Stokes–Darcy Equations: Analytic and Numerical Analysis [1st ed.] 978-3-030-02903-6, 978-3-030-02904-3

Table of contents :
Front Matter ....Pages i-viii
Introduction (Ulrich Wilbrandt)....Pages 1-7
Notation and Preliminary Results (Ulrich Wilbrandt)....Pages 9-25
Properties of Sobolev Spaces (Ulrich Wilbrandt)....Pages 27-56
Traces (Ulrich Wilbrandt)....Pages 57-82
Subproblems Individually (Ulrich Wilbrandt)....Pages 83-108
Stokes–Darcy Equations (Ulrich Wilbrandt)....Pages 109-151
Algorithms (Ulrich Wilbrandt)....Pages 153-174
Numerical Results (Ulrich Wilbrandt)....Pages 175-199
Back Matter ....Pages 201-212

Citation preview

Lecture Notes in Mathematical Fluid Mechanics

Ulrich Wilbrandt 

Stokes–Darcy Equations Analytic and Numerical Analysis

Advances in Mathematical Fluid Mechanics Lecture Notes in Mathematical Fluid Mechanics Editor-in-Chief: Galdi, Giovanni P

Series Editors Bresch, D. John, V. Hieber, M. Kukavica, I. Robinson, J. Shibata, Y.

Lecture Notes in Mathematical Fluid Mechanics as a subseries of ‘Advances in Mathematical Fluid Mechanics’ is a forum for the publication of high quality monothematic work as well lectures on a new field or presentations of a new angle on the mathematical theory of fluid mechanics, with special regards to the NavierStokes equations and other significant viscous and inviscid fluid models. In particular, mathematical aspects of computational methods and of applications to science and engineering are welcome as an important part of the theory as well as works in related areas of mathematics that have a direct bearing on fluid mechanics. More information about this series at http://www.springer.com/series/15480

Ulrich Wilbrandt

Stokes–Darcy Equations Analytic and Numerical Analysis

Ulrich Wilbrandt Weierstrass Institute for Applied Analysis and Stochastics Berlin, Germany

ISSN 2297-0320 ISSN 2297-0339 (electronic) Advances in Mathematical Fluid Mechanics ISSN 2510-1374 ISSN 2510-1382 (electronic) Lecture Notes in Mathematical Fluid Mechanics ISBN 978-3-030-02903-6 ISBN 978-3-030-02904-3 (eBook) https://doi.org/10.1007/978-3-030-02904-3 Library of Congress Control Number: 2018965415 Mathematics Subject Classification (2010): 46E35, 65J10, 65N12, 76D07, 76M10, 76S05 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Acknowledgements

This research would not have been possible without the support of many people. First and foremost, I would like to express my great appreciation to my advisor Prof. Dr. Volker John for his constructive suggestions and useful critiques. His patience and enthusiasm always serve me as an invaluable guide. Additionally, I would like to thank all the colleagues at the Weierstrass Institute for Applied Analysis and Stochastics, especially the members of the research group “Numerical Mathematics and Scientific Computing,” which I am happy to be a part of. In particular I wish to acknowledge the help provided by Swetlana Giere and Alfonso Caiazzo, who shared an office with me, as well as Naveed Ahmed, Felix Anker, Clemens Bartsch, Laura Blank, Jürgen Fuhrmann, and Timo Streckenbach. I furthermore wish to express my gratitude to my beloved families and friends, for their understanding and endless love, through the duration of my studies. I am particularly grateful for the assistance given by my parents, grandparents, and especially by my wife and my daughter who made this research possible. This work originates from my dissertation at the “Freie Universität Berlin” which has been handed in 2018. I sincerely thank the reviewers Prof. Dr. Volker John and Prof. Dr. Paul Deuring for their time, as well as very helpful comments and corrections.

v

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Main Contributions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 List of Notations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 2 3 4

2 Notation and Preliminary Results . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Preliminaries from Functional Analysis . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Suitable Domains .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Lebesgue Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Lebesgue Spaces on the Boundary . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6 Sobolev Spaces on the Boundary .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

9 9 17 19 21 24 25

3 Properties of Sobolev Spaces.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Mollifications .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Products with Smooth Functions and Lipschitz Transformations .. . . 3.3 Extension from Ω to Rd . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 Density of Smooth Functions .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5 Equivalent Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

27 27 32 37 42 49

4 Traces. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 The Trace Operator .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Continuity of the Trace Operator on W 1,p (Ω) . . .. . . . . . . . . . . . . . . . . . . . 4.3 Characterization of the Kernel of the Trace Operator . . . . . . . . . . . . . . . . 4.4 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 A Right Inverse of the Divergence Operator . . . . . .. . . . . . . . . . . . . . . . . . . . 4.6 Equivalent Norms on H 1 (Ω) . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

57 57 60 68 74 75 78

vii

viii

Contents

5 Subproblems Individually. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Laplace Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Saddle Point Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 Application to the Stokes Problem . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

83 83 93 99

6 Stokes–Darcy Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 The Setting .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Weak Formulation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Weak Formulation Rewritten . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5 Linear Operators on the Interface .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6 Robin–Robin.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.7 The Finite Element Method for the Stokes–Darcy Problem.. . . . . . . . .

109 109 113 116 119 127 133 148

7 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 Classical Iterative Subdomain Methods . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Algorithms for Interface Equations .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3 Convergence Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4 Remarks on the Implementation and Cost . . . . . . . .. . . . . . . . . . . . . . . . . . . .

153 154 157 161 173

8 Numerical Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 General Remarks on Numerical Examples .. . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3 Computations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4 Summary and Conclusions .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

175 175 179 186 199

A Symbolic Computations to Find Numerical Examples . . . . . . . . . . . . . . . . . . 201 A.1 Verifying an Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 201 A.2 Finding an Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 204 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 207 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 211

Chapter 1

Introduction

1.1 Motivation Flows in domains which are partly occupied by a porous medium are of great interest and importance, noticeable examples include groundwater—surface water flow, as well as air and oil filters, blood filtration through vessel walls, and fuel cells. Simulations are often characterized by complex geometries and require the solution of large systems of equations. For this reason, efficient solvers are needed to tackle these types of problems in real-world scenarios. Since the domain of interest is composed of two parts, one allowing a free flow, and one being the porous matrix, two different models are used in the respective subdomains, namely Stokes and Darcy equations together with suitable coupling conditions on the common interface. The individual models are well known and tailored software is available to solve them. The coupled Stokes–Darcy model is somewhat different and it therefore is advisable to find solution strategies which use solutions to the individual models, rather than the coupled one. Inevitably, such strategies are iterative. In this monograph several such approaches are analyzed and their efficiency, especially with respect to the number of iterations, is shown theoretically as well as numerically. It turns out that the straightforward definition of such an iterative algorithm already works very well, however, only for values of viscosity and hydraulic conductivity which are physically unrealistic. For realistic values, it fails. Alternative schemes have been developed but suffer from drawbacks as well. Therefore, it is an open problem to find algorithms which are efficient for a wide range of values of viscosity and hydraulic conductivity. Several authors have studied the Stokes–Darcy coupled system and introduced algorithms which try to solve this problem. The following, while not a complete list, are important works on this subject: [DQ09, DQV07, CGHW11, JM00, Saf71, Ang11, CGHW10, CGH+ 10, GOS11a, LSY02, RY05].

© Springer Nature Switzerland AG 2019 U. Wilbrandt, Stokes–Darcy Equations, Advances in Mathematical Fluid Mechanics, https://doi.org/10.1007/978-3-030-02904-3_1

1

2

1 Introduction

The analysis of the schemes introduced in the above mentioned literature considers an L2 space on the interface. In order to develop a new algorithm however, the spaces and operators which contribute to the coupling have to be defined and studied in greater detail. It is essential to understand the notion of traces together with their kernels, image spaces, and existence of (right) inverses. This theory is particularly involved on domains which are not simple, e.g., with non-smooth interfaces. Therefore, a major part of this monograph is devoted to fully develop this theory with many proofs explicitly given.

1.2 Main Contributions A new kind of Robin–Robin formulation of the coupled Stokes–Darcy problem is analyzed theoretically and numerically. In particular, it is clarified what the smoothness of the interface and the boundary conditions adjacent to it mean for the choice of appropriate spaces in the two subdomains as well as on the interface itself. These include subspaces of H 1/2 which are related to the so-called Lions–Magenes spaces and are introduced as the image spaces of suitable trace operators. Since the standard literature concerning the Stokes–Darcy coupling pays little attention to these theoretical details, in this monograph most statements can be found with complete, rigorous proofs. The uniqueness and existence of solutions of the coupled Stokes–Darcy problem in the usual Neumann–Neumann formulation is given and shown to be equivalent to the newly introduced Robin–Robin formulation. Iterative schemes based on the former are analyzed with special focus on the effect of small viscosity and hydraulic conductivity. These include block-wise Gauss–Seidel, fixed point, as well as Steklov–Poincaré iterations. In this context, a modified Robin–Robin iteration from the literature is introduced, analyzed, and compared with iterations based on the new Robin–Robin formulation. Specifically, in the discrete setting, a closely related method called D-RR is proposed. All these formulations introduce Robin parameters which have to be chosen appropriately, such that the resulting algorithm is efficient. Even for the algorithms already proposed in the literature, different advice is given. The numerical studies in Chap. 8 show that a good choice is possible at least for the D-RR approach, which is therefore superior to previously existing approaches in the case of small viscosity and hydraulic conductivity. Thus, this monograph proposes the first subdomain iteration for the coupled Stokes–Darcy problem which is applicable in the practically relevant case. The numerical studies support the given analysis for each of the introduced algorithms, using two standard examples from the literature. Finally, an example which is closer to geoscientific applications is considered to study the behavior of the new Robin–Robin iterations with respect to small viscosity and hydraulic conductivity, as well as the Robin parameters, which are introduced with this approach. This last example is furthermore used to get an impression of the effects of different Robin parameters in terms of the coupling conditions on the interface.

1.3 Outline

3

1.3 Outline In order to define the involved operators in the Stokes–Darcy coupling, it is fundamental to exactly characterize traces in the context of Sobolev spaces. Starting with a chapter on notation and preliminary results, Chap. 2, Sobolev spaces and some of their important properties are studied in Chap. 3. The main result in this chapter is Theorem 3.4.5 showing that functions, which are smooth up to the boundary of a Lipschitz domain, are dense in certain Sobolev spaces. This is used in Chap. 4 to study trace operators on various domains, including their kernels and image spaces. The somewhat difficult Lions–Magenes spaces are introduced and characterized together with a careful analysis of surjectivity as well as existence of linear and continuous right inverses of traces. As a first application of these results, the two subproblems, Stokes and Darcy, are studied individually in Chap. 5. This includes existence and uniqueness theorems, as well as the definition of operators which are similar in nature to the ones needed in Chap. 6; where the coupling of Stokes and Darcy equations is studied in detail. Different formulations are derived and shown to be equivalent. The last, rather theoretical part, is Chap. 7 which proposes and analyzes several algorithms suitable to solve the coupled Stokes– Darcy problem. Finally, in Chap. 8 numerical examples are introduced and the efficiency of the studied algorithms is tested. A rough overview of the dependencies among the chapters is given in Fig. 1.1. Readers who are well experienced with the theory of Stokes and Laplace problems, might directly start reading in Chap. 6. All chapters except the last also provide Fig. 1.1 A graph showing the dependencies among all chapters in this monograph. Note that implicit dependencies are not always shown

2 Notation and preliminary results

3 Properties of Sobolev spaces

4 Traces

5 Subproblems individually

6 Stokes–Darcy equations

7 Algorithms

8 Numerical results

4

1 Introduction

a graph to help understand the dependencies among the involved definitions, theorems, algorithms and so on. Finally, a list of notations used throughout the monograph follows.

1.4 List of Notations Symbol a(·, ·) af (·, ·) ap (·, ·) afR (·, ·) apR (·, ·) A

α

B(x0 , α)

B

b(·, ·) bf (·, ·) c C C1 , C2 C ∞ (Ω) D (Ω) D

Meaning Bilinear form used in the weak form of the Laplace and Stokes equation and in the formulation of the abstract saddle point problem Stokes bilinear form used in the weak form of the Stokes–Darcy coupled problem Darcy bilinear form used in the weak form of the Stokes–Darcy coupled problem Stokes bilinear form used in the Robin–Robin weak form of the Stokes–Darcy coupled problem Darcy bilinear form used in the Robin–Robin weak form of the Stokes–Darcy coupled problem Bilinear form used in the proof of existence and uniqueness of the Stokes–Darcy coupled system Coercivity/positivity constant, the respective operator/bilinear form is in the index, for example αa , also used as a parameter without index in the Beavers–Joseph–Saffman condition (6.3c) Open ball around a point x0 ∈ X with radius α in a Banach space X, i.e., the set B(x0 , α) = {x ∈ X | x − x0 X < α} Bilinear form used in the proof of existence and uniqueness of the Stokes–Darcy coupled system Bilinear form used in the Stokes weak formulation and in the abstract saddle point theory Stokes bilinear form coupling the Stokes pressure and velocity space in the Stokes–Darcy coupling A continuity constant, typically with the operator or bilinear form as an index Generic constant, appears in many proofs Matrices coupling Stokes and Darcy subproblems Set of infinitely differentiable functions on the set Ω Space of test functions on Ω, i.e., smooth functions with compact support in Ω, here Ω may be Rd Deformation tensor of a vector field, symmetric part of its gradient

Chapter/section 5.1.1, 5.2.1, 5.3.1

6.2 6.2 6.6.1 6.6.1 6.3 5.1.2, 5.3.2, 6.2

2.1, 3.1, 3.4.2, 4.3.1

6.3 5.3.1, 5.2.1 6.2, 6.6.1

6.7, 7 2.2, 2.4, 3, 4 2.4 3.5.1, 5.3, 6.1 (continued)

1.4 List of Notations Symbol D E Ef , Ep F

F f

G

ΓI Γ

γ γf γp H k (Ω) H s (Ω) Hk (Ω) H1Γ (Ω) Hf,N , Hp,N

Hf,D , Hp,D

γ ,γp

Hf f

γ ,γf

, Hp p

Meaning Darcy matrix Extension operator, various occurrences, often with an index Extension operators mapping variables on the interface ΓI to the Stokes/Darcy subdomain Linear form used in the proof of existence and uniqueness of the Stokes–Darcy coupled system A Lipschitz transformation, also used as right-hand side in Theorem 2.1.13 Index indicating a quantity, set, or operator with respect to the Stokes (free flow) subdomain in the coupled Stokes–Darcy setting Linear form used in the proof of existence and uniqueness of the Stokes–Darcy coupled system The interface separating the Stokes and Darcy subdomains, Ωf and Ωp A part of the boundary ∂Ω, often with an index N , D , or R denoting Neumann, Dirichlet, or Robin boundary Parameter for Robin boundary conditions, in general a L∞ -function Robin parameter for the Stokes subproblem Robin parameter for the Darcy subproblem Sobolev space W k,2 (Ω), this is a Hilbert space Sobolev space W s,2 (Ω), this is a Hilbert space Vector valued Sobolev space with each component in H k (Ω), this is a Hilbert space Subspace of H 1 (Ω) with vanishing trace on Γ ⊂ ∂Ω, also denoted WΓ1,2 (Ω) Operators taking Neumann data on the interface returning the Dirichlet data (trace) of the Stokes/Darcy solution, respectively; Hf,N = Tf ◦ Kf,N , Hp,N = Tp ◦ Kp,N Operators taking Dirichlet data on the interface ΓI returning the Neumann data of the Stokes/Darcy solution, respectively; Hf,D = TfN ◦ Kf,D , Hp,D = TpN ◦ Kp,D Operators taking Robin data on the interface returning other Robin data of the Stokes/Darcy γ ,γ R ◦ K γf , solution, respectively; Hf f p = Tf,γ f p γp ,γ

Im ker

5 Chapter/section 6.7, 7 3.3, 4, 7 6.7, 7 6.3 3.2 6

6.3 6.1 6.1

5.1.1, 5.3.1 6.6 6.6 2.4 2.4.1 2.4 4.3 6.4.3

6.4.3

6.6.5

γp

R ◦K Hp f = Tp,γ p f Image/range of an operator Kernel of an operator, i.e., the preimage of {0} under that operator

2.1, 5.3.2, 6.4.2, 6.6.4 2.1, 4.3, 5.3.2 (continued)

6

1 Introduction

Symbol K KD

KR

Kf,N , Kp,N

Kf,D , Kp,D γ

γ

Kf f , Kp p Lp (Ω) p L0 (Ω) L1loc (Ω) Lf,N , Lp,N

Lf,D , Lp,D

Λf Λp f Λ p Λ  n ν Ω Ωf Ωp pf p

Meaning Hydraulic conductivity tensor, assumed to be scalar in Algorithms 7 and 8 Solution operator taking Dirichlet data on a part of the boundary, returning a Stokes/Darcy solution, respectively Solution operator taking Robin data on a part of the boundary, returning a Stokes/Darcy solution, respectively Solution operators taking Neumann data on the interface ΓI and returning a Stokes/Darcy solution, respectively Solution operators taking Dirichlet data on the interface ΓI and returning a Stokes/Darcy solution, respectively Solution operators taking Robin data on the interface ΓI and returning a Stokes/Darcy solution, respectively Lebesgue space, 1 ≤ p ≤ ∞ Lebesgue space with zero mean value, 1 ≤ p ≤ ∞ Lebesgue space of locally integrable functions Linear operators taking Neumann data on the interface returning the Dirichlet data (trace) of the Stokes/Darcy solution, respectively Linear operators taking Dirichlet data on the interface returning the Neumann data of the Stokes/Darcy solution, respectively Trace space on the interface ΓI , Λf = Tf (V f , Qf ) Trace space on the interface ΓI , Λp = Tp (Qp ) Trace space interface ΓI ,  on the   f × L2 (Ωf ) f = Tf V Λ   p p = Tp Q Trace space on the interface ΓI , Λ Right-hand side in a weak formulation, sometimes with an index The normal vector pointing out of Ω on ∂Ω and out of the Stokes subdomain on the interface ΓI (Kinematic) viscosity Open and bounded subset of Rd with Lipschitz continuous boundary ∂Ω The Stokes subdomain The Darcy subdomain Stokes pressure solution component in the coupled Stokes–Darcy setting Index indicating a quantity, set, or operator with respect to the Darcy (porous media) subdomain in the coupled Stokes–Darcy setting

Chapter/section 5.1, 6.1 5.1.3, 5.3.3

5.1.3, 5.3.3

6.4.2

6.4.2

6.6.4

2.3 2.3 2.3 6.5

6.5

6.4.1 6.4.1 6.4.1 6.4.1 5.1.1, 5.3.1, 6.2 6.1 5.3, 6.1 2.2, 5, 6.1 6.1 6.1 6 6

(continued)

1.4 List of Notations Symbol ϕp Qf Qp p Q Rf , Rp Rd+ supp

S T T Tf Tp TfN , TpN TΓR R , TR Tf,γ p,γ τi uf Vf f V W k,p (Ω) W s,p (Ω) 1,p WΓ (Ω) s,p

W00 (·, ·)X (·, ·)X ∗ ×X ·X ∂Ω ∇

7

Meaning Darcy pressure (hydraulic head) solution component in the coupled Stokes–Darcy setting Stokes pressure test space in the coupled Stokes–Darcy setting, Qf = L2 (Ωf ) Darcy pressure test space in the coupled Stokes–Darcy setting, Qp = HΓ1p,D (Ωp )

Chapter/section 6

Darcy pressure solution space in the coupled Stokes–Darcy setting, this is an affine linear space Restriction operators which map the solutions of the Stokes/Darcy subproblems to the interface ΓI Half space The support of a function, i.e., the closure of all points which are not mapped to zero under that function Stokes matrix, saddle point structure Cauchy stress tensor Trace operator, often with indices Normal trace operator onto the interface ΓI in the Stokes–Darcy coupling, Definition 6.2.3 Trace operator onto the interface ΓI in the Stokes–Darcy coupling, Definition 6.2.3 Neumann data of a Stokes/Darcy solution on the interface ΓI Robin data of a solution on Γ ⊂ ∂Ω Robin data of a Stokes/Darcy solution on ΓI

6.4.1

Tangential along the interface ΓI Stokes velocity solution component in the coupled Stokes–Darcy setting Stokes velocity test space in the coupled Stokes–Darcy setting, V f = H1Γf,D (Ωf )

6.1 6

Stokes velocity solution space in the coupled Stokes–Darcy setting, this is an affine linear space Sobolev space, k ∈ N ∪ {0}, 1 ≤ p ≤ ∞ Sobolev space, 0 < s < 1, 1 ≤ p ≤ ∞ Sobolev space with vanishing trace on Γ ⊂ ∂Ω, 1≤p≤∞ Lions–Magenes space, 0 < s < 1, 1 ≤ p ≤ ∞ Inner product in the Hilbert space X Dual product for a Banach space X and its dual X ∗ Norm in a Banach space X The boundary of the domain Ω Nabla operator, first (weak) derivative

6.4.1

6.2 6.2

6.7, 7 4.2 2.4, 3.1

6.7, 7 5.3, 6.1 4, 6.1 6.2 6.2 6.4.2 5.1.3, 5.3.3 6.6.4

6.2

2.4 2.4.1 4.3 4.3.1 2.1 2.1 2.1 2.2 2.2, 2.4

Chapter 2

Notation and Preliminary Results

In this chapter most of the notation used in this monograph is introduced; in particular, Lipschitz domains on which the so-called Lebesgue and Sobolev spaces are defined, together with a few basic inequalities. Furthermore, the important theorem of Lax–Milgram is shown. To begin with, some definitions and results from functional analysis are stated. All of the covered topics in this section can be found in many textbooks, including [AF03, Eva10, Gri85, Mac09, QV99, Tar07, Tri92].

