Accuracy of Mathematical Models: Dimension Reduction, Homogenization, and Simplification 9783037192061

Table of contents :
Preface
Introduction
Basic notation
Domains and operators
Spaces of functions
Convex functionals
Functional inequalities
Hölder type inequalities
Friedrichs and Poincaré inequalities
Inequalities for functions with zero mean traceson the boundary
Korn's inequalities
Inf–Sup condition
Computable bounds of constantsin functional inequalities
Constant in the Friedrichs inequality
Constants in Poincaré-type inequalities
Constants in trace-type inequalities
Estimates of constants based on domain decomposition
Distance to exact solutions
A class of boundary value problems
The main error identity
Error measure
Decomposition of the error measure
Problems with linear F
Error identities in vector form
Difference between the exact solutions of two problems
Linear problems
Error relations in the general form
Special case
Primal-dual norms of errors in VY*
Errors in the full primal-dual norm
Majorant as a source of new models
Non-homogeneous boundary conditions
Applications to particular mathematical models
Diffusion type models
Mixed boundary conditions
Problems with periodic boundary conditions
Advanced estimates based on domain decomposition
Elasticity
Variational functionals with power growth
Stokes problem
Bingham problem
Another error estimation method
Validation of mathematical models
Errors of numerical approximations
Two-sided estimates of approximation errors
Reduction of the set Q**
Transformation of "426830A R(y*h),eh "526930B
Using extra regularity of the exact solution
Using an auxiliary finite-dimensional problem
Applications to least squares type methods
Nonconforming approximations
Dimension reduction models
Dimension reduction
Second-order elliptic problems
Basic problem
Reduced problem
Error generated by dimension reduction
Particular cases
Examples
Dimension reduction in linear elasticity
The plane stress problem
The function
Behavior of the modelling error as t0
Example
Bending of elastic plates
Statement of the problem
The Kirchhoff–Love plate model
Reconstruction of 3D displacements
Reconstruction of 3D stresses
Error estimates for plate-type domains
Accuracy of the KL plate model
Estimates of the modelling error
Asymptotic behaviour of the error majorant
Model simplification
Model simplification based on the concept of energy
Simplification of coefficients
Second-order elliptic problems
General elliptic problem
Using extra regularity of
Asymptotic rate of convergence of the error estimator E()in terms of the measure of the non-resolved geometry
Geometrical simplification
Simplification of the Dirichlet boundary
Generalizations
Simplification of the Neumann boundary
Comments
Problems with ``rough'' coefficients
Modelling-discretization adaptation strategies
Problems with uncertain data
Elliptic homogenization
Setting
Mathematical homogenization via asymptotic expansions
Properties of the homogenized problem
Well-posedness of the homogenized equation
Regularity estimates for the homogenized equation
Regularity estimates for the cell problem
Convergence of the first-order approximation
Discretization
Error estimation
General comments
Estimates of the modelling error
Error of the fully discrete first-order approximation
Comments
Regularity and embedding constants
Modeling-discretization strategies
Multiscale problems
Conversion of models
Regularization of models
Adding a regularizing term
Smoothing
Prox-type regularization
Errors of penalty-type models
General approach
Variational problems defined in subspaces
Fictitious domain methods
Linearization
Errors of time-incremental models
bold0mu mumu W1,pW1,pdottedW1,pW1,pW1,pW1,p-regularity constant for second-order elliptic problemswith nonsmooth coefficients
Bibliography
List of Notations
Index

Citation preview

Sergey I. Repin Stefan A. Sauter

Sergey I. Repin Stefan A. Sauter

Tr a c ts i n M a t h e m a t ic s 3 3

Sergey I. Repin Stefan A. Sauter

Accuracy of Mathematical Models

This book presents a unified approach to the analysis of accuracy of deterministic mathematical models described by variational problems and partial differential equations of elliptic type. It is based on new mathematical methods developed to estimate the distance between a solution of a boundary value problem and any function in the admissible functional class associated with the problem in question. The theory is presented for a wide class of elliptic variational problems. It is applied to the investigation of modelling errors arising in dimension reduction, homogenization, simplification, and various conversion methods (penalization, linearization, regularization, etc.). A collection of examples illustrates the performance of error estimates.

ISBN 978-3-03719-206-1

https://ems.press

Repin_Sauter Cover | Font: Nuri_Helvetica Neue | Farben: Pantone 116, Pantone 287 | RB 32 (?) mm

Accuracy of Mathematical Models

The expansion of scientific knowledge and the development of technology are strongly connected with quantitative analysis of mathematical models. Accuracy and reliability are the key properties we wish to understand and control.

Tr a c ts i n M a t h e m a t ic s 3 3

Accuracy of Mathematical Models Dimension Reduction, Homogenization, and Simplification

EMS Tracts in Mathematics 33

EMS Tracts in Mathematics Editorial Board: Michael Farber (Queen Mary University of London, Great Britain) Michael Röckner (Universität Bielefeld, Germany, and Purdue University, USA) Vladimir Turaev (Indiana University, Bloomington, USA) Alexander Varchenko (The University of North Carolina at Chapel Hill, USA) This series includes advanced texts and monographs covering all fields in pure and applied mathematics. The Tracts will give a reliable introduction and reference to special fields of current research. The books in the series will in most cases be authored monographs, although edited volumes may be published if appropriate. They are addressed to graduate students seeking access to research topics as well as to the experts in the field working at the frontier of research. For a complete listing see https://ems.press. 14 Steffen Börm, Efficient Numerical Methods for Non-local Operators. 2-Matrix Compression, Algorithms and Analysis 15 Ronald Brown, Philip J. Higgins and Rafael Sivera, Nonabelian Algebraic Topology. Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids 16 Marek Janicki and Peter Pflug, Separately Analytical Functions 17 Anders Björn and Jana Björn, Nonlinear Potential Theory on Metric Spaces 18 Erich Novak and Henryk Woz´niakowski, Tractability of Multivariate Problems. Volume III: Standard Information for Operators 19 Bogdan Bojarski, Vladimir Gutlyanskii, Olli Martio and Vladimir Ryazanov, Infinitesimal Geometry of Quasiconformal and Bi-Lipschitz Mappings in the Plane 20 Hans Triebel, Local Function Spaces, Heat and Navier–Stokes Equations 21 Kaspar Nipp and Daniel Stoffer, Invariant Manifolds in Discrete and Continuous Dynamical Systems 22 Patrick Dehornoy with François Digne, Eddy Godelle, Daan Kramer and Jean Michel, Foundations of Garside Theory 23 Augusto C. Ponce, Elliptic PDEs, Measures and Capacities. From the Poisson Equation to Nonlinear Thomas–Fermi Problems 24 Hans Triebel, Hybrid Function Spaces, Heat and Navier–Stokes Equations 25 Yves Cornulier and Pierre de la Harpe, Metric Geometry of Locally Compact Groups 26 Vincent Guedj and Ahmed Zeriahi, Degenerate Complex Monge–Ampère Equations 27 Nicolas Raymond, Bound States of the Magnetic Schrödinger Operator 28 Antoine Henrot and Michel Pierre, Shape Variation and Optimization. A Geometrical Analysis 29 Alexander Kosyak, Regular, Quasi-regular and Induced Representations of Infinite dimensional Groups 30 Vladimir G. Maz’ya, Boundary Behavior of Solutions to Elliptic Equations in General Domains 31 Igor V. Gel‘man and Vladimir G. Maz’ya, Estimates for Differential Operators in Half-space 32 Shigeyuki Kondo– , K3 Surfaces

Sergey I. Repin Stefan A. Sauter

Accuracy of Mathematical Models Dimension Reduction, Homogenization, and Simplification

Authors: Sergey I. Repin Steklov Institute of Mathematics Russian Acadademy of Sciences Fontanka, 27 191023 St. Petersburg Russia

Stefan A. Sauter Institut für Mathematik Universität Zürich Winterthurerstr. 190 8057 Zürich Switzerland

[email protected]

[email protected]

2010 Mathematical Subject Classification (primary; secondary): 35-02; 35J20, 35J50, 35J60, 35J88, 49M29, 65N15, 65N85, 74K20 Key words: Modelling error, a posteriori error majorant, model simplification, dimension reduction, homogenization, conversion of models

ISBN 978-3-03719-206-1 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2020 European Mathematical Society

Contact address:

European Mathematical Society – EMS – Publishing House Institut für Mathematik Technische Universität Berlin Straße des 17. Juni 136 10623 Berlin Germany [email protected] https://ems.press Typeset using the authors’ TEX files: le-tex publishing services GmbH, Leipzig, Germany Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid-free paper 987654321

To our parents

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Basic notation . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Domains and operators . . . . . . . . . . . . . . . . . 1.1.2 Spaces of functions . . . . . . . . . . . . . . . . . . . 1.1.3 Convex functionals . . . . . . . . . . . . . . . . . . . 1.2 Functional inequalities . . . . . . . . . . . . . . . . . . . . . 1.2.1 H¨older type inequalities . . . . . . . . . . . . . . . . 1.2.2 Friedrichs and Poincar´e inequalities . . . . . . . . . . 1.2.3 Inequalities for functions with zero mean traces on the boundary . . . . . . . . . . . . . . . . . . . . . 1.2.4 Korn’s inequalities . . . . . . . . . . . . . . . . . . . 1.2.5 Inf–Sup condition . . . . . . . . . . . . . . . . . . . 1.3 Computable bounds of constants in functional inequalities . . . . . . . . . . . . . . . . . . . . 1.3.1 Constant in the Friedrichs inequality . . . . . . . . . . 1.3.2 Constants in Poincar´e-type inequalities . . . . . . . . 1.3.3 Constants in trace-type inequalities . . . . . . . . . . . 1.3.4 Estimates of constants based on domain decomposition

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2 Distance to exact solutions . . . . . . . . . . . . . . . . . . . . . . . 2.1 A class of boundary value problems . . . . . . . . . . . . . . . 2.2 The main error identity . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Error measure . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Decomposition of the error measure . . . . . . . . . . . 2.2.3 Problems with linear F . . . . . . . . . . . . . . . . . . 2.2.4 Error identities in vector form . . . . . . . . . . . . . . 2.2.5 Difference between the exact solutions of two problems . 2.3 Linear problems . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Error relations in the general form . . . . . . . . . . . . 2.3.2 Special case . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Primal-dual norms of errors in V  Y  . . . . . . . . . 2.3.4 Errors in the full primal-dual norm . . . . . . . . . . . . 2.3.5 Majorant as a source of new models . . . . . . . . . . . 2.3.6 Non-homogeneous boundary conditions . . . . . . . . . 2.4 Applications to particular mathematical models . . . . . . . . . 2.4.1 Diffusion type models . . . . . . . . . . . . . . . . . . 2.4.2 Mixed boundary conditions . . . . . . . . . . . . . . . . 2.4.3 Problems with periodic boundary conditions . . . . . . . 2.4.4 Advanced estimates based on domain decomposition . .

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2.5 2.6

2.4.5 Elasticity . . . . . . . . . . . . . . . . . . . 2.4.6 Variational functionals with power growth . . 2.4.7 Stokes problem . . . . . . . . . . . . . . . . 2.4.8 Bingham problem . . . . . . . . . . . . . . . 2.4.9 Another error estimation method . . . . . . . Validation of mathematical models . . . . . . . . . . Errors of numerical approximations . . . . . . . . . 2.6.1 Two-sided estimates of approximation errors  . . . . . . . . . . 2.6.2 Reduction of the set Qƒ  2.6.3 Transformation of hR.yh /; eh i . . . . . . . . 2.6.4 Using extra regularity of the exact solution . 2.6.5 Using an auxiliary finite-dimensional problem 2.6.6 Applications to least squares type methods . . 2.6.7 Nonconforming approximations . . . . . . .

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64 70 75 79 81 86 89 90 91 92 93 94 99 101

3

Dimension reduction models . . . . . . . . . . . . . . 3.1 Dimension reduction . . . . . . . . . . . . . . . . 3.2 Second-order elliptic problems . . . . . . . . . . . 3.2.1 Basic problem . . . . . . . . . . . . . . . . 3.2.2 Reduced problem . . . . . . . . . . . . . . 3.2.3 Error generated by dimension reduction . . 3.2.4 Particular cases . . . . . . . . . . . . . . . 3.2.5 Examples . . . . . . . . . . . . . . . . . . 3.3 Dimension reduction in linear elasticity . . . . . . 3.3.1 The plane stress problem . . . . . . . . . . 3.3.2 The function  . . . . . . . . . . . . . . . 3.3.3 Behavior of the modelling error as t ! 0 . 3.3.4 Example . . . . . . . . . . . . . . . . . . . 3.4 Bending of elastic plates . . . . . . . . . . . . . . 3.4.1 Statement of the problem . . . . . . . . . . 3.4.2 The Kirchhoff–Love plate model . . . . . . 3.4.3 Reconstruction of 3D displacements . . . . 3.4.4 Reconstruction of 3D stresses . . . . . . . 3.4.5 Error estimates for plate-type domains . . . 3.4.6 Accuracy of the KL plate model . . . . . . 3.4.7 Estimates of the modelling error . . . . . . 3.4.8 Asymptotic behaviour of the error majorant

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103 103 107 107 108 110 115 117 123 123 130 132 134 136 136 137 140 140 141 149 150 152

4

Model simplification . . . . . . . . . . . . . . . . . . . . 4.1 Model simplification based on the concept of energy 4.2 Simplification of coefficients . . . . . . . . . . . . . 4.2.1 Second-order elliptic problems . . . . . . . . 4.2.2 General elliptic problem . . . . . . . . . . . 4.2.3 Using extra regularity of uQ . . . . . . . . . .

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4.3

4.4

4.2.4 Asymptotic rate of convergence of the error estimator E .v/ O in terms of the measure of the non-resolved geometry . . . Geometrical simplification . . . . . . . . . . . . . . . . . . . . . 4.3.1 Simplification of the Dirichlet boundary . . . . . . . . . . 4.3.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Simplification of the Neumann boundary . . . . . . . . . Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Problems with “rough” coefficients . . . . . . . . . . . . 4.4.2 Modelling-discretization adaptation strategies . . . . . . . 4.4.3 Problems with uncertain data . . . . . . . . . . . . . . . .

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5 Elliptic homogenization . . . . . . . . . . . . . . . . . . . . . 5.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Mathematical homogenization via asymptotic expansions . 5.3 Properties of the homogenized problem . . . . . . . . . . 5.3.1 Well-posedness of the homogenized equation . . . 5.3.2 Regularity estimates for the homogenized equation 5.3.3 Regularity estimates for the cell problem . . . . . . 5.3.4 Convergence of the first-order approximation . . . 5.4 Discretization . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Error estimation . . . . . . . . . . . . . . . . . . . . . . 5.5.1 General comments . . . . . . . . . . . . . . . . . 5.5.2 Estimates of the modelling error . . . . . . . . . . 5.5.3 Error of the fully discrete first-order approximation 5.6 Comments . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Regularity and embedding constants . . . . . . . . 5.6.2 Modeling-discretization strategies . . . . . . . . . 5.6.3 Multiscale problems . . . . . . . . . . . . . . . .

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6 Conversion of models . . . . . . . . . . . . . . . . . 6.1 Regularization of models . . . . . . . . . . . . . 6.1.1 Adding a regularizing term . . . . . . . . 6.1.2 Smoothing . . . . . . . . . . . . . . . . 6.1.3 Prox-type regularization . . . . . . . . . 6.2 Errors of penalty-type models . . . . . . . . . . 6.2.1 General approach . . . . . . . . . . . . . 6.2.2 Variational problems defined in subspaces 6.3 Fictitious domain methods . . . . . . . . . . . . 6.4 Linearization . . . . . . . . . . . . . . . . . . . 6.5 Errors of time-incremental models . . . . . . . .

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Contents

A W 1;p -regularity constant for second-order elliptic problems with nonsmooth coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 283 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 List of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

Preface The expansion of scientific knowledge and the development of technology are strongly connected with quantitative analysis of mathematical models. Accuracy and reliability are the main properties we wish to know about when considering a mathematical model. Reliability of Mathematical Models is a fundamental interdisciplinary problem, which has many aspects and includes subproblems related to different areas (natural sciences, computational methods, mathematical analysis, etc.). It is not surprising that many experts in the field of mathematical modelling consider the question How accurate is a mathematical model ‹ as the most difficult to answer. Natural sciences have developed a number of well-tested principles (e.g., conservation laws) that allow us to reject deliberately wrong models and to develop new ones, which are consistent with the physical axioms/laws. For example, a model does not deserve attention if it violates well-established physical laws or if it is in contradiction with laws and principles in other sciences. This “physical consistency” leaves only those models that are relevant from the physical point of view. The second crucial step in the analysis of a mathematical model is verification of its mathematical correctness. It is commonly accepted that a mathematical model is well posed if it is free from mathematical contradictions (which arise, e.g., in the case of overdetermined systems or problems with inconsistent data) and possesses a solution that depends continuously on the problem data. As usual, we also need that the solution is unique within a certain suitable functional class. These questions together with the regularity theory (which states additional properties of solutions) are at the core of the modern theory of partial differential equations and there is an extensive literature devoted to this subject. However, while being necessary, the above mentioned stages of verification are in general not sufficient for an unambiguous judgement on the suitability or unsuitability of the chosen mathematical model. Only quantitative verification enables us to make the final decision about the validity and accuracy of a model. Indeed, a model may be perfectly consistent from the viewpoint of physics and mathematics, but practically invalid because it produces wrong results. This situation may arise for many different reasons, for example due to instability of the model or because the model does not properly account for some essential physical effects (e.g., turbulence). In many cases, a particular phenomenon or process can be modelled by several different computer simulation codes based on competing mathematical models (which yield different results). It is often difficult to say whether the major differences came from the models or it is generated by computational methods, approximations, and algorithms. A similar question arises when experimental data are compared with results of computer simulations. If they are different, then we must understand the reason: is it the model itself or are the errors mainly associated with computations? In fact,

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Preface

we cannot answer this question unless the “modelling error” and the “approximation error” are clearly separated. Notice that the separation is always necessary, even in those cases where numerical results are in good agreement with experimental data. This fact does not fully confirm the validity of a model because modelling errors might be compensated by numerical ones (if they have opposite signs). Effects of this type are indeed known for some approximation methods. This book presents a mathematical theory developed for explicit evaluation of modelling errors. It summarises our experience gained by studying this problem for about 20 years. Clearly, the theme in the title of the book is extremely wide and we are forced to confine ourselves to only one class of deterministic models. The main goal of our analysis is to create a unified approach to quantitative analysis of the accuracy of mathematical models associated with variational and saddle point problems in mathematical physics. Though the book is focused on the modelling errors, we also discuss approximation errors that arise if a mathematical model is approximated in order to obtain a numerical solution. For example, in the dimension reduction, homogenization, and simplification models, the overall error contains two major components: a modelling one and an approximation one. This fact leads to estimates of the difference between a numerical solution of a simplified mathematical model and the exact solution of the original (complicated) problem and makes the whole analysis fully practical. The reader will find several numerical examples of this kind. However, in the book we do not focus on estimates of approximation errors, which are well known and presented in numerous publications (including several monographs). Instead, we focus our attention on the errors that arise due to differences in mathematical models (“complicated” and “simplified”). The corresponding estimates of modelling errors contain only known data and solutions of “simplified” models. For example, the solution of a dimensionally reduced 2D model (or a suitable approximation of it) is used in order to approximate the solution a full-scale 3D problem and have a computable and guaranteed bound of the error arising in this process. Basically, the book deals with stationary models, but the proposed concepts and methods of analysis can be naturally generalized to many evolutionary models. The reader will find several comments and references that could help in studying accuracy of time-dependent models. Moreover, in Chapter 6 we deduce estimates of errors generated by semi-discrete (incremental) approximations of parabolic-type problems. As implied by the title, the main subject of the book is modelling error. Depending on the context, this term has two close but nontheless different meanings. First, the modelling error is a quantity (or a collection of quantities) that measures how a mathematical model differs from a physical phenomenon or process under consideration. This error is always present because mathematical models describe physical phenomena with a limited accuracy. Modelling error of this kind can be evaluated only by a systematic comparison of the data obtained in the respective physical and mathematical experiments. Here, the main mathematical problem to be solved is the fully guaranteed estimation of computational errors. We briefly discuss it at the end of Chapter 2 and refer the reader to the relevant literature. However, the book is mainly devoted to modelling errors of another kind, namely, those that arise if we compare solutions of different mathematical models related to one and the

Preface

xiii

same object. For example, in dimension reduction models, we compare solutions of three-dimensional (3D) models of an elastic body with solutions of simplified two-dimensional (2D) models (which contain errors caused by a dimension reduction method). Analogously, models arising in homogenization theory contain errors arising when the original periodic structure is replaced by the corresponding homogenized model. The book consists of six chapters. Chapter 1 contains all preliminary information subsequently used in different parts of the book. Also, it includes a special section related to functional inequalities and respective constants. Chapter 2 gives a fairly complete exposition of the mathematical theory that allows us to estimate the distance to exact solutions of elliptic boundary value problems. The main attention is paid to the error identities (such as (2.21), (2.26), or (2.29)) that are further used in error analysis. They are derived for a wide class of problems related to the variational functional J.v/ D G.ƒv/ C F .v/, where G and F are convex functionals and ƒ is a differential operator. This class encompasses the majority of linear and nonlinear elliptic problems, including variational inequalities. We show that error identities define error measures, which are natural for the problem in question and serve as a source of computable two-sided error bounds for these measures. We should emphasize that the derivation of error bounds is based on purely functional arguments and do not use special properties of approximations or properties of the method by which they were constructed. Besides, the error bounds do not contain analytic quantities, which could be uncontrollably large (or even unbounded) as, e.g., H 2 regularity constants of problems with smooth but oscillatory coefficients. This fact is rather essential for the analysis of modelling errors because we are able to keep maximal generality of the results and are not confined to considerations of some specially selected methods and models. We can apply the estimates to approximations of quite different types obtained by, e.g., reconstruction of dimensionally reduced models, regularisation, penalisation, linearisation, etc. General results are illustrated by several examples related to particular classes of problems. Special attention is paid to linear elliptic problems of divergence type. At the end of the chapter, we discuss possible applications of the estimates to validation of a mathematical model based on analysis of physical data and direct comparison of them with results of computational experiments. The estimates could guarantee the accuracy of computational data and confirm that they possess the same level of accuracy as in physical experiments. In this case, we are indeed able to draw reliable conclusions on the validity (accuracy) of a mathematical model within the selected range of physical data (parameters, geometry, other conditions). Chapter 3 is concerned with dimension reduction models. First, we consider diffusion-type equations in “thin” domains. Formally this means that one characteristic size of the domain is much smaller than the other. No other essential restrictions on the shape and other properties of the domain and coefficients are imposed and we can analyse different models of lower dimensions based on various representations of the solution in the “degenerate” direction. Explicitly computable bounds of the errors are obtained for plate-type domains with plane faces as well as for domains

xiv

Preface

with curvilinear faces. Finally, we investigate problems arising in linear elasticity. Simplification of models is widely used in applied analysis. It has various forms related to geometry, coefficients, boundary conditions, etc. Modelling errors of this type appear in defeaturing (simplification) of mathematical relations when complicated coefficients (boundary conditions, geometrical details) are approximated by simpler ones. In Chapter 4 we study the main question that always arises: What we loose in the process of simplification? If one is able to answer it, then we get a certain accuracy level within which a simplified model can be successfully used instead of the original one. We consider two main cases: simplification of the coefficients of a partial differential equation and simplification of the underlying geometry. Boundary value problems with periodic structures arise in various applications. Chapter 5 is devoted to mathematical models of homogenization theory, which is the major tool used to quantitatively analyse media with periodic structures (e.g., see [46, 92, 146] and other publications cited in the chapter). Typically, a homogenized boundary value problem is a problem with specially constructed smooth coefficients. It has been proved that the functions constructed by this procedure converge (with respect to some topology) to the exact solution as the cell size " tends to zero. Moreover, known a priori error estimates qualify the convergence rate in terms of ": However, a priori convergence estimates give only qualitative information on the behaviour of the error. In general, they are unable to provide a sharp and reliable quantitative information on the difference between the exact solutions of the original problem and the respective homogenized counterpart. In contrast, the goal in Chapter 5 is to develop error majorants which allow to evaluate the quality of the model and its discretization in a split and a posteriori way (but not to develop new advanced homogenization models, which is also an important topic of intensive research in applied analysis). To highlight the principal ideas in a most transparent way and to keep this exposition self-contained, we have chosen a relatively simple, first-order homogenization model and we obtain fully guaranteed and computable bounds of the difference between the exact solution of a particular elliptic boundary value problem with periodic coefficients and an abridged problem generated by homogenization. The difference is measured in terms of the energy norm of the basic problem and also in the combined primal-dual norm. Using the technique discussed in Chapter 2, we obtain two-sided bounds of the modelling error, which depend only on the solution of the homogenized problem, auxiliary problems defined on the periodicity cell, and known data. The estimates include only global constants associated with embedding-type inequalities in the domain and in the periodicity cell. Also, they contain constants in some new regularity type estimates for functions defined in convex domains. Formally, the estimates are valid for any number of cells and could be used for thin periodic structures as well as for coarse ones (though in the latter case errors generated by a homogenized model may be quite large). If the overall number of cells increases, then the respective modelling error decreases. We prove that the error majorant behaves accordingly, i.e., if

Preface

xv

the parameters of the majorant are selected in an appropriate manner, then it decreases with the same rate as the true error. Finally, in Chapter 6 we consider various situations where one model is converted into another one. Usually this is done in order to obtain a more convenient problem (from the computational point of view) that can be solved by well developed numerical methods and/or standard computational software. A typical example of this kind is provided by a wide collection of methods sharing the name penalization. In these methods, various additional conditions imposed on the exact solutions are accounted in a weaker sense (as penalties). Penalization is widely used in the analysis of applied problems (if the “exact” incorporation of some constraints is difficult) and the question How different are the solutions of the original and penalized problems? is the first to be answered. Certainly asymptotic type estimates expressed in terms of the penalization parameter(s) are known for a rather wide collection of problems. We present estimates of a different type: they show the distance to the exact solution in terms of a fully computable functional, which contains only known constants and quantities that come from the solution of the penalized problem (or its numerical approximation). The well-known fictitious domains method (see, e.g., [126]) uses related ideas in order to reduce problems with complicated geometry to problems defined in simple domains (e.g., rectangles) where numerical approximations use simple meshes. We present the respective estimates and examples that demonstrate the efficiency of the approach. Similar ideas of model transformation are used in the regularization method. We discuss it on the example of the Bingham problem, where the nondifferentiable term in the energy functional is replaced by a smoothed one. Linearization of nonlinear models is one of the most common approaches used in both theoretical and computational analysis of mathematical models. We consider applications of our error estimation method to this case and deduce error estimates in several examples. The estimates provide quantitative qualification of the linearized model, which enables the scientist to judge whether it can be successfully used as a replacement of the original one. The authors would like to thank their colleagues and collaborators B. Khoromskij, S. Matculevich, S. Meier-Rohr, T. Samrowski, A. Smolianski, and M. Weymuth for numerous discussions concerning the topics of this book. Some parts of the book are based on papers published in various journals. References and comments are provided in the respective parts of the book. We use this opportunity to thank all co-authors for cooperation and contributions they have made to these publications. Part of this work was carried out during mutual visits of the authors at the Steklov Mathematical Institute and the Euler International Mathematical Institute in Saint Petersburg, at the Department of Mathematical Information Technology of the University of Jyv¨askyl¨a, at the Institute for Mathematics at the University of Z¨urich, and at the Max-Planck Institute for Mathematics in Leipzig. We express our deep gratitude to these institutions for their support. We also thank the Institute for Mathematical

xvi

Preface

Research (FIM) at ETH Z¨urich and the Swiss National Science Foundation (SNSF) for financially supporting these visits. We also owe thanks to S. Fellmann and T. Hintermann from the EMS publishing house for their advice and friendly cooperation. Sergey Repin and Stefan Sauter Saint Petersburg–Z¨urich–Jyv¨askyl¨a, 2019

Chapter 1

Introduction This chapter contains a concise mathematical background. We present the basic notation and functional inequalities used in different parts of the book. Since the inequalities are used mainly for quantitative analysis, special attention is paid to computable estimates of the corresponding constants. Also, the chapter includes a literature overview and an outline of the material exposed in subsequent chapters.

