulation of I d contains a path-simplex, it is the Kuhn triangulation, and that if it contains a cube corner, it is the family from Theorem 11.8. Using counting arguments and arguments based on the shapes of the diverse orthogonal simplices as defined in Sect. 11.4, it can finally be shown that any nonobtuse triangulation of I d contains either a path-simplex or a cube corner.

Chapter 12

Tessellations of Maximally Symmetric Manifolds

12.1 Regular Polytopes In arbitrary space dimension d ≥ 2 there exist three maximally symmetric manifolds: the Euclidean space Rd , the hypersphere Sd , and the pseudosphere Hd , which remain the same after translations and rotations. According to [244, p. 721] and [295, Chap. 13], up to scaling there are no other maximally symmetric manifolds. A natural question arises: Can we tile these manifolds face-to-face by congruent regular simplicial cells? These cells are, of course, curved when the manifold is not Euclidean. Edges of these tiles are geodesics and their faces consist of families of geodesics. Regular simplicial tessellations (partitions) represent a certain mesh showing how the particular manifold is bent. For simplicity, we shall often omit the term “regular” from now on. Recall that the hypersphere Sd is defined as follows Sd = {x ∈ Rd+1 | x = 1}, where · denotes the Euclidean norm, i.e., the sectional curvature of Sd equals 1 at every point. The hyperbolic space Hd is the maximally symmetric, simply connected, d-dimensional Riemannian manifold whose sectional curvature equals (−1) at every point. Further, we recall all regular polytopes in the Euclidean space Rd . The only regular polytope in the trivial case d = 1 is any straight line segment. For d = 2 there are infinitely many regular polygons: equilateral triangles, squares, etc. However, for d = 3 there exist only five regular polyhedra (up to scaling) all of whose faces are congruent regular polygons with the same number of faces meeting at each vertex [18]. They are called Platonic bodies (solids), see Fig. 12.1 and Table 12.1.

© Springer Nature Switzerland AG 2020 J. Brandts et al., Simplicial Partitions with Applications to the Finite Element Method, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-55677-8_12

159

160

12 Tessellations of Maximally Symmetric Manifolds

Fig. 12.1 Platonic bodies and their partitions into path-tetrahedra Table 12.1 Platonic bodies in R3 . The symbol f denotes the number of faces, e the number of edges, v the number of vertices, n is the degree of each vertex, and α is the dihedral angle (rounded) between two neighboring √ faces. The last column shows the volume provided that each edge has the length a, where τ = ( 5 + 1)/2 is the golden section (cf. [71, p. 455]) Name f e v n α Volume √ 3 ◦ Tetrahedron 4 triangles 6 4 3 70.529 2a /12 Cube 6 squares 12 8 3 90◦ a3 √ 3 Octahedron 8 triangles 12 6 4 109.471◦ 2a /3 √ 3 4 Dodecahedron 12 pentagons 30 20 3 116.565◦ 5a τ /2 Icosahedron 20 tiangles 30 12 5 138.19◦ 5a 3 τ 2 /6

Denoting by f , e, and v the number of their faces, edges, and vertices, respectively (see Table 12.1), the well-known Euler formula for more general polyhedra v+ f =e+2

(12.1)

holds. The number n of edges coming together at each vertex is called the vertex degree. Hence, we have nv = 2e. (12.2)

12.1 Regular Polytopes

161

Table 12.2 Regular polytopes in R4 . The symbol c denotes the number of three-dimensional convex polyhedral cells on their surfaces and the other symbols have the same meaning as √ for d = 3. The last column shows the volume provided that each edge has length a, where τ = ( 5 + 1)/2 is the golden section Name c f e v n Volume √ 4-simplex 5 tetrahedra 10 10 5 4 10a 4 /192 4 4-cube 8 cubes 24 32 16 4 a 4-orthoplex 16 tetrahedra 32 24 8 6 a 4 /6 24-cell 24 octahedra 96 96 24 8 2a 4 √ 120-cell 120 dodecahedra 720 1200 600 4 15 5a 4 τ 8 /2 600-cell 600 tetrahedra 1200 720 120 12 25a 4 τ 3 /4

The following definition is based on induction. A convex polytope in Rd is called regular if all its (d − 1)-dimensional facets are congruent regular polytopes and all its vertices have the same vertex degree. For d = 4 there are exactly six regular polytopes (up to scaling), see Table 12.2 and [234]. They were discovered by Ludwig Schläfli (1814–1895). The Euler formula (12.1) can be then generalized as follows (see (5.2)) v+ f =e+c

: Euler–Poincaré formula

(12.3)

and (12.2) holds again. Here c stands for the number of three-dimensional polyhedral cells. The following theorem is well known [266]. Theorem 12.1 For any d ≥ 5 there are only three regular polytopes in Rd : the d-simplex, the d-cube, and the d-orthoplex.