2.1 Preliminaries from Functional Analysis Let X be a Hilbertspace with inner product (·, ·)X : X × X → R and corresponding norm xX := (x, x)X . The inner product is linear in each component and a continuous map, which follows from the well known Cauchy–Schwarz inequality: for all x, y ∈ X it is

(x, y)X ≤ xX yX .

(2.1)

Let Y be another Hilbert space with inner product (·, ·)Y : Y × Y → R and let the operator T : X → Y be linear and continuous. Note that such a linear operator T : X → Y is continuous if and only if there exists a constant cT ≥ 0 such that for all x ∈ X it is T xY ≤ cT xX . In the study of partial differential equations one often considers so-called bilinear forms on Hilbert spaces. These are maps a : X × Y → R which are bilinear, i.e., linear in each component. Similarly to linear operators, continuity of a is equivalent to the existence of a constant ca ≥ 0 such that for all elements x ∈ X, y ∈ Y it is |a(x, y)| ≤ ca xX yY . Furthermore, if the spaces X and Y coincide the bilinear

© Springer Nature Switzerland AG 2019 U. Wilbrandt, Stokes–Darcy Equations, Advances in Mathematical Fluid Mechanics, https://doi.org/10.1007/978-3-030-02904-3_2

9

10

2 Notation and Preliminary Results weak derivative Definition 2.4.1

Open mapping Theorem 2.1.10

Closed range Theorem 2.1.9

Sobolev Space Definition 2.4.2

Corollary 2.1.11

Localization Definition 2.2.1

Corollary 2.1.12

Hölder inequality Theorem 2.3.1

Fréchet–Riesz Theorem 2.1.1

Hahn–Banach Theorem 2.1.2

Corollary 2.1.5

Corollary 2.1.3

Corollary 2.1.6

Corollary 2.1.4

Lax–Milgram Theorem 2.1.13

Theorem 2.1.7

Proposition 2.1.8

Fig. 2.1 A graph showing the dependencies among all theorems, definitions, corollaries, as well as propositions in Chap. 2. Note that implicit dependencies are not always shown

form a is called coercive1 if there exists a constant αa > 0 such that for all x ∈ X it is a(x, x) ≥ αa x2X . The inner product is an example for such a coercive and continuous bilinear form. Furthermore, in Hilbert spaces the parallelogram identity holds: For all x, y ∈ X it is   x + y2X + x − y2X = 2 x2X + y2X . (2.2) If on the other hand in a Banach space X with norm ·X the parallelogram law  holds, then its norm is derived from an inner product, i.e., xX = (x, x)X for all x ∈ X with (x, y)X :=

 1 x + y2X − x − y2X . 4

(2.3)

The set X∗ of all linear and continuous functions mapping from X to R is called the dual of X. The action of an element x ∗ ∈ X∗ on an element x ∈ X is written as x ∗ (x) =: (x ∗ , x)X∗ ×X and often called the duality pairing between x ∗ and x. A norm on X∗ is defined as x ∗ X∗ = supxX ≤1 |x ∗ (x)|, with which X∗ is a Banach space. For sets M ⊂ X and N ⊂ X∗ , define the orthogonal complement M ⊥ ⊂ X

1 Sometimes also called elliptic; if the space is not clear it is often explicitly mentioned as Xcoercivity or X-ellipticity.

2.1 Preliminaries from Functional Analysis

11

and the annihilators M ◦ ⊂ X∗ and N◦ ⊂ X by M ⊥ := {x ∈ X | for all xm ∈ M: (x, xm )X = 0}, 

  M ◦ := x ∗ ∈ X∗ for all xm ∈ M: x ∗ , xm X∗ ×X = 0 , 

  N◦ := x ∈ X for all xn∗ ∈ N: xn∗ , x X∗ ×X = 0 . Note that M ⊥ , M ◦ , and N◦ are closed subspaces of X and X∗ , respectively. If M is viewed as a subset of X∗∗ ⊃ X it is M ◦ = M◦ . If additionally M is a subspace of X, define the quotient space X/M with the norm ·X/M = inf  · +xm X . xm ∈M

In the case of a single element set N = {x ∗ }, the annihilator N◦ is the kernel of x ∗ . The definition of the dual X∗ is used for Banach spaces X; however, Hilbert spaces admit an especially simple representation: Theorem 2.1.1 (Fréchet–Riesz) Let X be a Hilbert space. Then the map Z : X → X∗ , x → (·, x)X is linear, continuous, bijective, and isometric. Proof By definition of an inner product Z is linear and the Cauchy–Schwarz inequality (2.1) implies ZxX∗ ≤ xX , i.e., continuity. For x ∈ X (x = 0) it is  x ZxX∗ = sup (Zx, y)X∗ ×X = sup (x, y)X ≥ x, = xX xX X yX ≤1 yX ≤1 and hence Z is isometric and injective. Next, let x ∗ ∈ X∗ \ {0} be given. Denote the kernel of x ∗ by U which is a closed subspace of X. Let u⊥ be a nonzero element of the orthogonal complement U ⊥ within X, hence (x ∗ , u⊥ )X∗ ×X = 0. Next, for any (x ∗ ,x ) ∗ x in X define xU := x − (x ∗ ,u⊥ )X ∗×X u⊥ ∈ U . It holds X ×X

(x ∗ , x)X∗ ×X u⊥ 2X (x ∗ , u⊥ )X∗ ×X 

(x ∗ , u⊥ )X∗ ×X = x, u⊥ . u⊥ 2X X

0 = (xU , u⊥ )X = (x, u⊥ )X − ⇒



x ∗, x

 X ∗ ×X

Defining y to be the second argument in the last inner product in the previous equation shows x ∗ = (·, y)X = Zy, hence surjectivity of Z because x ∗ is arbitrary.  

12

2 Notation and Preliminary Results

Assigned to an operator T : X → Y is its dual2 operator T ∗ : Y ∗ → X∗ which is defined for all y ∈ Y and all x ∈ X as 

T ∗ y, x

 X ∗ ×X

:= (y, T x)Y ∗ ×Y .

In the case of Hilbert spaces the theorem of Fréchet–Riesz 2.1.1 admits the simplified definition of a Hilbert space dual operator T  : Y → X where in the above definition the duality pairings are replaced by inner products. The relation of the two types of dual operators in the Hilbert space setting is depicted in the following diagram: X

T

ZX X∗

Y ZY

T∗

−1 ◦ T ∗ ◦ ZY T = ZX

Y∗

The bidual T ∗∗ is then identified with T . In the following, a few results are stated which are used to prove existence and uniqueness of solutions of saddle point problems in Sect. 5.2. Theorem 2.1.2 (Hahn–Banach) Let X be a normed linear space, Y  X a nontrivial subspace, and y ∗ ∈ Y ∗ linear and continuous. Then there exists a normpreserving extension x ∗ ∈ X∗ , i.e., it holds x ∗ X∗ = y ∗ Y ∗ and x ∗ Y = y ∗ . A proof of this fundamental theorem can be found for example in [Mac09, Chapter 3.1] where also the following conclusions can be found. Corollary 2.1.3 Let X be a nonempty, normed, linear space and x ∈ X. Then there exists a functional x ∗ ∈ X∗ with x ∗ X∗ = 1 and (x ∗ , x)X∗ ×X = xX . Proof Define the subspace M := {αx | α ∈ R} of X and the functional x ∗ on M as (x ∗ , αx)M ∗ ×M = αxX . Then x ∗ has norm one and the desired value at x. Using the theorem of Hahn–Banach, Theorem 2.1.2, the functional x ∗ can be extended to all of X without changing its norm.  

2 Also

called adjoint.

2.1 Preliminaries from Functional Analysis

13

Another consequence of the theorem of Hahn–Banach is the following corollary: Corollary 2.1.4 Let X be a normed linear space, Y a closed subspace, and x0 ∈ X \ Y . Then there exists a separating functional x ∗ ∈ X∗ such that  ∗  and x , x0 X∗ ×X = 1. x ∗ Y = 0 Proof This proof is taken from the German book [Wer00, Korollar III.1.8]. Let ω : X → X/Y =: Q be the quotient map. Then it is ω(x0 ) = 0 and ω(y) = 0 for all y ∈ Y . According to Corollary 2.1.3 choose x˜ ∗ ∈ Q∗ such that x˜ ∗ Q∗ = 1 and (x˜ ∗ , ω(x0 ))Q∗ ×Q = ω(x0 )Q . Finally define x ∗ ∈ X∗ for all x ∈ X as 

x ∗, x

 X ∗ ×X

:=

which has the desired properties.

  ∗ 1 x˜ , ω(x) Q∗ ×Q , ω(x0 )Q  

In case the subspace is not closed, there is a very similar result which is proved for example in [Suh03], ¸ Corollary 6.8.4: Corollary 2.1.5 Let X be a normed linear space and Y a subspace of X. Further let x0 ∈ X be given such that dist(x0 , Y ) = δ > 0. Then there exists a functional x ∗ ∈ X∗ such that x ∗ Y = 0, (x ∗ , x0 )X∗ ×X = 1, and x ∗ X∗ = 1/δ. Proof Let Z be the subspace generated by the set Y ∪ {x0}, i.e., z ∈ Z if and only if there are y ∈ Y and λ ∈ R such that z = y + λx0 . Then, on Z, define x ∗ to be   ∗ x , y + λx0 Z ∗ ×Z = λ   such that x ∗ Y = 0. For all y ∈ Y and λ ∈ R it is y + λx0 X = |λ| yλ + x0 X ≥ δ|λ| and hence (x ∗ , y + λx0 )Z ∗ ×Z ≤ y + λx0 X /δ. This shows the bound x ∗ Z ∗ ≤ 1/δ. On the other hand for all y ∈ Y it is 1 = (x ∗ , x0 − y)Z ∗ ×Z ≤ x ∗ Z ∗ x0 − yX and this inequality therefore also holds for the infimum over Y , i.e., x ∗ Z ∗ ≥ 1/δ. Together with the previous bound, the equality x ∗ Z ∗ = 1/δ is shown. According to the theorem of Hahn–Banach, Theorem 2.1.2, there exists a norm-preserving extension of x ∗ which has the desired properties.   The previous Corollary 2.1.5 admits a characterization of a dense subspace as follows: Corollary 2.1.6 Let X be a normed linear space and Y a subspace of X. If any functional x ∗ ∈ X∗ which vanishes on Y also vanishes on all of X, then Y is dense in X. Proof Suppose Y is not dense and let x0 ∈ X be given such that dist(x0 , Y ) = δ > 0. According to Corollary 2.1.5 there exists a functional x ∗ ∈ X∗ which vanishes on Y and has a nonzero value at x0 . This contradicts the assumption that any such functional vanishes on all of X.  

14

2 Notation and Preliminary Results

In the following, a number of results are stated which are needed in later chapters. Theorem 2.1.7 Let X and Y be Banach spaces and T : X → Y be linear and continuous. Then it holds   ker T ∗ ◦ = Im T . (2.4) If X and Y are Hilbert spaces then additionally (ker T )◦ = Im T ∗

(2.5)

is true. Proof Let y = T x ∈ Im T be given. For any y ∗ ∈ ker T ∗ it holds  ∗      y , y Y ∗ ×Y = y ∗ , T x Y ∗ ×Y = T ∗ y ∗ , x X∗ ×X = 0. Therefore, y ∈ (ker T ∗ )◦ and Im T ⊂ (ker T ∗ )◦ . Since (ker T ∗ )◦ is closed the inclusion (ker T ∗ )◦ ⊃ Im T is verified. Given y ∈ / Im T it must be shown that y∈ / (ker T ∗ )◦ . Again since Im T is a closed subspace, according to Corollary 2.1.4 of the Theorem of Hahn–Banach 2.1.2, there exists a y ∗ ∈ Y ∗ such that (y ∗ , ·)Y ∗ ×Y is zero on Im T and nonzero at y. The former reads (y ∗ , T x)Y ∗ ×Y = 0 for all x ∈ X, that means y ∗ ∈ ker T ∗ . Due to (y ∗ , y)Y ∗ ×Y = 0 it holds y ∈ / (ker T ∗ )◦ . ∗∗ In case of a Hilbert space the bidual T can be identified with T and Eq. (2.4) with T ∗ instead of T yields (2.5).   Proposition 2.1.8 Let X and Y be Hilbert spaces and let T : X → Y be linear and continuous. Then it holds: a) b) c) d)

Is T surjective then T ∗ is injective. Is T ∗ surjective then T is injective. Is T ∗ injective then Im T is dense in Y . Is T injective then Im T ∗ is dense in Y ∗ .

Proof a) Assume T is surjective and let y ∗ ∈ ker T ∗ ⊂ Y ∗ be given. Then it holds     0 = T ∗ y ∗ ⇐⇒ 0 = T ∗ y ∗ , x X∗ ×X = y ∗ , T x Y ∗ ×Y ∀x ∈ X   ⇐⇒ 0 = y ∗ , y Y ∗ ×Y ∀y ∈ Y.

T surj.

That means y ∗ = 0 which implies that the kernel of T ∗ is trivial, ker T ∗ = {0}, i.e., T ∗ is injective. b) Note that T ∗∗ = T and apply a) to T ∗ . c) Let T ∗ be injective, i.e., ker T ∗ = {0}. Then it is (ker T ∗ )◦ = Y and Eq. (2.4) completes the proof. d) Note that T ∗∗ = T and apply c) to T ∗ .  

2.1 Preliminaries from Functional Analysis

15

The following well known theorems can be found e.g. in [Wer00], Theorem IV.51.1, or in [Bre11], Theorem 2.19. Theorem 2.1.9 (Closed Range Theorem) Let X and Y be Banach spaces and let T : X → Y be linear and continuous. Then the following four statements are equivalent: (i) (ii) (iii) (iv)

Im T is closed, Im T = (ker T ∗ )◦ , Im T ∗ is closed, Im T ∗ = (ker T )◦ .

Theorem 2.1.10 (Open Mapping Theorem) Let X and Y be Banach spaces and let T : X → Y be linear, continuous, and surjective. Then T is open (i.e., images of open sets are open under T ). Proof See also in [DDE12], Theorem 1.12. First note that for T to be open it suffices to show that3 there exists a constant ε > 0 such that B(0, ε) ⊂ T (B(0, 1)).

(2.6)

In fact, let O ⊂ X be an open set and x ∈ O, y = T x ∈ T (O). Furthermore, let r be small enough so that B(x, r) ⊂ O. Then it also is T (B(x, r)) ⊂ T (O) and, using the linearity of T , 1r T (B(0, 1)) + {y} ⊂ T (O). Now choose ε as in (2.6) and 1 1 r B(0, ε) = B(0, ε/r) ⊂ r T (B(0, 1)). This means B(y, ε/r) ⊂ T (O) is an open ball around y within the set T (O). Hence, T (O) is open and so is T . Next, it is shown that there exists ε > 0 such that B(0, ε) ⊂ T (B(0, 1)).

(2.7)

Writing X as the union of balls with increasing radius the surjectivity of T gives   Y = T (X) = T (B(0, n)) = T (B(0, n)). n∈N

n∈N

Because Y as a Banach space is also a Baire space, there exists n0 ∈ N such that T (B(0, n0 )) has nonempty interior. There therefore is a y0 ∈ T (B(0, n0 )) and an ε0 > 0 such that B(y0 , ε0 ) ⊂ T (B(0, n0 )) and, equivalently, B(y0 , ε0 /n0 ) ⊂ T (B(0, 1)). Together with y0 also −y0 fulfills the latter inclusion because T (B(0, 1)) is symmetric and so is its closure. With this it is B(0, ε0 /n0 ) ⊂ T (B(0, 1)) + {−y0} ⊂ T (B(0, 1)) + T (B(0, 1)) ⊂ T (B(0, 2)). Choosing ε = ε0 /2n0 results in (2.7). and in what follows B(x0 , α) ⊂ X denotes the ball around x0 with radius α, i.e., B(x0 , α) = {x ∈ X | x − x0 X < α}. Furthermore, for sets M, N ⊂ X and a real number a the product aM and the sum M + N have to be understood element-wise: aM = {ax | x ∈ M}, M + N = {m + n | m ∈ M, n ∈ N}.

3 Here

16

2 Notation and Preliminary Results

In the last part of the proof let ε be as in (2.7) and the goal is to show that with this ε it is B(0, ε) ⊂ T (B(0, 1)). Let y ∈ B(0, ε) and yY < ε0 < ε. Then define y˜ := εε0 y ∈ B(0, ε). Since y˜ ∈ T (B(0, 1)) and according to statement (2.7), there exists y0 = T x0 ∈ T (B(0, 1)) with y˜ − y0 Y < αε, where 0 < α < 1 is chosen 1 to be smaller than 1 − εε0 , i.e., εε0 1−α < 1. Because α1 (y˜ − y0 ) ∈ B(0, ε), again    y−y  ˜ due to (2.7), there exists y1 = T x1 ∈ T (B(1, 0)) with  α 0 − y1  < αε, or Y

equivalently y˜ − (y0 + αy1 )Y < α 2 ε. This in turn means α12 (y˜ − y0 − αy1 ) ∈ B(0, ε) and inductively a sequence yi = T xi with xi ∈ B(0, 1) can be constructed with 

n       i α xi  < α n+1 ε. y˜ − T   i=0

Y

n

i 4 to, The sequence ∞ ( i i=0 α xi )n∈N is a Cauchy sequence and therefore converges say, x˜ = i=0 α xi . Due to the construction it is T x˜ = y. ˜ Finally set x := εε0 x˜ so that T x = y and

xX =

∞ ∞ ε0 ε0  i ε0  i ε0 1 x ˜ X≤ < 1, α xi X ≤ α = ε ε ε ε 1−α i=0

i=0

i.e., x ∈ B(0, 1) and y ∈ T (B(0, 1)). Thus statement (2.6) is shown.