1.1 Basic notation 1.1.1 Domains and operators Throughout the book we denote domains by the letters  and !. They are assumed to be open, bounded, and connected sets in the Euclidean space Rd , where d 2 N>0 . Here, N WD f0; 1; : : :g is the set of natural numbers and N>0 WD N n f0g. By R and R>0 we denote the set of real numbers and the set of positive real numbers, respectively. In some cases, it is convenient to use the extended set R of real numbers, which contains 1 and C1. The vector space Rd is endowed with the cartesian coordinate system, so that a point x has coordinates .x1 ; x2 ; : : :; xd /. B.x; ı/ denotes the open ball of radius ı centered at x 2 Rd . All the domains are assumed to be bounded and have Lipschitz continuous boundary (denoted @, @!, or ), which may have several nonintersecting parts (e.g., 1 and 2 ). By n we denote the outward unit normal to . The diameter of the set  and its Lebesgue measure are denoted by diam  and jj, respectively. Latin letters (e.g., u, v, w) are typically used to denote scalar-valued functions. We use special (sans serif or bold) letters p; q; y;  to indicate that the object is a vector or a vector-valued function. The same rule is used for matrices and tensor-valued functions (e.g., A,  .x/, .x/). All the quantities are assumed to be real-valued. Calligraphic and capital Greek letters (e.g., B , ƒ) are used for the operators and functionals. L.X; Y / denotes the space of bounded linear operators acting from X to Y . The scalar product of vectors is denoted by the dot, i.e. p  q WD

d X i D1

pi qi ;

p; q 2 Rd ;

2

1 Introduction

where the symbol := means “equals by definition”. Analogously, the product of d  d matrices (or matrix-valued functions) is denoted by a colon, i.e., " W  WD

d X

"ij ij :

i;j D1

Norms of vectors and matrices are associated with the respective scalar products, namely, jqj WD .q  q/1=2 ;

j j2 WD . W  /1=2 :

Note that j j is called the Frobenius norm of  . The tensor product of two vectors a and b is denoted by a ˝ b. It is a matrix with entries fai bj g. By Md d we denote the space of real d  d matrices and 1 denotes the unit matrix (if d D 2 we use a special notation b 1). Symmetric matrices form the subspace Msd d . For A 2 Msd d , the smallest and largest eigenvalues are denoted by  .A/ and ˚ .A/, respectively. The trace and the deviator of  2 M d d are defined by the formulas tr  WD

d X i D1

i i

and  D WD  

1 tr  1: d

(1.1)

Since 1 W  D D 0, the above decomposition of  is orthogonal and for any tensor  we have the identity j j2 D j D j2 C d1 jtr  j2 . By Œg we denote the jump (difference of the left-hand side and right-hand side limits of the function g) on a line (surface)  .

1.1.2 Spaces of functions Spaces of functions are denoted by capital letters X; Y; V . By default, all of them are assumed to be reflexive Banach spaces over the field of real numbers. The respective topologically dual spaces (which consist of linear continuous functionals) are marked by an asterisk (e.g., X  ; Y  ; V  ) and the duality pairings are denoted by round or angle brackets (e.g., .y  ; y/ or hv  ; vi). If V is a Banach space, then V  can also be normed by setting hv  ; vi : kv  k WD sup v2V nf0g kvk For ˛ 2 Œ1; 1, we denote by L˛ ./ the usual Lebesgue space of functions with norm kk˛; . If ˛ D 2 then we may also use the simplified notation k  k and for the scalar product .; / . L2 .; Rd / is the Hilbert space of vector-valued functions, whose components are square integrable in . The analogous space of tensor-valued functions is denoted L2 .; M d d /. By fjgjg! we denote the mean value of g 2 L1 .!/ in !, i.e., Z 1 gd x; (1.2) fjgjg! WD j!j !

1.1 Basic notation

3

where j!j is the Lebesgue measure of the set !. For vectors and matrices, the symbol fjjg means componentwise averaging. e L2 ./ denotes a subspace of L2 ./ consisting of the functions with zero mean values and L1 ./ is the space of functions bounded almost everywhere in , endowed with the supremum norm k  k1; . If f is a vector-valued function, than kfk1;! WD ess sup jf.x/j ; x2!

where jfj is the Euclidean norm. Whenever a different norm will be used this fact will be specially mentioned. For 1  p  1, j  j`p denotes the discrete `p -norm in Rd . If p D 2, then we use j  j instead of j  j`2 . For any p 2 Œ1; 1, the conjugate number p0 is defined by the relation p1 C p10 D 1 (in some formulas the adjoint numbers are marked by stars, e.g., p? ). Analogously, for p 2 Œ2; 1, the number p00 2 Œ1; 1 satisfies the relation 2 C p100 D 1. p P k ./ is the space of polynomials of maximal degree k defined in . 2G @f and @x@1 @x ). In some parts, For derivatives, we use the standard notation (e.g., @x 1 2 we also apply a shortened notation, where the directions of differentiation are shown in subscripts (e.g., f;1 and G;12 ). Also, in expressions containing multiindexes, we use Einstein’s convention on summation over the repeated indices, e.g., ui vi (where P i 2 f1; 2; : : :; d g) means the sum diD1 ui vi . C k ./ denotes the space of k-times differentiable scalar-valued functions and k C0 ./ is the subspace consisting of the functions with compact support in . C01 ./ is the space of all infinitely differentiable functions with compact support in . In the book, we use standard differential operators: gradient (r), curl, and div. The divergence of a tensor-valued function  is denoted by Div . It is defined by the vector .div i /diD1 , where j is the j -th row of . S./ denotes the set of solenoidal (divergence-free) vector-valued functions deı

fined in  and S.; Rd / denotes the closure of the set of smooth divergence-free functions vanishing on the boundary with respect to the norm of H 1 .; Rd /. Standard Sobolev spaces of functions having in  generalized derivatives up to the order l in Lp ./ are denoted W l;p ./ and k  kl;p; denotes the respective norm. Similar notation is used for spaces of vector- and tensor-valued functions. Also, for a sufficiently smooth vector field v W  ! Rd we define the norms krvk21;2;

WD

krvk2

d X   v;lk 2 ; C  l;kD1

kvk22;2; WD

d X

kvi k22;2; D kvk2 C krvk21;2; :

i D1

If p D 2, then for Sobolev spaces we use the simplified notation H l ./. The subspace of H 1 ./ consisting of the functions that vanish on the boundary is denoted

4

1 Introduction

ı

ı

H 1 ./. H 1 ./ is the space dual to H 1 ./. The space W 1;p ./ is dual to ı

0

W 1;p ./. It is endowed with the standard dual norm k  k1;p; . H.div; / denotes the Hilbert space of square-integrable vector-valued functions with square-integrable divergence, endowed with the scalar product and the norm Z 1=2 .u; v/div WD .u  v C div u div v/d x; kvkdiv WD .v; v/div : 

Analogously, H.Div; / denotes the Hilbert space of square-integrable tensor-valued functions with square-integrable divergence and Z 1=2 .;  /Div WD . W  C div   div  /d x; k kDiv D . ;  /Div : 

  Let M 2 L1 ; Msd d . We define ! kM.x/k sup :

.M/ WD ess sup kk x2  2Rd nf0g

(1.3)

For p  2, we introduce the function m 2 L1 ./ by m WD

kM ./ k`p0 kk`p 2Rd nf0g sup

and the norm

jjj M jjjp00 ; D kmkp00 ; :

If p D 2 then p0 D 2, p00 D 1, and jjj M jjj1; D .M / : (1.4)   We say that a matrix function B 2 L1 ; Msd d is uniformly positive definite if B.x/ is positive definite for all x 2  and 1 1 0 < 1 jjj1;  jjj B jjj1; DW 1  .B/ WDjjj B ˚ .B/ < 1

(1.5)

and define the spectral condition number 1 B WD 1 ˚ .B/= .B/:

(1.6)

1.1.3 Convex functionals A set K  V is called convex if v1 C .1  /v2 2 K for any v1 ; v2 2 K and  2 Œ0; 1. Conv.K/ denotes a smallest convex set containing K. It is called the convex hull of K. Let K be a convex set in a Banach space V . A functional I W K ! R is called convex if I .1 v1 C 2 v2 /  1 I .v1 / C 2 I .v2/ (1.7)

1.1 Basic notation

5

for all v1 ; v2 2 K and all 1 ; 2 2 R0 such that 1 C 2 D 1. It is called strictly convex if for positive i , i D 1; 2 the inequality is strict. A functional I is called concave (resp., strictly concave) if the functional I is convex (resp., strictly convex). The characteristic functional of the the set K  0; if v 2 K; (1.8)

K .v/ D C1; if v 62 K; is convex if and only if K is a convex set. The functional I  W V  ! R defined by the relation

I  .v  / D sup fhv  ; vi  I .v/g

(1.9)

v2V

is called dual (or conjugate) to I (see, e.g., [103, 110, 280]). For example, the functional K conjugate to K is a cone in the space V  , called the support functional of the set K. If V D R and I is a smooth function, then I  coincides with the Legendre transform of I . The second conjugate is defined by the relation

I  .v/ WD sup fhv  ; vi  I  .v  /g: v  2V 

If V is a reflexive Banach space and I is convex, then I  coincides with I . By definition, hv  ; vi  I .v/ C I  .v  /:

(1.10) ˛

For example, if V D Rd and I .v/ D ˛1 jvj˛ , then I  .v / D ˛1 jv j , where ˛ and ˛  are positive real numbers such that ˛1 C ˛1 D 1 (these numbers are called conjugate). In this case (1.10) reads 1 1  (1.11) v  v  jvj˛ C  jv j˛ : ˛ ˛ This inequality is also known as the Young inequality. It also holds for the space V D M d d endowed with the Frobenius matrix norm. From (1.11) we deduce the inequality 1Cˇ jv1 C v2 j2  .1 C ˇ/jv1 j2 C (1.12) jv2 j2 ; ˇ valid for any ˇ > 0 and any pair of vectors v1 and v2 in Rd . Setting 1 D 1 C ˇ and 2 D 1Cˇ , we rewrite this inequality in the somewhat different form ˇ jv1 C v2 j2  1 jv1 j2 C 2 jv2 j2 :

(1.13)

Clearly (1.11), (1.12), and (1.12) can be extended to spaces of functions. Let V be a Hilbert space with the norm k  kV . For any v1 ; v2 2 V , we have kv1 C v2 k2V  1 kv1 k2V C 2 kv2 k2V ; where 1 and 2 are positive numbers such that

1 1

C

1 2

D 1.

(1.14)

6

1 Introduction

For a given v0 2 V , an element v  2 V  satisfying hv  ; v  v0 i C I .v0 /  I .v/

8v2V

(1.15)

is called a subgradient of I at v0 . The set of all subgradients of I at v0 forms the subdifferential @I .v0 /. By I 0 .v0 / we denote an element v  2 V  such that the derivative of I at v0 2 V in the direction w has the form hv  ; wi for any w 2 V . This element is called the Gˆateaux derivative of I at v0 (if this notation is used, then it is assumed that the derivative exists). The functional DI W V  V  ! R defined by the relation

DI .v; v  / WD I .v/ C I  .v  /  hv  ; vi; where I and I  are conjugate functionals, is called a compound functional. These functionals play an important role in error analysis of nonlinear problems. Throughout the book, we denote them by the letter D supplied with an index that shows the functional used to form it. In view of (1.10), DI is nonnegative. Moreover, it vanishes only if v and v  satisfy the subdifferential (duality) relations (see, e.g., [103]) v 2 @I  .v  / and v  2 @I .v/:

(1.16)

In general, compound functionals are not convex. However, they possess a certain property similar to convexity. Let 1 and 1 be real numbers in Œ0; 1 and 2 D 11 , 2 D 1  1 . For any y1 ; y2 2 Y and y1 ; y2 2 Y  , we have   DI 1 y1 C 2 y2 ;1 y1 C 2 y2  1 1 DI .y1 ; y1 / C 1 2 DI .y1 ; y2 / C 2 1 DI .y2 ; y1 / C 2 2 DI .y2 ; y2 /:

(1.17)

Indeed,   DI y; 1 y1 C 2 y2 D I .y/ C I  .1 y1 C 2 y2 /  .1 y1 C 2 y2 ; y/  I .y/ C 1 I  .y1 / C 2 I  .y2 /  .1 y1 C 2 y2 ; y/ D 1 DI .y; y1 / C 2 DI .y; y2 /: Analogously

  DI 1 y1 C 2 y2 ; y   1 DI .y1 ; y  / C 2 DI .y2 ; y  /:

(1.18) (1.19)

Therefore,

DI .1 y1 C 2 y2 ; 1 y1 C 2 y2 /  1 DI .y1 ; 1 y1 C 2 y2 / C 2 DI .y2 ; 1 y1 C 2 y2 // and (1.17) follows from (1.18) and (1.19). Remark 1.1.1. From (1.19) it follows that for any z1 ; z2 2 Y and y  2 Y        z1  z2   DI z1 C z2 ; y  1 DI ; y C 2 DI ;y : 1 2 Similarly, for any z1 ; z2 2 Y  and y 2 Y       z z DI y; z1 C z2  1 DI y; 1 C 2 DI y; 2 : 1 2

(1.20)

(1.21)

1.2 Functional inequalities

7

1.2 Functional inequalities For functions in Sobolev spaces, there exists a wide collection of so-called embedding inequalities (see, e.g., S. L. Sobolev [295], O. A. Ladyzhenskaya and N. N. Uraltseva [168], D. Gilbarg and N. S. Trudinger [120], R. A. Adams and J. J. Fournier [4]). They are of crucial importance for both qualitative and quantitative analysis of partial differential equations. For the convenience of the reader we discuss briefly below some of the results used in subsequent chapters. A systematic overview of sharp estimates of constants in various functional inequalities is presented in [162].

1.2.1 H¨older type inequalities The discrete H¨older inequality ja  bj 

d X

!1=˛ jai j˛

i D1

d X

!1=˛ jbi j

˛

(1.22)

i D1 

holds for a; b 2 Rd . For w 2 L˛ .!/ and v 2 L˛ .!/, ˛ 2 Œ1; C1, where ! is a bounded Lipschitz domain, the integral H¨older inequality reads Z w v d x  kwk˛;! kvk˛ ;! : (1.23) !

Similar inequalities hold for vector- and matrix-valued functions. For instance, if   2 L˛ .; M d d / and  2 L˛ .; M d d /, then Z  W  d x  k k˛;! kk˛ ;! : (1.24) !

We will also use the following multiplicative estimate, which is valid for scalar and vector valued functions. r/ Let 2 < r < t < C1 and .r; t/ WD 2.t 2 .0; 1/. For w 2 Lt ./, r.t 2/ / 1.r;t / : kwkr;  kwk.r;t 2; kwkt;

(1.25)

A similar inequality holds for vector-valued functions. Hence for w 2 W 1;t ./, we have .r;t /

1.r;t /

krwkr;  krwk2; krwkt;

:

(1.26)

1.2.2 Friedrichs and Poincar´e inequalities Let ` W W 1;p ./ ! R (p 2 Œ1; C1) be a linear continuous functional satisfying the condition: if `.w/ D 0 for any constant function w, then w D 0. In this case,

8

1 Introduction

the original norm of W 1;p ./ is equivalent to the norm j`.w/j C krwkp; (this fact is proved with the help of the compactness method). Since W 1;p ./ is embedded in Lp ./, we conclude that   (1.27) 8 w 2 W 1;p ./: kwkp;  C.p; ; d / j`.w/j C krwkp; Particular forms of (1.27) arise if w belongs to the subspace of W 1;p ./ defined by the condition `.w/ D 0. Then, (1.27) reads kwkp;  C.p; ; d /krwkp;

8 w 2 fW 1;p ./ j w 2 ker `g:

(1.28)

In our analysis, we need guaranteed and explicitly computable estimates of the constant C.p; ; d /. Henceforth, for simplicity we often use a shorter notation C./ for such type constants. The Poincar´e inequality If `.w/ D

R

w d x, then the set ker ` consists of the func-



tions satisfying the condition fjwjg D 0 and (1.28) yields kwkp;  CP ./krwkp; where

e 1;p ./; 8w 2 W

(1.29)

e 1;p ./ WD fW 1;p ./ j fjwjg D 0g: W

If p D 2, we obtain the classical inequality established by H. Poincar´e [239] (originally for convex domains with smooth boundaries). For piecewise smooth domains this inequality (and a similar inequality for functions vanishing on the boundary) was 1 independently established by V. Steklov [298], who proved that CP D  2 , where  is the smallest positive eigenvalue of the problem u D u in ; @u D0 on @: @n

(1.30) (1.31)

Getting guaranteed and computable bounds of CP (and other constants in various functional inequalities; see, e.g., S. Mikhlin [203]) is a question of utmost importance for quantitative analysis of partial differential equations. Sometimes this question can be answered fairly easily. The very first estimates of CP was actually obtained by H. Poincar´e (CP ./  0:5401 diam  for d D 2, where diam  denotes the diameter of ). In general, finding the constant is equivalent to finding a lower bound of the smallest positive eigenvalue associated with some differential problem (as in (1.10)– (1.11)). Such a problem may be rather difficult. Below we briefly discuss different results that help to overcome these difficulties. If  is a convex domain and p D 2, then for any d we have the following easily computable upper bound of the constant (see L. Payne and H. Weinberger [232]): CP ./ 

diam   0:3183 diam : 

(1.32)

9

1.2 Functional inequalities

A lower bound of CP ./ was derived in S. Cheng [82] (for d D 2): CP ./ 

diam   0:2079 diam : 2j0;1

(1.33)

Here j0;1  2:4048 is the smallest positive root of the Bessel function J0 . For isosceles triangles an improvement of the upper bound is due to R. S. Laugesen and B. A. Siudeja [174], who proved that 8 1 if ˛  3 ; ˆ < j1;1 ; 1 1 ; j0;1 .2.  ˛/ tan.˛=2//1=2g; if ˛ 2 . 3 ; 2 ; CP ./  diam  minf j1;1 ˆ : 1 .2.  ˛/ tan.˛=2//1=2; if ˛ 2 . 2 ; /; j0;1 (1.34) and j1;1  3:8317 is the smallest positive root of the Bessel function J1 . G. Acosta and R. Duran [3], have shown that for convex domains the constant in the L1 Poincar´e type inequality satisfies the estimate inf kw  ck1; 

c2R

diam  krwk1; : 2

(1.35)

Estimates of the constant for other p can be found in S.-K. Chua and R. L. Wheeden [87] (also for convex domains). The Friedrichs inequality Another important case is when the functional ` is deı

fined by the trace operator, so that the condition `.w/ D 0 defines a subspace H 1 ./ containing functions vanishing on @ (or a part of @ with positive boundary measure). Then we arrive at the Friedrichs inequality kwk  CF ./krwk

ı

8w 2 H 1 ./:

(1.36)

Analogous estimates hold for Lp norms p 2 Œ1; C1/ (see, e.g., [120]) provided that w is a function in W 1;p ./ vanishing on the boundary. It is easy to show that the constant in (1.36) is defined by the lowest eigenvalue of the operator , which satisfies the Rayleigh relation 1 D  WD 2 CF ./

inf

ı w2H 1 ./ w6D0

krwk2 : kwk2

(1.37)

Therefore, lower estimates of the minimal eigenvalue generate upper estimates of the Friedrichs constant, and vice versa. An upper bound of CF ./ is easy to find by means of monotonicity arguments if the homogeneous boundary condition is imposed on the whole boundary @. Let

10

1 Introduction ı

ı

  C . For any w 2 H 1 ./, we can define w b 2 H 1 .C / by setting w b D w in  and w b.x/ D 0 for any x 2 C n . Since kwkC  CF .C /krwkC

ı

8w 2 H 1 .C /;

we see that CF ./  CF .C /. This simple observation opens a way of deriving simple upper bounds for the Friedrichs constant by using known constants for some special domains. For example, if   C WD fx 2 Rd j ai < xi < bi ;

bi  ai D li ; i D 1; : : :; d g;

then 1 CF ./  CF .C / D 

d X 1 l2 i D1 i

!1 :

(1.38)

For problems with mixed boundary conditions, the monotonicity approach is not applicable. However, there exist numerical methods that generate lower bounds of eigenvalues (see [54, 79, 290, 312] and references therein) and upper bounds of the respective constants. Also, we note that discrete versions of the Friedrichs and Poincar´e inequalities valid for piecewise H 1 functions are established in [67]. They are often used in error analysis of various nonconforming approximations (e.g., see [216]).

1.2.3 Inequalities for functions with zero mean traces on the boundary In some cases, the following advanced forms of the Poincar´e estimate are useful. Let  be a measurable part of @ (we assume that the surface measure of  is positive) and   Z e 1 .; / WD w 2 V WD H 1 ./ j fjwjg D 1 w ds D 0 ; H jj  R It is clear that the linear functional ` .w/ WD  w ds satisfies the condition ` .w/ D 0 ) w D 0 for any w 2 P 0 : Therefore, we have the estimates kwk2;  C1 .; /krwk2; ; kwk2;  C2 .; /krwk2; ;

e 1 .; /; 8w 2 H e 1 .; /: 8w 2 H

(1.39) (1.40)

Exact constants C1 .; / and C2 .; / are known for some basic domains (rectangles, parallelepipeds, right triangles; see [209]). For example, if  is a rectangle …h1 h2 WD .0; h1 /  .0; h2 / and  D fx1 D 0; x2 2 Œ0; h2 g, then 1=2   maxf2h1 ; h2 g  h1 and C2 D C1 D tanh : (1.41)  h2 h2

1.2 Functional inequalities

11

If  is a parallelepiped …h1 h2 h3 WD .0; h1 /  .0; h2 /  .0; h3 / and  is the face defined by the condition x1 D 0, then C1 D

maxf2h1 ; h2 ; h3 g 

and C2 D . tanh.h1 //1=2;

(1.42)

where  D maxfh2 Ih3 g . If  D f0 < x2 < x1 < hg and  D fx1 D h, x2 2 Œ0; hg (i.e.,  is a cathetus of the right triangle), then C1 D h 1 , where   2:02876 is the unique root of the

1=2 equation cot  C 1 D 0 in .0; / and C2 D h tanh  , where   2:3650 is the unique root of the equation tan  C tanh  D 0 in .0; /. A wider class of domains is considered in [190], where estimates of CP , C1 .; /, and C2 .; / are deduced for convex polygonal and polyhedral domains. Applications of these type estimates to a posteriori error estimation for elliptic and parabolic problems are discussed on [259, 189, 191].

1.2.4 Korn’s inequalities Korn’s inequalities [157] (first and second) establish the coercivity of bilinear forms generated by the linearised deformation tensor in continuum mechanics. For a bounded Lipschitz domain , the second Korn inequality states that Z  2  (1.43) jwj Cj".w/j2 d x  CK ./kwk21;2; 8w 2 H 1 .; Rd /; 

  where CK ./ is a constant independent of w and ".w/ WD 12 rw C .rw/T . The kernel of ".w/ is the space of rigid motions R./. Any vector field w 2 R./ has the form w D w0 C !0 x, where w0 is a vector independent of x 2 Rd , and !0 is a skew-symmetric tensor with coefficients independent of x, dim R./ D d.d2C1/ . In general, finding the constant CK ./ may be a very difficult problem. One exception is related to the case of homogeneous Dirichlet boundary conditions. For ı

w 2 H 1 ./, it is easy to show that krwk 

p

2k".w/k:

(1.44)

The Korn inequalities are well studied. First, we mention the classical work of Friedrichs [113] and subsequent publications [142, 144, 159, 212, 222, 75, 210] (see also the monographs [89, 102]). Korn-type inequalities for piecewise H 1 vector fields (which are important for certain classes of numerical approximations) were established in [68] and some interesting generalizations of the Korn inequality have been recently presented in [213]. For analysis of models in continuum mechanics we often need certain analogues of the Friedrichs and Poincar´e inequalities valid for vector valued functions and the

12

1 Introduction

operator ". They are kwk  CF;" k".w/k inf kw  zk  CP;" k".w/k

z2R

8w 2 V0 ./;

(1.45)

8w 2 H .; R /; 1

d

(1.46)

where V0 denotes a subspace of H 1 .; Rd / consisting of the functions that vanish on the boundary of  or on some measurable part 0 with positive surface measure. The value of CF;" (or CP;" ) readily follows from the respective Friedrichs (Poincar´e) constant and CK . However, this method is applicable only provided that CK is known. In Chapter 4, related to dimension reduction models, we suggest a way to bypass this difficulty for 3D plate-type domains, where a simpler majorant of the constant is deduced by using separation of variables.