12.2 Regular Triangular Tessellations In this section we recall some well-known facts about regular triangular tessellations of R2 , S2 , and H2 . They will be applied in Sect. 12.3 to establish local properties of regular tessellations for d = 3. By a regular (equilateral) triangle in these manifolds we shall mean a bounded domain all three sides of which are geodesics of the same length and whose three angles have the same size. There is, up to scaling, translation, and rotation, just one regular tessellation of the Euclidean plane R2 by equilateral triangles. Consider now three-dimensional Platonic bodies with triangular faces all of whose vertices lie on S2 . The radial (i.e. central orthographic) projection from the center of S2 then maps all edges onto S2 . This produces three regular tessellations of S2 by equilateral spherical triangles. Their vertices are thus connected by geodesics and the sum of the angles around each vertex is 360◦ . We will call these regular tessellations

162

12 Tessellations of Maximally Symmetric Manifolds

Fig. 12.2 Regular simplicial tessellations of the sphere S2 : the projected tetrahedron, the projected octahedron, and the projected icosahedron. The corresponding vertex angles are 120◦ , 90◦ , and 72◦

the projected tetrahedron, the projected octahedron, and the projected icosahedron (see Fig. 12.2). Theorem 12.2 All regular triangular tessellations of the unit sphere S2 are obtained by the radial projection of the regular tetrahedron, octahedron, and icosahedron onto S2 . Proof Formulae (12.1) and (12.2) obviously also hold for any tessellation of S2 . Moreover, for triangular spherical tiles we get 2e = 3 f,

(12.4)

since any edge is shared by two triangles and any triangle has three edges. Multiplying (12.1) by 2n, we get 2nv + 2n f = 2ne + 4n. Substituting there (12.2) and (12.4), we find that 6 f + 2n f = 3n f + 4n. From this we obtain the following necessary condition (6 − n) f = 4n (12.5) which any regular triangular tessellation of S2 has to satisfy. We see that for n ≥ 6 Eq. (12.5) has no solution and for n = 1 there is no integer solution. If n = 2, the unique solution f = 2 corresponding to two hemispheres is not applicable, because they do not have well-defined vertices. By inspection we find that (12.5) has only three solutions n = 3, 4, 5 and f = 4, 8, 20, respectively. We observe that these three solutions can be obtained by the radial projection of the corresponding Platonic bodies onto S2 (cf. Fig. 12.2) and there are no other regular triangular tessellations of S2 . Remark 12.1 The size of the above three spherical space-fillers of S2 cannot be arbitrary (as in R2 ), since they have to fit to the unit sphere. By [249, p. 83], their edges have uniquely determined sizes π − arccos(1/3), π/2, and 2 arctan(1/τ ), respec√ tively, where τ = ( 5 + 1)/2 is the golden section. In higher-dimensional simplicial tessellations of Sd the edges also have uniquely determined sizes for any d ≥ 2.