 

Corollary 2.1.11 Let X and Y be Banach spaces and let T : X → Y be linear, continuous, and bijective. Then the inverse operator T −1 is continuous. Proof First note that the inverse operator is in fact linear. The inverse T −1 is continuous if and only if preimages of open sets are open under T −1 (this is a topological definition of continuous functions), i.e., if for any arbitrary open set −1  O ⊂ X the preimage T −1 (O) ⊂ Y is open. According to the open mapping −1  theorem, Theorem 2.1.10, the operator T is open. Therefore, T −1 (O) = T (O)   is an open set, and hence T −1 continuous. Corollary 2.1.12 Let X and Y be Banach spaces and let T : X → Y be linear, continuous, and injective. Then the image of T is closed if and only if T is bounded away from zero, i.e. there exists a constant c > 0 such that infx T xY /xX ≥ c.5 Proof Assume Im T is closed. Then Im T is a Banach space and T : X → Im T is bijective. According to Corollary  2.1.11  of the  open  mapping theorem the inverse T −1 is continuous. This implies T −1 y X ≤ T −1  yY for all y ∈ Im T . Writing  m  α n+1 −α m+1 n,m→∞ i  i that  m −−−−−→ 0. i=n α xi ≤ i=n α = 1−α 5 In this work it is always assumed that infima or suprema are taken with respect to a set for which the following expression is defined. In this case, this means x ∈ X and x = 0. 4 Note

2.2 Suitable Domains

17

  y = T x, this reads xX ≤ cT xY for all x ∈ X, with c = T −1  > 0. This proves the boundedness of T away from zero. Now assume there exists c > 0 such that T xY ≥ cxX for all x ∈ X. Let yn = T xn ∈ Im T be a converging sequence in the image of T , say yn → y ∈ Y . Then the assumption implies yn − ym Y = T (xn − xm )Y ≥ cxn − xm X . Therefore, xn is as well a Cauchy sequence in the complete space X. Denote its limit by x. Then continuity of T proves y ∈ Im T : T x = T (lim xn ) = lim T xn = lim yn = y.   Finally, the important theorem of Lax and Milgram is stated. It is very helpful in the analysis of elliptic problems. Theorem 2.1.13 (Lax–Milgram) Let X be a Hilbert-space, a(·, ·) : X × X → R a continuous and coercive bilinear form, and F : X → R linear and continuous. Then there exists a unique u ∈ X such that for all v ∈ X it holds a(u, v) = F (v). Proof For a fixed w ∈ X the map v → a(w, v) is an element of the dual X∗ . According to the theorem of Fréchet–Riesz 2.1.1 there is a unique T w ∈ X such that (T w, v)X = a(w, v) holds for all v ∈ X. The map w → T w is linear and continuous because a is in its first component. Furthermore, it is injective: T w = 0 ⇒ a(w, w) = 0 ⇒ w = 0, because a is assumed to be coercive. Due to Corollary 2.1.12 its image is closed. If T was not surjective, let y ∈ (Im T )⊥ , y = 0. Then for all x ∈ X it is 0 = (T x, y) = a(x, y) and for x = y the coercivity of a implies the contradiction y = 0. Consequently, T is bijective as well and, according to Corollary 2.1.11, the inverse T −1 is continuous. Then define u = T −1 (Z −1 F ) with the operator Z from the theorem of Fréchet–Riesz 2.1.1. The continuity constant of T −1 is the coercivity constant αa of a which gives a bound on u: uX ≤

1 F X∗ . αa

Similarly it is F X∗ ≤ ca uX , where ca is the continuity constant of a (and T ).  

2.2 Suitable Domains In order to appropriately introduce Stokes and Darcy as well as other partial differential equations a solution space is needed. It turns out that special subspaces of the Lebesgue spaces, the so-called Sobolev spaces, together with the concept of weak derivatives are suitable. These spaces are defined on a domain Ω ⊂ Rd . In the entire monograph the set Ω is open, bounded, and connected. The space dimension

18

2 Notation and Preliminary Results

d is either 2 or 3, however many proofs in the following chapters hold for arbitrary d. The boundary of Ω is denoted by ∂Ω and assumed to be Lipschitz continuous. In fact, in most situations it is enough if a weaker condition is fulfilled, namely the cone condition. For simplicity here only Lipschitz domains are considered. A domain Ω is said to be Lipschitz if, for each x 0 ∈ ∂Ω, the coordinate system can be rotated by a map R : Rd → Rd in such a way that within a neighborhood U of x 0 the boundary is described by a Lipschitz continuous function h : V → R, V ⊂ Rd−1 . Elements in Rd−1 are denoted with an apostrophe, e.g. y  ∈ V , and elements in Rd as y = (y  , yd ). More specifically R shall map the graph of h onto the boundary of Ω within U , i.e., R(V × h(V )) = U ∩ ∂Ω. For any point x ∈ U there exists a y  ∈ V and a yd ∈ R, |yd | < r, such that x = R(y  , yd + h(y  )). In other words it is U = R(W )

with



 W = y = (y  , yd + h(y  )) ∈ Rd y  ∈ V , |yd | < r .

Here the boundary shall be characterized by yd = 0, the interior U ∩Ω by yn > 0 and the exterior U \ Ω by yn < 0. In Fig. 2.2 a typical such situation is sketched. Since the boundary is compact, finitely many such neighborhoods Ω1 , . . . , ΩN cover a neighborhood U∂Ω of ∂Ω with a uniform parameter r > 0. Associated to each Ωi is a rotation Ri so that Ωi = Ri (Wi ). In order to cover all of Ω, additionally

0

R R −1 (Ω)

Ω U

2r

h R −1 (U ) V

Rd−1

Fig. 2.2 Sketch of some relevant objects in a localization in two space dimensions. The domain Ω is rotated by R such that a local part of the boundary can be represented as a graph of a function h

2.3 Lebesgue Spaces

19

define the open set Ω0 := {x ∈ Ω | dist(x, ∂Ω) > r/2}. Then indeed it is Ω⊂

N 

Ωi .

i=0

Adapted to the Ωi , i = 0, . . . , N, define a partition of unity φ0 , . . . , φN ∈ C ∞ (U∂Ω ∪ Ω) with N 

φi ∈ C0∞ (Ωi ),

φi (x) = 1 for all x ∈ U∂Ω .

i=0

The rotations Ri are usually not explicitly mentioned because they are linear and the notation becomes less verbose, in particular ∇Ri is constant, has determinant one, and preserves the norm in Rd , i.e., |∇Ri x| = |x|. Hence, a change of variables using Ri does not alter the value of integrals. Definition 2.2.1 (Localization) The triples (Vi , hi , φi ), i = 0, . . . , N, constructed above are referred to as a (Lipschitz) localization of (the boundary of) Ω. Such a localization is of importance to define Lebesgue spaces on the boundary. It is possible to allow only special sets for the Vi , for example the unit ball around zero in Rd−1 . This can facilitate the analysis but is not strictly necessary.

2.3 Lebesgue Spaces The majority of the spaces appearing in this monograph are based on the Lebesgue spaces which in turn depend on the Lebesgue measure on Rd . Introductions to measure theory and integration can be found in, e.g., [EG92, Fol99], and in many other text books. Important results include Lebesgue’s dominated convergence theorem, the theorem of Fubini, and the fact that measurable functions can be approximated by continuous ones. The Lebesgue spaces are defined as   Lp (Ω) := f : Ω → R f is Lebesgue-measurable and f Lp (Ω) < ∞ , with 1 ≤ p ≤ ∞ and the norm  f Lp (Ω) =

Ω |f (x)|

p

dx

1/p

ess supx∈Ω |f (x)|

if p < ∞, if p = ∞.

A short notation for the integral is sometimes used if the dependence of the integrand   on the variable is clear: Ω f = Ω f (x) dx. Strictly speaking Lp (Ω) consists of equivalence classes of functions, rather than functions by themselves. However

20

2 Notation and Preliminary Results

in this monograph this distinction is not made, that means any statement about a function in a Lebesgue space holds for all members of its equivalence class. Many text books introduce the Lebesgue spaces and show important properties which are not proved here, see, e.g., [Jos05, Wil13, AF03]. For example it can be shown that the Lebesgue spaces with the given norm are complete, i.e., they are Banach spaces. Furthermore, the continuous functions C(Ω) are dense in Lp . Since the considered domain is bounded, the Lebesgue spaces are nested: For p > q it is Lp (Ω) ⊂ Lq (Ω), which follows from the following frequently used theorem.6 Theorem 2.3.1 (Hölder Inequality) Let 1 ≤ p, q ≤ ∞ such that

1 p

+ q1 = 1 with

1 = 0. Further let f ∈ Lp (Ω) and g ∈ Lq (Ω) be given. Then the the convention ∞ product fg is in L1 (Ω) and it is

fgL1 (Ω) ≤ f Lp (Ω) gLq (Ω) . Proof In case p or q is ∞ the Hölder inequality follows directly  fgL1 (Ω) =

 Ω

|f (x)g(x)| dx ≤ f L∞ (Ω)

Ω

|g(x)| dx = f L∞ (Ω) gL1 (Ω) .

In case p, q < ∞ the proof is taken from [AF03, Theorem 2.4]. Because the exponential function is strictly convex (its second derivative is strictly positive), given A, B ∈ R, it is eA/p+B/q ≤

eB eA + . p q

For a, b > 0 set A = ln(a p ) and B = ln(bq ) yielding the so-called Young inequality ab ≤

ap bq + . p q

(2.8)

If f or g is zero the Hölder inequality is trivially true, otherwise it follows from the Young inequality setting a = |f (x)|/f Lp (Ω) and b = |g(x)|/gLq (Ω) and integrating over the domain Ω.   Of special interest is the space L2 (Ω), which is even a Hilbert space with inner product (·, ·)0 : L2 (Ω) × L2 (Ω) → R, defined by  (f, g)0 =

6 In

f (x)g(x) dx

for all f , g ∈ L2 (Ω).

Ω

the case p = q = 2 the Hölder inequality is the Cauchy–Schwarz inequality (2.1).

2.4 Sobolev Spaces

21

For clarity the norm is denoted by ·0 := ·L2 (Ω) . If only a subset M ⊂ Ω is considered, denote by (·, ·)0,M and ·0,M the corresponding inner product and norm, respectively. The same symbols are used in the case of vectors, matrices, and higher order tensors, i.e., (f , g)0 is defined also for f , g : Ω → Rd or f , g : Ω → Rd×d as the sum of the inner products over all components. The norm is then  defined in the standard way: f 0 = (f , f )0 . Functions in Lp (Ω) which have a vanishing mean value are often considered in the context of Stokes and Darcy equations:    p f (x) dx = 0 . L0 (Ω) := f ∈ Lp (Ω)

(2.9)

Ω

A larger space which is used to define weak derivatives in the next section is 

L1loc (Ω) = f ∈ L1 (Ω) f ∈ L1 (Ω0 ) for all Ω0 ⊂ Ω 0 ⊂ Ω and it consists of locally integrable functions.

2.4 Sobolev Spaces Sobolev spaces are subspaces of the Lebesgue spaces Lp (Ω) which consist of weakly differentiable functions up to a certain degree. A weak derivative is a generalization of the classical one. Its definition is partly based on the space of test functions   D(Ω) := f ∈ C ∞ (Ω) supp(f) ⊂ Ω is compact , where supp(f ) := {x | f (x) = 0} is the support of f . It can be shown that D(Ω) is dense in Lp (Ω), 1 ≤ p < ∞, see e.g., [AF03, Corollary 2.30]. Further results regarding density of smooth functions are shown in Sect. 3.4. The following definition of weak derivatives uses a so-called multiindex α ∈ Nd0 , where  N0 = N ∪ {0}. Its order is defined as |α| = di=1 αi , with the components αi of α. Definition 2.4.1 (Weak Derivative) Let α ∈ Nd0 be a multiindex and f ∈ 1 1 Lloc (Ω). A function g ∈ Lloc (Ω) is called the αth weak derivative of f if for all test functions v ∈ D(Ω) it is7   α |α| f (x)D v(x) dx = (−1) g(x)v(x) dx. Ω

7 The

derivative D α v is defined as D α v =

Ω

∂ |α| v ∂ α1 x1 ...∂ αd xd

.

22

2 Notation and Preliminary Results

It can be shown that such a weak derivative, if it exists, is unique. The usual rules such as the product and composition rule as well as the linearity of the differentiation remain valid in the weak sense. Furthermore, a (classically) differentiable function is weakly differentiable with the same derivative, which is hence denoted with the same symbol g = D α f , moreover ∇ is used for the weak gradient as well. The term (−1)|α|

 f (x)D α v(x) dx Ω

is called distributional derivative of f even if f is not weakly differentiable. It has to be understood as a functional on D(Ω). Definition 2.4.2 (Sobolev Space) given. Then

Let 1 ≤ p ≤ ∞ and k ∈ N0 := N ∪ {0} be

 for all multiindices α ∈ Nd , |α| ≤ k, there exists 0 (Ω) := f ∈ L (Ω) D α f in the weak sense and D α f ∈ Lp (Ω) 

W

k,p

p

is called a Sobolev space. Its norm ·W k,p (Ω) is defined by ⎞1 p   D α f p p ⎠ =⎝ L (Ω) ⎛

f W k,p (Ω)

for all f ∈ W k,p (Ω).

|α|≤k

Furthermore, define the semi-norm ⎛

|f |W k,p (Ω)

  D α f p p =⎝ |α|=k

L (Ω)

⎞1

p

⎠

for all f ∈ W k,p (Ω).

Remark 2.4.3 Sobolev spaces are introduced in different ways by many authors. Most notably is the definition as the completion of the set {f ∈ C ∞ (Ω) | f W k,p (Ω) < ∞} with respect to the norm ·W k,p (Ω) . These definitions coincide with the one given above, see Theorem 3.4.1 in Sect. 3.4.1. Many more related spaces with special properties are studied and used for particular problems. See for example [AF03] and [Gri85]. It can be shown that the Sobolev spaces are complete, i.e., Banach spaces, see for example [AF03, Theorem 3.3]. In case p = 2, H k (Ω) := W k,2 (Ω) is even a Hilbert space with inner product (·, ·)k : H k (Ω) × H k (Ω) → R, defined by (f, g)k =

  D α f, D α g 0

|α|≤k

for all f , g ∈ H k (Ω).

2.4 Sobolev Spaces

23

The norm is denoted by ·k = ·H k (Ω) , the semi-norm by |·|k = |·|H k (Ω) . As before the notation ·k,M and (·, ·)k,M is used for the norm and inner product where all integrals are taken over M ⊂ Ω instead of Ω. Note that W 0,p (Ω) = Lp (Ω) and H 0 (Ω) = L2 (Ω). The spaces consisting of vector valued functions are written in bold face, for example  d Hk (Ω) := H k (Ω) .

2.4.1 Fractional Order Sobolev Spaces So far only Sobolev spaces of nonnegative integer order are introduced, namely W k,p (Ω) with k ∈ N0 . It is however meaningful to define intermediate spaces W s,p (Ω) ⊂ W 0,p (Ω) = Lp (Ω) with real 0 < s < 1. Functions in W s,p (Ω) also have a finite Sobolev–Slobodeckij semi-norm, which is defined in Eq. (2.10) below. They can be understood as a generalization of Hölder spaces, in the sense that W s,∞ (Ω) = C 0,s (Ω), [DNPV12], see also [Eva10] in the case of s = 1 and a C 1 boundary. Fractional order Sobolev spaces can as well be defined through interpolation methods, see [BS08] or [AF03]. The Sobolev–Slobodeckij semi-norm is given as

 

|v(x) − v(y)|P

|v|W s,p (Ω) = Ω×Ω

|x − y|sp+d

1/p dx dy

.

(2.10)

It is also called Aronszajn or Gagliardo semi-norm and is introduced by these three authors. Then define   W s,p (Ω) := v ∈ Lp (Ω) |v|W s,p (Ω) < ∞ which is a Banach space with the norm  1/p p p vW s,p (Ω) = vLp (Ω) + |v|W s,p (Ω) , see [DDE12], Proposition 4.24. In case p = 2, H s (Ω) = W s,2 (Ω) is a Hilbert space with the inner product  (u, v)H s (Ω) = (u, v)0,Ω + Ω×Ω

(u(x) − u(y)) (v(x) − v(y)) dx dy. |x − y|2s+d

Equivalently, on all of Rd the fractional order Sobolev space H s (Rd ) can be conveniently defined using the Fourier transform, see [DDE12], Section 4.2, and also [Tar07, DNPV12].

24

2 Notation and Preliminary Results

Remark 2.4.4 (Sobolev Spaces with Negative Index) So far in this monograph the space W s,p (Ω) is defined for s ∈ [0, 1]. An extension to arbitrary positive s is discussed in Remark 3.4.7. Additionally, it is possible to define the above space for negative s. In Rd , these are the spaces with positive index:  duals of corresponding  

for s < 0 it is W s,p (Rd ) := W −s,p (Rd )

∗

with 1 = 1/p + 1/p . For a domain 

Ω = Rd , one instead uses a suitable subspace of W −s,p (Ω) in the definition of W s,p (Ω). In this monograph Sobolev spaces with negative index are not studied any further, but more information on them can be found in the literature, see for example [AF03, Eva98] (p = 2), or [BC84].

2.5 Lebesgue Spaces on the Boundary Consider a localization (Vi , hi , φi ), i = 0, . . . , N, of Ω as in Definition 2.2.1. A function u : ∂Ω → R is called measurable on ∂Ω if the functions ui : Vi ⊂ Rd−1 → R, i > 0, defined as     ui (y  ) = φi y  , hi (y  ) u y  , hi (y  ) are measurable. Furthermore, u is integrable if additionally the integrals     ui (x) dσx = ui (y ) 1 + |Dhi (y  )|2 dy  ∂Ω

Vi

exist in the Lebesgue sense. As Lipschitz continuous functions, the mappings hi are differentiable almost everywhere (Rademacher’s theorem) and ∇hi L∞ (Vi ) ≤ |hi |C 0,1 (Vi ) , see Theorem 6 in Section 5.8.3 of [Eva98].8 The term on the right-hand side is the semi-norm |hi |C 0,1 (Vi ) = sup

x,y∈Vi x=y

hi (x) − hi (y) . |x − y|

Therefore, the integration of ui over ∂Ω is well defined and the integral of u on the boundary is then  u(x) dσx = ∂Ω

8 Some

N   i=1

ui (x) dσx . ∂Ω

generalization on Rademacher’s theorem can be found in [EG92].