1.2.5 Inf–Sup condition Well-posedness of mathematical problems in the theory of viscous incompressible fluids is based on the following result. Lemma 1.2.1 ([22, 71, 167]). Let  be a bounded domain with Lipschitz continuous boundary. There exists a constant  > 0 (which depends only on  ) such that for any function f 2 L2 ./ satisfying the condition fjf jg D 0 one can find a vectorı

valued function wf 2 H 1 .; Rd / such that div wf D f and

in 

(1.47)

krwf k   kf k:

(1.48)

This lemma is also called the “stability lemma for the Stokes problem” or “existence of a bounded inverse to the operator div”. Also, (1.48) can be viewed as a form of the Neˇcas inequality [211] (for Lipschitz domains a simple proof of this fact can be found in [64]). Thanks to the paper by C. Horgan and L. Payne [144], it is known that for simply connected domains in d D 2 the constants  and CK ./ in (1.43) are joined by the relation 2  D CK ./ D 2.1 C L /;

(1.49)

where L is the constant in the Friedrichs type inequality kuk2  L kvk2 ; which holds for an analytic function u C iv provided that fjujg D 0 (see [112]). Lemma 1.2.1 can be extended to Lq spaces for 1 < q < C1 (see [56, 237, 238, ı

118]), namely, for f 2 Lq ./ satisfying fjf jg D 0, there exists wf 2W such that div wf D f

and

krwf kq  ;q kf kq :

1;q

.; Rd / (1.50)

1.2 Functional inequalities

13

Another form of Lemma 1.2.1 is known in the literature as the Inf–Sup (or Ladyzhenskaya–Babuˇska–Brezzi (LBB)) condition: there exists a positive constant c such that R  q div wd x inf sup (1.51)  c : 2 kqk krwk ı q2e L ./ 1 d q¤0

w2H .;R / w¤0

It is easy to show that (1.51) holds with c D .  /1 . Indeed, for arbitrary q 2 e L2 ./, we can find wq such that div wq D q and krwq k   kqk, which implies the required result. The condition (1.51) and its discrete analogues are used for proving the stability and convergence of numerical methods in various problems related to the theory of viscous incompressible fluids (e.g., in [70, 71] this condition was proved and used to justify the convergence of the so-called mixed methods, in which a boundary-value problem is reduced to a saddle-point problem). Estimates of  Estimates of  for various domains are important for the quantitative analysis of incompressible media problems. It is not difficult to see that the constant c in (1.51) is nonnegative and cannot exceed 1 (hence   1). Moreover, c > 0 for any bounded Lipschitz domain. For domains with cusps, c may be 2 2 equal to zero. For a ball in Rd , c D p1 and for the ellipse xa2 C yb 2 < 1, where d

2

2  a2aCb 2 . Estimates for a number of a < b, the constant satisfies the estimate c other domains can be found in [100, 85, 152]. The latter publication is mainly devoted to numerical computation of c (what may be not an easy task even for simple domains). A variational principle obtained for  in [262] can help in constructing numerical approximations of this constant. Estimates of c are also known for Lipschitz domains in R2 , which are starshaped with respect to a ball with center x0 . Let r be the ray from x0 crossing  at x. For almost all x 2 , there exists a unique tangent line, which forms a positive angle   =2 with the ray r. The quantity ‚ WD maxx2 .x/ generates the first guaranteed lower bound that can be computed by simple geometrical analysis (see [144]): ‚ c  sin : (1.52) 2

However, this bound may be rather pessimistic (e.g., for a square ‚ D 4 and, therefore, the estimate shows that c  sin 8  0:0069 and   146). In [98], a significant improvement of these estimates was obtained for domains in R2 which are contained in a disc of radiusR and are star-shaped with respect to a concentric disc of radius . Specifically, it was shown that

1=2 p c  p 1 C 1  2 ; 2 where D

 R.

(1.53)

14

1 Introduction

For d D 3, explicit bounds of  are known only for domains with sufficiently regular boundaries. In [231], it was shown that for star shaped domains in R3 with C 1 boundary presented in the form r D r0 .; /, where .r; ; / are spherical coordinates. Estimates of the constants  and c for exterior domains have been recently obtained in [230]. Distance to the set of divergence free fields The constant  arises in estimates ı

ı

of the distance between a function v 2 H 1 .; Rd / and the space S.; Rd / consisting of divergence-free (solenoidal) vector functions vanishing on the boundary if the distance is measured in terms of the H 1 norm. ı In view of Lemma 1.2.1, for f D div v there exists wf 2 H 1 .; Rd / such that krwf k   kf k and div wf D f . Hence the function w0 WD v  wf belongs to ı

the space S.; Rd / and kv  w0 k   kf k. Therefore, ı

dist.v; S.; Rd // WD kr.v  …ı v/k   k div vk ; S

(1.54)

ı

ı

where …ı W H 1 .; Rd / ! S.; Rd / is the orthogonal projector. This estimate S also follows from (1.51) (see [254]). Estimates of this type are important in the evaluation of the accuracy of numerical solutions, which satisfy the divergence-free conditions only approximately or in comparing solutions of models accounting for the incompressibility condition in different (weaker) forms (see Chapter 6). For domains with complicated boundaries and holes, it may be very difficult to find sharp and guaranteed majorants of the constant  (especially for 3D domains). Therefore, there arises the question of how to get practically applicable versions of (1.54) for domains of such a type. To answer it, we use ideas of domain decomposition. Below we briefly discuss the corresponding method referring for a more detailed exposition to [258, 260] and some other publications cited therein. Assume that  is decomposed into N non-overlapping Lipschitz subdomains i , i D 1; 2; : : :; N and f 2 Lq ./ (0 < q < 1) satisfy the conditions fjf jgi D 0;

i D 1; 2; : : :; N:

(1.55)

Using (1.50), we obtain the following result: ı

Lemma 1.2.2. If f satisfies (1.55), then there exists vf 2 W 1;q .; Rd / such that div vf D f

and krvf kq;q 

N X i D1

qi ;q kf kqi ;q ;

where i ;q are positive constants associated with subdomains i .

(1.56)

15

1.3 Computable bounds of constants in functional inequalities ı

To prove this estimate, note that by (1.50) there exists vf; i 2 W 1;q .i ; Rd / such that div vf; i D f in i and krvf; i ki ;q  i ;q kf ki ;q : ı

Define vf .x/ D vf; i .x/ if x 2 i . Then, vf 2 W 1;q .; Rd /, div vf D f , and krvf kq;q D

n X i D1

kvf;i kqi ;q 

n X i D1

qi ;q kf kqi ;q :

Lemma 1.2.2 yields an estimate of the distance from v 2 W 1;q .; Rd / to the set of divergence-free fields provided that v satisfy additional conditions fjdiv vjgi D 0

i D 1; 2; : : :; N: (1.57) R Since fjdiv vjg D 0, the vector-valued function v satisfies  v  n ds D 0 and, therefore, the nonhomogeneous boundary condition on  admits a divergence-free extension. Thus, by shifting we can reduce this case to the above discussed case with homogeneous boundary conditions. Notice that the integral conditions (1.57) do not lead to essential technical difficulties provided that N is not too large. Indeed, if v does not satisfy the conditions (1.57) exactly, then it is easy to correct it by changing values of v  n on ij D i \ j and 1 \ i . The corresponding procedure changes N parameters in the representation of v such that all the boundary integrals vanish. Lemma 1.2.3. Let v 2 W 1;q .; Rd / satisfy (1.57) and div v 2 L i .i ; Rd /, where i  q, i D 1; 2; : : :; N . Then, there exists v0 2 W 1;q .; Rd / such that div v0 D 0, v D v0 on , and !1=q N X q 1 kr.v  v0 /k;q  qi ;q ji j i k div vkqi ; i : (1.58) i D1

Remark 1.2.4. If div v is bounded almost everywhere (which is typical for piecewise polynomial approximations), then (1.58) yields the estimate kr.v  v0 /kq;q 

N X i D1

 q qi ;q ji j ess supi j div vj :

(1.59)

1.3 Computable bounds of constants in functional inequalities The computation of exact (minimal) constants in Poincar´e, Friedrichs, and other functional inequalities may be a very difficult problem, especially for multi-connected domains with complicated boundaries. However, for quantitative analysis it is usually

16

1 Introduction

enough to have guaranteed and realistic bounds of these constants. Here we discuss a method (suggested in [257, 261], see also [263]) capable of providing them. In general, the main idea of this method is similar to the one that was used for the derivation of a posteriori estimates of functional type: use the integration by parts formulas generated by a pair of adjoint differential operators in order to transform certain unknown integral expressions into computable ones. As a result, estimates of constants in functional inequalities contain “free functions” (i.e., have the same principal structure as the estimates derived for measuring distances to the exact solution of a problem). Any choice of such free functions (and of the supplementary parameters) provides a guaranteed upper bound, but, certainly, getting a good bound requires a rational selection (which can be done by the direct minimization of the majorant with respect to the set of free functions and parameters). Advanced forms of the method are based on ideas of domain decomposition.

1.3.1 Constant in the Friedrichs inequality Let  be a bounded domain in Rd whose boundary has two measurable non-intersecting parts 1 and 2 . Our first goal is to estimate integral quantities associated with a function v 2 V0 WD fv 2 W 1;˛ ./ j v D 0 on 1 g;

˛ > 1:

For this purpose, we use a vector-valued function  in the set n o  Q WD  2 L˛ .; Rd / j div  D .x/ 2 L1 ./;   n D 0 on 2 ; where the condition   n D 0 on 2 is understood in the sense that Z 8w 2 V0 : ` .w/ WD .rw   C w div /d x D 0

(1.60)



Notice that the set Q is not empty if the equation u D  with the boundary  conditions u D g on 1 and ru  n D 0 on 2 has a solution in W 1;˛ for some g. In view of (1.60), we have for any v 2 V0 ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇZ ˇ ˇZ ˇ ˇZ ˇ ˇ ˇ ˇ ˇ ˇ ˇ v d xˇ D ˇ v div  d xˇ D ˇ   rv d xˇ  kk˛ krvk˛ ; ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ 



which yields the estimate ˇ ˇ ˇZ ˇ ˇ ˇ ˇ v d xˇ  C.˛  ; ; /krvk˛ ; ˇ ˇ ˇ ˇ 



where

C.˛  ; ; / D inf kk˛ : 2Q

1.3 Computable bounds of constants in functional inequalities

17

If .x/  0, it can be viewed as a weight function. Certainly, the exact value of C.˛  ; ; / may be difficult to find. Nevertheless, each  2 Q yields a computable majorant of this constant, which can be used in quantitative estimates. In particular, for  D 1 we have an upper bound of the mean value jfjvjg j 

1 kk˛ krvk˛ : jj

(1.61)

Using (1.61) with ˛ D 2 and the identity kvk2 D kv  fjvjg k2 C jjfjvjg2 ;

(1.62)

we find that kvk  C krvk;

 1=2 where C WD CP2 ./ C jj1 kk2 :

(1.63)

If the Poincar´e constant CP (or a majorant of it) is known, then (1.63) easily yields computable majorants of the Friedrichs constant for problems with mixed boundary conditions defined on 1 and 2 . The condition div  D 1 (contained in the definition of Q ) can be weakened and replaced by fjdiv jg D 1. Indeed, let %./ D div  1. Since ˇ ˇ ˇ ˇ ˇZ ˇ ˇZ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ jj jfjvjg j  ˇ v%./ d xˇ C ˇ   rvd xˇˇ ; ˇ ˇ ˇ ˇ 



we obtain 1 .CP ./k%./k C kk/krvk; jj 2 e  krvk; C e 2 D CP2 ./ C .CP ./k%./k C kk/ : kvk  C jj jfjvjg j 

(1.64) (1.65)

1.3.2 Constants in Poincar´e-type inequalities The same method allows us to deduce estimates of the constants in (1.39) and (1.40). Theorem below presents an upper bound of the constant C1 .; /. Theorem 1.3.1. [261] Suppose  has a positive surface measure. Then for any e 1 .; /, v2H C12 .; /  CP2 ./ C

jj jj2

inf

 2Q ./; ˇ>0

E.; ˇ/;

where  2  1Cˇ 2 jj    ; E.; ˇ/ D .1 C ˇ/kk C CP ./ div   ˇ jj  2

(1.66)

18

1 Introduction

and

Q ./ WD f 2 H.; div/ j   n D 1 on ;

nD0

on @ n g:

e 1 .; / Proof. Notice that for any  2 Q ./ and v 2 H Z jj ` .v/ D v ds D 0 and fjdiv jg D : jj 

Therefore, ˇ ˇ ˇZ    ˇ  ˇ ˇ jj v d xˇˇ : .  rv C div   jj jfjvjg j D ˇˇ jj ˇ ˇ 

Estimating the terms in the right-hand side, we obtain jfjvjg j 

kk C CP ./k div   fjdiv jg k krvk: jj

(1.67)

Now (1.62) and (1.67) show that for any ˇ > 0, kvk2  CP2 ./krvk2 C

jj E.; ˇ/krvk2 : jj2 

This inequality implies (1.66). Remark 1.3.2. If  2 Q ./ is selected such that div  D (1.66) has the simplified form C12 .; /  CP2 ./ C

jj jj2

inf

2Q ./

jj , jj

then the estimate

kk2 :

(1.68)

Different  yield different upper bounds of the constant. The best  in (1.68) is defined as   D ru , where u solves the auxiliary Neumann problem u D

jj jj

in ;

ru  n D g on @;

where g D 0 on @ n  and g D 1 on . Example 1.3.3. We compare approximate values of the constant C1 .; / computed with the help of the above presented method with the exact ones (if they are known). Let  D h WD .0; h/  .0; 1/ and  D fx1 D 0; x2 2 Œ0; 1g:

1.3 Computable bounds of constants in functional inequalities

19

x1 ; 0g. h

Since

By (1.41) we find that C1 .; / D maxf2h; 1g 1 . Set  D f1 C jj kk2 D h3 and jj 2 D h, we use (1.68) and find that C12 .h ; /  CP2 .h / C

h2 DW C 1 .h ; / : 3

(1.69)

h ;/ changes from 1 to 1:35 if h 2 Here CP2 .h / D 12 maxfh2 ; 1g. The ratio CC11. .;/ .0; 0:5, from 1:35 to 1:04 in the interval Œ0:5; 1, and it is close to 1:04 for h > 1.

Example 1.3.4. By (1.69) we also obtain estimates for simplices. Let  D 4h WD Convf.0; 0/; .h; 0/; .0; 1/g and  D fx1 D 0; x2 2 Œ0; 1g: In this case, jj D h2 . Set  D f1 C xh1 ; xh2 g. Then,     h 1 1 1 1 1 2 1C C C 2 D h C kk D 6 4 4 2h 4 12h2 and h2 1 C12 .; /  CP2 .4h / C (1.70) C DW C 1 .4h ; /: 8 24 h , where J1 D 3:8317 is the first root of In [174], it was shown that CP .4h /  diam4 J1 the Bessel function J1 . Therefore, we obtain an upper bound of the constant in the form 1 C h2 3h2 C 1 2 D h2 C ı; C 1 .4h ; / D C (1.71) 24 J12 where  D

1 J12

C 18  0:1931 and ı D

1 J12

1 C 24  0:1098: If h D 1, then we find that

C 1 .4h ; / D 0:5504 (the exact constant is equal to 0:4929, see [209]). Hence we see that the above simple choice of  generates quite realistic bounds of the constant. Similar arguments can be applied to the simplex 4a;b WD Convf.0; 0/; .1; 0/; .a; b/g; where a 2 Œ0; 1 and  D f0  x1  1; x2 D 0g. In this case, diam2 .4a;b / a2  a C b 2 1 C C : (1.72) C12 .4a;b ; /  2 8 24 J1 If a D b D 12 , then C1 .4 1 ; 1 ; /  0:3314 (the exact constant is 0:2465). 2 2

Consider the reference simplex in R3 : 41;1;1 WD Convf.0; 0; 0/; .1; 0; 0/; .0; 1; 0/; .0; 0; 1/g and set  D Convf.0; 0; 0/; .1; 0; 0/; .0; 1; 0/g: In this case, jj D 16 , jj D 12 , and CP .41;1;1 / 

p

2 . 

We use (1.68) with  D fx1 ; x2 ; 1 C x3 g and find that

2 4 C : 2  45 Using affine-equivalent coordinate transformations we can deduce guaranteed bounds of the constants for various nondegenerate simplices in R2 and R3 (see [190]). 2

C 1 .41;1;1; / 

20

1 Introduction

1.3.3 Constants in trace-type inequalities Next we discuss briefly estimates of the constant C2 .; /, where  is a part of @ e 1 .; / (see Sect. 1.2.3). If the constant C1 .; / has been defined, then and v 2 H an upper bound of C2 .; / follows from the integral identity Z Z 2 v ds D .v 2 div  C   r.v 2 / d x; 



where  is selected so that   n D 1 on ,  2 L1 .; Rd /, and div  2 L1 ./. We have kvk2  k div k1; C12 .; /krvk2 C kk1; kr.v 2 /k : This inequality shows that C2 .; /  C 2 .; /, where 2

C 2 .; / WD k div k1; C12 .; / C 2C1 .; /kk1; :

(1.73)

Now we can obtain a computable bound of the constant in the trace estimate. Let v 2 H 1 ./ and  2 Q . Then Z jj jfjvjg j D .v div  C rv  /d x  .kvk2 C krvk2 /1=2 kkdiv; : 

Since kvk2 D kv  fjvjg k2 C jjjfjvjg j2  C22 .; /krvk2 C kvk21;2; kk2div; ; we conclude that kvk  Ctr .; /kvk1;2; ; where Ctr2 .; / D C22 .; / C kk2div; .

1.3.4 Estimates of constants based on domain decomposition The estimate (1.66) yields easily computable bounds of the constant for domains with complicated boundaries if we combine it with domain decomposition. In the simplest case,  is decomposed into two non-overlapping domains and   @\@1 (typical examples are depicted in Fig. 1.3.1). We define  such that div  D c 2 R in 1 ,  D 0 in 2 and  n D 0 on @1 n. If in addition  n D 1 on , then  2 Q ./. Since c D jj=j1 j and fjdiv jg D jj=jj, we find that k div  

jj2 j2 j jj 2 k D : jj jjj1 j

1.3 Computable bounds of constants in functional inequalities

 

21



 

Figure 1.3.1. Decomposition of  into non-overlapping subdomains.

Then (1.66) yields the estimate C12 .; /



CP2 ./

C

j2 j1=2 jj1=2 kk1 C CP ./ jj j1 j1=2

!2 :

(1.74)

If j2 j D 0, then (1.74) reduces to (1.68). In particular, if 1 D h and jj D 1 (see Fig. 1.3.1, left), then using (1.69) we obtain !2 r j2 j1=2 h 2 2 1=2 C1 .; /  CP ./ C jj : (1.75) C CP ./ p 3 h If 1 D 4a;b (i.e., b is the height of the triangle) and jj D 1 (Fig. 1.3.1, right), jj 2 then we can take the same  as for the triangle 4a;b . Then jj 2 kk1 is defined by / the last two terms of (1.72) and we can use (1.74) with C P ./ D .diam .  More complicated (e.g., multi connected domains) can be decomposed into a larger number of subdomains. Then estimates of C12 .; / can be deduced by obvious generalisations of the method discussed above. However, using (1.74), (1.75), and other similar estimates requires a computable bound of the constant CP ./. This question is considered next. Assume that  can be divided into N disjoint subdomains i such that S  D N i D1 i and the constants CP .i / associated with the subdomains i are known (e.g., if all the subdomains are convex, then we can use the estimate (1.32)). We wish to find a computable majorant of CP ./ using these known constants. Introduce the set of vector-valued functions fy.1/ ; y.2/ ; : : : ; y.N 1/ g such that

y.i / 2 H.; div/; Let !i D supp.y.i / /,  D N 1 X j D1

SN 1 i D1

y.i /  n D 0 on  D @:

(1.76)

!i and

kwk2!i  C! kwk2

8w 2 L2 ./;

(1.77)

22

1 Introduction

where C! is a positive constant (it depends on the maximal number of intersections between different sets !i ). One more requirement is that the matrix B WD fˇij gN i;j D1 , .i / where ˇij WD fjdiv y jgj , ˇNj D 1 for i D 1; : : :; N  1; j D 1; : : :; N , is nondegenerate, i.e., det B 6D 0: (1.78) Remark 1.3.5. It is not difficult to show that functions y.i / with the required properties exist. For example, we can set y.i / D rui , where ui is the solution of the problem ji j jN j

ui D 1

in i ;

ui D 

ui D 0

in  n .i \ N / ;

@ui D 0 on : @n

in N ;

This Neumann boundary-value problem is solvable for any i D 1; 2; : : :; N  1 and ji j the corresponding matrix B has the entries ˇi i D 1, ˇiN D  j , i D 1; : : : ; N 1, Nj ˇij D 0 if i 6D j and j 6D N , i D 1; : : : ; N  1; j D 1; : : : ; N , ˇNj D 1, j D 1; : : : ; N . Since det B D 1 C

N 1 X i D1

jj ji j D > 0; jN j jN j

we see that (1.78) holds. Certainly the above example has mainly a theoretical meaning and in a particular practical example the functions y.i / can be constructed in a simpler way without solving auxiliary boundary value problems (see [257]). Theorem 1.3.6 ( [257]). Let y.i / satisfy the conditions (1.76)–(1.78) and ˛ 2 RN 1 be a vector with positive components ˛i . Then, the following estimate holds CP2 ./

 max

1i N

CP2 .i /

C ˚ .D/

N 1 X

! .1 C

˛i /Ei2

C .˛; y/;

(1.79)

i D1

 T where D D B1 ‡ B1 , ‡ is a diagonal matrix with entries 1=ji j, i D 1; 2; : : :; N  1, N X Ei2 D CP2 .j /k div y.i /  ˇij k2j ; j D1

and .˛; y/ D C! max

1i N

n o  1 C ˛i1 ky.i / k2 :

1.3 Computable bounds of constants in functional inequalities

23

Corollary 1.3.7. If the functions y.i / satisfy the condition div y.i / D const on any j , j D 1; 2; : : : ; N (such vector fields can be constructed with the help of the Raviart– Thomas approximations [245]), then the majorant has the following simplified form: CP2 ./

 max

1i N

CP2 .i /

C ˚ .D/C!

N 1 X

ky.i / k2 :

j D1

Finally, we note that estimates of the constants C1 .; / and C2 .; / have been used in a posteriori error estimation methods for elliptic and parabolic problems (see [259, 189, 191]) and in special interpolation methods for polygonal domains (see [261, 263]). In these publications, the reader will find explicit bounds of the constants for a wide collection of domains.

Chapter 2

Estimates of the distance to exact solutions This chapter surveys mathematical methods developed to estimate the distance between the exact solution of the corresponding boundary value problem and a given/computed function in the energy space which is considered as an approximation of the exact solution. We are mainly concerned with the problems that admit a variational or saddle point formulation. This class of mathematical models includes all convex variational problems, variational inequalities, models in linear and nonlinear mechanics of solids and fluids, and many incremental models describing evolutionary processes. First we deduce error identities in the most general form. They show that computable quantities arising in a variational method generate certain measures of errors, which naturally characterise accuracy of approximations. For linear problems, the measures can be expressed in terms of squared norms in Banach spaces. The identities serve as a source of two-sided estimates of the respective errors. In the second part of the chapter, the general theory is applied to various mathematical models. Finally, we discuss applications of the theory to quantitative validation of mathematical models based on results of physical experiments. Our analysis uses results obtained in [250, 249, 247, 255, 271, 275].

2.1 A class of boundary value problems Henceforth, we operate with two reflexive Banach spaces V and Y consisting of real valued functions. The spaces V  and Y  are the respective dual counterparts with the duality pairings hv  ; vi and .y  ; y/. Elements of V are denoted by the letters u; v; w (or similar) and elements of Y by p; q; y. Symbols with stars are used for elements of the dual spaces (e.g., y  ; y2 ). The norms of Y and Y  are denoted by k  k and k  k , respectively. By .Y  /2 we denote the tensor space Y  ˝ Y  with the induced norm. Typically V is a Sobolev space of non-negative order (or a subspace of a Sobolev space) with the norm k  kV and V  is a Sobolev space of negative order, whose norm is defined as a supremum over the functions in V and, therefore, is not directly computable. This fact may create difficulties (especially in analysis of approximations produced by various numerical methods). To avoid them, we use another pair of dual Banach spaces V and V  (with the norms defined by integrals) such that V  V and V   V  . For any v 2 V and v  2 V  , the respective pairing satisfies hv  ; viV  kv  kV  kvkV .