12.2 Regular Triangular Tessellations

163

The hyperbolic plane is an unbounded two-dimensional manifold. It has a somewhat counter-intuitive geometry. In 1901, David Hilbert proved that the hyperbolic plane H2 cannot be isometrically imbedded into R3 (see [138]). By an isometry we mean a continuous mapping F : M → M, whose inverse exists and is also continuous, and which preserves distances on the manifold M. In other words, ρ(F(A), F(B)) = ρ(A, B) for all A, B ∈ M, where ρ is a metric on M. Similarly, the well-known Klein bottle is a two-dimensional manifold isometrically imbeddable in R4 , but not in R3 . Hilbert’s statement is usually formulated as follows: Theorem 12.3 There is no complete regular two-dimensional manifold of negative constant Gaussian curvature imbedded in R3 . Later, Danilo Blanuša [25] showed that the hyperbolic plane can be isometrically imbedded into R6 . Therefore, one should deform the hyperbolic plane to get some idea of what its regular tessellations look like. There are at least six basic ways1 to perform such a deformation (see [60]). One way is to use the well-known Poincaré disc2 in which all angles between geodesics in H2 are preserved. Here the hyperbolic plane is represented by the interior of the unit circle k in R2 . Geodesics are then circular arcs that are orthogonal to k (see Fig. 12.3). We can easily show that there exists just one circular arc passing through two different arbitrary points A and B that is perpendicular to k at its endpoints P ∈ k and Q ∈ k. It may degenerate to a straight line segment (diameter). The distance between A and B is then given by the relation (see [244, p. 36]) |AQ| · |B P| , (12.6) ρ(A, B) = ln |A P| · |B Q| where ln is the natural logarithm and |A P|, |AQ|, |B P|, and |B Q| denote the standard Euclidean distances along straight lines. The function ρ satisfies the four basic properties required for a metric: 1) it is nonnegative, 2) ρ(A, B) = 0 if and only if A = B, 3) ρ(A, B) = ρ(B, A), and 4) the triangle inequality. The circle in the Euclidean plane with radius R > 0 in the metric (12.6) has a larger circumference than 2π R. Since the boundary circle k is unit in the standard Euclidean norm, the concentric circle k’ with radius R’= 1 in the hyperbolic metric (12.6) has length = 7.384 . . . instead of the usual 2π = 6.283 . . . (see Fig. 12.3). This is similar to measuring the length of a “circle” around a saddle point on some surface. The length of any unit circle with an arbitrary center in the hyperbolic plane is always the same number in the metric (12.6). A curve that has a constant Euclidean distance from the hyperbolic line is not a line (i.e. geodesic) in the hyperbolic plane, see [60, p. 88]. ˜ 2 = {(x0 , x1 , x2 ) ∈ R3 | x 2 + x 2 − x 2 = −1} is not suitable for our purpose, since model H 1 2 0 the variables x0 , x1 , x2 are not equivalent and the used Minkowski-like pseudometric is not natural. Another representation of the hyperbolic plane as the graph of a saddle-point is also not suitable, since this graph has different curvatures at different points and different directions, in general. 2 This model comes from a stereographic projection of the hyperboloid in R d+1 onto the hyperplane x0 = 0. 1 The

164

12 Tessellations of Maximally Symmetric Manifolds

Fig. 12.3 The boundary circle k is infinitely far away from the center. The shortest connections are represented by circular arcs (or straight lines passing through the center). They are perpendicular to the boundary circle k at its endpoints. By Lambert’s formula the sum of the angles in the triangle ABC is α + β + γ < 180◦

Theorem 12.4 There are infinitely many different tessellations by equilateral hyperbolic triangles of the hyperbolic plane H2 . In particular, for every n ≥ 7 there exists one equilateral hyperbolic triangle with angles 2π/n that tessellates the hyperbolic plane. The cases n = 7 and n = 8 are illustrated in Fig. 12.4.

Fig. 12.4 The regular tessellations of the hyperbolic plane by equilateral hyperbolic triangles with vertex degree n = 7 and n = 8. All sides on the left (resp. right) picture have the same length in the metric (12.6). (Picture courtesy of P. K˚urka.)