2.6 Sobolev Spaces on the Boundary

25

Remark 2.5.1 On the boundary ∂Ω it is important to define the outer unit normal   vector n : ∂Ω → S d = x ∈ Rd |x| = 1 . Using the localization as above it is defined as  1 ∇hi (y  ) n(y) =  , −1 1 + |∇hi (y  )|2 where i is taken to be such that y = (y  , hi (y  )) ∈ Ωi ∩ ∂Ω. Finally, the Lebesgue space on the boundary ∂Ω is defined similarly as before for Ω:   Lp (∂Ω) = f : ∂Ω → R f is integrable and f Lp (∂Ω) < ∞ , with the norm p f Lp (∂Ω)

 |f (x)|p dσx

= ∂Ω

and the usual convention in the case of p = ∞. Remark 2.5.2 Since Lipschitz functions are almost everywhere differentiable, many classic results involving integrals of differentiable functions extend to Lipschitz functions. This includes a change of variables (substitution) under the integral as well as the fundamental theorem of calculus.

2.6 Sobolev Spaces on the Boundary Having introduced Lebesgue spaces on the boundary, the Sobolev–Slobodeckij spaces on the boundary can be defined analogously to what is done for domains:   W s,p (∂Ω) = v ∈ Lp (∂Ω) |v|W s,p (∂Ω) < ∞ , with the Sobolev–Slobodeckij semi-norm p |v|W s,p (∂Ω)



|v(x) − v(y)|p

= ∂Ω×∂Ω

|x − y|sp+d−1

Again, restrictions to parts of the boundary are possible.

dσx dσy .

Chapter 3

Properties of Sobolev Spaces

This chapter collects a number of important properties of Sobolev spaces. Almost every claim is provided together with a proof. The main result is the density of D(Ω) in W s,p (Ω) and proved in Theorem 3.4.5. The necessary tools to establish these proofs are introduced and intermediate results are presented in the following subsections. This entire chapter can be viewed as a preparation for subsequent ones on traces, Chap. 4, and to meaningfully define the weak forms of some partial differential equations in Chap. 5.

3.1 Mollifications Essential parts in many proofs in this chapter are based on mollifications, which are convolutions with an appropriately scaled smooth function, which has compact support. This smooth function is called the standard mollifier and is introduced in the following definition. Afterwards, the mollification is defined and its main properties are shown, namely that it provides a technique to approximate integrable functions. This is introduced in many textbooks. e.g., [Eva98], Appendix C.4, and the second chapter in [AF03]. Definition 3.1.1 (Standard Mollifier)  η(x) = where C is chosen so that

 Rd

Define η ∈ C ∞ (Rd ) as 2 −1)

Ce1/(|x|

if |x| < 1

0

else,

η(x) dx = 1. Then the standard mollifier ηε (x) := ε−d η(x/ε)

© Springer Nature Switzerland AG 2019 U. Wilbrandt, Stokes–Darcy Equations, Advances in Mathematical Fluid Mechanics, https://doi.org/10.1007/978-3-030-02904-3_3

27

28

3 Properties of Sobolev Spaces equivalent norm on W 1,p (Ω) Proposition 3.5.1

standard mollifier Definition 3.1.1

Mollification Definition 3.1.2

equivalent norm on W s,p (RK ) Proposition 3.5.2

Korn inequality Lemma 3.5.3

W s,p is of local type Proposition 3.2.1

Density of D (Rd ) in W s,p (Rd ) Proposition 3.4.3

shift operator Corollary 3.4.4

Density of C ∞ (Ω) in W s,p (Ω) Theorem 3.4.1

W s,p (Rd )-extensions Proposition 3.3.1

D (Ω) dense in W s,p (Ω) Theorem 3.4.5

Mollification properties Proposition 3.1.3

W s,p (Ω) and Lipschitz transformations Proposition 3.2.3

W s,p (∂Ω) and Lipschitz transformations Proposition 3.2.4

W s,p (∂Ω)-extensions Proposition 3.3.4

Fig. 3.1 A graph showing the dependencies among all theorems, definitions, corollaries, lemmas as well as propositions in Chap. 3. Note that implicit dependencies are not always shown

is also in C ∞ (Rd ) and it holds   ηε (x) dx = 1, supp(ηε ) = B(0, ε) = x ∈ Rd Rd



|x| ≤ ε .

Definition 3.1.2 (Mollification) For ε > 0 and a given f ∈ L1loc (Ω) its mollification fε := ηε ∗ f , with the standard mollifier ηε as in Definition 3.1.1, is defined on the slightly smaller set Ωε = {x ∈ Ω | dist(x, ∂Ω) > ε}.1 That means

1 In

this definition Ω may be Rd , then Ωε = Ω for all ε.

3.1 Mollifications

29

for each x ∈ Ωε it is   fε (x) = ηε (x − y)f (y) dy = Ω

ηε (y)f (x − y) dy. B(0,ε)

The set Ωε is chosen such that f is only evaluated inside of Ω in the above integral. Extending f by zero outside of Ω results in a mollification on all of Ω or even Rd . The same notation f is used for this extension if no confusion is possible. The advantage of mollifications is that they are smooth approximations which is the statement of the following theorem, see [Eva98], Theorem 6 in Appendix C and Theorem 1 in Chapter 5.3 as well as [AF03], Theorem 2.29. Proposition 3.1.3 (Mollification Properties) Let u ∈ L1loc (Ω) be given and uε its mollification as in Definition 3.1.2. Then it holds i) ii) iii) iv) v)

uε ∈ C ∞ (Ωε ) and for α ∈ Nd it is D α uε = D α (ηε ∗ u) = (D α ηε ) ∗ u, if supp(u) ⊂ Ω and 0 < ε <  := dist(supp(u), ∂Ω), then uε ∈ D(Ω), p p if u ∈ Lloc (Ω), p ≥ 1, then uε → u in Lloc (Ω) for ε → 0, k,p k,p if u ∈ Wloc (Ω), p ≥ 1, k ∈ N, then uε → u in Wloc (Ω) for ε → 0, s,p s,p if u ∈ Wloc (Ω), p ≥ 1, s ∈ (0, 1), then uε → u in Wloc (Ω) for ε → 0.

Proof i) Let x ∈ Ωε , i = 1, . . . , d, and h > 0 small enough so that x + hei ∈ Ωε . Then it is     1 uε (x + hei ) − uε (x) 1 x + hei − y x−y = d η −η u(y) dy. h ε Ωh ε ε The support of the integrand above is a compact subset of Ω in which     ∂ηε 1 x + hei − y x−y 1 ∂η x − y (x − y) η −η → = εd h ε ε ε ∂xi ε ∂xi converges uniformly for h → 0. According to the dominated convergence theorem the derivative in ei direction of uε exists and equals ∂uε ∂ηε (x) = ∗ u(x) = ∂xi ∂xi

 Ω

∂ηε (x − y)u(y) dy. ∂xi

Similarly, it can be shown that any derivative of uε exists and for all multiindices α ∈ Nd0 it is D α uε = D α (ηε ∗ u) = D α ηε ∗ u. Therefore, uε ∈ C ∞ (Ωε ). ii) According to i), uε is an element of D(Ωε ). Due to the choice of ε, its extension by zero to all of Ω fulfills supp(u) ⊂ supp(uε ) ⊂ Ω−ε ⊂ Ω and hence is contained in D(Ω).

30

3 Properties of Sobolev Spaces

iii) Let U ⊂ Ω be an open set whose closure is also contained in Ω, i.e., U ⊂ Ω, and choose ε > 0 such that ε < dist(U, ∂Ω). Then, using Hölder’s inequality, Theorem 2.3.1, it is  |uε (x)| = ηε (y)u(x − y) dy B(0,ε)



1−1/p 

≤

1/p ηε (y)|u(x − y)|p dy

ηε (y) dy B(0,ε)

B(0,ε)

for any x ∈ U . The first factor above is one and therefore the Lp -norm of uε can be bounded by p uε Lp (U )



  |uε (x)| dx ≤

=

ηε (y)|u(x − y)|p dy dx

p

U

U



B(0,ε)

=

 |u(x − y)|p dx dy

ηε (y) 

B(0,ε)

≤



U

|u(x)|p dx dy

ηε (y) B(0,ε)

V

p

= uLp (V ) ,   where Fubini’s theorem is applied and V = y ∈ Rd dist(U, y) < ε ⊂ Ω. Next, let δ > 0 be given and choose a continuous v ∈ C(V ) such that u − vLp (V ) ≤ δ/3 and hence uε − vε Lp (U ) = (u − v)ε Lp (U ) ≤ u − vLp (V ) ≤ δ/3. Furthermore, for x ∈ U it is  |vε − v|(x) = ηε (x − y)|v(y) − v(x)| dy ≤ sup |v(y) − v(x)|, B(x,ε)

y∈B(x,ε)

which tends to zero independently of x because v is uniformly continuous on U  V . Therefore, upon possibly further reducing ε it is vε − vLp (V ) ≤ δ/3. Combining all these results yields uε − uLp (U ) ≤ uε − vε Lp (U ) + vε − vLp (U ) + v − uLp (U ) ≤ δ. k,p

iv) Let u ∈ Wloc and U an open set whose closure is contained in Ω. According to iii), uε converges to u in Lp (U ). Additionally, the derivatives D α uε , |α| ≤ k,

3.1 Mollifications

31

belong to Lp (U ) and similarly to i) it is D α uε = ηε ∗ D α u. Using again iii) also D α uε converges to D α u in Lp (U ), which completes the proof. v) This step is shown in a very similar manner as iii) but in R2d . As before, that its closure is a subset of Ω and V = an open set such  let U be y ∈ Rd dist(U, y) < ε ⊂ Ω, i.e., U ⊂ V . Let δ > 0 be given. For all x, y ∈ V denote fu (x, y) =

u(x) − u(y) , |x − y|s+d/p

so that |fu (x, y)|p is the integrand in the Sobolev–Slobodeckij semi-norm, see Eq. (2.10). Since u ∈ W s,p (V ) it is fu ∈ (Lp (V ))2 . Now choose g ∈ C(V 2 ) close to fu such that fu − g(Lp (V ))2 ≤ δ/3. Instead of a regular mollification in R2d , a modified one is used. For this purpose define  η : R2d → R using the d regular mollification η in R :  η(x, y) = η(x)η(y). The properties of η are inherited with slight modifications: √ • supp( η) = B(0, 1)2 ⊂ B(0, 2) ⊂ R2d and  2   • R2d  η(x, y) dx dy = B(0,1) η(x) dx = 1. η(x/ε, y/ε) = ηε (x)ηε (y) is then a mollifier in The scaled version  ηε (x, y) = ε−2d  R2d and therefore gε :=  ηε ∗ g an approximation to g. In particular, it is  (gε − g)(x, y) = ≤

B(0,ε)2

ηε (z)ηε (w)(g(x − z, y − w) − g(x, y)) dz dw

sup

|g(x − z, y − w) − g(x, y)|,

(z,w)∈B(0,ε)2

which tends to zero uniformly, i.e., independently of (x, y) ∈ U 2 , because g is uniformly continuous on the closure of U 2 as it is continuous on the larger set V 2 . This implies gε − g(Lp (V ))2 ≤ δ/3 for a small enough ε. Furthermore, fuε and gε are close as well. First note that for x, y ∈ U it is fuε (x, y) − gε (x, y) (ηε ∗ u)(x) − (ηε ∗ u)(y) − ( ηε ∗ g)(x, y) |x − y|s+d/p  u(x − z) − u(y − z) = ηε (z) dz |x − y|s+d/p B(0,ε)

=

32

3 Properties of Sobolev Spaces

 −

B(0,ε)2

 =

B(0,ε)2

 



u(x − z) − u(y − z) − g(x − z, y − w) dz dw |x − y|s+d/p 1−1/p ηε (z)ηε (w) dz dw

ηε (z)ηε (w)

  ≤

ηε (z)ηε (w)g(x − z, y − w) dz dw

B(0,ε)2

p 1/p u(x −z) − u(y −z) dz dw ηε (z)ηε (w) − g(x −z, y −w) |x − y|s+d/p B(0,ε)2

where Hölder’s inequality, Theorem 2.3.1, is applied. The first factor above is one. Hence, the Lp -norm of fuε − gε is bounded by   fu − gε p ε (Lp (U ))2 p   u(x −z) − u(y −z) ≤ ηε (z)ηε (w) − g(x − z, y − w) dz dw dx dy s+d/p 2 2 |x − y| U B(0,ε) p  u(x) − u(y) − g(x, y) dx dy ≤ s+d/p V 2 |x − y| p (Lp (V ))2

= fu − g

≤ δ/3,

where Fubini’s theorem and the fact U 2 ⊂ V 2 are applied similarly to iii). Finally, the results can be combined (using ii) for the Lp -part)  p p p uε − uW s,p (U ) = uε − uLp (U ) + fuε − fu (Lp (U ))2  p p ≤ δ + fuε −gε (Lp (U ))2 + gε −g p

(L (U ))2

p (Lp (U ))2

+ g−fu 

≤ 2δ.   Remark 3.1.4 The above properties remain valid in the case Ω = Rd .

3.2 Products with Smooth Functions and Lipschitz Transformations The Hölder inequality, Theorem 2.3.1, shows that the product of two functions in suitable Lebesgue spaces is in L1 . In the following it is shown that the product of a function in a Sobolev space with a smooth one is in that same Sobolev space. Sometimes such spaces are called to be of local type, see [DDE12],

3.2 Products with Smooth Functions and Lipschitz Transformations

33

Proposition 4.26. A proof for unbounded domains can also be found in [DNPV12], Lemma 5.3. Proposition 3.2.1 (W s,p is of Local Type) Let Ω ⊂ Rd be a bounded domain and p > 1, as well as s ∈ [0, 1]. Then for any u ∈ W s,p (Ω) and ϕ ∈ D(Ω) their product ϕu is also in W s,p (Ω) and it holds ϕuW s,p (Ω) ≤ cuW s,p (Ω) for a constant c which depends on ϕ∞ and, if s > 0, also on ∇ϕ∞ . Proof In case s = 0, i.e., u ∈ W 0,p (Ω) = Lp (Ω) the claim follows directly from the Hölder inequality, Theorem 2.3.1, because ϕ p ∈ L∞ (Ω) and up ∈ L1 (Ω):         p p ϕuLp (Ω) = ϕ p up L1 (Ω) ≤ ϕ p L∞ (Ω) up L1 (Ω) = ϕ p L∞ (Ω) uLp (Ω) . For s = 1 the same reasoning shows that ϕ∇u is in (Lp (Ω))d . Similarly it is ∇ϕ ∈ (D(Ω))d and hence ∇ϕu is also in (Lp (Ω))d . Therefore, ∇(ϕu) = ∇ϕu + ϕ∇u is an element of (Lp (Ω))d , i.e., ϕu ∈ W 1,p (Ω). In the case of intermediate s ∈ (0, 1), observe that the Sobolev–Slobodeckij semi-norm can be written in two terms using the inequality (a + b)p ≤ 2p−1 (a p + b p ): 

p

|ϕu|W s,p(Ω) =

|(ϕu)(x) − (ϕu)(y)|p Ω2

≤

2



dx dy

|ϕ(x)(u(x) − u(y))|p

p−1

 +2

|x − y|sp+d

|x − y|sp+d

Ω2

p−1

|(ϕ(x) − ϕ(y))u(y)|p |x − y|sp+d

Ω2

p

dx dy dx dy. p

The first term can be bounded by the product of ϕL∞ (Ω) and |u|W s,p (Ω) . In order to bound the second term note that, due to the mean value theorem, it is ϕ(x) − ϕ(y) ≤ ∇ϕL∞ (Ω) |x − y| and therefore   Ω

|(ϕ(x) − ϕ(y))u(y)|p Ω

|x − y|sp+d

dx dy p

≤ ∇ϕL∞ (Ω)



 |u(y)|p Ω

Ω

1 |x − y|(s−1)p+d

dx dy.

The inner integral is finite because |x − y| ≤  := diam(Ω) and (s − 1)p + d < d. In fact, using polar coordinates it can be bounded by ωd p(1−s)/(p(1 − s)), where ωd is the measure of the unit ball’s surface (i.e., the unit sphere). Concluding, p p |ϕu|W s,p (Ω) ≤ cuW s,p (Ω) where c depends on ϕL∞ (Ω) , ∇ϕL∞ (Ω) , , s, p, and ωd .  

34

3 Properties of Sobolev Spaces

Remark 3.2.2 In case a function u ∈ W s,p (Γ ), Γ ⊂ ∂Ω, 0 ≤ s < 1, is multiplied with a smooth function φ, the same proof as in the previous Proposition 3.2.1 also shows that uφ ∈ W s,p (Γ ). Many proofs involving Sobolev spaces on a bounded domain Ω ⊂ Rd use a localization as introduced in Definition 2.2.1 to show the claim locally first. In a second step the local results are suitably combined to conclude the statement. The previous Proposition 3.2.1 shows that the products of the smooth functions φi , which formed the partition of unity, with a W s,p -function u essentially keep the norm of the factor u. The transformations to the local setting from a global one and vice versa have to preserve the necessary norms, too. This is the core statement of the following two propositions: Proposition 3.2.3 (W s,p (Ω) and Lipschitz Transformations) Let Ω and Ω  be two Lipschitz domains. Furthermore, let F : Ω  → Ω be a bijective Lipschitz map whose inverse is also Lipschitz continuous. The composition u ◦ F : Ω  → R of u : Ω → R inherits its properties from u as follows: If u ∈ W s,p (Ω), s ∈ [0, 1], then u ◦ F is in W s,p (Ω  ) and there exists a constant c such that u ◦ F W s,p (Ω  ) ≤ cuW s,p (Ω) . Proof A similar proof for s = 0 and s = 1 can be found in [DDE12], Lemma 2.22 and 2.21. The transformation formula (change of variables) yields p

u ◦ F Lp (Ω  ) =



 Ω

|u(F (y))|p dy = Ω

|u(x)|p det(∇F −1 (x)) dx.

Since the determinant is a continuous the above integral can be bounded by   operator, p cuLp (Ω) , where c depends on ∇F −1 L∞ (Ω) . In the case of s = 1 the same reasoning for the derivative shows the desired bound. However, it remains to show that  u◦F is in W 1,p (Ω  ). Let φ ∈ D(Ω  ) be given and let ε < dist ∂Ω, F −1 (supp(φ)) . 1,p The mollification uε of u is in Wloc (Ωε ), see Proposition 3.1.3. Since uε is smooth it is  −

Ω

 (uε ◦ F )(x)∇φ(x) dx =  =

Ω

Ω

∇(uε ◦ F )(x)φ(x) dx ∇uε (F (x)) · ∇F (x)φ(x) dx.

Taking the limit ε → 0 shows that (∇u ◦ F ) · ∇F is the weak derivative of u ◦ F .