26

2 Distance to exact solutions

In the analysis of linear problems, this scheme is simplified and it is sufficient to assume the existence of an intermediate Hilbert space V supplied with the norm k  kV and the scalar product .; /V such that V  V  V  . The symbol ƒ is used for a bounded linear operator acting from V to Y . The conjugate operator ƒ W Y  ! V  is defined by the relation .y  ; ƒw/ D hƒ y  ; wi;

8w 2 V; y  2 Y  :

(2.1)

If y  2 Y ƒ WD f y  2 Y  j ƒ y  2 V  g  Y  ; then instead of (2.1) we use .y  ; ƒw/ D .ƒ y  ; w/V ;

8w 2 V:

(2.2)

Let G W Y ! R be a convex and lower semicontinuous (l.s.c.) functional and G  W Y  ! R be the dual to G (see (1.9)). Analogously, for a convex l.s.c. functional F W V ! R, we define the dual counterpart   F  .v  / WD sup hv  ; vi  F .v/ : v2V

In physical problems, the functionals G and G  are associated with the primal and dual energies, respectively. Throughout the book we assume that G is nonnegative and vanish at the zero element 0Y of Y (what is natural for the energy functional). Then G  is also nonnegative and vanishes at 0Y  . The functional F may take both positive and negative values and we also assume that F .v/ D 0 if v D 0V . Notice that the latter condition is not essential and if necessary it can be bypassed by a suitable “shifting” procedure (similar to the one used in Sect. 2.3.6). We need to impose one more (coercivity) condition on G and G  , namely, that

G .y/ ! C1 G  .y  / ! C1

if kyk ! C1; if ky  k ! C1:

(2.3) (2.4)

The conditions presented above yield well posedness of the related variational problems provided that the operator ƒ generates an equivalent norm in V , i.e., there exist positive constants c1 and c2 such that c1 kwkV  kƒwk  c2 kwkV ;

8v 2 V:

(2.5)

For G and F , we define two nonnegative functionals (see Sect. 1.1.3)

DG .y; y  / WD G .y/ C G  .y  /  .y  ; y/; DF .v; v  / WD F .v/ C F  .v  /  hv  ; vi;

(2.6) (2.7)

which are widely used in our analysis. In view of (1.16), DG .y; y  / D 0 if and only if y  and y satisfy the relations y  2 @G .y/ and y 2 @G  .y  /:

(2.8)

2.1 A class of boundary value problems

27

The functional DF .v; v  / has quite analogous properties: it is nonnegative and DF .v; v  / D 0 if and only if v  2 @F .v/

and v 2 @F  .v  /:

(2.9)

These properties of compound functionals make them components of the natural error measure adapted to the structure of the problem. We consider boundary value problems that can be reduced to finding a saddle point of the Lagrangian L.vI y  / WD F .v/ C .y  ; ƒv/  G  .y  /;

(2.10)

where v 2 V , y  2 Y  . L.vI y  / satisfies conditions that guarantee the existence of a saddle point .uI p / 2 V  Y  such that L.uI y  /  L.uI p /  L.vI p /

(2.11)

for any v 2 V and any y  2 Y  . From the general theory (e.g., see Chapter 6 of [103]), we know that in addition to the convexity and semicontinuity conditions1, one needs to assume that at least one of the following two conditions hold: there exists v 2 V such that L.vI y  / ! 1

if ky  k ! C1 I

(2.12)

there exists y  2 Y  such that L.vI y  / ! C1 if kvkV ! C1:

(2.13)

In view of (2.4) and properties of the functionals G , G  , and F , the condition (2.12) is satisfied if we set v D 0V 2 . From the left inequality in (2.11), we conclude that3 .   .p ; ƒu/  G  .p /  sup y  ; ƒu/  G  .y  / D G .ƒu/: y  2Y 

Since G .ƒu/ C G  .p /  .ƒu; p /  0; this inequality implies the first relation that holds for the exact solutions u and p :

DG .ƒu; p / D 0:

(2.14)

1 The semicontinuity conditions read: for any v 2 V , the functional y  ! L.v; y  / is concave and upper semicontinuous and for any y  2 Y  , the functional v ! L.v; y  / is convex and lower semicontinuous. 2 In general, the conditions (2.12) and (2.13) are not necessary and the derivation of error estimates for certain problems can be based on weaker assumptions (as, e.g., for the Henky plasticity problems; see [115]). 3 Notice that G is a convex functional and G  D G.

28

2 Distance to exact solutions

The right-hand side of (2.11) implies

F .u/ C .p ; ƒu/  inf fF .v/ C .p ; ƒv/g v2V

D  sup fhƒ p ; vi  F .v/g D F  .ƒ p /: v2V

Hence, we obtain

DF .u; ƒ p / D 0:

(2.15)

Let us emphasize that the relations (2.14) and (2.15) determine completely the solutions u and p and play an important role in the subsequent analysis. The saddle point problem (2.11) generates two variational problems associated with two functionals J .v/ and I  .y  /, as follows The primal functional J is defined by the formula

J .v/ WD G .ƒv/ C F .v/ D sup L.vI y  /: y  2Y 

Hence, the first variational Problem P is to find u 2 V such that

J .u/ D inf J .v/:

(2.16)

v2V

The dual functional I  is defined by taking infimum with respect to the variable v:   I  .y  / D inf L.v; y  / D inf F .v/ C .y  ; ƒv/  G  .y  / v2V v2V   D  sup .y  ; ƒv/  F .v/  G  .y  /: v2V

Hence,

I  .y  / D F  .ƒ y  /  G  .y  / and the second (dual) variational Problem P  is to find p 2 Y  such that

I  .p / D sup I  .y  /:

(2.17)

y  2Y 

Directly from the definition of these problems it follows that

I  .y  /  J .v/

8v 2 V; y  2 Y  :

(2.18)

Moreover, if the saddle point exists, then sup I  .y  / D I  .p / D J .u/ D inf J .v/:

y  2Y 

v2V

(2.19)

2.2 The main error identity

29

2.2 The main error identity 2.2.1 Error measure First, we prove the main identity, which motivates a measure natural for analysis of errors4 . Theorem 2.2.1. Let .u; p / 2 V  Y  be such that (2.11) is satisfied. Then for any y  2 Y  and v 2 V , it holds that

.v; y  I u; p / D J .v/  I  .y  /;

(2.20)

where

.v; y  I u; p / WD DG .ƒv; p / C DG .ƒu; y  /

C DF .v; ƒ p / C DF .u; ƒ y  /:

Proof. It is easy to see that

J .v/  I  .y  / D G .ƒv/ C G  .y  / C F .v/ C F  .ƒ y  / D G .ƒv/ C G  .p / C G .ƒu/ C G  .y  / C F .v/ C F  .ƒ y  / C F .u/ C F  .ƒ p / D DG .ƒv; p / C .p ; ƒv/ C DG .ƒu; y  / C .y  ; ƒu/ C DF .v; ƒ p /  hƒ p ; vi C DF .u; ƒ y  /  hƒ y  ; ui: 

By recalling (2.1) we obtain (2.20).

The quantity .v; y  I u; p / is a sum of four nonnegative terms. Theorem 2.2.1 shows that it is a natural measure of the distance between v; y  , (approximations) and u; p , (exact saddle point). Indeed, .v; y  I u; p /  0 and .v; y  I u; p / D 0 if and only if J .v/ D I  .y  /. In other words, the measure vanishes if and only if v and y  coincide with the minimizer u and maximizer p , respectively (see (2.18) and (2.19)). This fact also follows directly from the definition of . If .v; y  I u; p / D 0, then

DG .ƒv; p / D 0; DF .v; ƒ p / D 0;

DG .ƒu; y  / D 0; DF .u; ƒ y  / D 0:

These relations mean that v 2 @F  .ƒ p / and y  2 @G .ƒu/ (cf. (2.8) and (2.9)). Hence, v D u and y  D p . It is worth making a number of comments on the components of the measure . The quantity DG .ƒv; p / can be viewed as a measure of the distance from v to the 4 Some parts of this section are based on results obtained in [214, 250, 249, 247, 255, 271, 275] and other publications cited in subsequent sections.

30

2 Distance to exact solutions

minimizer u (which is defined by the relation p 2 @G .ƒu/). In particular, if G is Gˆateaux differentiable, then p D G 0 .ƒu/ and DG .ƒv; G 0 .ƒu//  0 is a nonlinear measure of the distance between u and v, which vanishes if and only if ƒv coincides with ƒu. The functional DG .ƒu; q  / has a similar meaning. If G  is differentiable, then DG .ƒu; q  / D DG .G 0 .p /; q  /  0. It vanishes if q  D p . In the general case, DG .ƒu; q  / vanishes if q  coincides with a function in @G .ƒu/. Hence we view it as a measure of the distance between q  and p . Thus the first two functionals are measures of the distance from v to u and from q  to p expressed via the functional G and the operator ƒ. Analogously, the functionals DF .v; ƒ p / and DF .u; ƒ y  / are measures of the distance from v to u and from q  to p expressed via the functional F and the operator ƒ . Later we will see that for quadratic functionals (and only for them) the four terms of .v; y  I u; p / are represented by norms (cf. Sect. 2.3.1). Let us show that the measure  generates a collection of convex neighbourhoods of .u; p /5 . For  > 0, consider the set T WD f.v; y  / 2 V  Y  j .v; y  I u; p /   g : Let .v1 ;y1 / 2 T and .v2 ;y2 / 2 T . Since J and I  are convex, for any 1 ; 2  0 with 1 C 2 D 1 we have

.1 v1 C 2 v2 ; 1 y1 C 2 y2 /

 1 J .v1 / C 2 J .v2 /  1 I  .y1 /  2 I  .y2 / D 1 .v1 ; y1 I u; p / C 2 .v2 ; y2 I u; p /  :

Hence .1 v1 C 2 v2 ; 1 y1 C 2 y2 / 2 T . Remark 2.2.2. Theorem 2.2.1 shows that every variational method based on minimization of the gap J .v/  I  .y  / approximates .u; p / in terms of the measure . Theorem 2.2.3. Let .u; p / 2 V  Y  satisfy (2.11). Then for any y  2 Y  and v 2 V , it holds

.v; y  I u; p / D DG .ƒv; y  / C DF .v; ƒ y  /:

(2.21)

Proof. Since

G .ƒv/ C F .v/ C F  .ƒ y  / C G  .y  / D DG .ƒv; y  / C DF .v; ƒ y  / the identity (2.21) follows from (2.20).



We consider (2.21) as the main error identity. If v and y  are functions computed in a numerical experiment, then (2.21) can be also viewed as the a posteriori error identity. It states the equality between the error measure and the quantity in the 5

We may say that  generates a locally convex topology at the vicinity of .u; p  /.

31

2.2 The main error identity

right-hand side of (2.21), which is fully defined by approximations. The right-hand side of (2.20) is also defined by approximate solutions, but the identity (2.21) is more informative, because its right-hand side explicitly shows two different components of the error associated with the conditions generated by the functionals G and F , respectively. Knowledge on their values may be useful for the selection of a computational strategy. In this chapter, we discuss particular forms of (2.21). Later they will be used for the evaluation of various modeling errors. Remark 2.2.4. We have

DF .v; ƒ p / C DF .u; ƒ y  /  DF .v; ƒ y  / D hƒ p ; vi C hƒ y  ; ui  hƒ y  ; vi C F .u/ C F  .ƒ p / D hƒ p ; ei  hƒ y  ; ei D hƒ e ; ei; where e D v  u and e D y   p . Analogously,

DG .ƒv; p / C DG .ƒu; y  /  DG .ƒv; y  / D G .ƒv/ C G  .p /  .p ; ƒv/ C G .ƒu/ C G  .y  /  .y  ; ƒu/  G .ƒv/  G  .y  / C .y  ; ƒv/ D .y   p ; ƒe/ D .e ; ƒe/: Therefore, (2.21) consists of the following two identities:

DG .ƒv; p / C DG .ƒu; y  / D DG .ƒv; y  / C .e ; ƒe/

(2.22)

and

DF .v; ƒ p / C DF .u; ƒ y  / D DF .v; ƒ y  /  hƒ e ; ei:

(2.23)

It is easy to see that summing (2.22) and (2.23) yields (2.21). These abridged error identities contain the errors e and e in the right hand sides (which vanish only after summation) They cannot be directly used for the error estimation, but are useful for theoretical analysis.

2.2.2 Decomposition of the error measure We can decompose the error measure  into two independent parts, related to the primal and dual solutions (for them we use abbreviated notation .v/ and  .y  /). Set y  D p in (2.20). Then two components of .v; y  I u; p / vanish and we obtain the measure

.v/ WD DG .ƒv; p / C DF .v; ƒ p / related to the primal variable. In view of Theorem 2.2.1 and (2.19),

.v/ D J .v/  J .u/: Hence .v/ is nonnegative and vanishes if and only if v D u.

(2.24)

32

2 Distance to exact solutions

Set v D u in (2.20). Then another two components of .v; y  I u; p / vanish and the remaining terms form the error measure

 .y  / WD DG .ƒu; y  / C DF .u; ƒ y  /: Since

 .y  / D I  .p /  I  .y  /;

(2.25)

this measure vanishes if and only if y  D p . Now (2.20) and (2.21) imply a somewhat different form of the main error identity:

.v/ C  .y  / D DG .ƒv; y  / C DF .v; ƒ y  /:

(2.26)

Remark 2.2.5 (convergence of a minimizing sequence). The identity (2.24) yields a simple answer to the question: in what sense a sequence minimizing a convex and continuous functional J W V ! R converges to the corresponding minimizer? Notice that the sets

O WD fv 2 V j .v/ < g form a local topology (system of open neighbourhoods) around the minimizer u. Let uk 2 V be a minimizing sequence, i.e., J .uk / ! J .u/ as k ! C1. In view of (2.24), every element uk of this sequence belongs to some O k where k ! 0, what means that a minimizing sequence converges to u in this topology. Assume that the sequence fvk g is constructed by minimisation of J .v/ over a finite dimensional subspace Vk so that J .uk / D infv2Vk J .v/ (i.e., uk is the Galerkin approximation of u). If the spaces Vk satisfy the condition Vk  VkC1  V and S 6 the set 1 kD1 Vk is dense in V , then (2.24) guarantees that uk converges to u with respect to the measure . Indeed, for any  we can find a number k and a function uk 2 Vk such that J .uk /  J .u/  . Since J .uk /  J .uk /, we use (2.24) to conclude that .uk / D J .uk /  J .u/   for k  k . Hence, the sequence uk converges to u with respect to the measure . It is clear that for a minimizing sequence no stronger convergence can be guaranteed 7 . Therefore, we consider this measure as the most relevant one and, in a sense, natural for the class of variational problems under consideration. In order to illustrate the above statements, we consider a simple example, where properties of the measure  can be visualized. Example 2.2.6. Let V D Y D R, G .y/ D ƒv D v. Then ƒ y  D the problem is to minimize 6 This

1 jyj˛ , F .v/ D ˇ1 jvjˇ , ˛; ˇ > 1, and ˛   y  , G  .y  / D ˛1 jy  j˛ , F  .v  / D ˇ1 jv  jˇ , and the function J .v/ D ˛1 j vj˛ C ˇ1 jvjˇ (u D 0 is the

condition is often called the “limit density property”. we use only properties of the sequence and do no input additional information, which does not follow from the variational method. 7 If

2.2 The main error identity 

equal to zero. Then DG .ƒv; y  / D

DG .ƒu; y  / D



 j jˇ jy  jˇ , the maximizer ˇ  1 j vj˛ C ˛1 jy  j˛  vy  , and ˛

minimizer). Since J  .y  / D  ˛1 jy  j˛  1  ˛ jy j ; ˛

DG .ƒv; p / D

33

p is also

1 j vj˛ : ˛

Analogously, 1 ˇ 1  jvj C  j  y  jˇ C y  v ˇ ˇ

DF .ƒ y  ; v/ D and

DF .ƒ p ; v/ D

1 ˇ 1  jvj ; DF .ƒ y  ; u/ D  j  y  jˇ : ˇ ˇ

Hence, the measure is given by the formula 

j j˛ ˛ 1 1 j jˇ   .v; y I u; p / D jvj C jvjˇ C  jy  j˛ C  jy  jˇ : ˛ ˇ ˛ ˇ 



Fig. 2.2.1 shows the neighbourhoods generated by the measure  around the exact solution: the point .0; 0/. The picture at the top right corner corresponds to the case where ˛ D ˇ D 2. In this (and only this) case the level lines are circles. In general, the level lines may have various forms, but they always bound convex sets. While this example is related to an elementary variational problem, it nevertheless provides intuition on how the measure  defines neighbourhoods of the exact solution.

2.2.3 Problems with linear F An important equation in mathematical physics is the Euler equation for variational problems, where F has the form

F .v/ D h`; vi and ` 2 V  . In this case, the respective conjugate functional is rather special  0; if v  D `;    F .v / D sup hv  `; vi D C1; if v  6D ` v2V and, therefore,

F  .ƒ q  / D



0; C1;

if ƒ q  C ` D 0; if ƒ q  C ` 6D 0:

The fact that the functional F  is degenerate leads to difficulties and requires a special consideration for this case. Recalling (2.7), we see that both sides of (2.21) and (2.26)

34

2 Distance to exact solutions =2,

=2 =1

=3,

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

−0.1

−0.1

−0.2

−0.2

−0.3

−0.3

−0.4

−0.4

−0.4

−0.3

−0.2

−0.1

=1.3,

0

0.1

0.2

0.3

0.4

0.5

−0.4

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

−0.1

−0.1

−0.2

−0.2

−0.3

−0.3

−0.4

−0.4

−0.2

−0.1

0

0.1

−0.1

=4, 0.5

−0.3

−0.2

=2 =1

0.5

−0.4

−0.3

0.2

0.3

0.4

0.5

−0.4

−0.3

−0.2

−0.1

=2 =3

0

0.1

0.2

0.3

0.4

0.5

0.2

0.3

0.4

0.5

=1.5 =1

0

0.1

Figure 2.2.1. Level lines of  for ˛ D 2, ˇ D 2, D 1 (top left), ˛ D 3, ˇ D 2, D 3 (top right), ˛ D 1:3, ˇ D 2, D 1 (bottom left) and ˛ D 4, ˇ D 1:5, D 1 (bottom right)

are finite only if the dual variable is subject to certain conditions, namely, y  D q  , where q  2 Q` WD fq  2 Y  j ƒ q  C ` D 0g :

(2.27)

This fact explains why all boundary value problems in divergence form (without lower order terms) require additional differential relations for the corresponding dual variable. Error identity It is clear that the main error identity (2.21) has a quantitative meaning only for q  2 Q` (otherwise, it holds in the sense C1 D C1). If q  2 Q` ,

2.2 The main error identity

then

35

DF .v; ƒ q  / D DF .u; ƒ q  / D DF .v; ƒ p / D 0:

Hence, (2.21) takes a reduced form, in which the additive parts of the error measure are defined by the formulas8

G .v/ WD DG .ƒv; p / and G .q  / WD DG .ƒu; q  /:

(2.28)

Now, the error identity (2.26) reads

G .v/ C G .q  / D DG .ƒv; q  /:

(2.29)

From (2.28) and (2.29), it follows that

G .v/ D inf  DG .ƒv; q  /;

(2.30)

q 2Q`

G .q  / D inf DG .ƒv; q  /:

(2.31)

v2V

Hence

G .v/ D DG .ƒv; q  /  inf DG .ƒw; q  /;

8q  2 Q` ;

(2.32)

G .q  / D DG .ƒv; q  /  inf  DG .ƒv;  /;

8v 2 V:

(2.33)

w2V

 2Q`

From (2.30)–(2.33), we deduce two-sided bounds for G .v/ and G .q  /: for any w 2 V and  2 Q` ,

DG .ƒv;  /  DG .ƒw;  /  G .v/  DG .ƒv;  /; DG .ƒw; q  /  DG .ƒw;  /  G .q  /  DG .ƒw; q  /:

(2.34) (2.35)

Other measures of errors Assume that G is Gˆateaux differentiable and uniformly convex in the vicinity of the minimizer. Usually, uniform convexity9 of G in a ball B  Y means that there exists a nonnegative function ' such that '.0/ D 0, './ > 0 if  6D 0, and for all y1 ; y2 2 B,   1 1 y1 C y2 G (2.36) C '.ky1  y2 kY /  G .y1 / C G .y2 /: 2 2 2 Below we use a more general definition of uniform convexity. Assume that there exists a nonnegative functional ˆ W Y  Y ! R0 such that ˆ.y; y/ D 0 for any y 2 Y and   1 1 y1 C y2 G 8y1 ; y2 2 B: (2.37) C ˆ.y1 ; y2 /  G .y1 / C G .y2 / 2 2 2 8 Notice that in general these two measures differ from the measures .v/ and  .q  / defined by (2.24) and (2.25). 9 This kind of definitions can be found in, e.g., [123, 206, 325]

36

2 Distance to exact solutions

We define two nonnegative quantities

C .v/ WD hG 0 .ƒv/  G 0 .ƒu/; ƒv  ƒui and

 .v/ WD 2ˆ.ƒv; ƒu/:

The measure C .v/ is induced by the monotonicity10 and  .v/ is generated by extra convexity properties of G . It is clear that the largest possible functional ˆ in (2.37) is defined by the formula

2 G .y1 / C G .y2 /  2G y1 Cy 2 : ˆ.y1 ; y2 / D 2 It is readily seen that ˆ.y1 ; y2 / vanishes if y1 and y2 lie on an affine part of G (if such a part exists), and it is large if y1 and y2 belong to a part where the “curvature” of G is large. If the functional G is sufficiently smooth and y1 is close to y2 , then y2  y1

y2  y1  hG 0 .y1 /; G .y1 /  G y1 C i; 2

2 y1  y2 y1  y2  hG 0 .y2 /; G .y2 /  G y2 C i: 2 2 Hence,    1 1 y1 C y2 G .y1 / C G .y2 /  G  hG 0 .y2 /  G 0 .y1 /; y2  y1 i 2 2 4 and we see that ˆ.y1 ; y2 /  14 hG 0 .y2 /  G 0 .y2 /; y2  y1 i, so that under these conditions  .v/  12 C .v/. We can also establish more general relations. Theorem 2.2.7. If G is differentiable and satisfies (2.37), then

 .v/  G .v/  C .v/

8v 2 V:

Proof. We have

G .v/ D DG .ƒv; p / D G .ƒv/ C G  .p /  hp ; ƒvi

 hG 0 .ƒv/; ƒ.v  u/i C G .ƒu/ C G  .p /  hG 0 .ƒu/; ƒvi:

Notice that DG .ƒu; p / D 0. Therefore,

G .v/  hG 0 .ƒv/; ƒ.v  u/i C hG 0 .ƒu/; ƒui  hG 0 .ƒu/; ƒvi D C .v/: 10 If G is a convex differentiable functional, then the corresponding Gˆ ateaux derivative defines a monotone mapping, see, e.g. [103].

2.2 The main error identity

37

Now we use another representation of G .v/:

G .v/ D G .ƒv/ C h`; vi C G  .p / D J .v/  G .ƒu/ C hp ; ƒui D J .v/  G .ƒu/ C .ƒ p ; u/ D J .v/  J .u/:

From (2.37) and the linearity of F it follows that

  1 1 uCv J .v/ C J .u/  J : 2 2 2   J.v/  J uCv , we conclude that 2

ˆ.ƒv; ƒu/  Since J .u/  infv2V

ˆ.ƒv; ƒu/ 

1 1 .J .v/  J .u// D G .v/: 2 2



Example 2.2.8. We illustrate Theorem 2.2.7 by a simple example, where G ,  , and C can be explicitly computed. Let Y D Y  D R, G .v/ D ˛1 jvj˛ , F .v/ D `v for some ` > 0. The minimizer satisfies the relation juj˛2 u D `. Let ` > 0; then 1  u D ` ˛1 and J .u/ D  ˛1 `˛ . In this case, 1 ˛

G .v/ D jvj˛  `v C

1 ˛ ` : ˛

Since G 0 .v/ D jvj˛2 v, the monotonicity measure is defined by the relation

C .v/ D .jvj˛2 v  juj˛2 u/.v  u/:

ˇ ˇ˛

ˇ is generated by uniform conThe measure  .v/ D ˛1 jvj˛ C juj˛  2 ˇ uCv 2 vexity of the functional G . Fig. 2.2.2 presents the respective bounds for ` D 1 and different ˛. Due to the Clarkson inequality for ˛  2 (see, e.g., [214]), the functional ˆ satisfies the estimate 1 ˇˇ y1  y2 ˇˇ˛ ˆ.y1 ; y2 /  ˇ ˇ : ˛ 2 This yields a simple lower bound  .v/  simistic if ˛ is not close to 2.

1 ˛2˛1

jv  uj˛ , which is rather pes-

Error identity for a wider class of y  For practical purposes, it is necessary to deduce error relations valid for a wider set of y  that may not satisfy (2.27). Notice that (2.29) implies the identity

G .ƒv/ C G  .p /  .p ; ƒv/ C G .ƒu/ C G  .y  /  .q  ; ƒu/ D G .ƒv/ C G  .y  /  .q  ; ƒv/;

38

2 Distance to exact solutions =2

=4

0.03

0.03

 +

0.025

−

0.02

0.015

0.01

0.01

0.005

0.005

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

−

0.02

0.015

0 0.75

 +

0.025

0 0.75

1.25

0.8

0.85

0.9

0.95

=1.5

1

1.05

1.1

1.15

1.25

=1.2

0.03

0.03

 + 

0.025

 + 

0.025

−

−



0.02

0.015

0.01

0.01

0.005

0.005

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2



0.02

0.015

0 0.75

1.2

1.25

0 0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

Figure 2.2.2. Two-sided bounds of G .v/.

where q  2 Q` and y  2 Y  . We recast it as

G .v/C G .y  / D DG .ƒv; y  / C .q  y  ; ƒ.u  v//;

(2.38)

which yields the identity

G .v/ C G .y  / D DG .ƒv; y  / C hƒ y  C `; vui:

(2.39)

This identity can be viewed as an extended version of (2.29) valid for any y  2 Y  . Since ` D ƒ p , it also follows from (2.22). We see that the difference between G .v/ C G .y  / and DG .ƒv; y  / (which contains approximate solutions only) is equal to the product of the error v  u related to the primal variable and ƒ y  C` 2 V  (residual of the dual variable). If y  2 Q` then this term vanishes and we arrive at (2.29).