12.3 Regular Tetrahedral Tessellations

165

12.3 Regular Tetrahedral Tessellations Regular simplicial face-to-face tessellations of maximally symmetric manifolds in an arbitrary space dimension are formed by congruent bounded regular simplicial cells, possibly curved. These cells can be defined by induction, i.e., its (d − 1)dimensional facets are congruent regular simplicial cells (possibly curved) and all its vertices have the same vertex degree. This condition is necessary for regularity. For example, the surface of the polyhedron consisting of two adjacent congruent regular tetrahedra has equilateral triangular faces, but its vertex degrees are 3 and 4. By Theorem 5.10 the regular tetrahedron does not tile R3 . Consider now all six Schläfli four-dimensional regular polytopes from Table 12.2. Assume that their vertices lie on the hypersphere S3 . Then S3 can be tessellated by means of the radial projection of any Schläfli regular polytope into S3 . Hence, 5 or 8 or 16 or 24 or 120 or 600 points can be uniformly placed on the sphere S3 . From Table 12.2 we find that S3 can be tessellated by 5, 16, 600 spherical tetrahedral cells with vertex degree n = 4, 6, 12, respectively.3 In this case, m = 4, 8, 20 cells meet at each vertex and each edge is surrounded by p = 3, 4, 5 cells, respectively. The above three regular tetrahedral tessellations of S3 can be visualized in R3 by means of the standard nonlinear stereographic projection. Theorem 12.5 All regular tetrahedral tessellations of the unit hypersphere S3 are obtained by the radial projection of the regular 4-simplex, 4-orthoplex, and 600-cell onto S3 . Proof Obviously, (12.2) and the Euler–Poincaré formula (12.3) also hold for any spherical tessellations of S3 . Moreover, for tetrahedral spherical tiles we get 2 f = 4c,

(12.7)

since every face is shared by two tetrahedra and every tetrahedron has 4 faces. Multiplying (12.3) by n, we get nv + n f = ne + nc. Substituting there (12.2) and (12.7), we find that 2e + 2nc = ne + nc. From this we obtain the following necessary condition dc = (d − 2)e (12.8) which any regular triangular tessellation of S3 has to satisfy. Using Theorem 12.2 locally around every vertex, we find that n ∈ {4, 6, 12} and no other n is allowed (see Fig. 12.2). Hence, 6c = 3e or 6c = 4e or 6c = 5e. 3 Let

(12.9)

us point out that each vertex of the 600-cell in R4 is a common vertex of six regular decagons passing through its edges (cf. (12.2) and Table 12.2). All 120 vertices can be divided into the following 9 classes containing 1 (North Pole), 12 (Arctic Sphere), 20, 12 (Tropic of Cancer), 30 (Equatorial Sphere), 12 (Tropic of Capricorn), 20, 12 (Antarctic Sphere), and 1 (South Pole) vertices.

166

12 Tessellations of Maximally Symmetric Manifolds

Consequently, each edge in any regular tetrahedral tessellation has to be surrounded by 3, 4 or 5 regular spherical tetrahedra, because each spherical tetrahedron has 6 edges. The corresponding dihedral angles are clearly 120◦ , 90◦ , and 72◦ , respectively. Since S3 is locally Euclidean, the dihedral angle is taken in the intersection of the particular tetrahedral cell with the plane orthogonal to any of its edges (geodetics). The above three cases (12.9) correspond to the radial projection of the regular 4simplex, 4-orthoplex, and 600-cell onto S3 and there exist no other regular tetrahedral tessellations of S3 . There are several substantial differences between the geometry of two-dimensional and three-dimensional manifolds (see [277]). Blanuša’s statement about isometrical imbedding of the hyperbolic plane H2 into R6 (see Sect. 12.2) was generalized by David Brander [32] as follows: Theorem 12.6 For d > 1 the pseudosphere Hd can be isometrically imbedded into R6d−6 . It is not known whether the exponent 6d − 6 can be reduced. Brander [33] however proves that there exists a local isometric imbedding from Hd to R2d−1 . From Sect. 12.2 we know that there are infinitely many different regular hyperbolic triangular space-fillers that tile H2 . On the other hand we have the following theorem that states that there are no regular tessellations of H3 such that each edge is surrounded by p ∈ {6, 7, 8, . . . } hyperbolic tetrahedra (cf. also [77, p. 157]). Theorem 12.7 There is no regular hyperbolic tetrahedral space-filler of H3 . Proof The manifold H3 is locally almost Euclidean at every point, i.e., the intersection of H3 with an arbitrarily small two-dimensional sphere centered at any vertex of the tessellation of H3 leads to a two-dimensional regular spherical tessellation (see Fig. 12.2). Since there is only a finite number of point groups (that are also finite), we shall also have only a finite number of different face-to-face regular tetrahedral tessellations for d = 3 (up to translation, rotation, reflection, and scaling) in contrast to the two-dimensional case. Let Si stand for the symmetric group of all permutations and Ai for the alternating group of all even permutations of i elements (see [18, p. 86]). A necessary condition for the existence of a regular tessellation of a threedimensional maximally symmetric manifold is that local symmetries represented by the point groups A4 or S4 or A5 must be preserved at every vertex. This means that there are no other point groups for d = 3. However, the symmetries of these three groups have the tessellations from Theorem 12.5.