3.2 Products with Smooth Functions and Lipschitz Transformations

35

Assume now 0 < s < 1. Then again using a change of variables the Sobolev– Slobodeckij semi-norm can be bounded as well: |u ◦ F |W s,p (Ω  )  u(F (y)) − u(F (y  )) p = dy dy  |y − y  |sp+d Ω 2  u(F (y)) − u(F (y  )) p sp+d ≤ LF dy dy  sp+d Ω 2 |F (y) − F (y  )| p   u(x) − u(x  ) sp+d −1 −1  = LF (x)) (x )) det(DF dx dx  det(DF sp+d  2 |x | −x Ω = c|u|W s,p (Ω) , where LF is the Lipschitz constant of F and the constant c depends on LF and on  ∇F −1  ∞ .   L (Ω) The previous result, Proposition 3.2.3, extends to the case of a function u defined on the boundary of a domain Ω. Then it is u ◦ F W s,p (∂Ω  ) ≤ cuW s,p (∂Ω) : Proposition 3.2.4 (W s,p (∂Ω) and Lipschitz Transformations) Let Ω, Ω  , and F be as in Proposition 3.2.3 and u ∈ W s,p (∂Ω), s ∈ [0, 1). Then the composition u ◦ F ∈ W s,p (∂Ω  ) and there exists a constant c such that u ◦ F W s,p (∂Ω  ) ≤ cuW s,p (∂Ω) . Proof Consider a localization (Vi , hi , φi ) of Ω  as in Definition 2.2.1 and let  {Ωi }N i=0 be the corresponding covering of Ω . Then the transformed sets Ωi =  F (Ωi ) cover Ω and the transformed functions φi = φi ◦ F −1 form a partition of unity adapted to Ωi . Note however that the φi are no longer in C0∞ (Ωi ) but only Lipschitz continuous with compact support, φi ∈ C00,1 (Ωi ), the following argument still holds though. Alternatively, they can be approximated uniformly with smooth functions. Next, consider the Lp -norm of u ◦ F on ∂Ω  : 

p

u ◦ F Lp (∂Ω  ) = =

∂Ω 

|u(F (x))|p dσx =

=

i=1

N   i=1

∂Ω  ∩Ωi

N   i=1

N  

Vi

∂Ω  ∩Ωi

φi (x)|u(F (x))|p dσx

φi (F (x))|u(F (x))|p dσx

p  2 φi (F (x, hi (x))) u(F (x, hi (x))) 1 + ∇hi (x) dx.

36

3 Properties of Sobolev Spaces

For each y ∈ Vi ⊂ Rd−1 there exists an x ∈ Vi ⊂ Rd−1 such that F (x, hi (x)) = (y, hi (y)) and it is unique because F is bijective. Therefore, the map κi : Vi → Vi , y → x is well defined and bijective, even Lipschitz continuous: κi (y 1 ) − κi (y 2 ) ≤ (κi (y 1 ), h (κi (y 1 ))) − (κi (y 2 ), h (κi (y 2 ))) i i = F −1 (y 1 , hi (y 1 )) − F −1 (y 2 , hi (y 2 )) ≤ LF −1 (y 1 , hi (y 1 )) − (y 2 , hi (y 2 ))  2 2 = LF −1 y 1 − y 2 + hi (y 1 ) − hi (y 2 )  ≤ LF −1 1 + L2hi y 1 − y 2 , whereL are the Lipschitz constants of the functions in the subscript, i.e., it is Lκi ≤ LF −1 1 + L2hi . In particular the determinant of the derivative of κi is bounded by

a constant C which only depends on the domain Ω and the transformation F −1 , |det(∇κi )| ≤ C. In the integrals over Vi ⊂ Rd−1 above a change of variables x = κi (y) is done. For this purpose it is necessary to understand what happens to the term in the square root above. In fact, it is the norm of a suitable vector whose behavior under the desired change of variables is easier to study. Let gi : Rd → R be such that ∂Ω ∩ Ωi = gi−1 ({0}), i.e., gi (y, ξ ) = hi (y) − ξ and gi (y, hi (y)) = 0 for all y ∈ Vi . Similarly define gi and note that it is gi = gi ◦ F . With this notation it is  2 1 + ∇hi (x) = ∇gi (x, hi (x)) = ∇gi (F (x, hi (x)))∇F (x, hi (x)) ≤ ∇F L∞ (Ω  ) ∇gi (F (y, hi (y))) .   Consequently, using the two identities F κi (y), hi (κi (y)) = (y, hi (y)) and  |∇gi (y, hi (y))| = 1 + |∇hi (y)|2 , a change of variables x = κi (y) in the Lp norm of u ◦ F on ∂Ω  yields p

u ◦ F Lp (∂Ω  ) ≤ C∇F L∞ (Ω  )

N   Vi

i=1

= C∇F L∞ (Ω  )

 φi (y, hi (y))|u(y, hi (y))|p 1 + |∇hi (y)|2 dy

N  

φi (y, hi (y))|u(y, hi (y))|p dσy ∂Ω∩Ωi

i=1 p

= C∇F L∞ (Ω  ) uLp (∂Ω) .

3.3 Extension from Ω to Rd

37

Next, consider the Sobolev-Slobodeckij semi-norm of u: 

|u(F (x)) − u(F (y))|p ∂Ω 2

=

|x − y|sp+d−1

N  N   i=1 j =1

Vi2

dσx dσy

φi (F (x, hi (x)))φj (F (y, hi (y)))

u(F (x, h (x))) − u(F (y, h (y))) p i i (x, h (x)) − (y, h (y)) sp+d−1 i i  2  2 1 + ∇h (x) 1 + ∇h (y) dx dy. i

i

Again, changing variables using κi and κj as before and using the Lips −1 chitz continuity of the transformation F , i.e., (x, hi (x)) − (y, hi (y)) ≤ −1   LF F (x, hi (x)) − F (y, hi (y)) , gives the desired result for the SobolevSlobodeckij semi-norm, |u ◦ F |W s,p (∂Ω  ) ≤ c|u|W s,p (∂Ω) . Summarizing, there is a constant c depending on the domain Ω and the transformation F such that u ◦ F W s,p (∂Ω  ) ≤ cuW s,p (∂Ω) .  

3.3 Extension from Ω to Rd The following proposition states that functions defined on a Lipschitz domain can be continuously extended to all of Rd . The proof is taken from [DDE12], Proposition 4.43 and [DNPV12], Theorem 5.4, for 0 < s < 1. The case s = 1 is shown in [DDE12], Proposition 2.70, as well as in [Eva98], Theorem 1 in Section 5.4. Proposition 3.3.1 (W s,p (Rd )-Extensions) Let Ω ⊂ Rd be Lipschitz, p > 1 and s ∈ [0, 1]. Then there exists a linear and continuous extension operator E : W s,p (Ω) → W s,p (Rd ), i.e., for all u ∈ W s,p (Ω) and almost every x ∈ Ω it is Eu(x) = u(x). Proof Note that for s = 0 a function u ∈ W 0,p (Ω) = Lp (Ω) can be extended by zero outside of Ω. Then the norm is not changed, uLp (Rd ) = uLp (Ω) . Let (Vi , hi , φi ) be a localization of Ω, see Definition 2.2.1, and u ∈ W s,p (Ω). Then the product uφi is in W s,p (Ω) as well with uφi W s,p (Ω) ≤ cuW s,p (Ω) , see Proposition 3.2.1. For any point x = (x  , xd ) of Ωi outside of Ω, i.e., xd < hi (x  ), define its reflexion P (x) = (x  , 2hi (x  ) − xd ). Next, define the local

38

3 Properties of Sobolev Spaces

extension uφi of uφi as

uφi (x, xd ) =

⎧ ⎪ ⎪ ⎨0

if (x  , xd ) ∈ / Ωi ,

(uφi )(x  , xd ) ⎪ ⎪ ⎩(uφ )(P (x  , x )) i d

if xd ≥ hi (x  ), if xd < hi (x  ).

  Clearly, uφi is in Lp (Rd ) with uφi Lp (Rd ) ≤ 2uφi Lp (Ω) . The cases s = 1 and 0 < s < 1 are treated separately. 0 < s < 1: In order to show that this local extension is in fact in W s,p (Rd ), p the double integral in the Sobolev–Slobodeckij semi-norm |uφi |W s,p (Rd ) is split into four parts J1 , J2 , J3 , and J4 which in turn are double integrals over the sets Ω × Ω, Ω × Ωc , Ωc × Ω, and Ωc × Ωc with Ωc = Rd \ Ω. The first part J1 is |uφi |W s,p (Ω) . The second and the third are equal due to the Theorem of Fubini and can be bounded as follows. For x ∈ Ω and y ∈ Ωc (i.e., xd ≥ hi (x  ) and yd ≤ hi (y  )) it is2 xd − 2hi (y  ) + yd ≤ xd − yd ≤ |x − y|, xd − 2hi (y  ) + yd ≥ 2(hi (x  ) − hi (y  )) − xd + yd ≥ −2∇hi ∞ x  − y  − |xd − yd | ≥ −(1 + 2∇hi ∞ )|x − y|. It therefore is |x − P (y)| = x  − y  + xd − 2hi (y  ) + yd ≤ C|x − y| where C depends on ∇hi ∞ which in turn can be bounded independently of i. Then with a different C it is p p     uφi (x) − uφi (y) uφi (x) − uφi (y) dx dy ≤ C dx dy J2 = |x − y|sp+d |x − P (y)|sp+d Ω Ωc Ω Ωc   |(uφi )(x) − (uφi )(P (y))|p =C dx dy |x − P (y)|sp+d Ω Ωc  |(uφi )(x) − (uφi )(y)|p =C dx dy, |x − y|sp+d Ω2 where in the last step a change of variables is done. Hence, J2 = J3 ≤ CJ1 . In the last integral J4 note that for x ∈ Ωc and y ∈ Ωc it is |P (x) − P (y)| = x  − y  + 2hi (x  ) − xd − 2hi (y  ) + yd ≤ (1 + 2∇hi ∞ )|x − y|.

2 For

simplicity here the norm on Rd is the 1-norm, i.e., |x| =



|xi |.

3.3 Extension from Ω to Rd

39

Hence, the integral can be bounded, similarly as above, by CJ1 . Together it is shown that uφi ∈ W s,p (Rd ) and that |uφi |W s,p (Rd ) ≤ 4C|uφi |W s,p (Ω∩Ωi ) . Combining these results, it is  p uφi  s,p W

p

(Rd )

p

≤ 2uφi Lp (Ω) + 4C|uφi |W s,p (Ω∩Ωi ) .

Finally, these local extensions are combined to a global one, namely Eu =  N i=0 uφi , which is linear and continuous: EuW s,p (Rd ) ≤

N    uφi  s,p d ≤ CuW s,p (Ω) W (R ) i=0

with a constant C which does not depend on u. s = 1: First, a result very similar to the definition of weak derivatives is shown, see also Lemma 2.58 in [DDE12]. With v ∈ W 1,p (Rd+ ), Rd+ is the half space {x = (x  , xd ) ∈ Rd | xd > 0}, and ϕ ∈ D(Rd )3 it is  Rd+

∂v (x)ϕ(x) dx = − ∂xj

 Rd+

v(x)

∂ϕ dx ∂xj

(3.1)

for 1 ≤ j < d and this equation is also true for j = d provided that ϕ(x  , 0) = 0 for all x  ∈ Rd−1 . To see this, let vn ∈ C ∞ (Rd+ ) ∩ W 1,p (Rd+ ) converge4 to v in W 1,p (Rd+ ). For the vn the stated equation (3.1) is true (integration by parts for i = d) and taking the limit also shows it for v. For convenience denote v := uφi and transform the space Rd such that the boundary of Ω is straightened out, i.e., define the bi-Lipschitz continuous transformation F : Rd → Rd to be F (x  , xd ) = (x  , xd + hi (x  )). Then, according to Proposition 3.2.3, the function w := v ◦ F is in W 1,p (Rd+ ). Assume that w % :=  v ◦ F is a W 1,p -extension of w with % w W 1,p (Rd ) ≤ 2wW 1,p (Rd ) , then +

Proposition 3.2.3 states that  v ∈ W 1,p (Rd ) and

 v W 1,p (Rd ) ≤ c% ˜ w W 1,p (Rd ) ≤ 2cw ˜ W 1,p (Rd ) ≤ cvW 1,p (Ωi ∩Ω) , +

where c is a constant depending on hi ∞ . It remains to show that w % is a W 1,p extension of w. Note that it is  for yd ≥ 0 w(y  , yd ) w %(y  , yd ) =  w(y , −yd ) for yd < 0.

contrast to the definition of the weak derivative, the test function ϕ is not in D (Rd+ ) but in D (Rd ). 4 It is shown in Theorem 3.4.1 and Remark 3.4.2 that such a sequence exists. 3 In

40

3 Properties of Sobolev Spaces

A change of sign in the last component yields p

% w Lp (Rd ) =



 Rd+

|w(x)|p dx +

Rd−

w(x  , −xd ) p dx = 2wp p

L (Rd+ )

.

To show the (weak) differentiability of w %, let ϕ ∈ D(Rd ) be given. Together with ϕ    also (x , xd ) → ϕ(x , xd ) + ϕ(x , −xd ) is in D(Rd ) and Eq. (3.1) yields  Rd

w %(x)

∂ϕ (x) dx = ∂xj

 Rd+

w(x)



 ∂  ϕ(x) + ϕ(x  , −xd ) dx ∂xj

  ∂w (x) ϕ(x) + ϕ(x  , −xd ) dx Rd+ ∂xj   ∂w ∂w (x)χRd (x) + (x, −xd )χRd (x) ϕ(x) dx, =− + − ∂xj Rd ∂xj

=−

where χM is the indicator function of the set M. Hence, the weak derivative in xj direction of w % is given by the function in the parentheses in the above integrand and is an element of Lp (Rd ). Finally, consider the derivative in xd -direction. The function (x  , xd ) → ϕ(x  , xd ) − ϕ(x  , −xd ) is in D(Rd ) and vanishes on {xd = 0}. Therefore, Eq. (3.1) shows  Rd

w %(x)

∂ϕ (x) dx = ∂xd

 Rd+



w(x)

 ∂  ϕ(x  , xd ) − ϕ(x  , −xd ) dx ∂xd

  ∂w (x) ϕ(x) − ϕ(x  , −xd ) dx Rd+ ∂xd   ∂w ∂w =− (x)χRd (x) − (x, −xd )χRd (x) ϕ(x) dx. + − ∂xd Rd ∂xd

=−

Again the term in the parentheses in the above integrand is in Lp (Rd ) and therefore the weak derivative of w %. The explicit derivatives also show the required estimate % w W 1,p (Rd ) ≤ 2wW 1,p (Rd ) .   +

Remark 3.3.2 In the previous Proposition 3.3.1, the support of the extension Eu of u can be chosen: Let V ⊂ Rd be a bounded set such that Ω ⊂ V . Then define a cutoff function φ ∈ D(Rd ) with φ(x) = 1 for x ∈ Ω, supp(φ) ⊂ V . Then the product v = φEu is an element of W s,p (Rd ) due to Proposition 3.2.1 and its support is in V . Furthermore, it is an extension of u and it holds vW s,p (V ) ≤ CuW s,p (Ω) , where C additionally depends on V , in particular on dist(V , Ω) > 0.

3.3 Extension from Ω to Rd

41

Remark 3.3.3 In [AF03], Definition 5.17, extension operators on W k,p (Ω), k ∈ N are further distinguished: They are called simple-k, p-extension operators if the properties in the above Proposition 3.3.1 are fulfilled for given k and p. Furthermore, a strong-m-extension operator is a simple-k, p-extension operator for all p ≥ 1 and k ≤ m. Moreover, a total extension operator is a strong-m-extension operator for each m ∈ N. In the proof of Proposition 3.3.1 the extension is constructed independently of p, therefore it is even a strong-1-extension-operator, but not a total extension operator. In [AF03] the authors refer to [Ste70] for a proof of the existence of a total extension operator for Ω possessing a “strong local Lipschitz condition”. In applications one faces boundary conditions on a part Γ of the boundary ∂Ω. In order to proof existence and uniqueness of solutions of some partial differential equations, it is necessary to extend functions on Γ to all of ∂Ω. This in turn poses a condition on the boundary of Γ because in general it is not possible to extend a W s,p (Γ )-function continuously to W s,p (∂Ω). The next theorem therefore requires the boundary ∂Γ of Γ to be Lipschitz as well. This means that locally after transforming Ωi ∩ ∂Ω to Rd−1 , the set originating from Γ is Lipschitz in Rd−1 . This allows for an extension of a function in W s,p (Γ ) to W s,p (∂Ω): Proposition 3.3.4 (W s,p (∂Ω)-Extensions) Let Ω ⊂ Rd be a Lipschitz domain and Γ ⊂ ∂Ω a relatively open subset of its boundary. Furthermore, let Γ itself be of Lipschitz type and s ∈ [0, 1). Then there exists a linear and continuous extension operator E : W s,p (Γ ) → W s,p (∂Ω), i.e., for all u ∈ W s,p (Γ ) and almost every x ∈ Γ it is Eu(x) = u(x). Proof Let u ∈ W s,p (Γ ) be given and let (Vi , hi , Φi ), i = 1, . . . , N, be a localization of Ω as in Definition 2.2.1. The extension is composed of local contributions with respect to the sets Ωi ∩ ∂Ω. If Γ and ∂Ω coincide locally, i.e., if Γ ∩Ωi = ∂Ω ∩Ωi , the function Φi u has compact support in ∂Ω ∩Ωi and belongs to W s,p (∂Ω). In case the boundary of Γ intersects Ωi , define the local version of u on Rd−1 as vi (x  ) := Φi u(x  , hi (x  )) whenever x  ∈ Γ˜i := {y  ∈ Vi | (y  , hi (y  )) ∈ Γ }. Due to Propositions 3.2.1 and 3.2.4 together with Remark 3.2.2 the function vi is in W s,p (Γ˜i ). According to the assumption, the set Γ˜i ⊂ Rd−1 is itself of Lipschitz type. Proposition 3.3.1 therefore gives an extension vi ∈ W s,p (Rd−1 ) of the function vi , where for simplicity the same name is used. It is furthermore assumed that supp(vi ) ⊂ Vi , see Remark 3.3.2. Transforming this back to the boundary of Ω, ui (x  , hi (x  )) = vi (x  ), yields an element in W s,p (Ωi ∩ ∂Ω)  which extends Φi u. Next, define the desired extension operator as the sum Eu = N i=1 ui . Since the transformations to and from Rd−1 as well as the extension operator on Rd−1 are linear and continuous, the global extension E is, too: EuW s,p (∂Ω) ≤ C

N 

Φi uW s,p (Γ ) ≤ CNuW s,p (Γ ) .

i=1

 

42

3 Properties of Sobolev Spaces

Remark 3.3.5 A subset Γ of the boundary of a domain Ω which itself has a Lipschitz boundary is also called admissible patch. In this context the so-called creased domains are studied. These are Lipschitz domains together with two disjoint subsets of its boundary which are separated by a Lipschitz continuous interface, i.e., which are both admissable patches, see [MM07, AF03].

3.4 Density of Smooth Functions The elements in Lebesgue and Sobolev spaces are only defined almost everywhere, i.e., unlike continuous functions, cannot be evaluated at points or other sets of zero measure. However in this section it is shown that certain smooth functions are dense in W s,p -spaces. The main property of Sobolev spaces shown here is the density of the test functions D(Ω) if the domain Ω is Lipschitz.