2.2 The main error identity

39

Two-sided estimates The identity (2.39) yields two-sided bounds of the error. Let „ W V ! R and „ W V  ! R be two conjugate functionals. Then (2.39) leads to the following lower bound of the error:

G .v/ C G .y  / C „.v  u/  DG .ƒv; y  /  „ .ƒ y  C `/:

(2.40)

In particular, for „.v/ D ˛2 kvk2V (˛ > 0) we find that ˛ 2

G .v/ C G .y  / C ku  vk2V  DG .ƒv; y  / 

1 kƒ y  C `k2V  : 2˛

(2.41)

The left-hand side of (2.41) is a measure of the distance between .u; p / and .v; y  / and the right-hand side is fully defined by approximate solutions. In the next sections, we discuss these estimates in the context of various problems. The identity (2.39) also yields various upper bounds of the error. For example, analogous arguments lead to the estimate

G .v/ C G .y  /  „.v  u/  DG .ƒv; y  / C „ .ƒ y  C `/:

(2.42)

The estimate (2.42) is useful if the measure G .v/ dominates over „.v  u/. To satisfy this condition, we can estimate the last term in (2.39) by the quantity kƒ y  C `k kv  ukV : If the measure G .v/ is bounded from below by kv  ukV multiplied by a positive constant, then we easily obtain an error majorant (see examples in the next section). Here we discuss another way. Let us represent the right-hand side of (2.29) in the form

DG .ƒv; q  / D G  .q  / C G .ƒv/  .q  ; ƒv/ (2.43) D DG .ƒv; y  / C G  .y  C q   y  /  G  .y  / C .y   q  ; ƒv/ D DG .ƒv; y  / C G  .y  C q   y  /  G  .y  / C .y  ; ƒv/  h`; vi: Since G  is convex,

  y DG .ƒv; q  /  DG .ƒv; y  / C G   G  .y  /      q y C .1  / G C hƒ y  C `; vi; 1

where  2 .0; 1/. Hence 

 y  G  .y  / C hR.y  /; vi      q y C .1  / inf  G ; (2.44) 1 q  2 Q`

G .v/ C G .q  /  DG .ƒv; y  / C G 

40

2 Distance to exact solutions

where R.y  / D ƒ y  C ` is an element of V  , i.e., R.y  / W V ! R is a linear functional defined by the relation hR.y  /; wi WD .y  ; ƒw/ C h`; wi: Then, the set Q` is defined as the collection of elements y  2 Y  such that hR.y  /; wi D 0 for all w 2 V . To estimate the last term we need an estimate of the distance to Q` expressed in terms of the functional G  . Lemma 2.2.9. Let H W V ! R0 be a continuous functional such that

G .ƒw/  H.w/

8w 2 V:

(2.45)

Then, for any y  2 Y  and ˇ > 0, inf

q  2 Q`

G  .ˇ.y   q  //  H .ˇ R.y  //;

(2.46)

where H .v  / WD sup fhv  ; wi  H.w/g. w2V

Proof. We have inf

q  2 Q`

G  .ˇ.y   q  //

D inf  sup f G  .ˇ.y   q  // C .q  ; ƒw/ C h`; wig q 2Y

w2V

D inf  sup f G  .ˇ / C .y    ; ƒw/ C h`; wig ;  2Y

w2V

where  D y   q  . Since G  is coercive on Y  , the functional in curly brackets is coercive with respect to  for w D 0. Hence inf

q  2 Q`

G  .ˇ.y   q  // D sup inf  f G  .ˇ / C .y    ; ƒw/ C h`; wig w2V  2Y

D sup

 sup

w2V

We obtain inf

q  2 Q`

 2Y 



!   1      ; ƒw  G . / C hR.y /; wi : ˇ

    1  ƒw C hR.y /; wi G .ˇ.y  q // D sup G ˇ w2V 





D sup fG .ƒw/ C hˇ R.y  /; wig w2V

 sup fH.w/ C hˇ R.y  /; wig D H .ˇ R.y  //:  w2V

Lemma 2.2.9 and (2.44) yield the following result.

41

2.2 The main error identity

Theorem 2.2.10. For any y  2 Y  and  2 .0; 1/,   R.y  / G .v/  DG .ƒv; y  / C .1  /H C 0 and ˇ2 > 1. Another way of getting bounds for the combined error norm is based on the identity kƒek2A C ke k2A1 D kƒek2A C kp  q  k2A1

C 2.A1.p  q  /; q   y  /U C kq   y  k2A1

D kAƒv 

q  k2A1

C 2.A

1









(2.90) 

.p  q /; q  y /U C kq 

y  k2A1

;

where e D v  u, e D p  y  , q  2 Q` , and we have used (2.71). We set q  D y   Arwy , recall (2.75), and find that .A1 .p  q  /; q   y  /U D .A1 .p  q  /; Aƒwy /U D .p  q  ; ƒwy  /U D 0 : The third term in (2.90) in the last row of (2.90) is equal to kAƒwy k2A1 . By (2.74), this quantity can be estimated from above by the squared norm j ` C ƒ y  j 2ƒ . Combining the results, we obtain  2 kƒek2A C ke k2A1  kAƒv  y  kA1 C j ` C ƒ y  j ƒ C j ` C ƒ y  j 2ƒ : (2.91) From (2.91), it follows that (notice that for any nonnegative real numbers a and b we p have the algebraic inequality .a C b/2 C b 2  .a C 2b/2 ) p k.e; e /kV Y   kAƒv  y  kA1 C 2 j ` C ƒ y  j ƒ (2.92)

52

2 Distance to exact solutions

 and for ` 2 V and y  2 Qƒ 

k.e; e /kV Y   kAƒv  y  kA1 C

p

2 C k` C ƒ y  kV :

(2.93)

2.3.4 Errors in the full primal-dual norm Now we discuss estimates in terms of norms that include ƒ e . First, we introduce the norm k.e; e /k2V U WD kƒek2A C ke k2A1 C j ƒ e j 2ƒ : The upper bound follows from (2.91). Indeed, j ƒ e j ƒ D j ` C ƒ y  j ƒ and we find that  2 k.e; e /k2V U  kAƒv  y  kA1 C j ` C ƒ y  j ƒ C 2 j ` C ƒ y  j 2ƒ : This estimate implies k.e; e /kV U  kAƒv  y  kA1 C

p

3 j ` C ƒ y  j ƒ :

(2.94)

To obtain the lower bound, we first observe that kAƒv  y  kA1 C j ` C ƒ y  j ƒ  kAƒ.v  u/kA1 C ke kA1 C j ` C ƒ y  j ƒ D kƒekA C ke kA1 C j ƒ e j ƒ : Since a C b C c 

p p 3 a2 C b 2 C c 2 (for all a; b; c  0), we obtain the estimate

 1  p kAƒv  y  kA1 C j ` C ƒ y  j ƒ  k.e; e /kV U : 3

(2.95)

In what follows, we use special notation for the quantities that bound the deviations from exact solutions in the full product norm, namely, p M˚ .v; y/ WD kAƒv  y  kA1 C 3 j ` C ƒ y  j ƒ  1  M .v; y  / WD p kAƒv  y  kA1 C j ` C ƒ y  j ƒ : 3 Then, (2.94) and (2.95) reads

M .v; y  /  k.e; e /kV U  M˚ .v; y  /:

(2.96)

It is easy to observe that if y  ! p in U and v ! u in V , then M .v; y  / and M˚ .v; y  / tend to zero, precisely with the same rate as the exact error in the full norm k.e; e /kV U does.

2.3 Linear problems

53

p Remark 2.3.5. Notice that the second term of M˚ has the multiplier 3, which cannot be diminished. To show this, we set v D 0 and y  D 0. Then the estimate (2.94) takes the form

2kƒuk2A C j ` j 2ƒ

1=2



p

3 j ` jƒ ;

which is equivalent to the energy estimate kƒukA  j ` j ƒ (the latter one cannot be improved in the sense that no factor less than 1 can be inserted in the right-hand side). The efficiency of the two-sided estimates (2.96) is estimated by the efficiency index. For this purpose we introduce the number p p kAƒv  y  kA1 C 3 j ` C ƒ y  j ƒ M˚ .v; y  / ieff WD  3  3: (2.97) k.e; e /kV U kAƒv  y  kA1 C j ` C ƒ y  j ƒ Estimate (2.97) provides a guaranteed upper bound for this index. However, this bound may be coarse and in the majority of the cases ieff will be strictly less than   3. It is easy to see that if the term p j ` C ƒ y j ƒ is essentially smaller than the first one, then ieff would be close to 3 (which was observed in computations; see [275]). On the other hand, (2.94) shows that ieff  1. Hence the bounds are quite realistic and the only difficulty is the norm j ` C ƒ y  j ƒ , which should be replaced by an integral-type norm.  Let y  2 Qƒ  and ` 2 V . We combine (2.84), (2.85) with (2.79) and obtain   1  2 kƒek2A  kAƒv  y  k2A1 C ˇC 2 k` C ƒ y  k2V ; (2.98) ke kA1 C 1  ˇ   1  2 kƒek2A  kAƒv  y  k2A1  ˇC 2 k` C ƒ y  k2V : (2.99) ke kA1 C 1 C ˇ In particular, (2.98) implies the estimate ke k2A1  kAƒv  y  k2A1 C C 2 k` C ƒ y  k2V :

(2.100)

Since k` C ƒ y  kV D kƒ e kV , we also obtain estimates for the case, where the errors are measured in terms of the norm

1=2    2  2   2  e; e  WD kƒek C C e : (2.101) k k ke kƒ A V A1 V Q ƒ

For ˇ  1, we have ke k2A1



1 C 1 ˇ



kƒek2A C ˇkƒ e k2V

 kAƒv  y  k2A1 C ˇ.1 C C 2 /k` C ƒ y  k2V

(2.102)

54

2 Distance to exact solutions

and   1 kƒek2A C ˇC 2 kƒ e k2V  kAƒv  y  k2A1 : ke k2A1 C 1 C ˇ

(2.103)

A different approach relies on (2.93), which yields kƒek2A C ke k2A1 C k` C ƒ y  k2V p  .kAƒv  y  kA1 C 2 C k` C ƒ y  kV /2 C k` C ƒ y  k2V :

(2.104)

Hence  1=2 k.e; e /kV Q   kAƒv  y  kA1 C 1 C 2C 2 k` C ƒ y  kV ƒ

DW M˚ .v; y  /:

(2.105)

Quite analogously to (2.95), we deduce a somewhat different (with respect to (2.103)) lower bound  1  M .v; y  / WD p kAƒv  y  kA1 C k` C ƒ y  kV  k.e; e /kV Q  : ƒ 3 (2.106) Thus, for the error norm k.e; e /kV Q  we deduce two-sided estimates containing ƒ the same terms but supplied with different multipliers. It is easy to see that the estimates are indeed efficient and robust. By (2.105) and (2.106), for all v 2 V and  y  2 Qƒ , the efficiency index Ieff of the majorant (2.105) satisfies the double inequality p  1=2 : 1  Ieff  3 1 C 2C 2 Remark 2.3.6. If C is generated by the smallest positive constant in (2.80), then the majorant in (2.105) has the minimal weight. This fact follows from the same arguments as in Remark 2.3.5. Remark 2.3.7. The constant C is formed by the constants  in (2.61) and Cƒ . The first one is often known. Moreover, it can be made equal to 1, if one performs the corresponding rescaling of the operator A and of the functional ` (i.e. the multiplication of the linear problem .P / by 1= ). The constant Cƒ depends only on the operator ƒ and the spaces V and V . It depends on the domain  and can be evaluated by the methods discussed in Chapt. 1.

2.3.5 Majorant as a source of new models The majorant M˚ .v; y  / is Consistent, i.e., M˚ .v; y  / ! 0;

as v ! u; y  ! p I

2.3 Linear problems

55

Reliable, i.e., it provides a guaranteed bound of the error for any v and y  in the respective energy spaces. Efficient, i.e., the quantity c M˚ .v; y  / with an explicitly known constant c independent on v and y  provides a lower bound of the error. Therefore, the functional M˚ serves not only as an error control tool, but also as a basis for the construction of simplified models related to the class of linear problems. For example, assume that we have a collection of different “reduced” spaces b  V and Y b  Y  that can be used to generate a simplified model (as in (b) V of Sect. 2.2.5). The “sharpest” simplified problem can be defined as the one minimizbY b /, the quantity ing M˚ . For each pair .V b; Y b / WD inf inf M˚ .b v ;b y / emod .V   b v2b V b y 2b Y

(2.107)

b; Y b /, which depends on the condefines the corresponding modeling error emod .V struction of reduced spaces and shows the best quality of the model that could be b; Y b / depends achieved within the framework of this ansatz. The quantity emod .V on the data of a particular problem P , so that by computing it we can investigate how the accuracy depends on the domain, boundary conditions, coefficients, etc., and select a relatively simple one, which guarantees an acceptable value of emod . For exb and Y b used ample, assume that Problem P is three-dimensional and the spaces V for dimension reduction (with respect to x3 ) are defined by linear combinations of functions gi .x1 ; x2 / hk .x3 /, where hk are some given (e.g., polynomial) functions and gi are functions defined in a two-dimensional domain, which have to be determined by solving a simplified model. We substitute these functions into the majorant M˚ and integrate over x3 . Then we obtain a two-dimensional functional with known coefficients that defines the simplified mathematical model. Substituting the corresponding solutions b v and b y  (or good approximations of them) into the majorant we b; Y b /. Since the majorant M˚ is consistent and efficient, emod .V b; Y b / obtain emod .V is an adequate measure of the error, whose asymptotic behaviour is the same as of the true error. We consider such type estimates in Chapter 3. Also, it is worth noting that the variational problem in (2.107) is interesting from the analytical point of view because it induces the respective Euler equations, which describe the reduced problem. Therefore, it suggests a formal way of deriving various dimension reduction models avoiding heuristic a priori hypotheses.

2.3.6 Non-homogeneous boundary conditions In many cases, boundary value problems are defined on an affine set V0 C u0 , where u0 2 V is a given function and V0 is a subspace of V consisting of functions that satisfy homogeneous Dirichlet boundary conditions on the whole boundary or on a certain part of the boundary. The relations derived above remain valid if we consider V0 (equipped with the norm induced from V ) instead of V . Indeed, we can represent

56

2 Distance to exact solutions

the problem (2.59) in the form u C ıe u Ce ` D 0; ƒ Aƒe

(2.108)

where e ` D ` C ıu0 C ƒ Aƒu0 2 V  WD .V0 / . Then the solutions of the original and “shifted” problems ((2.59) and (2.108), respectively) satisfy the relations u De u C u0

and p D e p  C Aƒu0 :

We apply (2.67) (and (2.68) for the case ı D 0) to the problem (2.108). Let e v 2 V0 and e y  2 U be approximations of e u and e p  , respectively, and v D e v C u0 , y  D e y C Aƒu0 . We have   1 1  f 2  2  2 2 f kAƒe (2.109) e kƒ ;ı D v  y kA1 C kƒ y C ıe v C `kV ke ekı C ke 2 ı p  e y  D p  y  D e . It is easy to see that where e e De u e v D e, e e D e

Aƒe v e y  D Aƒ.v  u0 /y  C Aƒu0 D Aƒv  y  : Analogously, ƒe y  C ıe v Ce ` D ƒ y   ƒ Aƒu0 C ı.v  u0 / C ` C ıu0 C ƒ Aƒu0 D ƒ y  C ıv C `: Consequently, kek2ı

Cke k2ƒ ;ı D

  1 1    2 2 kAƒvy kA1 C kƒ y CıvC`kV : 2 ı

(2.110)

Other equalities and respective estimates for the problem (2.108) (in terms of e v, e y ,    e e, and e e ) yield similar estimates in terms of v, y , e, and e . More sophisticated arguments show that analogous extensions of results to the case of non-homogeneous boundary conditions are possible for a wide class of convex variational problems (see [214]). Remark 2.3.8. It may happen that ı (which may be associated with, e.g., a chemical reaction acting together with a diffusion process) takes drastically different values in different parts of . In particular, ı may attain large values in !1   and be zero in !2  . In this case, the identities and estimates containing ı1 become invalid. Here, we need to apply the advanced estimates suggested in [266]. Remark 2.3.9. By means of the identity (2.67), we can estimate the difference between exact solutions for ı D 0 and ı > 0. Let uı solve the problem ƒ Aƒuı C ıuı C ` D 0 and uı denote the solution of ƒ Aƒuı C ` D 0. From (2.67), it follows that kek2ı Cke k2ƒ ;ı D

1 k.ƒ pı Cıuı C`/k2V : 2ı

(2.111)

2.4 Applications to particular mathematical models

57

where the norms are defined in (2.65) and (2.66), e D uı  uı ;

e D pı  pı ;

and pı D Aƒuı :

Since ƒ pı C ` D 0, we arrive at the identity 1 kƒek2A C ıkek2V Cke k2A1 C kƒ e k2V D ıkuı k2V ; ı

(2.112)

which shows the behaviour of different components of the error with respect to ı. Estimates of this type of errors for more complicated models are discussed in Chapt. 6.

2.4 Applications to particular mathematical models 2.4.1 Diffusion type models In these models, ƒ is the gradient operator, ı

V0 C u0 WD fw D w0 C u0 j w0 2 V0 ./ WD H 1 ./g; Y D U D L2 .; Rd /, V D L2 ./, V  D H 1 ./, A is a symmetric matrix with real values (A 2 Msd d ),  > 0 and ˚ are the minimal and maximal eigenvalues of A, respectively. The adjoint operator is defined by the divergence operator ƒ D div and (2.59) is the equation  div Aru C ıu D f;

(2.113)

where f 2 L2 ./ is a given function. The relations (2.63) and (2.64) read12 p D Aru;

div p D ıu  f:

(2.114)

  Let v 2 V and y 2 Qƒ  D H.; div/ be approximations of u and p , respectively.    Define the errors e D v  u and e D y  p and use the main error identity (2.67) (see also comments in Sect. 2.3.6). We have the identity

1 krek2A C ıkek2 C ke k2A1 C k div e k2 Zı 1  2 D kArv  y kA1 C j div y  ıv C f j2 d x: ı

(2.115)



The left-hand side of (2.115) contains errors generated by approximations. The righthand side depends only on the problem data (i.e., A, ı, ) and the approximations v and y . It is directly computable. 12

Notice that now ` D f and ƒ y  C ` D .div y C f / D R.y /.

58

2 Distance to exact solutions

For the case ı D 0, we use (2.68) and obtain krek2A

C

ke k2A1

D kAru 

y k2A1 C

Z 2

R.y /.uv/d x;

(2.116)

 



where R.y / WD div y C f . Estimation of the last integral (see (2.76) and (2.81)) yields the estimate krek2A   kArv  y k2A1 C  ? j R.y /jj 2ƒ where  > 1 and  ? D (2.117) is replaced by

  1

8y 2 U;

(2.117)

are two conjugate numbers. If y 2 H.; div/, then

krek2A   kArv  y k2A1 C  ? C 2 kR.y /k2 :

(2.118)

Both terms in (2.118) are represented by integrals, C D CF =1=2  (cf. (1.36). Minimization with respect to  transforms (2.118) into krekA  kArv  y kA1 C C kR.y /k

8y 2 H.; div/:

(2.119)

By (2.84) and (2.85), we obtain two-sided estimates for the combined norms (where ˇ  1): ˇ1 krek2A  kArv  y k2A1 C ˇC 2 kR.y /k2 ; ˇ ˇC1 C krek2A  kArv  y k2A1  ˇC 2 kR.y /k2 : ˇ

ke k2A1 C

(2.120)

ke k2A1

(2.121)

We can obtain sharper estimates, but this requires additional efforts and lead to more sophisticated majorants and minorants (see Sect. 2.6.1). Remark 2.4.1. Assume that we know a priori that u.x/ 2 Œu.x/; u.x/ for almost all x 2 . Define the quantities Z Z            uRC y  uR y d x and  WD uRC y  uR y d x ;

C WD 



which are formed by the negative and positive parts of R .y /. Then (2.116) yields the estimates Z   (2.122) krek2A C ke k2A1  kArv  y k2A1  2 R y v d x C 2 C 

and krek2A

C

ke k2A1

 kArv 

y k2A1

Z 2 

  R y v d x C 2  :

(2.123)

59

2.4 Applications to particular mathematical models

2.4.2 Mixed boundary conditions Estimates for problems with mixed boundary conditions can be deduced in a similar way. Consider the problem div Aru C f D 0 u D u0 n  Aru D F

in ; on 1 ; on 2 ;

(2.124) (2.125) (2.126)

where 1 and 2 are two nonintersecting parts of the boundary (it is assumed that measd 1 f1 g > 0) and the function u0 2 H 1 ./ defines the Dirichlet boundary condition (in the sense of traces). If F 2 L2 .2 / then the functional ` has an integral representation. In the case of mixed boundary conditions, V0 WD fw 2 H 1 ./ j w D 0 on 1 g; the operator ƒ is represented by the pair f div; n  j2 g, and V is a product space formed by the pairs fw; g, with w 2 L2 ./ and  2 L2 .2 / (since F 2 L2 .2 /,  we can define  in this class). Hence the set Qƒ  is defined by    2  2 Qƒ  WD fy 2 U j div y 2 L ./; y  n 2 L .2 /g:

We use (2.76) (and recall Sect. 2.3.6), which now has the form krekA  kArv  y kA1 C j ` C div yj ƒ :

(2.127)

 Here ` is generated by the source terms f and F . If y 2 Qƒ  , then R R R.y /wd x C .y  n  F /wds

j ` C div yj ƒ D

sup



2

krwkA

w2V0 nf0g

 C1 kR.y /k C C2 ky  n  F k2 ; where C1 D

CF , 1=2  .A/

C2 D

inequalities (cf. Sect. 1.2.2)

Ctr 1=2  .A/

(2.128)

and the constants CF and Ctr are generated by the

kwk  CF krwk kwk2  Ctr krwk

8w 2 V0 ; 8w 2 V0 :

(2.129) (2.130)

Evidently these constants depend on , 1 , and 2 . By (2.128)–(2.130), we obtain krekA  kArv  y kA1 C C1 kR.y /k C C2 ky  n  F k2 :

(2.131)

For the primal-dual error norm we have the error identity (2.68), which reads 0 1 Z Z    B C y  n F eds A : k.e; e /k2V Y  D kArv y k2A1 C2 @ R.y /ed xC 

2

(2.132)

60

2 Distance to exact solutions

Remark 2.4.2. Note that for y D p the identity (2.132) reduces to Z   2 krekA D Arv  rv C A1 p  p  2rv  p d x:

(2.133)



R R Since  p  rv d x D  f v d x C 2 F vds, the identity (2.133) can be rewritten R R in terms of the energy functional J .v/ WD 12  .Arv  rv  f v/d x  2 F vds associated with the problem. We have Z Z 2 krekA D .Arv  rv  2f v C Aru  ru/d x  2 F vds (2.134) R





Z

D 2J .v/ 

2

Z

Aru  ru d x C 2 

Z F uds D 2.J .v/  J .u//:

f u dx C 2 

2

2.4.3 Problems with periodic boundary conditions Assume that  has a geometric shape that admits periodic boundary conditions (e.g.,  D .0; 1/d ). The problem is to minimize the functional  Z  1 J .v/ D Arv  rv  f v d x 2 

over the set

n o Vper ./ WD v 2 H 1 ./ j fjvjg D 0; v ji D v j C ; i D 1; 2; : : :; m ; i

where fjf jg D 0, i˙ denote the opposite parts of the boundary @ and the equality is understood in the sense of traces. The flux p D Aru generated by the minimizer u satisfies similar periodicity conditions and fjdiv p jg D 0. We use (2.68) with y 2 H.; div/ that satisfies the periodic boundary conditions (since p meets these conditions this imposes no restrictions). Notice that for any ˇ > 0, ˇ ˇ ˇZ ˇ ˇ ˇ  ˇ 2 ˇ R.y /.v  u/ d xˇˇ  2C kR.y /k kr.v  u/kA ˇ ˇ 

 ˇC 2 kR.y /k2 C Here C D

CP ./ .  .A/

1 kr.v  u/k2A : ˇ

Hence we obtain (for ˇ 2 .0; 1)

.1  ˇ 1 /krek2A C ke k2A1  kAru  y k2A1 C ˇC 2 kR.y /k2 :

(2.135)

A slightly different argument leads to the estimate (2.118), where C is defined as above and y satisfies the periodicity conditions.