12.4 Regular Simplicial Tessellations By [266] the 4-cube, 4-orthoplex and the 24-cell are the only space-fillers of R4 (cf. Table 12.2), i.e., there is no regular simplicial space-filler for R4 .

12.4 Regular Simplicial Tessellations Table 12.3 The number of regular simplicial tessellations of the maximally symmetric manifolds

167

d

Rd

Sd

Hd

1 2 3 4 5 .. .

1 1 0 0 0 .. .

1 3 3 1 1 .. .

Not def. ∞ 0 0 0 .. .

Using again the radial projection of the regular simplex in Rd+1 into Sd , we find by Theorem 12.1 that for any d ≥ 4 the hypersphere Sd can be tessellated by d + 2 hyperspherical d-simplices. The following theorem shows that there is no analogy with Fig. 12.4 in higherdimensional cases. Theorem 12.8 There are no regular simplical tessellations of Rd and Hd for d ≥ 3. The case Rd with d ≥ 3 was proved in Theorem 5.10. The proof for Hd , can be done similarly as in Theorem 12.7. The main results of this chapter are summarized in Table 12.3.

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Index

0–9 0/1-equivalence, 141 0/1-equivalence class, 143 0/1-equivalent, 141 0/1-matrix, 142 0/1-polytope, 141 0/1-simplex, 141 0/1-triangles, 143 120-cell, 75 16-cell, 53 4-orthoplex, 53 600-cell, 53, 60, 75, 76, 165

A Acute 0/1-simplex, 146 d-simplex, 71 partition, 52, 77 simplex, 51 triangle, 144 Adjacent (dihedral angles), 79 Adjacent simplices, 18 Algebraic DMP, 117 Algorithm longest-edge bisection, 106 refinement, 102 tent-pitcher, 69 Angle dihedral, 51 solid, 45 trihedral, 45 Antipodal, 148 family, 154 simplices, 147

Applications in numerical mathematics, 66 Approximate solution, 133, 134 Associated function, 139 Averaged gradient, 50

B Barycentric coordinates, 10 Bilinear form, 21 Blue refinement, 43

C Cauchy–Bunyakowski–Schwarz inequality, 70 Cayley–Menger determinant, 10 Céa’s lemma, 23 Center of gravity, 6 Centroid, 6 rule, 124 Checkerboard pattern, 147 Circumcenter, 9 Circumradius, 9 Coefficients (numerical quadrature formula), 123 Coloring of a partition, 18 simplicial partitions, 18 Conforming triangulation, 13 Congruence, 141 Coning off to a vertex, 156 Conjecture Hadamard, 151 Hadamard maximum determinant, 152 Hadwiger, 65

© Springer Nature Switzerland AG 2020 J. Brandts et al., Simplicial Partitions with Applications to the Finite Element Method, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-55677-8

183

184 Convex polytope, 13 regular, 161 Coordinates barycentric, 10 tetrahedral, 11 triangular, 11 Cosine theorem, 6 Cube-corner tetrahedron, 146 D d-cube symmetries, 141 (d − 1)-dimensional facet, 51 d-dimensional maximum angle condition, 41 d-dimensional sine, 29 Delaunay triangulation, 56 Dihedral angle, 51 Dihedral angle bounds Gaddum, 82 nonobtuse simplices, 84 Dirichlet regions, 55 Discrete Maximum Principle (DMP), 116 Discretization parameter, 15, 26 Doubly stochastic matrix, 150 d-simplex, 9 acute, 71 d-sine, 29 dual d-simplex, 101 E Elements, 9 Elliptic problem, 115 Euler formula, 74, 161 Euler line, 10 Euler–Poincaré formula, 59, 75, 77, 78, 161 F Facet, 9 (d − 1)-dimensional, 51 face-to-face longest-edge bisection algorithm, 108 face-to-face triangulation, 13 Family of partitions, 16 regular, 16 regularity, 26 strongly regular, 16 Family of partitions into tetrahedra regular, 89 Family of partitions of a polyhedron into triangular prisms regular, 112 strongly regular, 112 Family of partitions of a polytope into simplices