3.4.1 Density of Smooth Functions In the proof of Proposition 3.1.3, iii), it is shown that uε Lp (U ) ≤ uLp (V ) where U  V . Extending u by zero outside of Ω allows a meaningful definition of uε on all of Ω and gives uε Lp (Ω) ≤ uLp (Ω) . Furthermore the rest of that proof also extends to Lp (Ω), i.e., uε → u in Lp (Ω). However for u ∈ W s,p (Ω), s > 0, an extension by zero is in general not in W s,p (Rd ), and the result above cannot be extended this way. However the local approximations enable also global ones, that means not only for every compact subset, i.e., in the spaces with index loc , but also on all of Ω. The proof is taken from [Eva98], Theorem 2 in 5.3.2, where only the case s ∈ N ∪ {0} is treated, see also Proposition 2.12 in [DDE12]. Theorem 3.4.1 (C ∞ (Ω) is Dense in W s,p (Ω)) Let u ∈ W s,p (Ω), p ≥ 1, s ∈ [0, 1] be given. Then there is a sequence of functions ui ∈ C ∞ (Ω)∩W s,p (Ω) which converges to u in W s,p (Ω). Proof The idea is to partition the domain Ω into suitable subsets on which the local result from Proposition 3.1.3 can be applied. Let {Ui }i∈N be a sequence of subsets of Ω which approach Ω from the inside, namely Ui := {x ∈ Ω | dist(x, ∂Ω) > 1/i}. Then the union of all Ui is Ω and Ui ⊂ Uj whenever i < j . For convenience, denote U−1 = U0 = ∅. Next, define the differences i ∈ N ∪ {0}. & Vi := Ui+2 \ U i for ∞ ∞ Then also the union of the Vi contains Ω, i.e., ∞ i=0 Vi = Ω. Let {ζi }i=0 ⊂ C (Ω)

3.4 Density of Smooth Functions

43

be a partition of unity adapted to the sets Vi , so it holds ζi ∈ C0∞ (Vi ),

∞ 

0 ≤ ζi ≤ 1,

ζi = 1.

i=0

As u ∈ W s,p (Ω) also the products ζi u are elements of W s,p (Ω), see Proposition 3.2.1, and their support is contained in Vi , that is supp(ζi u) ⊂ Vi . Let δ > 0 be given. For each i, due to Proposition 3.1.3, one can choose εi > 0 small enough so that the mollification ui := ηεi ∗ (ζi u) approximates ζi u with ui − ζi uW s,p (Ω) ≤

δ 2i+1

and

supp(ui ) ⊂ Wi := Ui+3 \ U i−1 ⊃ Vi . ∞ Next, define v = i=0 ui and note that at any point x ∈ Ω only finitely many summands are nonzero due to the choice of the partition of Ω. Hence, v ∈ C ∞ (Ω) and it is  ∞  ∞      v − uW s,p (Ω) =  ui − ζi uW s,p (Ω) (ui − ζi u) ≤   i=0

≤δ

W s,p (Ω)

i=0

∞  1 = δ. i+1 2

 

i=0

Prior to a famous paper by Meyers and Serrin, another definition of W k,p , k ∈ N, existed, namely as the closure of {f ∈ C ∞ (Ω) | f W k,p < ∞} which is usually denoted by H k,p . The previous Theorem shows that these two definitions are equivalent, even if Ω is not Lipschitz continuous. Both notations are still in use, however in this monograph the letter H is only used in the case of p = 2. Remark 3.4.2 For unbounded domains Ω, define the subsets Ui = {x ∈ Ω | |x| < i, dist(x, ∂Ω) > 1/i} in the previous proof and possibly choose a subsequence such that U1 = ∅. This is how one can show that Theorem 3.4.1 holds true even for non-Lipschitz domains.

3.4.2 Density of Smooth Functions with Compact Support Using mollifications it is shown in Theorem 3.4.1 that in bounded Lipschitz domains Ω smooth functions are dense in W s,p (Ω). An improved result is the density of test functions on Ω in W s,p (Ω). For this purpose it is shown that the space of smooth functions with compact support are dense in W s,p (Rd ). Proofs can be found in [DDE12], Theorem 1.91, Proposition 2.29 (s = 1), and Proposition 4.27 (0
0 and the inner integral over Bn can be bounded by 



1 Bn

|x − y|

sp+d

dx ≤

−sp−d

Rd \B(0,ε)

|z|

 dz = ωd

∞

r −sp−1 dr =

ε

ωd −sp ε , sp (3.3)

where polar coordinates are used and ωd is the surface area of ∂B1 . In consequence it is  |φn (y) − 1|p ωd |u(y)|p dy. In ≤ ||y| − n|sp sp Bnc The fraction in the integrand above is bounded on Bnc which can be seen as follows: sup y∈Bnc

|φn (y) − 1|p 1 = sp ||y| − n|sp n

|φ(z) − 1|p 1|sp z∈Rd \B(0,1) '||z| − () * sup

=:f (z)

46

3 Properties of Sobolev Spaces

and the function f : Rd \ B(0, 1) → R is bounded by 1 on Rd \ B(0, 2) and by p ∇φL∞ (Rd ) otherwise, i.e., |z| ≥ 2

⇒ f (z) =

1 ≤ 1, ||z| − 1|sp

1 ≤ |z| ≤ 2

⇒ f (z) =

|φ(z) − φ(z/|z|)|p p ≤ ∇φL∞ (Rd ) ||z| − 1|(1−s)p ||z| − 1|sp p

≤ ∇φL∞ (Rd ) , where in the second case the mean value theorem is applied. Hence, it is In ≤



1 ωd n→∞ p p ∇φ max 1, ∞ (Rd ) uLp (Rd ) −−−→ 0. sp L n sp

The other integral Jn in Eq. (3.2) can be decomposed using the inequality (a +b)p ≤ 2p−1 (a p + b p ):  Jn = ≤2

 Bnc

|(un − u)(x) − (un − u)(y)|p

Bnc



|x − y|sp+d



|un (x) − un (y)|p

p−1 Bnc

|x − y|sp+d

Bnc

dx dy 

dx dy + 2



|u(x) − u(y)|p

p−1 Bnc

Bnc

|x − y|sp+d

dx dy.

The second term on the right-hand side converges to zero because u ∈ W s,p (Rd ). Denote the first term by 2p−1 Kn and decompose it into  Kn(1) =

An

 Kn(2) =

An

 

|un (x) − un (y)|p |x − y|sp+d

An

|un (x) − un (y)|p c B2n

|x − y|sp+d

dx dy, dx dy,

c . Note that u is zero in Rd \ B(0, 2n) = B c where An = B2n \ Bn = Bnc \ B2n n 2n (1) (2) (2) and hence Kn = Kn + 2Kn , again using Fubini’s Theorem to get a second Kn . (2) c Since un (x) = 0 for x ∈ B2n , Kn is bounded by

 Kn(2) =

 An

|un (y)|p c B2n

|x − y|sp+d

dx dy ≤

ωd sp

 An

|un (y)|p dy, ||y| − 2n|sp

3.4 Density of Smooth Functions

47

where the inner integral is handled very similarly to the inner integral of In in Eq. (3.3). As φ(2y/|y|) = 0 and un = φn u it is 

|(φ(y/n) − φ(2y/|y|))u(y)|p dy ||y| − 2n|sp An   |(φ(y/n) − φ(2y/|y|))|p ωd |u(y)| dy sup ≤ ||y| − 2n|sp sp y∈An An  |(φ(z) − φ(2z/|z|))|p 1 ωd p uLp (An ) sup ≤ sp ||z| − 2|sp n sp z∈A1

Kn(2) ≤

ωd sp

p

As before with f the supremum above is bounded by ∇φL∞ (Rd ) and, accordingly, (2)

(1)

Kn converges to zero with n → ∞. The term Kn also converges to 0 which is shown using the same techniques as above, see also the proof of Proposition 3.2.1. In the numerator of its integrand one adds and subtracts u(y)φn (x), uses the inequality (a + b)p ≤ 2p−1 (a p + b p ), and applies the mean value theorem to yield Kn(1) ≤ 2p−1



|φ(y/n) − φ(x/n)|p |u(y)|p A2n

 +

≤

|x − y|sp+d

dx dy

|u(y) − u(x)|p |φ(x/n)|p |x − y|sp+d

A2n

p ∇φL∞ (Rd ) 

np

dx dy

 |u(y)|p An

p

|x − y|(1−s)p−d dx dy An

p

+ φL∞ (Rd ) |u|W s,p (An ) . ' () * =1

The inner integral is finite for all y ∈ An , in fact using polar coordinates it is  |x − y|(1−s)p−d dx ≤ ωd An

(2n)p(1−s) p(1 − s)

and consequently

Kn(1)

≤2 ≤

p−1



p

∇φL∞ (Rd ) ωd 2p(1−s) nps

p(1 − s)

p uLp (An )

p + |u|W s,p (An )

c uW s,p (An ) . nps

Hence, Kn = Kn(1) + Kn(2) converges to zero with n → ∞ and therefore un → u in W s,p (Rd ).  

48

3 Properties of Sobolev Spaces

As an example, the previous result is used to show that small shifts in the argument of an Lp function only have a small effect. Corollary 3.4.4 (Shift Operator) Let h ∈ Rd be given. Then the shift operator τh : Lp (Rd ) → Lp (Rd ), defined for all x ∈ Rd as τh u(x) = u(x + h), preserves the norm, i.e., τh uLp (Rd ) = uLp (Rd ) , and it holds τh u → u in Lp (Rd ) for |h| → 0. Proof A change of variables y = x − h yields p



τh uLp (Rd ) =

 Rd

|u(x − h)|p dx =

Rd

p

|u(y)|p dy = uLp (Rd ) .

To show the convergence, let ε > 0 be given. According to Proposition 3.4.3 there exists a function v ∈ D(Rd ) such that u − vLp (Rd ) < ε/3. Since v is continuous and has compact support, it is even uniformly continuous, hence there exists a h0 ∈ Rd such that for all h ∈ Rd with |h| < |h0 | it is τh v − vL∞ (Rd ) ≤

ε (2 |supp(v)|)−1/p . 3

This implies τh v − vLp (Rd ) ≤ τh v − vL∞ (Rd ) (2|supp(v)|)1/p ≤ ε/3. Finally, using τh u − τh vLp (Rd ) = τh (u − v)Lp (Rd ) = u − vLp (Rd ) ≤ ε/3 and the triangle inequality, it is τh u − uLp (Rd ) ≤ τh u − τh vLp (Rd ) + τh v − vLp (Rd ) + v − uLp (Rd ) ≤ ε.  

3.4.3 Density of Smooth Functions up to the Boundary With the density of test functions in W s,p (Rd ), Proposition 3.4.3, it is possible to proof density of smooth functions in W s,p (Ω), see [DDE12], Corollary 2.71 (s = 1) and Proposition 4.52 (s ∈ (0, 1)). This is Proposition 3.4.3 above but Rd replaced by Ω. Another proof can be found in [Tar07], Lemma 12.3. Theorem 3.4.5 (D(Ω) is Dense in W s,p (Ω)) Let Ω ⊂ Rd be a bounded Lipschitz domain, p ≥ 1 and s ∈ [0, 1]. Then D(Ω) is dense in W s,p (Ω). Proof Let u ∈ W s,p (Ω) be given and according to Proposition 3.3.1 it has an extension Eu ∈ W s,p (Rd ). Since D(Rd ) is dense in W s,p (Rd ), according to Proposition 3.4.3, there is a sequence un ∈ D(Rd ) which converges in W s,p (Rd ) to

3.5 Equivalent Norms

49

Eu. Then restricting each un to Ω yields a sequence in D(Ω) which converges to u in W s,p (Ω): n→∞

u − un W s,p (Ω) = Eu − un W s,p (Ω) ≤ Eu − un W s,p (Rd ) −−−→ 0.

 

This subsection finishes with two interesting remarks which however have no particular impact on the rest of the monograph. Remark 3.4.6 The spaces W s,p (Ω) are nested. In [DNPV12] (Propositions 2.1  and 2.2) it is shown that for a Lipschitz domain Ω the space W s ,p (Ω) is s,p  continuously embedded into W (Ω) if 0 < s ≤ s ≤ 1. However unless u ∈ W 1,p (Ω) is constant the limit of |u|W s,p (Ω) with s → 1 does not exist and in fact approaches infinity. However if one considers (1 − s)|u|W s,p (Ω) instead, the respective limit can be bounded by the W 1,p (Ω) semi-norm: p

lim (1 − s)|u|W s,p (Ω) =

s1

Kp,d p |u|W 1,p (Ω) p

with a constant Kp,d which depends on p and d, see [Bre02]. Remark 3.4.7 So far s is in the unit interval. For larger s the space W s,p (Ω) is not defined by simply inserting s ≥ 1 into the definition of the Sobolev–Slobodeckij semi-norm. One reason is that  |u(x) − u(y)|p dx dy < ∞ |x − y|d+1 Ω×Ω for a measurable function u : Ω → R implies that u must be constant, see [Bre02], Proposition 1. Instead let s = m + σ with m ∈ N and σ ∈ (0, 1), then W s,p (Ω) is defined to be the subspace of W m,p (Ω) whose elements and all their derivatives up to order m are in W σ,p (Ω).

3.5 Equivalent Norms Later proofs use a localization of the domain Ω, see Definition 2.2.1. Then on W 1,p (Ω), one can define an equivalent norm using the sum of the local contributions as follows, see [DDE12], Proposition 2.68. Proposition 3.5.1 (Equivalent Norm on W 1,p (Ω)) Let Ω be a Lipschitz domain which is covered by a finite set of domains Ωi , i ∈ {0, . . . , N}, and u ∈ Lp (Ω). Then u is in W 1,p (Ω) if and only if u ∈ W 1,p (Ωi ∩ Ω) for each i. The map u →

N  uW 1,p (Ωi ∩Ω) i=0

is an equivalent norm to uW 1,p (Ω) .

50

3 Properties of Sobolev Spaces

Proof Let u ∈ W 1,p (Ω) and i ∈ {0, . . . , N} be given. Clearly u ∈ Lp (Ωi ∩ Ω) and any given ϕ ∈ D(Ωi ∩ Ω) can be extended by zero to Ω so that the weak derivative of u exists also on Ωi ∩ Ω, i.e., u ∈ W 1,p (Ωi ∩ Ω). Furthermore it N is i=0 uW 1,p (Ωi ∩Ω) ≤ CuW 1,p (Ω) , where C is bounded by the maximum number of subdomains Ωi having a nonempty intersection (at most N). On the other hand let u ∈ W 1,p (Ωi ∩ Ω) for all i ∈ {0, . . . , N} be given. In order to show that u is also in W 1,p (Ω) let ϕ ∈ D(Ω) be given. Writing φ = N i=0 Φi ϕ for a partition of unity {Φi } subordinate to the Ωi gives   N   ∂(Φi ϕ) ∂ϕ u(x) (x) dx u (x) dx = ∂x ∂xj j Ω Ωi ∩Ω i=0

=−

N   i=0

=−

Ωi ∩Ω

  N Ω

i=0

∂u(x) (Φi ϕ)(x) dx ∂xj

 ∂u Φi (x)ϕ(x) dx, ∂xj

where the second equality uses the weak differentiability on Ωi ∩ Ω (Φi ϕ ∈ D(Ωi ∩ Ω)).  Hence, u is also weakly differentiable on Ω with its derivaN ∂u tive given by N i=0 Φi ∂xj . Finally, using u = i=0 Φi u, it is uW 1,p (Ω) ≤  C N i=0 uW 1,p (Ωi ∩Ω) , where C depends on the Φi and its derivatives, i.e., on the cover {Ωi }.   A similar approach is possible for s < 1 as well, however another equivalent norm on W s,p (RK ) is useful with K = d − 1. The proof is mainly that of [DDE12], Lemma 3.27, where the case s = 1 − 1/p is shown. However a shorter proof for general s < 1 can be found in the same book, Lemma 4.33, [DDE12]. Proposition 3.5.2 (Equivalent Norm on W s,p (RK )) Then the following two statements are equivalent:

Let u ∈ Lp (RK ), p > 1.

(i) u ∈ W s,p (RK ), (ii) for all i ∈ {1, . . . , K} it is p |u|i,s,p

 :=

 RK

R

|u(x + tei ) − u(x)|p dt dx < ∞. t sp+1

Moreover, an equivalent norm to ·W s,p (RK ) is given by u := uLp (RK ) + K i=1 |u|i,s,p . Proof Note that in the case of K = 1 this is essentially a change of variables and the statement of this proposition does not need further proof. Therefore, assume K > 1. (ii) ⇒ (i): Let u ∈ Lp (RK ) and assume |u|i,s,p < ∞ for all i ∈ {1, . . . , K}. Furthermore, for all x = (x1 , . . . , xK ), y = (y1 , . . . , yK ) ∈ RK define x+ y i :=

3.5 Equivalent Norms

i

j =1 xj ej

y and x+ yK

51

 + K + y0 = j =i+1 yj ej = (x1 , . . . , xi , yi+1 , . . . , yK ). In particular it is x = x. Then using a telescoping sum, it is K−1 



u(y) − u(x) =

 u(+ x y i ) − u(+ x y i+1 ) ,

i=0

hence the Sobolev–Slobodeckij semi-norm can be bounded as follows: 

 RK

|u(x) − u(y)|p |x − y|sp+K

RK

dx dy ≤

K−1 

Ii ,

i=0

with  Ii :=

 RK

RK

p u(+ x y i ) − u(+ x y i+1 )  2 q dx dy, K xj − yj j =1

where q = (sp + K)/2 ≥ 1. In order to estimate the denominators above consider the constants C1 , . . . , C4 , depending only on q and K, such that ⎛ ⎛ ⎞q ⎛ ⎞q ⎛ ⎞2q ⎞2q K K K K K      |ηi |2q ≤ C3 ⎝ |ηi |2 ⎠≤ C4 ⎝ |ηi |⎠ . C1 ⎝ |ηi |⎠ ≤ C2 ⎝ |ηi |2 ⎠≤ j =1

j =1

j =1

j =1

j =1

The existence of the constants Cj is assured because these are all norms in the finite dimensional space RK . In fact, one can choose C3 = C4 = 1. Next, consider IK−1 and in particular the integration with respect to y1 : p   u(+ x y K−1 ) − u(x)  q dy1 ≤ K 2 R R j =1 xj − yj

p u(+ x y K−1 ) − u(x) 2q dy1 1 1 K 2q j =2 xj − yj C3 |x1 − y1 | + C3 

= 2C3

∞ x1

 ≤ 2C3

∞ x1

p u(+ x y K−1 ) − u(x) 2q dy1  |x1 − y1 |2q + K j =2 xj − yj p u(+ x y K−1 ) − u(x)  2q dy1 , K |x1 −y1 |2q + C1 j =2 xj − yj

 −1 K and applying the change of variables z = (x1 − y1 ) yields j =2 xj − yj p  ∞ u(+ x y K−1 ) − u(x) 1 = 2C3   2q−1 0 z2q + C1 dz. K j =2 xj − yj

52

3 Properties of Sobolev Spaces

The integral over z above is finite because 2q > 1, hence combining all the constants in M1 , it is p p  u(+ u(+ x y K−1 ) − u(x) x y K−1 ) − u(x)  2 q dy1 ≤ M1  sp+K−1 . K K R xj − yj x −y j =1

j =2

j

j

Next, integrate this estimate over R with respect to y2 and the same computations yield p p   u(+ u(+ x y K−1 ) − u(x) x y K−1 ) − u(x)  2 q dy1 dy2 ≤ M1 M2  p+K−2 . K K R R − y x j j x − y j =1 j j =3 j In these computations it is q = (sp + K − 1)/2 such that 2q > 1 assures the finiteness of the above integral over z.5 By induction it is IK−1 ≤

K−1 ,

Mj

j =1

=

K−1 ,

p  u(+ x y K−1 ) − u(x)

 RK

 Mj

j =1



RK

|xK − yK |sp+1

R

dyK dx

|u(x + teK ) − u(x)|p |t|sp+1

R

dt dx,

which is finite due to the assumption (ii). The other integrals Ii , i ∈ {1, . . . , K − 2}, can be handled the same way proving (i). Furthermore, it is shown that there exists a constant C which does not depend on u such that p uW 1−1/p,p (RK )

≤

p uLp (RK )

+C

K   i=1

 RK

|u(x) − u(x + tei )|p R

|t|sp+1

dt dx.