2.4 Applications to particular mathematical models

61

2.4.4 Advanced estimates based on domain decomposition There are other ways to estimate the norm (2.128) that may use constants other than CF and Ctr . For example, if fjR.y /jg D 0;

(2.136)

then instead of (2.129) we apply the Poincar´e inequality for functions with zero mean in . If, in addition, fjy  n  F jg2 D 0;

(2.137)

then we can replace (2.130) by the trace inequality (1.40) e 2 .2 ; /krwk kwk2  C

8w 2 H 1 ./; fjwjg2 D 0:

(2.138)

Then (2.131) holds with 1=2

e2 1=2 : and C2 D C 

C1 D CP 

More sophisticated methods (see [256]) are based on the ideas of domain decomposition and on the replacement of global constants by local ones. Reducing the set Qƒ A simple way to obtain estimates with constants smaller than C1 and C2 is based on the assumption that y satisfies additional conditions fjR.y /jgi D 0

for i D 1; 2; : : :N;

where i , are nonintersecting Lipschitz subdomains such that  D constants CP .i / are known, then we can apply the estimate Z

R.y /wd x 

N X

(2.139) S

i . If the

CP .i /kR.y /ki krwki

i D1





N X i D1

!1=2 CP2 .i /kR.y /k2i

krwk :

(2.140)

Analogously, if 2 consists of the parts 2k , k D 1; 2; : : :; K and fjy  n  F jg2k D 0;

(2.141)

62

2 Distance to exact solutions

then Z

K Z X

.y  n  F /wds D

.y  n  F /wds

kD1 2k

2



K X

e 2k .2k ; !k /ky  n  F k krwk! C 2k k

kD1



K X

!1=2 e 2 .2k ; !k /ky C 2k

n

F k22k

krwk : (2.142)

kD1

e 2k .2k ; !k / is as in (1.40), 2k D @!k \ 2 , and !k   is a Lipschitz Here C subdomain used in this trace type inequality. It is assumed that !k \ !j D ; if k 6D j . It is not required that !k coincides with one of the subdomains i (but such a construction is possible). In particular, if 2k is a plane part of the boundary, then !k may be a simplex or a polygonal domain having 2k as a face. In other words, there are various ways to define !k and the best one is selected by analysing a particular problem. Obviously, the main requirement is that the corresponding constant in (1.40) is easy to compute, see [190, 209]. By (2.128), (2.140), and (2.142), we obtain 

j ` C div y j ƒ 

N X i D1

!1=2 C1i2 kR.y /k2i

C

K X

!1=2 2 C2k ky

n

F k22k

;

kD1

(2.143) e 2k .2k ; !k /1=2 . Using (2.143) we rewhere C1i D CP .i /1=2 and C2k D C   place (2.131) by the estimate krekA  kArv  y kA1 C

N X i D1

C

!1=2 C1i2 kR.y /k2i

K X

!1=2 2 C2k ky

 n

F k22k

:

(2.144)

kD1

Other estimates of the previous section are modified quite analogously. For example, (2.92) reads p 1=2   kArv  y kA1 C 2 j ` C div y j ƒ ; krek2A C ke k2A1

(2.145)

where the last term is estimated by (2.128) (or by (2.144) provided that the conditions (2.139) and (2.141) hold). Other estimates follow from (2.84)–(2.89). Also, we note

2.4 Applications to particular mathematical models

63

that estimates for the full primal-dual norms k.e; e /k2V U WD krek2A C ke kA1 C j div e j 2ƒ ; k.e; e /k2V Q WD krvk2A1 C ke k2A1 C k div e k2 C ke  nk22 ƒ

follow from (2.94), (2.95), (2.105), and (2.106). Comments Embedding-type inequalities (2.129) and (2.130) contain the constants CF and Ctr , which may be difficult to find. At the same time, the constants CP and e tr often have easily computable majorants (see Chapter 1). Therefore, (2.144) (and C other similar estimates) are convenient for numerical approximations. For elliptic problems this estimate was deduced in [259, 209]. For parabolic problems estimates of this type were obtained and numerically studied in [189, 191, 169, 170, 171] and other publications. In [256], error majorants based on decomposition of  into a collection of simple subdomains were derived in the general framework and studied in application to various problems (elasticity, reaction-convection-diffusion, Stokes, and others). Estimates of this type have shown high efficiency in the context of finite element, finite volume, discontinuous Galerkin, and other methods (see [184] and references cited in this book). Nevertheless, comparing (2.144) and (2.131) we emphasize that the latter estimate has one essential advantage: it holds for the widest possible set of y . Indeed, a computable majorant should be based on computation of integrals and avoid negative norms defined by means of supremum taken over spaces (subspaces) of infinite dimension. The space U contains vector-valued functions, whose divergence is not integrable (it belongs to H 1 ). Therefore, this space is too wide to satisfy the above  conditions. Thus, the restrictions encompassed in the definition of Qƒ  are in a sense minimal (cf. (2.77)). The conditions (2.139) and (2.141) can be called weak equili bration of y . They reduce the set Qƒ  . If N and K in (2.139) and (2.141) are not large, then these extra conditions do not lead to serious complications. However, if the number of subdomains is very large (e.g., if each i is identified with a finite element), then exact satisfaction of these additional conditions may lead to extra computations (post processing of fluxes), which have to be performed after any new refinement with a growing cost. First of all, this is related to the conditions (2.139) related to weak equilibration in  (the conditions (2.141) related to the boundary are much easier to satisfy). Advanced estimates that do not use the mean value conditions (2.139) and (2.141) We can avoid the aforementioned difficulties related to the weak equilibration of y if the integral in (2.140) is represented in a somewhat different form. Let i WD fjR.y /jgi . Then ˇ ˇ ˇZ ˇ Z N N ˇ ˇ X X ˇ ˇ   R.y /wd x  CP .i /kR.y /  i ki krwki C ji j ˇ w d x ˇ : ˇ ˇ i D1 i D1 ˇ i ˇ  (2.146)

64

2 Distance to exact solutions

Hence,

R

R.y /wd x



sup

krwkA

w2V0 nf0g



N X

!1=2 CP2 .i /kR.y /



i D1

C CF .; 1 /

N X

i k2i !1=2

ji j2 ji j

(2.147)

i D1

and instead of (2.144) we obtain N X

j ` C div yj ƒ 

i D1

C

!1=2 CP2 .i /kR.y /  i k2i

N CF .; 1 / X 1=2



!1=2 ji j2 ji j

i D1

C

K X

!1=2 2 C2k ky  n  F k22k

:

kD1

(2.148) Now (2.127) yields an upper bound of krekA , which does not use the conditions (2.139). If these conditions are satisfied, then (2.148) coincides with (2.144). However, the error bounds based on (2.148) are valid even if y is equilibrated only approximately and i 6D 0 in i . Certainly the quantities ji j should not be large if we wish to have practically efficient estimates. Further discussion of advanced error estimates is presented in Sect. 2.6.1, where we consider errors of numerical (e.g., FEM) approximations.

2.4.5 Elasticity Although the application of the theory to the problem of linear elasticity is similar to the case of the diffusion problem, it is worthwhile considering elasticity problem as a special important case of the theory developed above. Dirichlet boundary conditions As before,  denotes a bounded domain in Rd with Lipschitz boundary @. We define the spaces V0 WD fv 2 V WD H 1 .; Rd / j v D 0 on @g; U D L2 .I Msd d /;   and the operator ƒv WD ".v/ WD 12 rv C .rv/T . For any  2 U the adjoint operator ƒ  D Div  has values in V0 and13 Z hƒ ; wi D W ".w/ d x 8w 2 V0 :  13 In this section, we replace y and p by  and  , respectively, in order to make the exposition closer to the notation commonly used in the mechanics of solids.

2.4 Applications to particular mathematical models

65

The operator A is defined by the 4th order tensor L D fLij kl g, which satisfies the conditions Lij kl D Lj i kl D Lklij ;

Lij kl 2 L1 ./ ;

 j  j  LW   ˚ j  j 2

2

8 2

Msd d

Then the norms k  kA and k  kA1 are given by Z Z 2 2 LW  d x ; kkL1 WD L1 W  d x kkL WD 

:

(2.149) (2.150)

8 2 U



and (2.61) holds. The inequality (2.5) follows from the Korn inequality, which also implies (2.151) kwk  CF; " k".w/k 8w 2 V0 : Notice that the constants in Friedrichs inequality and in the inequality (1.44) yield a majorant of CF; " (see Sect. 1.2.4). Henceforth, we assume that u0 2 H 1 .I Rd / and f 2 L2 .I Rd / are given. The variational problem for u 2 V0 C u0 leads to the integral identity Z Z L".u/W ".w/ d x D f  w d x 8w 2 V0 (2.152) 



and the corresponding mixed formulation defines the weak solution as the pair .u;  / 2 .V0 C u0 /  U such that Z .".u/  L1  /W  d x D 0 8 2 U; (2.153) 

Z .".w/W   f  w/ d x D 0

8w 2 V0 :

(2.154)



Now  d Qƒ  D f 2 U j Div  2 L2 .I R /g;  1=2 kkƒ WD kk2L1 C kDiv k2 ;

and

 1=2 k.v; /kV Q  WD k".v/k2L C kk2ƒ ƒ

 8.v; / 2 V  Qƒ  :

 Let .v; / 2 .V0 C u0 /  Qƒ  denote approximations of .u;  /. By (2.81), we obtain

k".e/kL  DL1=2 .".v/; / C

CF ;" 1=2 

kDiv  C fk ;

(2.155)

66

2 Distance to exact solutions

where e D u  v and Z DL .".v/; / D .".v/  L1 / W .L ".v/  / d x 

Z

D

  L".v/W ".v/ C L1 W   2".v/W  d x D kL".v/  k2L1 :



All other estimates in Sect. 2.3 can be applied analogously. For example, (2.98) and (2.99) read as follows (ˇ  1, e D    ):   1  2 ke kL1 C 1  (2.156) k".e/k2L  DL .".v/; / C ˇC 2 kDiv C fk2 ; ˇ   1 ke k2L1 C 1 C (2.157) k".e/k2L  DL .".v/; /  ˇC 2 kDiv C fk2 : ˇ In particular, for ˇ D 1 the estimate (2.156) has the form ke k2L1  DL .".v/; / C

2 CF;"



kDiv C fk2

and ke k2ƒ

 DL .".v/; / C 1 C

2 CF;"



(2.158)

! kDiv C fk2 :

(2.159)

Estimates in the combined stress–strain norm follow from (2.105) and (2.106): !1=2 2 C F ;" 1=2 kDiv C fk ; (2.160) k.e; e /kV Q   DL .".v/; / C 1 C 2 ƒ 

1 k.e; e /kV Q   p DL1=2 .".v/; / C kDiv C fk : (2.161) ƒ 3 Mixed Dirichlet–Neumann boundary conditions Let @ consist of two disjoint parts 1 and 2 (meas.1 / > 0) where two different boundary conditions are imposed: for given u0 2 V and F 2 L2 .; Rd /, u and  must satisfy u D u0 n D F

on 1 ; on 2 ;

(2.162)

F 2 L .; R /: 2

d

(2.163)

Now V0 WD fw 2 V j w D 0 on 1 g and the solution u 2 V0 C u0 satisfies the identity Z Z Z L ".u/ W ".w/ d x D f  w d x C F  w ds 8w 2 V0 : (2.164) 



2

2.4 Applications to particular mathematical models

67

The operator adjoint to "./ is defined (in the case of mixed boundary conditions) by the relation Z hƒ ; wi D  W ".w/ d x; 8w 2 V0 ; (2.165) 

If

  j Div  2 L2 .; Rd /; n 2 L2 .2 ; Rd /g;  2 Qƒ  WD f 2 Y

then

Z



Z .n  F/  w d  

h` C ƒ ; wi D 2

.f C Div /  w d x

(2.166)



and Q` consists of tensor-valued functions satisfying the equilibrium conditions Div  C f D 0 and the Neumann condition (2.163). For  2 Q` and v 2 V0 C u0 , we can use (2.71), which yields ke k2L1 C k".e/ k2L D kL".v/  k2L1 :

(2.167)

However, this relation is difficult to use because  must satisfy exactly (2.163) and a system of three differential equations appearing in the definition of Q` . By (2.76) we obtain a majorant of the error in terms of displacements k".e/kL  kL".v/  kL1 C j ƒ  C ` j ; where j ƒ  C ` j D

(2.168)

hƒ  C `; wi : k".w/kL w2V0 nf0g sup

 If  2 Qƒ  , then

R 

j ƒ  C ` j WD

sup

.f C Div /  w d x C



w2V0 nf0g

R

.n  F/  w ds

2

k".w/kL

:

In view of Korn’s inequality (1.43) and the trace inequality, there exists a constant C.; 2 / such that Z Z 2 jwj d x C jwj2 ds  C 2 .; 2 / k".w/k2 ; 8w 2 V0 : (2.169) 

2

Therefore, j ƒ  C ` j ˚ 1=2  kDiv  Cfk2 CkF nk22

sup

.kwk2 Ckwk22 /1=2

k".w/kL w2V0 nf0g ˚ 1=2  C.; 2 /1=2 : kDiv  C fk2 C kF  nk22 

68

2 Distance to exact solutions 1=2

 . Using (2.168) for  2 Qƒ  we obtain ˚ 1=2 k".e/kL  kL".v/kL C CL kDiv  Cfk2 C kF nk22 :

Let CL WD C.; 2 /

Analogously, (2.98) and (2.99) for ˇ  1 imply the estimates   1 ke k2L1 C 1  k".e/k2L ˇ    kL".v/  k2L1 C ˇ C2L kDiv  C fk2 C kF  nk22 and

(2.170)

(2.171)

  1 k".e/k2L ke k2L1 C 1 C ˇ

   kL".v/  k2L1  ˇ C2L kDiv  C fk2 C kF  nk22 :

(2.172)

In particular, (2.171) yields

  ke k2L1  kL".v/  k2L1 C C2L kDiv  C fk2 C kF  nk22 :

(2.173)

In general, finding the constant CL may be difficult. However, for the plate-type domains considered in Chapter 4, a majorant of CL can be found fairly easily. Isotropic media In the case of isotropic media, the elasticity relations read (see [102, 309])

L " D KE tr " 1 C 2E "D ; 1 1 D L1  D tr 1 C  : 9KE 2E

(2.174) (2.175)

In (2.174)–(2.175), KE and E are positive (elasticity) constants and  D is the deviator ı

of  (cf. (1.1)). Consider the case where  D 1 , V0 D H 1 .; Rd /, and the energy functional J W V0 C u0 ! R is defined by the relation  Z  Z KE J .v/ D (2.176) .div v/2 C ˆ.j"D .v/j/ d x C f  vd x: 2 



The linear elasticity problem is generated by ˆ.t/ D E t 2 . First, we consider this case. The structure of the functional J is such that it is convenient to use estimates that follow from Sect. 2.2.4. Define Z Z KE G1 .y1 / D jy1 j2 d x; ƒ1 v D div v; F .v/ D f  vd x; 2   Z G2 .y2 / D E jy2 j2 d x; ƒ2 v D "D .v/; 

and take y1 2 Y1 D L2 ./, y2 2 Y2 D L2 .; Msd d / D Y2 .

2.4 Applications to particular mathematical models

Z

Since

.div v/ y1 d x D 



Z

69

ry1  v d x;



the operator conjugate to ƒ1 is defined by the relation ƒ 1 y1 D ry1 and Y1 consists of scalar-valued functions such that ry1 is in the space dual to V0 . Next, Z Z Z Z D  D D " .v/ W y2 d x D ".v/ W y2 d x D rv W y2 d x D  Div yD 2 vd x; 

 

and we see that ƒ

 2 y2

D



Div yD 2





2 V . We have

1 1 ky  k2 ; DG1 .ƒ1 v; y1 / D ky   KE .div v/k2 ; 2KE 1  2KE 1 1 1 G2 .y2 / D ky2 k2 ; DG2 .ƒ2 v; y2 / D ky  2E "D .v/k2: 4E 4E 2

G1 .y1 / D

Since 2 D ;, the set Q` contains pairs .y1 ; y2 / satisfying the identity Z Z  D .y1 1 C y2 /W ".v/ d x D f  vd x; 8v 2 V0 : 

(2.177)



We use Sect. 2.2.4 and notice that

D G .ƒv; y / D DG1 .ƒ1 v; y1 / C DG2 .ƒ2 v; y2 /; KE D G .ƒv; p / D k div ek2 C E k"D .e/k2 DW G .v/; 2 1 1 D G .ƒu; y / D ke1 k2 C ke k2 DW G .y /; 2KE 4E 2  where p1 D KE div u, p2 D 2E "D .u/, e D u  v, e1 D p1  y1 , and e2 D p2  y2 . For .y1 ; y2 / 2 Q` , (2.52), (2.53) yield the error identity

G .v/ C G .y / D

1 1 ky1  KE .div v/k2 C ky  2E "D .v/k2 : (2.178) 2KE 4E 2

For .y1 ; y2 / 62 Q` , we use (2.55) where Z   hƒ y C `; vui D ..y1 1 C yD 2 /W ".v  u/  f  .v  u//d x: 

Hence this identity comes in the form

G .v/ C G .y / D

1 ky   KE .div v/k2 (2.179) 2KE 1 Z 1 ky2  2E "D .v/k2 C .f  e  .y1 1 C yD C 2 /W ".e//d x: 4E 

70

2 Distance to exact solutions

Other forms of ˆ arise in models of the so-called “deformation plasticity theory” and more complicated models of elasto-plastic media (e.g., see [116, 137, 141, 291, 270]). For the case where Z ˆ." / WD D

."D / d x; 

the corresponding estimates are derived in much the same way. Indeed, we can use the relations of Sect. 2.2.4. If the functional ˆ is explicitly defined, then the quadratic compound functional DG2 .ƒ2 v; y2 / is replaced by the functional associated with ˆ,14 i.e., Z D  DG2 ." .v/; y2 / D .."D .v// C   .y2 /  "D .v/W y2 /d x: 

Then, the error measures are defined by the relations

G .v/ D

1 kp  KE .div v/k2 C DG2 ."D .v/; p2 /; 2KE 1

G .y / D

1 ky   KE .div u/k2 C DG2 ."D .u/; y2 /: 2KE 1

From (2.34) and (2.35), we find that for any q D .q1 ; q2 / 2 Q` and any v 2 V0 Cu0 , the measures G .v/ and G .q / are bounded from above by the quantity

DG ."D .v/; q / D

1 kq   KE .div v/k2 C DG2 ."D .v/; q2 /: 2KE 1

2.4.6 Variational functionals with power growth ˛-Laplacian This problem is a typical representative of the class of nonlinear variational problems generated by convex energy functionals with power growth. In the case of homogeneous boundary conditions, the corresponding variational problem is ı

to find u 2 W 1;˛ ./ such that

J .u/ D

inf

ı

v2W 1;˛ ./

J .v/;

J .v/ D

1 ˛

Z

Z jruj˛ d x  

f vd x;

(2.180)



14 The relation was derived under the condition that u and  exists. In the case of Hencky plasticity, the minimizer u may not exist in a reflexive Banach space. However,  exists provided that certain ”safe load” conditions are satisfied. For this special case, we can use our relations for measuring errors in terms of the dual variable (analysis of this case can be found in [115]).

2.4 Applications to particular mathematical models

71

where ˛ 2 .1; C1/. The minimizer u satisfies the equation div.jruj˛2 ru/ C f D 0: Let ˛  WD

˛ . ˛1

Then G .y/ D ˛1 kyk˛˛ , G  .y / D Z 



DG .ƒv; y / D

 1 ky k˛˛ ˛

, and

 1  ˛ 1 ˛  jrvj C  jy j  rv  y d x: ˛ ˛

 

The set Q` consists of elements q 2 Y  D W 1;˛ .; Rd / such that Z

   q  rw  f w d x D 0

ı

8w 2 V ./ WD W 1;˛ ./:

(2.181)



Then, (2.29) yields the error identity

G .v/ C 

  G .q /

Z  D

 1 1  ˛ ˛  jrvj C  jq j  rv  q d x; ˛ ˛

(2.182)



valid for q 2 Q` . Consider the measures in the left-hand side of (2.182). By definition,  Z  1 1  G .v/ D jrvj˛ C  jp j˛  rv  p d x: ˛ ˛  0

Notice that G .y/ D jyj

G .v/ D

˛2

Z 

y and in view of (2.8), p D jruj˛2 ru: Hence

 1 1 ˛ ˛ ˛2 d x: jrvj C  jruj  rv  rujruj ˛ ˛



It is easy to see that G .v/ is nonnegative and vanishes if v coincides with the exact solution u. For ˛ D 2 it coincides with 12 kr.uv/k2 (and this is the only case, where such a simple representation via a norm is available). However, G is not symmetric (with respect to changing u to v and v to u) and, in general, does not generate a metric. We should regard it only as a measure of the distance to the exact solution u (notice that the sets fv 2 V j G .v/ < g generate a local convex topology in the vicinity of the exact solution). The measure G .v/ differs from ˛1 kr.u  v/k˛˛; , which at a first glance seems to be perfectly adequate to the problem in question (see Example 2.2.8). The error identity shows that G .v/ provides a more realistic characteristic of the quality of v (as an approximation of u) than the norm.

72

2 Distance to exact solutions

Analogously, the functional  Z  1 1  jruj˛ C  jq j˛  ru  q d x G .q / D ˛ ˛ 

is nonnegative and vanishes if q coincides with p . Hence G .q / (as in (2.28)) is a measure of the distance between q and the exact solution of the dual problem (presented by the gradient ru). We can represent it in a different form by noticing  that jruj˛ D jp j˛ . Then ( 0; if jruj D 0; ru  q D 2˛ jruj2˛ p  q D jp j ˛1 p  q ; if jruj 6D 0; and we rewrite the dual measure as  Z  1  ˛ 1  ˛     G .q / D jp j C  jq j  %.x/p  q d x; ˛ ˛ 

where

( %.x/ D

if x 2 0 WD fx 2  j jp .x/j D 0g;

0; 2˛

jp j ˛1 ;

if x 62 0 :

For the special case ˛ D 2, the measure G is given by the norm: 1 2

G .q / D kp  q k2 : We now turn to the more general case where y is selected in a set wider than Q` . For ˛ > 1, we have a generalised version of the Friedrichs inequality: kwk˛  CF krwk˛ ; Therefore,

1 G .rw/ D ˛

8w 2 V:

Z jrwj˛ d x  

(2.183)

1 kwk˛˛ ˛CF˛

and (2.48) holds for 1 g.t/ D ˛ Hence, g  .t/ D G 



y 



1 jtj ˛

˛

ˇ ˇ t ˇ ˇC

F

ˇ˛ ˇ ˇ : ˇ

˛

CF and 

 G  .y / D



1˛ 1  ˛   ˛   ˛ ky k ky k ky k˛˛ :    D ˛ ˛ ˛ ˛ ˛  ˛

73

2.4 Applications to particular mathematical models 

Thus, if R.y / WD div y C f 2 L˛ ./, then (2.50) yields the estimate 1 G .v/  DG .rv; y / C  ˛ 





  ˛  C  ˛ ky k˛˛ ˛ 

˛  CF  kR.y /k˛ 1 Z C .y  rv  f v/d x:

(2.184)



It is easy to see that this estimate has no gap between the left- and right-hand sides. To show this fact, if suffices to set y D p and let  tend to 1. Finally, we note that a posteriori estimates for variational problems associated with the ˛-Laplacian are presented in [251, 51] and for the case where ˛ depends on x they are derived in [228]. Nonlinear problem with friction As another example, we consider the functional 1 J .v/ D ˛

Z

Z jrvj d x C ı 

Z jvj d x 

˛



f v d x; 

where 1 < ˛ < C1, ı > 0, and f is a bounded real-valued function. If ˛ D 2, then this problem is often called “a model problem with friction” [124]. The space V , the functionals G .v/, G  .y /, and the operators ƒ, and ƒ are defined exactly as in the case of the ˛-Laplacian. We define Z

F .v/ D ı

Z jvjd x 



f v d x: 

As before, p D jruj˛2 ru and 

Z 

DG .y; y / D

 1  ˛ 1 ˛  jyj C  jy j  y  y d x: ˛ ˛



However, the form of DF differs from the previous case. Indeed, for any real-valued  function v  2 L˛ ./, 

Z



F .v / D sup v2V

( 

..v C f /v  ıjvj/d x D 

0;

if jv  C f j  ı a:e: in ;

C1; otherwise: (2.185)

74

2 Distance to exact solutions

Therefore, the functional F  .ƒ y  / reduces to ( 0; if jR.y /j  ı a:e: in ;   F .div y / D C1; otherwise: Hence

DF .v; ƒ y / D

8 Z ˆ < .ıjvj  v.div y C f //d x;

if jR.y /j  ı a:e: in ;

ˆ :

otherwise:



C1;

We see that the measure DF is finite if and only if y 2 Qı WD fy 2 Y  j jR.y .x//j  ı

for a:a: x 2 g :

(2.186)

The set Qı is a certain analog of the set Q` for variational problems with F .v/ D h`; vi. However, there is an essential difference between these two cases. Now the error identities are well defined (in the sense that both sides of the relevant identity are finite) not for only those y that belong to the affine manifold Q` , but for y in a wider set Qı (a “strip”, whose width depends on the parameter ı). Notice that the exact solution p belongs to Qı (variation of the functional J shows that u div p C f D ı juj ). We use (2.26), where for any y 2 Qı and v 2 V , the left-hand side is now defined by the formula  Z  1 1   ˛ ˛ ˛2 jrvj C  jruj  rv  rujruj dx .v/ C  .y / D C ˛ ˛   Z  1  ˛ 1 ˛    ˛  2 C dx jp j C  jy j  p  y jp j ˛ ˛  Z   C ıjuj  uR.y / C ıjvj  v R.p / d x: 

The right-hand side of (2.26) is the sum DG .ƒv; y  / C DF .v; ƒ y  /. Hence we obtain the following error identity:  Z  1  ˛ 1   ˛  .v/ C  .y / D jrvj C  jy j  rv  y d x ˛ ˛  Z   C (2.187) ıjvj  v R.y / d x: 

The second term in the right-hand side is nonnegative (because y 2 Qı ). It reflects the nonlinear differential equation. The second term penalises possible violations of the nonlinear dependence between ru and p . We see that the left- and right-hand parts of the identity vanish if u D v and p D y .

2.4 Applications to particular mathematical models

75

2.4.7 Stokes problem ı

In this case, the basic space V ./ D S.; Rd / is the closure of smooth solenoidal ı

fields vanishing on @ with respect to the norm of H 1 .; Rd / and v is a vectorvalued function (velocity). The stationary Stokes problem The classical formulation of the problem is to find a vector field u (velocity) and a scalar-valued function g (pressure) such that Div ".u/ D f  rg div u D 0 u D u0

in  in  on @;

(2.188) (2.189) (2.190)

where u0 is a divergence-free field in H 1 .; Rd / and  > 0 (the viscosity parameter, which is a constant or a function whose values lie between two positive constants  and ˚ ). Since the pressure is defined up to an additive constant, it is commonly accepted to set fjgjg D 0. It is easy to see that the solution u is the minimizer of the variational problem Z

 J .u/ D inf J .v/; J .v/ D (2.191) j".v/j2  f  v d x: ı 2 v2S.;Rd / 

Since the problem is defined for divergence-free fields vanishing on the boundary, the linear part of the functional can be equivalently written as Z e hf; vi D .f  v C g div v/ d x; 

where e f D f  rg 2 H 1 ./. To apply the general scheme, we define ƒ as the symmetric part of the gradient operator, and set ı

ƒ D Div; V D S.; Rd /; V D L2 .; Rd /; Y D U D Y  D L2 .; Msd d /; and

Z

A D 1;

F .v/ D

f  v d x;

G ./ D

 kk2 : 2

(2.192)



Let v 2 V and  2 U be approximations of the exact velocity u and stress  , respectively. By e D v  u and e D    we denote the errors (notice that, as in Sect. 2.4.5, the notation  is used instead of y ). By (2.76), we obtain k".e/k WD k 1=2 ".e/k  k".v/  k 1 C jje f C Div jj ƒ : 

(2.193)

76

2 Distance to exact solutions

Let e g2e L2 ./ be an approximation of the exact pressure field g. Set e  WD   e g1 and notice that R .f  w C e g div w   W ".w// d x  je f C Divjj ƒ D sup D j f C Dive j ƒ : k".w/k w2V nf0g Now we rewrite (2.193) in the form  e g 1k 1 C j f C Div e j ƒ k".e/k  k".v/  e 

(2.194)

and obtain the following form of (2.68): k".e/k2

Z

 2

2





C ke k 1 D k".v/  e   g1k 1  2

.Dive  C f/  ed x: 

By (2.92), we obtain an upper bound in the combined norm:

2 p  e g 1k 1 C 2 j f C Div e j ƒ ; k".e/k2 C ke k21  k".v/  e 



(2.195)

where e  is an approximation of   g1. If e  2 H.; Div/, then  k ; j f C Div e j ƒ  C kf C Dive where C D

CF 1=2 . 