Index regular, 34 semiregular, 35 Family of simplicial partitions, 26 regular, 28 strongly regular, 108 Fibonacci numbers, 147 Finite element discretization, 116 method, 21 solution, 22 Four color theorem for tetrahedral partitions, 19 Friedrichs’ inequality, 48 Functional DMP, 117 Fundamental domain, 69 G Gaddum’s dihedral angle bounds, 82 Galerkin solution, 22 Geodesic distance, 81 Girard’s theorem of spherical excess, 82 Global points (on unit sphere), 81 Green refinement, 89

H Hadamard conjecture, 151 inequality, 151 maximum determinant conjecture, 152 Hadwiger conjecture, 65 Half-perimeter, 5 Hat element, 135, 136 Height, 9 Heron’s formula, 5 Higher-dimensional simplices, 9 Highly distorted partitions, 41 Hyperbolic plane, 163 space, 159 Hypersphere, 159

I Inradius, 10 Isometry, 163 Isoparametric quadratic elements, 128 Isosceles tetrahedron, 70

K Kepler’s Tree of Fractions, 147 k-facet, 9 Klein bottle, 163 Kuhn partition, 62

Index L Lax–Milgram lemma, 21 Lexicographically minimal matrix, 143 Local nonobtuse refinements, 102 Longest-edge bisection algorithm, 106

M Matrix representation (of a 0/1-simplex), 142 Maximum angle condition, 32, 41 d-dimensional, 41 Maximum principle for an elliptic problem, 115 Mesh size, 15 Minimal matrix representation, 143 Minimal representative (of a 0/1equivalence class), 143 Minimum angle condition, 25, 31 in higher dimensions, 29 M-matrix, 120 Multiindex, 3

N Natural extension, 139 n-colorable partition, 18 n-coloring of a partition, 18 Neighbor theorems for nonobtuse simplices, 148 Nested partition, 89 Nodal points (numerical quadrature formula), 123 Nodes (numerical quadrature formula), 123 Nonacute simplex, 51 Nonobtuse partition, 52, 60 simplex, 51

O Obtuse simplex, 51 One-neighbor conjecture for nonobtuse 0/1simplices, 151 Orthocenter, 9 Orthocentric, 9 Orthogonal 0/1-simplex, 144 simplex, 144 Orthogonality condition, 48 Orthoscheme, 61 Ortho-simplex, 61, 144 Outward normals, 81 unit length, 81

185 P Partition, 15 acute, 52, 77 into tetrahedra, 132 into well-centered simplices, 100 nonobtuse, 52, 60 of a closed polytopic domain, 52 simplicial, 13 tetrahedral, 13 uniform, 49 Path-simplex, 60, 61 Path-tetrahedron, 61, 146 Platonic bodies (solids), 159 Poincaré disc, 163 Polygon, 13, 51 Polyhedron, 13, 51 Polytope, 51 Polytopic domain, 51 Post-processing, 46 Prism, 110 Projected icosahedron, 162 octahedron, 162 tetrahedron, 162

Q Quadrature formulae on simplices, 123 Quality factor, 92

R Red refinement, 58, 89, 97 properties, 93 Refinement blue, 43 green, 89 into path-tetrahedra, 98 local nonobtuse, 102 of tetrahedra, 89 of triangular prisms, 110 red, 58, 89, 93, 97 yellow, 98 Refinement algorithm, 102 Refinements of unstructured partitions, 87 Regular convex polytope, 161 family of partitions, 16 polytopes, 159 simplicial tessellations, 166 tetrahedral tessellations, 165 triangular tesselations, 161 Regular family of partitions

186 into simplices, 28 into tetrahedra, 89 of a polyhedron into triangular prisms, 112 of a polytope into simplices, 34 Regularity of a family of partitions, 26 Row complementation, 143 stochastic matrix, 150