(i) ⇒ (ii): Let u ∈ W s,p (RK ). The statement (ii) is shown for i = K, for other i the same steps work as well, with a slightly more inconvenient notation. Set  J1 :=

RK



∞ 0

|u(x + tK eK ) − u(x)|p sp+1

tK

dtK dx.

5 Of course this is only true for K ≥ 2, for K = 1 there is no integration over a second component y2 . Similarly in the following steps.

3.5 Equivalent Norms

53 p

Note that it is J1 = 12 |u|K,1−1/p,p because 



RK

0

|u(x + tK eK )−u(x)|p

−∞

|tK |sp+1



 dtK dx =

RK

∞

|u(y)−u(y + sK eK )|p sp+1

sK

0

dsK dy,

where sK = −tK and y = x + tK eK . Therefore, statement (ii) is true if J1 is finite. Generalize J1 to Jk , 2 ≤ k ≤ K as follows: p + tK eK ) − u(x) dtK−1 dtK dx, J2 := (tK−1 + tK )sp+2 RK (0,∞)2 p   K   u x + j =K−k+1 tj ej − u(x) Jk := dtK−k+1 . . . dtK dx,  sp+k K RK (0,∞)k t j =K−k+1 j p   K   u x + j =1 tj ej − u(x) dt1 . . . dtK dx. JK :=  sp+K K RK (0,∞)K j =1 tj 



u(x + tK−1 ee

K−1

The sum in the denominator of JK can be bounded from below by  6 with y = x and z = x + K j =1 tj ej it is  JK ≤



RK

 ≤



RK

|u(z) − u(y)|p (0,∞)K +{y}

|z − y|sp+K

|u(z) − u(y)|p RK

|z − y|sp+K



K 2 j =1 tj

and

dz dy

dz dy

p

≤ uW s,p (RK ) . p

Due to the assumption (i) this is finite. In order to prove J1 ≤ C1 uW s,p (RK ) , p

p

it is shown that Jk+1 ≤ Ck+1 uW s,p (RK ) ⇒ Jk ≤ Ck uW s,p (RK ) for all k ∈ {1, . . . , K − 1} with suitable constants Ck and Ck+1 , which do not depend on u. p Therefore, assume Jk+1 ≤ Ck+1 uW s,p (RK ) . The fundamental theorem of calculus

K set (0, ∞)K + {y}   has to be understood element-wise, i.e., it is defined as (0, ∞) + {y} = x + y x ∈ (0, ∞)K , see also Footnote 3 on page 15.

6 The

54

3 Properties of Sobolev Spaces

 −sp−k  applied to the function t  → t  + K shows j =K−k+1 tj  K



1

sp+k = (sp + k)

j =K−k+1 tj

0

∞

1   sp+k+1 dt .  K  t + j =K−k+1 tj

Hence, Jk can be reformulated as  Jk = (sp + k)

RK

 (0,∞)k+1

p   K u x + j =K−k+1 tj ej − u(x)  sp+k+1  t + K t j j =K−k+1 dt  dtK−k+1 . . . dtK dx.

Using the inequality |a + b|p ≤ 2p−1 (|a|p + |b|p ) and a point x ∗ , which is to be chosen, yields Jk ≤ (sp + k)2p−1 p   K   u x + j =K−k+1 tj ej − u(x ∗ ) + |u(x ∗ ) − u(x)|p  sp+k+1  RK (0,∞)k+1 t + K t j j =K−k+1 dt  dtK−k+1 . . . dtK dx. treated Write the last term as (sp + k)2p−1  (B + A) where A and B are  separately.  + Choose x ∗ = x + 12 t  eK−k + 32 K t e , then using t tj ≥ 23 ( 12 t  + j j j =K−k+1  3 tj ) leads to 2 p   K  sp+k+1  u x + 12 t  eK−k + 32 j =K−k+1 tj ej − u(x) 3 A≤  sp+k+1 2 1  3 K RK (0,∞)k+1 j =K−k+1 tj 2t + 2 dt  dtK=k+1 . . . dtK dx. Changing the variables to τj = 32 tj , j ∈ {K − k + 1, . . . , K}, and τK−k = 12 t  the integral above reads A≤

 sp+k+1  k−1 3 2 2 2 3 p   K   u x + j =K−k τj ej − u(x) dτK−k . . . dτK dx  sp+k+1 K RK (0,∞)k+1 τ j =K−k j

3.5 Equivalent Norms

55

 sp+2 3 Jk+1 2  sp+2 3 p ≤2 Ck+1 uW s,p (RK ) . 2 =2

To bound B, a change of variables to y = x + {K − k + 1, . . . , K}, and τK−k = 12 t  yields 

B=2

K

j =K−k+1 tj ej ,

τj =

1 2 tj ,

j ∈



−sp−k−1

RK

(0,∞)k+1

    p K K u x + j =K−k+1 tj ej − u x + 12 t  + 32 j =K−k+1 tj ej  sp+k+1 1  1 K t + t j j =K−k+1 2 2 dt  dtK−k+1 . . . dtK dx = 2−sp−k−1 2k



 RK

  p K u(y) − u y + j =K−k τj ej dτK−k . . . dτK dx  sp+k+1 K (0,∞)k+1 j =K−k τj

= 2−sp−1 Jk+1 ≤ 2−sp−1 Ck+1 uW s,p (RK ) . p

p

Together this shows that Jk ≤ Ck uW s,p (RK ) with Ck = (sp + k)2(1−s)p−2(1 + p

3sp−2 )Ck+1 . By induction over k it follows that uK,s,p = 2J1 is bounded by p CuW s,p (RK ) , where C only depends on p and K, but not on u. Proceeding  p the same way to bound ui,s,p , i = K, results in the estimate K i=1 |u|i,s,p ≤ CuW s,p (RK ) with a different C.  

3.5.1 Further Properties The following important inequality bounds the gradient of a vector by its symmetric part, also called the deformation tensor, D(v) :=

 1 ∇v + ∇v  . 2

56

3 Properties of Sobolev Spaces

Lemma 3.5.3 (Korn Inequality)

For all v ∈ H1 (Ω) it is

∇v0 ≤ C(D(v)0 + v0 )

(3.4)

where C does not depend on v. This is remarkable because while the gradient ∇v has in general d 2 different entries, the deformation tensor only has (d 2 + d)/2 due to its symmetry. The proof can be found for example in [KO88, Theorem 5.13]. Furthermore, there is also a version for p = 2, an entire chapter in [DDE12] considers Korn’s inequality in Lp . Some important and well known properties of Sobolev spaces are not shown in this monograph, most notably the Sobolev embedding theorems, which are stated here for completeness and can be found for example in [DDE12], Theorem 3.5.4 ([DDE12], Theorem 2.72 and Corollary 4.53) Let s ∈ (0, 1] and p > 1 be given. Further let Ω ⊂ Rd be an open bounded Lipschitz set. Then it holds: • If sp < d, then W s,p (Ω) is continuously embedded in Lq (Ω) for all q ≤ dp/(d − sp). • If sp = d, then W s,p (Ω) is continuously embedded in Lq (Ω) for all q ≤ ∞, • If sp > d, then W s,p (Ω) is continuously embedded in L∞ (Ω) and even in a suitable Hölder space.

Chapter 4

Traces

One important property of certain Sobolev spaces is the fact that there is a well defined restriction onto the boundary, even though the boundary has measure zero. Such restrictions are known as traces and allow for prescribed boundary data of solutions of partial differential equations. A notion of trace extends that of the restriction of continuous functions to Sobolev spaces. A first result shows that such an operator exists and that it continuously maps into a Lebesgue space on the boundary of a Lipschitz domain. This is what many textbooks also cover, especially in the finite element community. A second theorem is more involved and states that the trace operator in fact maps into a proper subspace, namely a suitable Sobolev–Slobodeckij space, on the boundary. This is shown in steps; at first for the half space, then on all of ∂Ω, and finally on a part of the boundary. Of special interest are the kernels of these trace operators, i.e., the sets which have zero boundary values in a sense which is to be made precise. This kind of analysis leads to the so-called Lions–Magenes spaces which are studied in Sect. 4.3.1. Together with traces, the questions of inverses arises. Every trace operator in this chapter is introduced with a suitable right inverse which is of importance to define weak formulations of some partial differential equations in Chap. 5. Of course to define right inverses the trace operator needs to be surjective which is the main motivation in studying the image spaces of the trace operators.

4.1 The Trace Operator Theorem 4.1.1 (Trace Operator) Let Ω be a bounded Lipschitz domain and p > 1. Then there exists a linear and continuous map T : W 1,p (Ω) → Lp (∂Ω) which extends the restriction of continuous functions, i.e., for all u ∈ C(Ω) ∩ W 1,p (Ω) the function T u is the restriction onto the boundary: T u(x) = u(x) for all x ∈ ∂Ω. © Springer Nature Switzerland AG 2019 U. Wilbrandt, Stokes–Darcy Equations, Advances in Mathematical Fluid Mechanics, https://doi.org/10.1007/978-3-030-02904-3_4

57

58

4 Traces

Fig. 4.1 A graph showing the dependencies among all theorems, definitions, corollaries, lemmas as well as propositions in Chap. 4. Note that implicit dependencies are not always shown

Proof This proof mainly follows [Eva10], Theorem 1 in Chapter 5.5, however there it is only shown for C 1 domains. See also [DDE12], Theorem 2.86, for the Lipschitz case. Consider a localization (Vi , hi , Φi ), i ∈ [0, N], of Ω as in Definition 2.2.1 and, at first, assume u ∈ C ∞ (Ω). Let vi be the local representation of u, i.e., for all (x  , xd ) ∈ Rd−1 × R+ define vi (x  , xd ) = (Φi u)(x  , xd + hi (x  )). The function vi is an element of W 1,p (Rd+ ), see Propositions 3.2.1 and 3.2.3, and is Lipschitz continuous because hi is. According to the fundamental theorem of calculus it is  Rd−1

vi (x  , 0) p dx  = −

 

=−

Rd+

Rd+

∂|vi |p (x) dx ∂xd p|vi (x)|p−1 sgn(vi (x))

∂vi (x) dx ∂xd

4.1 The Trace Operator

59

p ∂vi |vi (x)| dx + ≤ (p − 1) ∂x (x) dx d d d R+ R+ 



p

p Lp (Rd+ )

≤ (p − 1)vi 

p , Lp (Rd+ )

+ ∇vi 

where the first estimate follows from the Young inequality, Eq. (2.8), with ∂vi a = | ∂x (x)| and b = |vi (x)|p−1 . It therefore is vi (·, 0)Lp (Rd−1 ) ≤ d max{(p − 1)1/p , 1}vi W 1,p (Rd ) . Since vi is a Lipschitz transformation of Φi u, + Proposition 3.2.3 extends this estimate to Φi u on ∂Ω ∩ Ωi : Φi uLp (∂Ω∩Ωi ) ≤ CΦi uW 1,p (Ωi ∩Ω) , where C depends on Ω through hi and can be chosen independently of i. Combining all the local contributions and Proposition 3.5.1 yields uLp (∂Ω) ≤

N N   Φi uLp (∂Ω∩Ωi ) ≤ C Φi uW 1,p (Ωi ∩Ω) ≤ CuW 1,p (Ω) . i=1

i=1

(4.1) Thus the Lp norm of the restriction to the boundary of the continuous function u is bounded by its W 1,p norm inside the domain. Now assume u ∈ W 1,p (Ω) and let un ∈ D(Ω) be a sequence of smooth function converging to u in W 1,p (Ω), according to Theorem 3.4.5. Then, due to Eq. (4.1) applied to un − um , un ∂Ω is a Cauchy sequence in Lp (∂Ω) and hence it converges. Define the trace of u to be its limit: T u := lim un ∂Ω . n→∞

In case u ∈ C(Ω) ∩ W 1,p (Ω) note that the un above converge uniformly to u, because the domain is bounded, i.e., T u = u ∂Ω as claimed.   Remark 4.1.2 In case one only considers a nonzero relatively open part Γ of ∂Ω, a trace TΓ : W 1,p (Ω) → Lp (Γ ) can be defined restricting the trace T = T∂Ω to Γ . Hence, such a trace operator is also continuous and identical to a restriction on the set C(Ω) ∩ W 1,p (Ω). Remark 4.1.3 In the case of vector valued spaces such as H1 (Ω) a trace operator is defined by the vector of traces for each component. The same name T with possible subscripts is used here.

60

4 Traces

4.2 Continuity of the Trace Operator on W 1,p (Ω) In this section it is shown that the trace operator T from Theorem 4.1.1 maps not into the entire Lp (Γ ), but only into the Sobolev–Slobodeckij space W 1−1/p,p (Γ ). A rather lengthy proof is taken from the book [DDE12]. First, the half space 

Rd+ = Rd−1 × (0, ∞) = x ∈ Rd x = (x1 , . . . , xd ), xd > 0 is considered. Later a more general open set Ω with Lipschitz continuous boundary is studied. Theorem 4.2.1 (Theorem 3.9, [DDE12]) For d ≥ 2 the image of the trace operator T : W 1,p (Rd+ ) → Lp (Rd−1 ) from Theorem 4.1.1 is the Sobolev– Slobodeckij space on the boundary, i.e.,      T W 1,p Rd+ = W 1−1/p,p Rd−1 .   Furthermore, as a map from W 1,p (Rd+ ) to W 1−1/p,p Rd−1 it is continuous   and possesses a linear and continuous right inverse E : W 1−1/p,p Rd−1 → W 1,p (Rd+ ). The proof uses another result which is introduced first, namely an inequality due to Hardy: Lemma 4.2.2 (Hardy’s Inequality) Let f ∈ Lp (R+ ), p > 1, R+ = (0, ∞), and  1 t define g : R+ → R as g(t) = t 0 f (s) ds. Then g is in Lp (R+ ) and it holds gLp (R+ ) ≤

p f Lp (R+ ) . p−1

Proof This proof is from Lemma 13.4 in [Tar07]. In [DDE12], Lemma 3.14, a more general version is proved but with a larger constant. Using the Hölder inequality, Theorem 2.3.1, it is 1 |g(t)| ≤ t ⇒

t|g(t)|p ≤



t

1/p |f (s)| ds

0 p f Lp ((0,t )).

p

t 1−1/p = t −1/p f Lp ((0,t ))

In particular t|g(t)|p → 0 for t → 0. Because p > 1 the definition of g shows t|g(t)|p → 0 for t → ∞. Furthermore, the derivative of g is given by g  (t) = −

1 t2



t 0

1 1 f (s) ds + f (t) = − (g(t) − f (t)). t t

4.2 Continuity of the Trace Operator on W 1,p (Ω)

61

Next, g is shown to be in Lp ((0, M)) for any M ∈ R+ . Using the fact t|g(t)|p → 0 for t → 0 from above and integration by parts gives 

M



M

|g(t)|p dt = −p

0

t|g(t)|p−2 g(t)g  (t) dt + M|g(M)|p

0



M

=p

|g(t)|p−2 g(t)(g(t) − f (t)) dt + M|g(M)|p

0



M

=p  ⇒ (1 − p)

 |g(t)|p − p

0 M

|g(t)|p−2 g(t)f (t) dt + M|g(M)|p

0



M

|g(t)|p dt = −p

0

M

|g(t)|p−2 g(t)f (t) dt + M|g(M)|p .

0

Taking the absolute value followed by an application of the Hölder inequality, Theorem 2.3.1, gives (p

p − 1)gLp ((0,M))

 ≤p

M

|g(t)|p−1 |f (t)| dt + M|g(M)|p

0



M

≤p

|g(t)|(p−1)p/(p−1) dt

0

1−1/p f Lp (0,M) + M|g(M)|p

p−1

= pgLp (0,M) f Lp (0,M) + M|g(M)|p M→∞

⇒ gLp (R+ ) ≤

p f Lp (R+ ) p−1

as claimed. Note that in the last step it is used that t|g(t)|p → 0 for t → ∞.

 

Having proved Hardy’s inequality, Theorem 4.2.1 can be shown: Proof of Theorem 4.2.1 In this proof it is assumed that d ≥ 2. In [DDE12] the case d = 2 is treated separately. The first step in this proof is to show that W 1−1/p,p (Rd−1 ) is continuously embedded in T (W 1,p (Rd+ )). To this end, let u ∈ W 1−1/p,p (Rd−1 ) be given. The aim is to construct a function Eu = v ∈ W 1,p (Rd+ ) such that T v = u and vW 1,p (Rd ) ≤ cuW 1−1/p,p (Rd−1 ) . Then the extension operator has the claimed + properties. Let ϕ ∈ D(R) be a test function with ϕ(0) = 1 and ϕL∞ (R) = 1, then define v : Rd+ → R as v(x, t) :=

ϕ(t) t d−1



 (0,t )d−1

u(x + y) dy = ϕ(t)

(0,1)d−1

u(x + ty) dy.

62

4 Traces

Using Hölder’s inequality, Theorem 2.3.1, and Fubini’s Theorem to exchange the order of integration shows that v is in Lp (Rd+ ): 

p v p d L (R+ )

=  ≤

 Rd−1



Rd−1

 =

(0,∞)

 ϕ(t)

(0,1)



|ϕ(t)|p (0,∞)



p u(x + ty) dy dt dx d−1

(0,1)d−1

|u(x + ty)|p dy dt dx



(0,1)d−1 (0,∞)

|ϕ(t)|p

Rd−1

|u(x + ty)|p dx dt dy

p

≤ uLp (Rd−1 ) ϕL∞ (R)|supp(ϕ)|. For almost all x ∈ Rd−1 and all t ∈ R it is  |v(x, t) − u(x)| = ϕ(t)u(x + ty) − u(x) dy (0,1)d−1

 ≤ ϕL∞ (R)

(0,1)d−1

u(x + ty) − u(x) dy + |u(x)| |ϕ(t) − 1|.