(2.196)

Hence in (2.194) and (2.195) we can replace the norm j  j ƒ by the

L2 -norm. In particular, instead of (2.194), we have  e g 1k 1 C C kDiv e  C fk : k".e/k  k".v/  e 

(2.197)

Now our goal is to extend these estimates to a wider class of approximations. ı

Assume that v 2 H 1 .; Rd /, i.e., the condition div v D 0 may not hold. For any vı 2 V , we have k".v  u/k  k".vı  u/k C k".vı  v/k . Therefore,  e g 1k 1 C C kDiv e  C fk C 2 inf kr.vı  v/k : k".v  u/k  k".v/  e vı 2V



Thus, the problem is reduced to a projection on the space V consisting of divergencefree fields. This question is discussed in Sect. 1.2.5. In the general case, we use (1.54) and obtain 1=2

 e g 1k 1 C C kDiv e  C fk C 2˚  k div vk : k".v  u/k  k".v/  e  (2.198) If the velocity field is subject to the additional integral conditions fjdiv vjgi D 0;

i D 1; 2; : : :; N;

77

2.4 Applications to particular mathematical models

where i is a collection of non-overlapping subdomains, then we use estimates presented in Sect. 1.2.5. In particular, Lemma 1.2.3 with q D 2 yields the estimate k".v  u/k  k".v/  e  e g 1k 1



N X

C C kDiv e  C fk C 2 ˚

i D1

!1=2 2  k div vk2i i

;

(2.199)

which contains local constants i related to the subdomains i . If v possesses additional regularity and div v 2 L i .i /, i > 2, i D 1; 2; : : :; N , then (2.199) can be replaced by  e g 1k 1 k".v  u/k  k".v/  e



C C kDiv e  C fk C 2 ˚

N X i D1

!1=2 ji j

1 1 2  i

2  k div vk2 i ;i i

: (2.200)

Estimates for the pressure Estimates of kg  e gk can be also derived by employing Lemma 1.2.1. Since .g  e g/ 2 e L2 ./, we know that div e w D g e g; ı

and kre wk   kg  e g k ;

g k2 D for some e w 2 H 1 .; Rd /. Hence kg  e

R

(2.201)

div e w.g  e g / d x.



Notice that u and g satisfy the integral identity that follows from (2.188) with e w as a test function, so that Z Z .g  e g / div e w d x D .".u/ W ".e w/  f  e w e g div e w/ d x (2.202) 



Z

D

Z .".v/ W ".e w/  f  e w e g div e w/ d x:

".u  v/ W ".e w/ d x C 



A Cauchy–Schwarz inequality combined with (2.201) leads to Z 1=2 ".u  v/ W ".e w/ d x   ˚ k".u  v/k kg  e gk : 

We decompose the second integral by means of the integral identity Z .e  W ".e w/ C Div e  e w/ d x D 0: 

(2.203)

78

2 Distance to exact solutions

Using (2.201) one more time, we obtain Z .".v/ W re wfe w e g div e w/ d x 

Z D

Z .Dive  C f/  e w dx

.".v/  e  e g 1/ W re w dx  



 k".v/  e  e g1k 1 kre wk C kDiv e  C fk CF kre wk 

1=2  k".v/  e  e g 1k 1 C C kDiv e  C fk  ˚ kg  e g k ; 

(2.204)

where C is the same as in (2.198). Now (2.202)–(2.204) yield the estimate 1 1=2 2˚ 

1=2

kg  e g k  k".v/    e g 1k 1 C C kDiv  C fk C ˚  k div vk : 

(2.205) Generalized Stokes problem We are dealing with Div ".u/ C ıu D f  rg div u D 0 u D u0

in ; in ; on @:

(2.206) (2.207) (2.208)

Usually, the parameter ı > 0 appears in the equation when the evolutionary Stokes problem is discretized with respect to time (then ı1 is proportional to the time step). In this case, we change (2.192) and set  Z  ı 2 F .v/ D jvj  f  v d x: 2 

Let v 2 V and  2 U be approximations of the exact velocity u and exact stress  2 U , respectively. The errors e D u  v and e D    are measured in terms of the norms  Z Z    1 2 1 2 2 2  2  2 kek;ı D j".e/j C ıjej d x and ke k 1 ; 1 WD je j C jDive j d x;  ı  ı 



respectively. By (2.67), we obtain the error identity 1 kek2;ı C ke k21 ; 1 D k 1=2 ".v/   1=2 k2 C kDiv   ıv C f  rgk2 :  ı ı (2.209)

2.4 Applications to particular mathematical models

79

2.4.8 Bingham problem A classical model in the theory of variational inequalities is connected with the problem of the flow of a visco–plastic incompressible fluid in a pipe (e.g., see [102, 124]). It is reduced to minimization of the functional Z

 J .v/ D (2.210) jrvj2 C kjrvj  f v d x 2  ı

over the space V0 D H 1 ./. Here  and k are positive constants that define physical properties of the fluid and  is the cross section of the pipe. We use the relations (2.51) and (2.53), where m D 2. In this case, Z J .v/ D G1 .ƒ1 v/ C G2 .ƒ2 v/ C F .v/; F .v/ D  f v d x; f 2 L2 ./; 

V  D H 1 ./; ƒ1 D ƒ2 D r; Z  jy1 j2 d x; G1 .y1 / D 2

Y1 D Y2 D L2 .; R2 /; ƒ1 D ƒ2 D  div; Z G2 .y2 / D kjy2 jd x:



We see that    F .v / D

G1 .y1 /



0; C1; Z D

v  C f D 0; v  C f 6D 0; 1 2 jy j d x; 2 1





DF .v; v / D

G2 .y2 /

 D

0; C1;

v  C f D 0; v  C f 6D 0;

jy2 j  k a:e: in ; : else:

0; C1;



The compound functionals take the form

DG1 .y1 ; y1 /

1 D ky1  y1 k2 ; 2

DG2 .y2 ; y2 /

Z D

.kjy2 j  y2  y2 /d x:



We use the integral identity (2.53), which holds for the functions y1 and y2 such that (cf. (2.54)) .y1 ; y2 / 2 Q` WD f div.y1 C y2 / C f D 0 g and jy2 j  k. The measure is defined by the relation

.v; y I u; p / WD DG1 .ƒ1 v; p1 / C DG2 .ƒ2 v; p2 / C DG1 .ƒ1 u; y1 / C DG2 .ƒ2 u; y2 / C DF .v; ƒ 1 p1  ƒ 2 p2 / C DF .u; ƒ 1 y1  ƒ 2 y2 /;

80

2 Distance to exact solutions

where the last two terms vanish and, therefore,

Z 1 1  2  2 .v; y I u; p / D krv  p1 k C kru  y1 k C .kjrvj  p2  rv/d x 2 2  Z C .kjruj  y2  ru/d x: 





Since

p1

D ru, we find that  2

.v; y I u; p / D kr.u  v/k2 C Z C

1  kp  y1 k2 C 2 1

Z

.kjrvj  p2  rv/d x



.kjruj  y2  ru/d x:



In order to see the meaning of the last two terms we introduce two sets: u0 WD fx 2  j jru.x/j D 0g ; v0 WD fx 2  j jrv.x/j D 0g : The set u0 is the so-called stagnation zone associated with the exact solution (notice that this set is a priori unknown). Analogously, the set v0 is an approximation of u0 defined by means of v. Let uC WD  n u0 and vC WD  n v0 . Since  ru k jruj ; if x 2 uC ;  p2 .x/ D if x 2 u0 ;   ; for any j  j  k; we find that   Z Z Z ru  rv  .kjrvj  p2  rv/d x D k jrvj  d x C .kjrvj     rv/d x: jruj u C



u 0

The first integral in the right-hand side is always nonnegative. It vanishes if v D u in the set uC where the exact solution does not stagnate (actually uC is the “flow zone”). The second integral is equal to zero. Indeed, if jrvj D 0, then the integral vanishes. On the other hand, if jrvj > 0 (i.e., in vC \ u0 ), then we can set rv   D k jrvj and the integrand also vanishes. In uC , we have ru  y2 D Hence,

Z .kjruj  

y2

1 jrujp2  y2 k Z  ru/d x D u C

and jp2 j2 D k 2 :  jruj  2 k  p2  y2 d x: k

2.4 Applications to particular mathematical models

81

It is easy to see that this term is also nonnegative and vanishes if y2 D p2 . Finally, we employ (2.53) to derive the identity  1 kr.u  v/k2 C kp1  y1 k2 2 2 Z Z  k jruj  2 C .jrujjrvj  ru  rv/ d x C k  p2  y2 d x jruj k u C

D

1 krv  y1 k2 C 2

Z

u C

  kjrvj  y2  rv d x:

(2.211)



2.4.9 Another error estimation method The method presented in previous sections is applicable for stationary problems that admit a variational (or saddle-point) formulation. For other problems (such as nonsymmetric elliptic and parabolic equations) we can use another method, based on transformations of the corresponding integral identities (see [255] and a systematic exposition in [256]). Below we collect some results which are helpful for subsequent chapters. Convection-diffusion problem Consider the stationary convection-diffusion problem  div Aru C a  ru D f uD0

in ; on :

(2.212) (2.213)

Here a is a given vector-valued function satisfying the conditions a 2 L1 .; Rd /;

div a 2 L1 ./;

div a  0

a:e: in :

(2.214)

Hence, ı2 D  12 div a  0 for almost all x 2 . For the error norm defined by the relation jŒu  vj2 WD kr.u  v/k2A C kı.u  v/k2 ; we have the first error majorant: jŒu  vj2  kı1 R.v; /k2 C k  Arvk2A1 ;

(2.215)

where R.v; / WD f  a  rv C div  and  is a vector valued function in H.; div/. For very small values of ı or very large values of a this estimate may be useless. In this case, we can use a different error majorant (see [256]), jŒu  vj  k  ArvkA1 C C kR.v; /k :

(2.216)

82

2 Distance to exact solutions

If  is decomposed into a collection of subdomains i and fjR.v; /jgi D 0, then we have estimates similar to (2.144), e.g., v uN uX C1i2 kR.v; /k2i : (2.217) jŒu  vj  k  ArvkA1 C t i D1

Linear parabolic equations Estimates of the distance to exact solutions of parabolic problems were derived in [253, 191, 189]. We refer the reader to these and other publications cited therein for proofs and a systematic exposition. Below, we sketch the idea of the method with the paradigm of a general linear parabolic equation generated by a uniformly elliptic operator ƒ Aƒ (see notation in Sect. 2.1). Let u solve the problem ut C ƒ Aƒu D f .x; t/ in QT WD   .0; T /; x 2 ; u.x; 0/ D u0 .x/; u.x; t/ D 0; .x; t/ 2 ST WD @  .0; T /:

(2.218) (2.219) (2.220)

It is assumed that f and u0 are such that the problem has a unique solution u in a suitable function space V .QT /, which is defined by the following integral relation: Z

ZT .Aƒu; ƒw/ dt  0

uwt d x dt

(2.221)

QT

Z

Z

.u.x; T /w.x; T /  u.x; 0/w.x; 0//d x D

C 

f w d x dt

8w 2 V .QT /;

QT

where the product .; / is associated with the spatial part of the equation. So far we do not specify the space V .QT /, which depends on the choice of ƒ and A (below we consider an example, where the space V .QT / is explicitly defined). To justify formal operations, we assume that any function w 2 V .QT / has finite integral RT 2 2 0 kƒwkA dt and for almost all t 2 T it has a trace w.; t/ 2 L ./ so that all integrals in the above relation exist and all necessary integration by parts formulas hold. For the error e D u  v, we have Z Z Z ZT .Aƒe; ƒw/ dt  ewt d x dt C e.x; T /w.x; T /d x  e.x; 0/w.x; 0/d x 0

QT

ZT

Z



ZT .Aƒv; ƒw/ dt:

.f w  vt w/ d x dt 

D 0 

0



2.4 Applications to particular mathematical models

83

Set w D e. Then ZT

1 kƒek2A dt C ke.; T /k2 2

0

Z D

ZT .f e  vt e/d xdt 

1 .Aƒv; ƒe/ dt C ke.; 0/k2 : 2

(2.222)

0

QT

Here e.; 0/ WD u0  v.; 0/ denotes the error at t D 0. Let R be a vector-valued function that admits the integral representation . ; ƒw/ D ƒ w d x (e.g., for the 

evolutionary diffusion problem ƒ D  div and this relation holds if 2 H.; div/ for almost all t 2 .0; T /). Now we rewrite (2.222) as follows: ZT

1 1 kƒek2A dt C ke.; T /k2  ke.; 0/k2 2 2

0

ZT Z D .  Aƒv; ƒe/ dt C .f  ƒ  vt /e d xdt: 0

(2.223)

Qt

Notice that .  Aƒv; ƒe/  kƒekA k  AƒvkA1 and Z

ZT



.f  ƒ  vt /e d xdt  C

kf  ƒ  vt k kƒekA dt ;

0

Qt

where the constant C is such that kwk  C kƒwkA . Let ˛1 .t/ and ˛2 .t/ be positive functions. By Young’s inequality, Z

  f  ƒ  vt e d xdt C

ZT .  Aƒv; ƒe/ dt 0

QT



1 2

ZT



˛1 C2 kf  ƒ vt k2 C ˛2 k  Aƒvk2A1 dt

0

1 C 2

ZT 0

˛1 C ˛2 kƒek2A dt: ˛1 ˛2

84

2 Distance to exact solutions

Now (2.223) yields the estimate ZT .2  ~/ kƒek2A dt C ke.; T /k2  ke.; 0/k2 0



ZT

˛1 C2 kf  ƒ vt k2 C ˛2 k  Aƒvk2A1 dt;

(2.224)

0 2 where ˛1 .t/ and ˛2 .t/ are selected such that ~.t/ WD ˛˛11C˛ ˛2  2. Consider the diffusion problem as an example. In this case, ƒ is the gradient operator and A is a positive definite matrix. Assume that f 2 L2 .QT / and u0 2 H 1 ./. Then we can set ˚  1 .QT / WD v 2 H 1 .QT / j v.x; t/ D 0 for .x; t/ 2 ST V .QT / D W2;0

and (2.224) reads ZT .2  ~/ krek2A dt C ke.; T /k2  ke.; 0/k2 0

ZT

˛1 C2 kf C div vt k2 C ˛2 k  Aƒvk2A1 dt; 

(2.225)

0

˚   where 2 Ydiv .QT / WD 2 L2 .QT ; Rd / j div y  2 L2 .QT / . Applications of these estimates to modeling errors generated by semi-discrete approximations are discussed in Sect. 6.5. Problems in the theory of incompressible viscous fluids First we discuss a generalized Oseen problem Div C Div.a ˝ u/ C ıu D f  D ".u/  g1 div u D 0 u D u0

in ; in ; in ; on @:

(2.226) (2.227) (2.228) (2.229)

This problem arises in quantitative analysis of the corresponding Navier–Stokes problem (see, e.g., [121, 179, 244]) if it is solved by means of incremental-type iteration schemes, e.g., ukC1  uk  Div  kC1 C Div.uk ˝ ukC1 / D f  rg kC1 ; k D 1; 2; 3; : : : Mt  kC1 D ".ukC1 /  g kC1 1; div ukC1 D 0; ukC1 D u0 on @:

85

2.4 Applications to particular mathematical models

This system is equivalent to (2.226)–(2.229) if uk is considered as a given function 1 and ı WD Mt > 0. Then, ukC1 and g kC1 are functions to be found. We assume that Z u0  n ds D 0; (2.230) f 2 L2 .; Rd /; u0 2 H 1 .; Rd /; and @

and div a D 0: (2.231) a 2 L1 .; Rd /; p It is convenient to set ˛ D ı and use this function instead of ı. The generalized ı

solution u 2 S./ C u0 satisfies Z Z ı   ".u/ W ".w/ C ˛ 2 u  w  .a ˝ u/ W rw d x D f  wd x; 8w 2 S./: 



(2.232) It is known (see, e.g., Chapter 2 of [165] ) that under the above assumptions u exists and is unique. Estimates of the error e D v  u in terms of the norm Z 1=2 2 2 2 2 .j".e/j C ˛ jej / d x kek;˛ WD 

were derived in [256, 258, 260]. The summary of these results is as follows. ı

g 2 e L2 ./, and  2 H.; Div/, the Theorem 2.4.3. For any v 2 S./ C u0 , e following estimate holds: kek;˛  CF k1=2 R.v; /k C k 1=2 .  ".v/ C e g 1/k ;

(2.233)

where .x/ WD

1 .x/ C CF2 ˛ 2 .x/

and R.v; / WD Div   a  rv  ˛ 2 v C f:

The right-hand side of (2.233) vanishes if and only if v D u, g D e g, and  D  . ı

If v 2 H 1 ./ C u0 and satisfies the additional conditions fjdiv vjgi D 0 in the subdomains i , then we have an analog of the estimate (2.199): !1=2 N X 2 g / C .2 C 3 /  k div vk2i ; (2.234) kek;˛  M˚ .v; ;e i i D1

g / denotes the right-hand side of (2.233) and 2 and 3 are defined where M˚ .v; ;e by the relations  1=2  1=2 1 D ˚ kak1 C k˛k21 CF ; 2 D ˚ C 1 CF ; 32 D ˚ C CF2 k˛k21 : The corresponding proofs and other estimates related to the generalized Oseen problem can be found in [258] (Theorem 2.1 and (2.24)), where the equivalence of the error and the majorant is also shown.

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2 Distance to exact solutions

Remark 2.4.4. The method discussed in Sect. 2.4.9 can be applied to evolutionary models of viscous incompressible fluids. In particular, for a divergence-free approximation v.x; t/ of the evolutionary Stokes problem, we obtain the estimate ZT .2  ~/ k".e/k2 dt C ke.; T /k2  ke.; 0/k2 0

ZT

˛1 k  ".v/ C e C g 1k21 C C 2 ˛2 kf  vt C Divk2 dt;

(2.235)



0

where ~ 2 .0; 2, e g.x; t/ is a square integrable function with zero mean, C is the same constant as in the case of the stationary problem, and .x; t/ is a matrix-valued function with square integrable spatial divergence.

2.5 Validation of mathematical models by comparison with experimental data Any mathematical model gives only an approximate description of a physical phenomenon. The final judgment on the suitability of a model is impossible without a comprehensive comparison of the results obtained in real life experiments with the data of computer simulation. At a first glance, the verification of a model is a simple matter and can be done straightforwardly by solving the mathematical problem numerically and comparing the results with those obtained in physical experiments. This approach is based upon the statement: if all (or almost all) results of computational and physical experiments are close, then the model is considered as “good”. Intuitively, this statement seems flawless and it is widespread among engineers. However, in reality there exists several obstacles that make such as straightforward method doubtful. In the vast majority of cases, numerical solutions contain various errors. Approximation errors arise when a continuous (differential) problem is replaced by a finite-dimensional one. The results may be also polluted by errors of numerical integration and differentiation, truncation of slowly converging iteration processes, instability of a numerical method, and also by bugs and computational defects that might be hidden in a numerical code. If the above errors are significant, then a direct comparison of the computational and physical data may lead to wrong conclusions, although the physical/mathematical model in use could be sufficiently accurate. To make verification of a mathematical model reliable, we must exclude computational errors or separate them from the modeling errors. Estimates of the distance to exact solutions of variational problems considered in this chapter provide such a possibility.

2.5 Validation of mathematical models

87

If we have a computable measure of the distance to the exact solution of the mathematical model being tested, then a suitable reconstruction of experimental data can be viewed as an approximation and directly compared with the solution. This method does not require explicitly finding the corresponding exact solutions. The data confirm the validity of the model if in all experiments the errors are smaller than the desired tolerance level. Below we discuss briefly this approach with the paradigm of the variational problems considered above.  Let u~ and p~ be the functions (in the energy space) constructed by experimental measurements (or obtained by processing of experimental data) related to a real life object (or a process). We compare them with the solutions generated by a mathematical model, which is supposed to provide an adequate mathematical counterpart of the object. As before, the exact solutions of the model are denoted by u 2 V and p 2 Y  , where V and Y  are the respective (energy) spaces. By using (2.21) (or (2.26) and other relations exposed in this chapter) we can directly use experimental data and verify the model in terms of the error measure  .u; p I u~ ; p~ /, which shows the difference between the theoretical solutions u; p  and experimental data u~ ; p~ . We have    .u; p I u~ ; p~ / D DG .ƒu~ ; p~ / C DF .u~ ; ƒ p~ /:

(2.236)

Notice that the right-hand side of (2.236) contains only known functions. All the data related to the mathematical model (domain, boundary conditions, material constants, boundary conditions) are encompassed in G , F , ƒ, and in the functional sets used in (2.11). They determine completely the functions u and p , which, however, do not enter the right-hand side of (2.236) explicitly. / .i / Let the functions .u.i ~ ; p~ /, i D 1; 2; : : :; N~ be obtained in N~ different experiments related to one and the same object. We wish to know how accurately the object is described by the model (2.11) and respective variational problems (2.16) and (2.17). It is natural to characterize the validity of our model by means of the averaged modeling error e e mod

N~

1 X / .i / .i /  .i / D DG .ƒu.i ~ ; p~ / C DF .u~ ; ƒ p~ / : N~

(2.237)

i D1

Another option is to use the maximal modeling error

/ .i / .i / max  .i / D max DG .ƒu.i ; p /C D .u ; ƒ p / : emod F ~ ~ ~ ~ i D1;2;:::;N~

(2.238)

Hence a direct comparison with experimental data by (2.237) shows how accurate the model is “in the average” and (2.238) shows the “worst case scenario” error. Certainly, (2.237) and (2.238) provide a trustable validation only if all the experimental data are reliable and if N~ is sufficiently large.

88

2 Distance to exact solutions

Validation of a mathematical model by comparison with experimental data is always faced with limits of our knowledge on the physical object studied. In practice, the amount of reliable experimental data is often very limited, so that in reality it is impossible to validate a model with absolute reliability. Therefore, a meaningful discussion of a model is possible only within a certain set of data (parameters, conditions, etc.). In the majority of cases, inexactness of experimental data should be also taken into account (e.g., see [23, 30, 198, 199]). Then, the formulas (2.237) and (2.238) should be modified accordingly. For example, instead of (2.237) we use e e mod

N~

1 X .i / .i / .i / .i / .i / D

.i / DG .ƒu~ ; p~ / C DF .u~ ; ƒ p~ / C ~ ; N~

(2.239)

i D1

.i /

where ~ is the error related to measurements in the i -th physical experiment. These errors always exist, but usually their values are known. The modified formula (2.239) PN~ has weights .i / > 0, i D1

.i / D 1 that reflect the impact (importance) of different experiments. Possible application of formulas discussed above to identification of the best mathematical model is illustrated by the example below. Example 2.5.1. Suppose one wants to examine whether the diffusion model (2.113) / is consistent with an available set of experimental data (concentration values u.i ~ .i / and fluxes p~ obtained in N~ experiments with different source terms f .i / and .i / boundary conditions u0 ). Moreover, one wants to find the most suitable coefficients / of the diffusion matrix A and reaction parameter . Let u.i / (where u.i / D u.i 0 on ) .i / denote the exact solutions of (2.113). From (2.115) one derives the identity and p / 2 / 2 .i /  u.i kr.u.i /  u.i ~ /kA C k.u ~ /k 1 .i / 2 .i / 2 C kp.i /  p~ kA1 C k div.p.i /  p~ /k

1 .i / .i / .i / .i / D kAru~  p~ k2A1 C k div p~  u~ C f .i / k2 :

(2.240)

The left-hand side of (2.240) is the error (evaluated in terms of the full primal–dual norm) of the mathematical model (2.113) in the i -th experiment. The right-hand side .i / .i / contains only known experimental data u~ and p~ and, therefore, it is directly computable. Certainly, it depends on A and which characterize a particular model within the class of equations (2.113). Summarising the results of all experiments (as in (2.237) or (2.239)), we can find an averaged modeling error e e mod for this series of experiments. It depends of the values of A and . Finding those values that minimize e e mod , we identify the most adequate mathematical model (within the selected class of diffusion type models).

2.6 Errors of numerical approximations

89

2.6 Errors of numerical approximations Errors of numerical approximations can be viewed as a special class of modeling errors described in the item (b) of Sect. 2.2.5. These errors arise when a continuous infinite-dimensional setting is replaced by a finite-dimensional one and the space V (energy space of a mathematical problem) is replaced by a finite-dimensional space Vh . Typically, Vh is associated with a certain mesh T and h is a positive small parameter proportional to the size of mesh cells. In this section, we denote mesh cells by i and assume thatSthey are open convex domains (e.g., simplices or convex polygons) such that  D i . Modern mesh-adaptive numerical technologies operate with successively refined meshes that satisfy certain geometrical regularity conditions (related to the size and geometry of the cells in Th ). Analysis of approximation errors generated by different numerical methods is one of the main research directions actively developing in the last decades. There exists a vast literature devoted to this question (the main approaches to error control and mesh-adaptive computational methods are presented in [5, 6, 8, 25, 24, 36, 37, 77, 76, 108, 147, 219, 240, 250, 300, 330, 329] and in the monographs [7, 35, 60, 69, 71, 57, 88, 121, 224, 245, 256, 279, 288, 316, 318], supplied with numerous references to other publications). Our focus here is of a different kind. In this section, we restrict ourselves with a concise overview of some principal questions and discuss advanced forms of the estimates that are adapted to numerical (e.g., FEM) approximations. Applications of the theory presented above to measuring the distance between the exact solution and a numerical approximation have been studied in depths. The estimates of Sects. 2.1– 2.4 have been derived on the functional level and do not exploit specific features of approximations (e.g., Galerkin orthogonality). Therefore, we can use them directly by setting v and y  equal to the corresponding approximations (or to the functions obtained by suitable post-processing of v and y  ). In this way we obtain estimates, that are called a posteriori error estimates of functional type. Three important properties make them essentially different from other a posteriori estimators of approximation errors: (a) they are fully guaranteed in the sense that the reliability of the estimates is strict (they do not contain “higher order terms”, and they are not only “error indicators”); (b) the estimates do not contain mesh dependent constants. The constants used (e.g., CF , CP ,  , ˚ ) can be sharply estimated; (c) the estimates provide error bounds for any conforming approximation regardless of the method by which it has been constructed. An exposition of the theory, numerical examples, and references can be found in the monographs [184, 214, 256] and in the papers [247, 248, 250, 271, 272, 249, 266]. Among other publications we mention [176, 81, 215, 268] related to nonconforming methods and [253, 169, 189, 191, 170, 171] devoted to approximations of parabolic equations. Error estimates for mixed-type approximations of elliptic problems were

90

2 Distance to exact solutions

studied in [12, 163, 275, 256]. A posteriori estimates of functional type for various nonlinear problems were derived in [15, 115, 160, 270, 269] and some other sources cited therein. In [153] estimates of this type are applied to rank-structured approximation method for quasi–periodic elliptic problems and in [183, 229, 230] they are derived for problems in exterior domains.