S Semiperimeter, 5 Semiregular family of partitions of a polytope into simplices, 35 Simplex, 9 acute, 51 nonacute, 51 nonobtuse, 51 obtuse, 51 well-centered, 100 Simplicial partition, 13 Sine theorem, 6 Slice element, 135, 136 Sobolev space, 3 Solid angle, 45 Sommerville tetrahedron, 16, 52, 65, 94, 101 Spatial altitude, 9 Spherical excess, 45, 82 Stieltjes matrix, 120 Strict Delaunay triangulation, 69 Strong law of small numbers, 153 Strongly regular family of partitions, 16 of partitions of a polyhedron into triangular prisms, 112 of simplicial partitions of a bounded polytope, 108 Subdivision (of a partition), 89 Subpartition, 89 Sums of dihedral angles, 81 Supercloseness, 48 Superconvergence, 46, 47 Surround (a point), 72 Synge’s condition, 33

T Tent-pitcher algorithm, 69 Tessellation regular simplicial, 166 regular tetrahedral, 165 regular triangular, 161

Index Tetrahedral coordinates, 11 Tetrahedral partition, 13 Tetrahedron, 6 Thales circle, 62 Triangle, 5 Triangular coordinates, 11 numbers, 35, 127 prism, 110 Triangulation, 13 conforming, 13 Delaunay, 56 face-to-face, 13 of a closed polytopic domain, 52 of I d using 0/1-simplices, 154 strict Delaunay, 69 Trihedral angle, 45

U Uniform partition, 49 Unit length outward normals, 81

V Variational crimes, 121 formulation, 21, 121 V -ellipticity condition, 21 Vertex degree, 160 Vertices, 9 Vh -interpolant, 24 Volumic regularity conditions, 26 Voronoï cells, 55

W Weak formulation, 116 solution, 22, 132 Weights (numerical quadrature formula), 123 Well-centered simplex, 100

Y Yellow refinement, 98

Z Zlámal’s condition, 25 Zlámal’s minimum angle condition, 25

Author Index

A Appel, K., 18 Aristotle, 54 Averroës, 54 Aziz, K., 42

B Babuška, I., 42, 46, 87 Baidakova, N. V., 41 Blanuša, D., 163, 166 Böhm, J., 66 Bossavit, A., 69 Brander, D., 166 Burago, Y. D., 53 Burman, E., 69

C Cassidy, C., 52 Ciarlet, P. G., 67, 116, 118 Courant, R., 1 Coxeter, H. S. M., 65, 69, 70, 104

D D’Azevedo, E., 42 Delone, B., 56 Descartes, R., 55 Dirichlet, P. G. L., 55

E Eppstein, D., 55 Eriksson, F., 29 Ern, A., 69

F Feng, K., 35 Fiedler, M., 52, 83, 144 Freudenthal, H., 62, 97 G Gaddum, J. W., 81, 82 Gardner, M., 52 Gerver, J. L., 53 Goldberg, M., 64 H Hadamard, J., 151, 152 Hadwiger, H., 65 Haken, W., 18 Hannukainen, A., 42 Harazišvili, A. B., 66 Hilbert, D., 163 Hošek, R., 101 J Jamet, P., 33 K Kaiser, H., 66 Kalai, G., 78 Kobayashi, K., 33 Kuˇcera, V., 34, 45 L Leibniz, G. W., 1 Lenhard, H.-C., 65, 66

© Springer Nature Switzerland AG 2020 J. Brandts et al., Simplicial Partitions with Applications to the Finite Element Method, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-55677-8

187

188 Lord, G., 52

M Maehara, H., 53 Mara, P. S., 154

N Neˇcas, J., 113

O Oganesjan, L. A., 47, 48 Oswald, P., 34

P Penrose, R., 52 Práger, M., 46

R Rajan, V. T., 62 Raviart, P.-A., 67 Ruhovec, L. A., 47, 48

S Schläfli, L., 53, 161 Simpson, B., 42 Sobolev, S. L., 3 Somerville, D. M. Y., 65 Stynes, M., 106, 107

Author Index Subbotin, Y. N., 41 Sullivan, J. M., 55 Synge, J. L., 1, 32

T Tschirpke, K., 66 Tsuchiya, T., 33

U Üngör, A., 54, 55, 62

V Varga, R., 116, 118 Vatne, J. E., 101 Vitásek, E., 46 Voronoï, G., 55

X Xu, J., 67

Z Zalgaller, V. A., 53 Zamfirescu, T., 53 Zeníšek, A., 26 Zhang, S., 58, 93, 96 Ziegler, G., 153 Zikatanov, L., 67 Zlámal, M., 1, 25