2.6.1 Two-sided estimates of approximation errors Let vh 2 Vh  V and yh 2 Yh  Y  be approximations of u and p computed using finite-dimensional subspaces Vh and Yh , respectively. From the main error identity (2.21) it readily follows that

.vh / C  .yh / D DG .ƒvh ; yh / C DF .vh ; ƒ yh /:

(2.241)

The right-hand side of (2.241) contains only known approximate solutions and can be used as a natural measure of their accuracy. Hence (2.241) is the error identity for approximation errors. It is applicable except for the case where F .v/ D h`; vi (because yh may not satisfy the equation ƒ yh C ` D 0). Using (2.39), for F .v/ D h`; vi, we obtain

.vh / C  .yh / D DG .ƒvh ; yh / C hƒ yh C `; vh ui:

(2.242)

For linear problems ƒ Aƒu C ` D 0, the error measures  and  are reduced to norms (see (2.68)) and we obtain kƒeh k2A C keh k2A1 D kAƒuh  y  h k2A1 C 2hR.yh /; eh i:

(2.243)

Here eh D vh  u, eh WD yh  p are the errors generated by approximations uh 2 V and yh 2 Y  , respectively, and

R.yh / WD ƒ yh C ` 2 V  : Now the main difficulty is related to the term hR.yh /; eh i, which contains the unknown numerical error eh . In Sect. 2.3.2, we have already discussed several ways   to estimate this term. In particular, if yh belongs to Qƒ  (which is a subspace of Y ,   cf. (2.77)), then R.yh / belongs to a pivot space intermediate between V and V (see Sect. 1.1) and the estimates (2.98)–(2.99) yield two-sided estimates of approximation errors   1  2 keh kA1 C 1  kƒeh k2A  kAƒvh  yh k2A1 C ˇC 2 kR.yh /k2V ; (2.244) ˇ   1  2 kƒeh k2A  kAƒvh  yh k2A1  ˇC 2 kR.yh /k2V ; (2.245) keh kA1 C 1 C ˇ  where ˇ > 1. In practice, the condition yh 2 Qƒ  is not very restrictive. Moreover, many numerical methods (e.g., dual mixed or discontinuous Galerkin methods)

2.6 Errors of numerical approximations

91

produce dual approximations that automatically satisfy this condition. If it is not the case (as in the classical finite element method where numerical fluxes may not belong  to Qƒ  ), then there exist relatively cheap (averaging type) procedures that generate  a more regular approximation e y h 2 Qƒ  (e.g., see [6, 8, 25, 60, 77, 161, 316, 318, 327, 330, 329]). Estimates (2.244) and (2.245) produce sharp bounds of errors if the first term dominates the term kR.yh /kV . An essential quantitative difference between the error measure and above two-sided bounds may arise if the product hR.yh /; eh i is strongly overestimated by the product kR.yh /kV keh kV . Below we consider more sophisticated estimates of this product that generate sharper error bounds.

2.6.2 Reduction of the set Qƒ  The idea of this method has been discussed briefly in Sect. 2.4.4 with the paradigm of diffusion type equations. It is based on the observation that imposing additional conditions on the variable y  may yield estimates with constants smaller than the constant C (which is defined by the smallest eigenvalue of the positive definite operator ƒ Aƒ). A general form of this approach is considered in Sect. 7.1 of [256]. If  is decomposed into N non-overlapping Lipschitz subdomains i , then fjvjgi should be defined as the orthogonal projector of v on ker ƒ and the constants Ci satisfy the inequality kv  fjvjgi kV .i /  Ci kƒvkU.i / : Additional conditions to be satisfied are fjR.yh /jgi D 0

8 i D 1; 2; : : :; N:

In particular, if v is a scalar-valued function and ƒv D rv, then fjvjgi is reduced to finding the mean value and we arrive at the relations considered in Sect. 2.4.4, where the mean type conditions (2.139) lead to estimates containing Poincar´e constants associated with subdomains. Other variants may include the constants in (1.39) and (1.40) (e.g., see [259, 189]). The method is efficient unless the amount of additional conditions is not too large. Sometimes these conditions are satisfied automatically (as the mean conditions in locally conservative approximation methods). Then the corresponding estimates can be successfully used (e.g., see [106, 107, 105]) if the subdomains i are defined as the finite elements and it is guaranteed that the approximation is the exact solution of the discrete system. In other cases, special “equilibration” procedures are required. They may also generate errors and could be expensive if the number of elements is very large. Therefore, we believe that for real life problems this method is useful as a way to overcome difficulties associated with complicated geometry of  when the domain is decomposed into a certain (not very large) number of simple subdomains i for which the constants Ci are known.

92

2 Distance to exact solutions

2.6.3 Transformation of hR.yh /; eh i Assume that we know a function # such that ƒ Aƒ# D g 2 V . Then for any  2 R the last term in (2.243) can be rearranged as follows: hR.yh /; vh ui D hR.yh / C g; vh ui  .Aƒ#; ƒ.vh  u// (2.246)

D hR.yh / C g; vh ui   .Aƒvh ; ƒ#/  h`; #i

 j R.yh / C g j ƒ kƒ.vh  u/kA   .Aƒvh ; ƒ#/  h`; #i :  Since g 2 V and yh 2 Qƒ  , we have

jj R.yh / C g j ƒ D 

.R.yh / C g; w/V kƒwkA w2V nf0g sup

kR.yh / C gkV kwkV  C kR.yh / C gkV ; kƒwk A w2V nf0g sup

C D

Cƒ 1=2

;



where the constants appear due to (2.5) and (2.80). Setting  D  j R.yh /

 .R.yh /;g/V

kgk2 V

2

C g j ƒ

, we find that

C2  ƒ 

kR.yh /k2V



.R.yh/; g/2V kgk2V

! :

Now (2.243), (2.246), and (2.247) yield the estimates   1 keh k2A1 C 1  kƒeh k2A  kAƒvh  yh k2A1 ˇ C

.R.yh /; g/V ..Aƒvh ; ƒ#/  h`; #i/ C ˇC 2 kR.yh /k2V  kgkV

(2.247)

(2.248) ! .R.yh /; g/2V kgk2V

and keh k2A1 C



1 C 1C ˇ



kƒeh k2A  kAƒvh  yh k2A1

.R.yh /; g/V ..Aƒvh ; ƒ#/  h`; #i/  ˇC 2 kgkV

(2.249) ! .R.yh /; g/2V  2 kR.yh /kV  ; kgk2V

which in general are sharper than (2.244) and (2.245). The reason is that a possible overestimation in (2.248) and underestimation in (2.249) is associated only with the last term, which is smaller than the last term in (2.244) and (2.245).

2.6 Errors of numerical approximations

93

Generalizations of this method are rather obvious. Let #k , k D 1; 2; : : :; n be a collection of functions in V such that ƒ Aƒ#k D gk 2 V . If #k are linearly independent functions, then gk possess the same property. Indeed, assume the opposite, i.e., there exist a vector  D .1 ; 2 ; : : :; n / 2 Rn , whose components are not all P equal to zero and nkD1 k gk D 0. Then ! n X k #k D 0 ƒ Aƒ kD1

P and in view of the unique solvability of the problem, we have nkD1 k #k D 0. The latter equality cannot hold P because #k are linearly independent. Therefore, we use the representation g D nkD1 k gk and find k by solving the problem15   n !2 ( ) n  X X   max k .R.yh /; gk /V ; where M WD  2 Rn ;  k gk  D 1 :   2M kD1

kD1

V

Now the corresponding g and # (constructed as linear combinations) can be used in the above presented estimates.

2.6.4 Using extra regularity of the exact solution It is often known a priori that the exact solution possess an additional regularity. We can use this fact in order to derive a posteriori estimates (for finite element approximations) with better weights than in the general estimates (2.98) and (2.99). Below this idea is discussed with the paradigm of the equation (2.113) and Dirichlet boundary conditions. The exact solution u is defined by the relation Z Z Aru  rw d x D f w d x 8w 2 V0 : (2.250) 



Assume that u 2 H 2 ./ and we know the constant creg in the regularity estimate, i.e., juj2;2;  creg kf k :

(2.251)

For example, if we consider the Poisson equation with homogeneous Dirichlet boundary conditions in a convex domain , then creg D 1 (see [166, 168]). Estimates for some other cases are presented in Proposition 5.3.8. We use standard interpolation operators h defined for a Lipschitz domain ! (usually ! is a convex polygon) that map the functions in V C WD V \ H 2 .!/ to a finite-dimensional subspace Vh .!/ (which is typically formed by piecewise affine functions). The corresponding interpolation estimates read k u  h u k!  Cint;1 .!/juj2;2;! kr.u  h u/k!  Cint;2 .!/juj2;2;! : 15

(2.252) (2.253)

For moderate values of n this problem can be easily solved by standard numerical optimization methods.

94

2 Distance to exact solutions

in which the constants Cint;1 and Cint;2 depend on the diameter h of the domain !. For domains typically used in finite element approximations (e.g., simplices and convex polygons), these constants are usually defined by affine equivalent mappings of basic domains (e.g., see [88]). As before, let vh 2 Vh denote an approximation of u, yh 2 L2 .; Rd / by a nu merical flux, and e yh 2 Qƒ D H.; div/ be a regularisation of yh (we need to use e yh   instead of yh only in the case where yh does not have square integrable divergence). By (2.116), we know that the errors e D vh  u and e D e yh  p satisfy the identity Z yh k2A1  2 R.e yh /.vh u/d x: (2.254) krek2A C ke k2A1 D kArvh  e 

Assume that (a) vh coincides with the Galerkin approximation uh , (b) the regularity requirements necessary for the validity of the Aubin–Nitsche lemma (see, e.g., [88]) hold, i.e., ku  uh k  CAuNi hkr.u  uh /k ; and (c) the constant CAuNi is known. This constant depends on creg , interpolation constants associated with elements in Th , and properties of the adjoint boundary value problem. Then an obvious way to estimate the last integral is as follows: ˇ ˇ ˇZ ˇ ˇ ˇ  ˇ R.e yh /.uh  u/ d xˇˇ  kR.e yh /k kuh  uk  kR.e yh /k CAuNi hkr.uh  u/k : ˇ ˇ ˇ 

If Cint;2 .Th / is proportional to h then (2.254) implies yh k2A1 CC h2 kR.e yh /k ; krek2A Cke k2A1  kAruh  e

(2.255)

where C depends on CAuNi , Creg and kf k . If this constant is known, then the estimate (2.255) may be very efficient. However, the conditions (a)–(c) are very specific and, therefore, the estimate (2.255) could be useful only in some special cases. Below we consider other ways to estimate the term containing R.yh /, which use much weaker additional conditions.

2.6.5 Using an auxiliary finite-dimensional problem The main idea of the approach considered in this section is to reform the functional yh /; eh i with the help of an auxiliary finite-dimensional problem. hR.e Let us define another finite-dimensional subspace VH  V0 . In particular, VH may be a subspace of Vh generated by coarsening of the mesh Th , but in general this space is independent on Vh . The cells (elements) of the mesh TH associated with VH are denoted by i , i D 1; 2; : : :; NH . Consider the problem: find uH 2 VH such that Z Z ruH  rwH d x D R.e yh /wH d x 8wH 2 VH : (2.256) 



2.6 Errors of numerical approximations

95

Notice that the problem (2.256) is generated by the Laplace operator16. It is simpler than the original finite-dimensional problem (solved to find uh ) and can be used for the error analysis of the problem (2.250) with various A D A.x/. It is easy to see that R R ruH  rwH d x R.e yh /wd x   kruH k D sup  sup DW j R.e yh /jj : krw k krwk H   wH 2VH nf0g w2V0 nf0g (2.257) Hence, the energy norm of uH is dominated by a weak norm of Rh .e yh /, which tends to zero if e yh tends to p in L2 .; Rd /. Moreover, since Z Z R.e yh /wH d x D .p  e yh /  rwH d x; 



we use (2.256) and find that yh k : kruH k  kp  e

(2.258)

Assume that e yh is obtained by a post-processing procedure applied to the numerical flux Aruh (which is typical for the classical finite element method), i.e., e yh D Gh Aruh , where Gh is a suitable averaging operator.17 If the exact solution u possesses extra regularity and the meshes satisfy certain additional conditions, then there exist efficient averaging methods mapping yh 2 L2 .; Rd / to e yh 2 H.; div/   such that e yh 2 H.; div/ converges to p with a higher rate compared to yh 18 . The estimate (2.258) shows that in this case kruH kA will tend to 0 with a superconvergence rate (with respect to h). We have Z R.e yh /.uh  u/ d x D J1 C J2 C J3 ; (2.259) 

where

Z

J1 D

R.e yh /.uh  H uh / d x;



Z

J2 D

R.e yh /.Huh  H u/ d x;



Z

J3 D

R.e yh /.Hu  u/d x;

 16 We

may use another positive definite elliptic operator. operators are well studied in the theory of finite element approximations, e.g., see [77, 78, 161, 316, 318, 327, 330, 329]. 18 This phenomenon is known as superconvergence [77, 86, 109, 161, 175, 329, 318, 322, 327, 331]. 17 Averaging

96

2 Distance to exact solutions

and H W V ! VH is an interpolation operator (we assume that H wH D wH for any wH 2 VH ). The first integral can be computed directly. In view of (2.256), we get for the second one Z Z J2 D R.e yh /.Huh H u/d x D ruH r.H uh  H u/d x (2.260) 

Z

Z

D

ruH r.H uh  uh /d x C 

Z ruH  r.uh  u/d x C



ruH  r.u  H u/d x: 

In this representation, the first integral can be computed directly. The second integral is estimated by Young’s inequality Z 1 ˇ ruH  r.uh  u/d x  kruH kkrek  krek2A C kruH k2 : 2ˇ 2 

For further analysis, we introduce the quantities Z Z

i2 D jR.e yh /j2 d x ; &i2 WD jruH j2 d x ; i

S1 D

NH X

i

!1=2 2

i2 Cint;1 .i /

S2 D

;

i D1

NH X

!1=2 2 &i2 Cint;2 . i /

;

i D1

and apply (2.252) and (2.253). We have Z ruH  r.u  H u/d x 

NH X

&i Cint;2 .i /juj2;2;i  S2 creg kf k :

i D1



Analogously,

J3 

X

kR.e yh /ki kH uuki

i



X

!1=2

i2

creg kf k  S1 creg kf k :

i

Now (2.254) and (2.259) imply (for ˇ  

2 Cint;1 .i /

1 1 ˇ



1 ) 

krek2A Cke k2A1  kAruh e yh k2A1 C I .uh ;e y h /C E .uH;e yh ; ˇ/: (2.261)

Here

I .uh;e y h /

Z WD 2 

  R.e y h /.Huh  uh /  ruH  r.H uh  uh / d x

97

2.6 Errors of numerical approximations

and

E .uH ;e yh ; ˇ/ WD ˇkruH k2 C 2creg .S1 C S2 / kf k :

It should be noted that the first two terms in the right-hand side of (2.261) can be comy h / puted directly and contain no overestimation. If VH D Vh , then the term I .uh ;e vanishes. The identity (2.259) also yields a lower bound:   1 1C krek2A Cke k2A1  kAruh e yh k2A1 C I .uh ;e y h / E .uH ;e yh ; ˇ/: ˇ (2.262) It is clear that the quality of the estimates (2.261) and (2.262) depends on the reminder term E .uH ;e yh ; ˇ/ containing interpolation estimates. Let us analyse its behaviour assuming for simplicity that VH D Vh and that this finite-dimensional space is based on regular meshes and standard interpolation operators, whose interpolation constants satisfy the estimates (for P 1 finite elements) Cint;1 . i /  c1 h2

and Cint;2 . i /  c2 h;

(2.263)

with some positive constants c1 and c2 . Then

S1  c1 h2 kR.e yh /k ;

S2  c2 hkruH k ;

and we find that   E .uh ;e yh ; ˇ/  ˇkruH k2 C 2creg c2 hkruH k C c1 h2 kR.e yh /k kf k : yh kA1 tends to If e yh D Gh Aruh possesses superconvergence properties, then kjpe 1Cı zero as h with some ı > 0. In view of (2.258), kruH k decreases with the same rate. Then the first two terms are of the order h2Cı as h ! 0. If dive yh converges to f in L2 , then the last term also converges faster than h2 . On the other hand, the first squared term of the combined error norm is (in general) decreasing with the rate h2 . Since kr.uh u/kA  kp e yh kA1  kAruh e yh kA1 kr.uh u/kA C kp e yh kA1 ; we see that the first term of the majorant (2.261) also decreases with the rate h2 and, therefore, it dominates over E for small h. In any case, this simply computable majorant (which is based on a post-processed flux e yh and a solution uH of a finitedimensional problem (2.256)) has the same asymptotic convergence rate as the true error. Moreover, we can expect that for superconvergence operators the remainder term E becomes negligible for small h. Another method uses the dual mixed formulation (e.g., see [245]) of the auxiliary problem (2.256). Let  and all the cells i be polygonal (in the simplest case i are simplices). We define two finite-dimensional subspaces: QH  H.; div/ and VH  L2 ./ constructed by means of the lowest order approximations, i.e., in i

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2 Distance to exact solutions

the functions in VH have constant values and the vector-valued functions in QH have constant values of div y . For simple simplicial meshes such types of approximations are formed by the Raviart-Thomas (RT0 ) finite elements of the lowest order. In the case of more complicated polygonal cells, one can use the approximations suggested in [163]. We solve the following finite-dimensional problem: find pH 2 QH and uH 2 VH such that Z   (2.264) uH div yH C yH  pH d x D 0 8yH 2 QH ; 

Z

.div pH C R.e yh //wH d x D 0

8wH 2 VH :

(2.265)



Z

Since kpH k2

D

uH div pH

Z dx D



kpH k

R.e yh /uH d x;



R.e yh /

the norm tends to zero if tends to zero weakly in L2 ./ as h ! 0. We use (2.265) in order to represent the integral containing R.e yh / in the following form: Z Z Z   R.e yh /e d x D R.e yh /H e d x C R.e yh /.e  H e/ d x 



Z

D



div pH H e d x



Z

C

R.e yh /.e  H e/ d x:

(2.266)



In any subdomain i , the Rinterpolation operator H is defined by the relation R H .w/ji D fjwjgi , so that i H .w/d x D fjwjgi ji j D i w d x. Since div pH is a constant on any i , we have ˇ ˇ ˇ ˇ ˇ ˇZ ˇ ˇZ ˇ ˇ ˇ ˇ ˇ div p H e d xˇ D ˇ div p .uh  u/ d xˇ H H ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ 



 kpH k kr.uh  u/k  Let

S .e yh /2

D

NH X i D1

1 ˛ krek2A C kpH k2 : (2.267) 2˛ 2

CP2 .i /kR.e yh /  i k2i ;

i D fjR.e yh /jgi :

If all i are convex and their diameters are bounded from above by H, then

S .e yh /2 

H

NH 2 X

2

i D1

kR.e yh /  i k2i :

2.6 Errors of numerical approximations

99

We have ˇ ˇ ˇ NH ˇZ ˇ X ˇ  ˇ ˇ R.e y /.e   e/ d x CP .i /kR.e yh /  i ki kreki H h ˇ ˇ ˇ i D1 ˇ 

 S .e yh /krek 

1 ˇ krek2A C S .e yh /2 : 2ˇ 2

The relations (2.266) – (2.268) yield a simple estimate ˇ ˇ ˇZ ˇ ˇ ˇ    ˇ R.e ˇ  ˛ C ˇ krek2 C 1 ˇ S .e y /e d x yh /2 C ˛kpH k2 ; A h ˇ ˇ 2ˇ˛ 2 ˇ ˇ

(2.268)

(2.269)



where the positive constants ˛ and ˇ satisfy the condition ˇ1 C ˛1   . Now instead of (2.261) and (2.262) we obtain   ˛Cˇ krek2A Cke k2A1  kAruh e yh k2A1 Cˇ S .e yh /2 C ˛kpH k2 ; 1 ˇ˛ (2.270)   ˛Cˇ 1C krek2A Cke k2A1  kAruh e yh k2A1 ˇ S .e yh /2  ˛kpH k2 : ˇ˛ (2.271) In particular, for ˇ D ˛ D

2 

we have

ke k2A1  kAruh e yh k2A1 C

 2   2 yh /2 : kpH k C S .e 

(2.272)

These estimates require no extra regularity of u. Also, they do not need any equilibration of e yh . Notice that the term S .e yh /2 is, in general, essentially smaller than the y h /k2 in (2.244) and (2.245) and kpH k2 is small if the reconstructed term C 2 kR.e  flux e yh approximates p in a weak sense only (this condition usually holds for commonly used averaging operators). Moreover, if the fluxes are weakly equilibrated (i.e., fjR.e yh /jgi D 0 on all i ), then pH D 0 and the term S .e yh /2 penalizes only 2

deviations from mean values weighted with the factor H2 . It is clear that in general these estimates are much sharper than (2.244) and (2.245).

2.6.6 Applications to least squares type methods The right-hand side of (2.119) and of other estimates derived for linear elliptic problems in Sect. 2.3 (e.g., (2.81) and (2.93)) have the same principal structure as the functionals used in the least-squares finite element methods studied in [19, 65, 285] and some other publications. These methods are often used for getting numerical

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2 Distance to exact solutions

approximations of u and p (this form is known as the mixed least squares finite element method, e.g., see [73, 74, 233] and many other publications). This method is unconditionally stable and generates converging approximations of linear elliptic problems provided the additional least squares stabilization terms are linearly coupled with arbitrary positive coefficients. In the majority of publications related to mixed least squares methods, the weights are simply set equal to one. The estimate (2.118) shows that being supplied with proper weights the quantities kArv  y kA1 and kR.y /k provide a guaranteed upper bound of the energy error norm and a more general estimate (2.93)) yields a similar conclusion for the combined primal-dual norm. Let us emphasize that the terms of these estimates are supplied with weights and it is known that with other weights they may not hold. Also, it is worth noticing that the weights make the terms unidimensional (in the sense that they have same physical dimension). This facts suggest that the theory presented in this chapter generates weights, that are indeed optimal if least square type constructions are used to get approximations of boundary value problems19. For example, the estimate (2.198) derived for the Stokes problem shows that the functional 2 k div vk M.v; ; g/ WD krv    g1k21 C C 2 kDiv  C fk2 C 4˚  

majorates the quantity 3kr.v  u/k2 and, therefore, minimization of (2.6.6) over ı

certain finite-dimensional subspaces of H 1 .; Rd /  H.; Div/  e L2 ./ always generates a sequence that converges to the exact solution in terms of the energy error norm. Notice that this numerical scheme is unconditionally stable and does not require divergence-free approximations. The weight 1 in the first integral and the 2 depending on the Friedrichs constant and the constant in multipliers C 2 and 4˚  Lemma 1.2.1 properly balance the terms. Moreover, the above presented theory gives a clear answer to the question on the correct structure of the least square type functionals to be used for nonlinear problems. It directly follows from the error identity (2.26), which shows that the corresponding functional is M.v; y  / WD DG .ƒv; y  / C DF .v; ƒ y  /:

(2.273)

Indeed, this functional is equal to the natural error measure associated with the problem, so it is logical to minimize it for finding approximations of the primal and dual solutions. Certainly, in this case the term “least square” sounds strange because the terms DG and DF are not quadratic. For example, consider the problem with friction discussed in Sect. 2.4.6. The functional M is generated by the error identity (2.187), whose right-hand side contains nonlinear terms  Z  Z   1  ˛ 1 ˛  ıjvj  v R.y / d x; jrvj C  jy j  rv  y d x and ˛ ˛ 



19 Some comments related to minimization of least squares type functionals for linear elliptic equations with and without weights can be found in [275].

2.6 Errors of numerical approximations

101

ı

which should be minimized on W 1;˛ ./  Qı . For the Bingham problem, an analogous functional bound follows from the error identity (2.211). If the functional F is linear, then the corresponding functional M follows from (2.47). In this case the functional contains the nonlinear terms DG .ƒv; y  /, H , and 0. The first term in the right-hand side of (2.276) is related to a conforming approximation, e.g., we can set v D Pb v . This measure

is estimated by the methods discussed above. The remaining term b  12 .b v  Pb v/

represents the error generated by the nonconformity of b v . The factors 1 and 2 may be used to balance two parts of the error bound. Finally, we note that the reader interested in a more systematic analysis of this question may consult Chapt. 9 of the book [256] and papers [81, 176, 268, 271, 272], which also contain results of numerical experiments.

Chapter 3

Dimension reduction models This chapter is devoted to the analysis of dimension reduction errors for several well-known models in continuum mechanics. Estimates of the respective modelling errors are derived by means of the error relations deduced in the previous chapter.

3.1 Dimension reduction The systematic study of mathematical models associated with the reduction of dimension began in the 19th century motivated mainly by the development of solid mechanics. Probably the theory of elastic thin-walled structures (connected with the names of Kirchhoff, Love, Timoshenko, Reissner, Mindlin, and others) was the one that stimulated the earliest steps in this direction. The modern theory of beams, plates and shells ([310, 181, 246, 217]) provides a rich collection of dimension reduction models (a survey is found in [90]). Other important examples arise in the theory of fluids (e.g., see [179]). In the majority of cases, dimension reduction is applied to a 3D problem, which is then replaced by a simplified 2D (or 1D) one. Typically, the models are studied in the context of the following two main directions: proving that a sequence of solutions associated with a sequence of 3D bodies with deteriorating thickness d tends (in a certain sense) to a limit function, which can be found as a solution of a “reduced” 2D problem supplied with a suitable 3D recovery operation, establishing the asymptotic rate of convergence of the above sequence with respect to d . Here we refer to [9, 14, 26, 63, 90, 91, 196, 197, 205, 286, 287] and works of many other authors cited in these publications. Schematically, the concept of dimension reduction is presented in Fig. 3.1.1. The original problem P (which has the exact solutions u and p ) is replaced by a reb (whose solutions b b duced 2D (or 1D) problem P u and b p  belong to the spaces V  b ). Typically, the reduced problem is solved numerically using a certain finiteand Y b of V b and Y b , respectively. The respective solub h and Y dimensional subspaces V h  tions b v and b p h are supposed to provide a quantitative information about u and p . In general, the functions b v and b q  do not belong to V and Y  , respectively. For this reason any comparison with the exact solutions should use specially designed reconstruction operators b ! V