In the limit t → 0 both terms above approach zero, the first due to the continuity of the shift operator on Lp (Rd−1 ), see Corollary 3.4.4, the second because ϕ is continuous and ϕ(0) is chosen to be 1. Taking the p-th power and integrating with respect to x over Rd−1 therefore shows v(·, t) → u(·) in Lp (Rd−1 ) for t → 0, which implies T v = u. It remains to show that v is in fact an element in W 1,p (Rd+ ). Consider the derivative of v in xi direction. The integral is split into two where the inner one integrates over the i-th variable zi while the outer one integrates over the remaining d − 2 variables. For i ∈ [1, d − 1] and y = (y1 , . . . , yd−1 ) ∈ (0, 1)d−1 define y˘ i = (y1 , . . . , yi−1 , 0, yi+1 , . . . , yd−1 ), i.e., y = y˘ i + yi ei . Furthermore, the inner integral is transformed such that xi no longer enters the argument of u. This avoids having to differentiate u in the application of the Leibniz integral rule, which would be impossible because in general it is only in W 1−1/p,p (Rd−1 ) rather than W 1,p (Rd−1 ). ∂v ∂ (x, t) = ∂xi ∂xi



ϕ(t) = t ϕ(t) = t = ϕ(t)

ϕ(t) t 

(0,1)d−2 (xi ,xi +t )

(0,1)d−2

 







(0,1)d−2

(0,1)d−2

∂ ∂xi

u(x˘ i + t z˘ i + zi ei ) dzi d˘zi



(xi ,xi +t )

u(x˘ i + t z˘ i + zi ei ) dzi d˘zi

u(x˘ i + t z˘ i + (xi + t)ei ) − u(x˘ i + t z˘ i + xi ei ) d˘zi u(x + t z˘ i + tei ) − u(x + t z˘ i ) d˘zi . t

4.2 Continuity of the Trace Operator on W 1,p (Ω)

63

Integrating with respect to x and t over Rd−1 and R+ followed by a change of variables y = x + t z˘ i leads to (note that ϕL∞ (R) = 1 is assumed) p      ∂v u(x + t z˘ i + tei ) − u(x + t z˘ i ) p dt dx ≤ (x, t) t Rd−1 R ∂xi Rd−1 R+ (0,1)d−2 d˘zi dt dx 

 = =

Rd−1

u(y + tei ) − u(y) p dt dy t R+

1 p |u| , 2 i,1−1/p,p

where the last semi-norm is defined in Proposition 3.5.2. The derivative of v with respect to t can be bounded in a similar fashion:   ∂v ϕ  (t) ϕ(t) (x, t) = d−1 u(x + z) dz + d (1 − d) u(x + z) dz ∂t t t (0,t )d−1 (0,t )d−1  ϕ(t) ∂z + d−1 u(x + z) · n dz. t ∂t ∂ ((0,t )d−1)   In the last integral, for almost every z ∈ ∂ (0, t)d−1 , the normal n is either −ei and zi = 0 or n = ei and zi = t for some i. Therefore, the term ∂z ∂t · n vanishes whenever zi = 0 and reduces to 1 otherwise, hence, leaving a sum of integrals over all boundary faces with z = z˘ i + tei : ϕ  (t) ∂v (x, t) = d−1 ∂t t



d−1  ϕ(t)  u(x + z) dz u(x + z) dz − d−1 d−1 d−1 t t (0,t ) (0,t ) i=1

d−1  

ϕ(t) u(x + z˘ i + tei ) d˘zi d−2 t d−1 i=1 (0,t )  ϕ  (t) = d−1 u(x + z) dz t (0,t )d−1 +

d−1  ϕ(t)  u(x + z˘ i + tei ) − u(x + z) dz d−1 d−1 t t i=1 (0,t )  ϕ  (t) = d−1 u(x + z) dz t (0,t )d−1

+

+ ϕ(t)

d−1  i=1

u(x + t y˘ i + tei ) − u(x + ty) dy, t (0,1)d−1

64

4 Traces

where in the last step it is ty = z. The first term on the right-hand side above clearly is a function in Lp (Rd+ ). Taking the p-th power and integrating over Rd+ followed by a change of variables x  = x + ty and s = t (1 − yi ) in the second term leads to d−1   i=1

=

 Rd−1

 R+

(0,1)

u(x  + sei ) − u(x  ) p (1 − yi )p−1 dy ds dx  d−1 s

 d−1  u(x  + sei ) − u(x  ) p 1 ds dx  p s Rd−1 R+ i=1

=

1 p |u| . 2p i,1−1/p,p

Due to Proposition 3.5.2 all partial derivatives of u can be bounded by the Sobolev– Slobodeckij norm of u with a constant c which depends on p, d, and ϕ but not on u. It therefore is vW 1,p (Rd ) ≤ cuW 1−1/p,p (Rd−1 ) , +

as claimed. With this construction the extension operator E is a linear and continuous right inverse of the trace T . Conversely, now let v ∈ W 1,p (Rd+ ) be given and it must be shown that the trace u = T v, defined as u(x) = v(x, 0) for almost all x, is a function in W 1−1/p,p (Rd−1 ) whose norm can be bounded by that of v. According to Proposition 3.5.2 it suffices to prove |u|i,1−1/p,p ≤ cvW 1,p (Rd ) for all i ∈ {1, . . . , d − 1}. Using (a + b)p ≤ 2p−1 (a p + bp ) it is p |u|i,1−1/p,p

+

 v(x + tei , 0) − v(x, 0) p dt dx = t Rd−1 R   t t p v(x + tei , 0) − v(x + 2 ei , 2 ) p−1 ≤2 dt dx t Rd−1 R   t t p v(x + 2 ei , 2 ) − v(x, 0) p−1 +2 dt dx. t Rd−1 R 

Consider the first of the last two terms, the second can be handled in the same way. First, a change of variables (y = x + tei , τ = 12 t) is followed by an application of

4.2 Continuity of the Trace Operator on W 1,p (Ω)

65

the fundamental theorem of calculus:   t t p v(x + te , 0) − v(x + e , ) i i 2 2 2p−1 dt dx d−1 t R R   v(y, 0) − v(y − τ ei , τ ) p dτ dy = 2p 2τ Rd−1 R    1 p ∂ dτ dy v(y − se = , s) ds i Rd−1 R τ (0,τ ) ∂s    1 p dτ dy |∇v(y − se , s)| ds ≤ i Rd−1 R τ (0,τ )   |∇v(y − τ ei , τ )|p dτ dy Hardy’s inequality, Lemma 4.2.2, ≤ c(p) Rd−1

 = c(p) ≤

R

|∇v(z)|p dz

Rd p c(p)v 1,p d . W (R+ )

Combining these results yields the bound |u|i,1−1/p,p ≤ 2c(p)1/p vW 1,p (Rd ) with + c(p) being the constant from Hardy’s inequality, Lemma 4.2.2. Therefore, with an application of Proposition 3.5.2 it is u ∈ W 1−1/p,p (Rd−1 ), i.e., the continuity of the trace T : W 1,p (Rd+ ) → W 1−1/p,p (Rd−1 ) holds as claimed.   To prove such a result for a trace on Lipschitz domains, a partition of unity is used and the local results are suitably combined. The proof of the following theorem is mainly from [DDE12], Proposition 3.31. Theorem 4.2.3 (Trace on Lipschitz Domains) Let Ω ⊂ Rd be a Lipschitz domain. Then the image of the trace map T : W 1,p (Ω) → Lp (∂Ω) introduced in Theorem 4.1.1 is the Sobolev–Slobodeckij space on the boundary, i.e.,   T W 1,p (Ω) = W 1−1/p,p (∂Ω). Furthermore, as a map from W 1,p (Ω) to W 1−1/p,p (∂Ω) it is continuous and possesses a linear and continuous right inverse E : W 1−1/p,p (∂Ω) → W 1,p (Ω). Proof Let u ∈ W 1,p (Ω) be given and let (Vi , hi , Φi ), i = 0, . . . , N, be a (Lipschitz) localization of Ω, see Definition 2.2.1. Let vi : Rd+ → R be the local part of u transformed by hi :   vi (x  , xd ) := (Φi u) x  , hi (x  ) + xd ,

66

4 Traces

where Φi is extended by zero outside of its domain. Note that vi has compact support in Rd+ and is an element of W 1,p (Rd+ ) due to Propositions 3.2.3 and 3.2.1 applied to the product Φi u and the transformation (x  , xd ) → (x  , hi (x  ) + xd ), respectively. Additionally, the estimate vi W 1,p (Rd ) ≤ CΦi uW 1,p (Ωi ∩Ω) ≤ CuW 1,p (Ωi ∩Ω) +

holds. Theorem 4.2.1 implies that its trace is in the respective Sobolev–Slobodeckij space, i.e., T vi ∈ W 1−1/p,p (Rd−1 ), with the bound T vi W 1−1/p,p (Rd−1 ) ≤ vW 1,p (Rd ) . +

According to Proposition 3.2.4 the product Φi u is in W 1−1/p,p (Ωi ∩ ∂Ω) because it is a Lipschitz transformation of T vi and it holds Φi uW 1−1/p,p (Ωi ∩∂Ω) ≤ T vi W 1−1/p,p (Rd−1 ) . Combining all these partial results leads to T uW 1−1/p,p (Γ ) ≤

N 

Φi uW 1−1/p,p (Ωi ∩∂Ω)

i=1

≤ C

N 

uW 1,p (Ωi ∩Ω)

i=1

≤ CuW 1,p (Ω) , where the last inequality is due to Proposition 3.5.1, showing the continuity of the trace operator. Conversely, let u ∈ W 1−1/p,p (∂Ω) be given. The aim is to construct an extension Eu ∈ W 1,p (Ω) such that T ◦ E = id on W 1−1/p,p (∂Ω). First, a local extension is constructed. Let vi : Rd−1 → R be defined by vi (x) = (Φi u)(x, hi (x)). According to Propositions 3.2.4 and 3.2.1, the function vi belongs to W 1−1/p,p (Rd−1 ) with the estimate vi W 1−1/p,p (Rd−1) ≤ cΦi uW 1−1/p,p (∂Ω∩Ωi ) , where c depends on Ω. Next, in the setting of the half space, there is a continuous extension Evi ∈ W 1,p (Rd+ ) such that Evi W 1,p (Rd ) ≤ Cvi W 1−1/p,p (Rd−1) , +

4.2 Continuity of the Trace Operator on W 1,p (Ω)

67

see Theorem 4.2.1. This extension can be transformed to W 1,p (Ωi ∩ Ω) applying Proposition 3.2.3 which yields Vi ∈ W 1,p (Ωi ∩Ω) with Vi (x  , xd ) := Evi (x  , xd − hi (x  )) for almost all (x  , xd ) ∈ Ωi ∩ Ω, and Vi W 1,p (Ωi ∩Ω) ≤ Evi W 1,p (Rd ) . +

Due to the construction it is T Vi = vi . The desired extension operator is then the sum of the local ones, Eu := N i=1 Vi . Combining these results yields EuW 1,p (Ω) ≤

N  Vi W 1,p (Ωi ∩Ω) i=1

≤C

N  Φi uW 1−1/p,p (∂Ω∩Ωi ) i=1

≤ CuW 1−1/p,p (∂Ω) , hence the continuity of this extension operator.

 

In practice, solutions of partial differential equations have prescribed boundary values often only on a subset Γ of the boundary ∂Ω. While the trace can be further restricted to Γ without difficulties, the extension operator in general does not exist. Additionally to the Lipschitz continuity of the boundary ∂Ω, another such condition on the boundary ∂Γ of Γ is needed. Theorem 4.2.4 Let Ω ⊂ Rd be a Lipschitz domain and Γ ⊂ ∂Ω a relatively open subset of its boundary. Moreover, assume the boundary ∂Γ of Γ to be Lipschitz. Then the image of the trace map TΓ : W 1,p (Ω) → Lp (Γ ) introduced in Remark 4.1.2 is the Sobolev–Slobodeckij space on that boundary part, i.e.,   TΓ W 1,p (Ω) = W 1−1/p,p (Γ ). Furthermore, as a map from W 1,p (Ω) to W 1−1/p,p (Γ ) it is continuous and possesses a linear and continuous right inverse E : W 1−1/p,p (Γ ) → W 1,p (Ω). Proof Let u ∈ W 1,p (Ω). Then, using Theorem 4.2.3, its trace T u onto ∂Ω is in W 1−1/p,p (∂Ω). Restricting T u onto Γ to yield TΓ u is then in W 1−1/p,p (Γ ) and it holds TΓ uW 1−1/p,p (Γ ) ≤ T uW 1−1/p,p (∂Ω) ≤ CuW 1,p (Ω) . Conversely, let u ∈ W 1−1/p,p (Γ ) be given. Due to Proposition 3.3.4 there is an extension u ∈ W 1−1/p,p (∂Ω) which in turn can be extended to W 1,p (Ω), according to Theorem 4.2.3, where the extensions keep the same name Eu for simplicity.

68

4 Traces

These extensions are linear and continuous, i.e., there exists a constant C which does not depend on u such that EuW 1,p (Ω) ≤ CuW 1−1/p,p (Γ ) .

 

4.3 Characterization of the Kernel of the Trace Operator In Sect. 3.4 it is shown that certain smooth functions are dense in some Sobolev spaces. In particular D(Rd ) is dense in W s,p (Rd ) and so is D(Ω) in W s,p (Ω), see Proposition 3.4.3 and Theorem 3.4.5, respectively. However D(Ω) is dense in W s,p (Ω) if, and only if, s ≤ 1/2, see [LM72], Section 11.1. Yet, for bounded Lipschitz domains Ω, the case which is most important for this monograph, and s = 1 the two aforementioned spaces do not coincide. In fact the closure 1,p

W 1,p (Ω)

W0 (Ω) := D(Ω)

(4.2)

is a particular subspace of W 1,p (Ω) which can be characterized as the kernel of the trace operator: Proposition 4.3.1 The kernel of the trace operator from Theorem 4.2.3, T : 1,p W 1,p (Ω) → W 1−1/p,p (∂Ω), is W0 (Ω). Proof This proof can be found in [Eva98], Theorem 2 in Section 5.5, as well as in [Tar07], Lemma 13.7, and [Tri92], Theorem 1, Section 1.5.5. 1,p Let u ∈ W0 (Ω) be given together with an approximating sequence un ∈ D(Ω), according to Eq. (4.2). Because all un have compact support, their restrictions onto the boundary are zero, hence so are their traces. The continuity of the trace operator T now ensures that T u = 0, i.e., u ∈ ker(T ). Now let u ∈ ker(T ) ⊂ W 1,p (Ω) be given and let (Vi , hi , Φi ), i = 0, . . . , N, be a (Lipschitz) localization of Ω, see Definition 2.2.1. As in previous proofs let vi be the local part of u transformed by hi : vi (x  , xd ) := (Φi u)(x  , hi (x  ) + xd ). Due to Propositions 3.2.1 and 3.2.3 it is vi ∈ W 1,p (Rd+ ) with compact support in

Rd+ and T vi = 0 on Rd−1 . Next, consider a smooth function ζ ∈ C ∞ (R) such that ζ = 1 on [0, 1],

ζ = 0 on [2, ∞),

and

0 ≤ ζ ≤ 1.

Let ζn be a scaled version of ζ , ζn (x) = ζ (nx), and define1 wn ∈ W 1,p (Rd+ ) as wn (x) = vi (x)(1 − ζn (xd )). As 1 − ζn converges to 1 point-wise in Rd+ the wn 1 The

index i in the definition of wn is omitted for better readability.

4.3 Characterization of the Kernel of the Trace Operator

69

converge to vi in Lp (Rd+ ). To show that the derivatives also converge in Lp (Rd+ ) note that   p |∇(vi − wn )(x)|p dx ≤ 2p−1 |∇vi (x)ζn (xd )|p + nvi (x)ζ  (nxd ) dx. Rd+

Rd+

The first integral above approaches 0 with n → ∞ because the support of ζn (i.e., [0, 2/n]) intersected with that of vi vanishes. The second can be bounded by 

2 n

B := Cnp

 Rd−1

0

vi (x  , xd ) p dx  dxd ,

where C depends on the derivative of ζ and p. To bound this, further note that according to Theorem 3.4.5 for any f ∈ W 1,p (Rd+ ) there is a sequence fn ∈ D(Rd+ ) which converges to f in W 1,p (Rd+ ). The continuity of the trace operator yields n→∞

in W 1−1/p,p (Rd−1 ).

Tfn −−−→ Tf

For fn and (x  , xd ) ∈ Rd+ the fundamental theorem of calculus yields fn (x  , xd ) ≤ fn (x  , 0) +





xd ∂f n 0

∂x

d

(x  , t) dt.

The inequality (a + b)p ≤ 2p−1 (a p + bp ) together with the Hölder inequality, Theorem 2.3.1, imply   fn (x  , xd ) p ≤ 2p−1 fn (x  , 0) p + x p−1 d



p ∂x (x , t) dt . d

xd ∂f n 0



Integrating this equation over Rd−1 gives  Rd−1

  fn (x  , xd ) p dx  ≤ 2p−1 Tfn p p d−1 + x p−1 ∇fn p p d−1 , d ) ×[0,x ]) L (R L (R d

which also holds for f because fn as well as its trace converge appropriately. Hence, this result can be inserted for the inner integral of B with f = vi . Noting that T vi = 0 it is 

2 n

B ≤ Cn 2

p p−1 0

≤C

p−1

xd

p

∇vi Lp (Rd−1 ×[0,x

22p−1 p ∇vi Lp (Rd−1×[0,2/n]) . p

d ])

dxd

70

4 Traces

The norm above converges to zero for n → ∞, therefore so does B. Thus, it is shown that the wn converge to vi in W 1,p (Rd+ ). By definition wn (x) vanishes whenever xd < 1/n and therefore has compact support. Then mollifications (wn )ε with ε < 1/n converge to vi in W 1,p (Rd+ ), have compact support and are smooth, 1,p 1,p see Proposition 3.1.3. This shows vi ∈ W0 (Rd+ ) and also uΦi ∈ W0 (Ω) for N 1,p   each i = 1, . . . , N. Since u = i=0 uΦi it is u ∈ W0 (Ω). In case one only considers a part Γ of the boundary ∂Ω and the corresponding trace operator, there is a very similar result to the previous one. First, similarly to Eq. (4.2) define 1,p

WΓ (Ω) := D(Ω ∪ ∂Ω \ Γ )

W 1,p (Ω)

(4.3)

.

The support of a function in D(Ω ∪ ∂Ω \ Γ ) does not intersect with Γ but it may do so with the rest of the boundary. The proof of the following proposition is not given here. Basically, the same considerations as in the previous proof suffice. Proposition 4.3.2 The kernel of the trace operator TΓ 1,p W 1−1/p,p (Γ ) from Theorem 4.2.4 is WΓ (Ω).

:

W 1,p (Ω)

→

s,p

4.3.1 The Lions–Magenes Spaces W00

Together with traces on Γ one can as well consider traces on the rest ∂Ω \ Γ . In the following the range of the trace operator TΓ restricted to the kernel of T∂Ω\Γ is studied. The resulting spaces are strictly smaller than W 1−1/p,p (Γ ) and typically associated with Lions and Magenes, [LM72] (only p = 2), and denoted 1−1/p,p 1/2 by W00 (Γ ) (and H00 (Γ ) for p = 2). Definition 4.3.3 Let Ω ⊂ Rd be a Lipschitz domain, 0 < s < 1, and p > 1. For a given u ∈ W s,p (Ω) denote its extension by zero to all of Rd by u. ˜ Then define   

s,p W00 (Ω) := u ∈ W s,p (Ω) u˜ ∈ W s,p Rd with norm uW s,p (Ω) := u ˜ W s,p (Rd ) . In case u ∈ W s,p (Γ ) for some Γ ⊂ ∂Ω and 00 u˜ is its extension by zero to all of ∂Ω define   s,p W00 (Γ ) := u ∈ W s,p (Γ ) u˜ ∈ W s,p (∂Ω) ˜ W s,p (∂Ω) . again with norm uW s,p (Γ ) := u 00

4.3 Characterization of the Kernel of the Trace Operator

71

Next, a connection to an appropriate range of traces can be established: Theorem 4.3.4 Let Ω ⊂ Rd be a Lipschitz domain and Γ ⊂ ∂Ω be a part of its boundary which is itself Lipschitz continuous. Then it is   1,p 1−1/p,p TΓ W∂Ω\Γ (Ω) = W00 (Γ ). 1,p

1−1/p,p

Furthermore, as a map from W∂Ω\Γ (Ω) to W00

1−1/p,p

exists a continuous right inverse EΓ : W00

(Γ ) it is continuous and there 1,p

(Γ ) → W∂Ω\Γ (Ω).

1,p

Proof Let u ∈ W∂Ω\Γ (Ω) be given. Note that with v = TΓ u it is v˜ = T∂Ω u ∈ W 1−1/p,p (∂Ω). The continuity of the trace map T∂Ω shows TΓ uW 1−1/p,p (Γ ) = T∂Ω uW 1−1/p,p (∂Ω) ≤ CuW 1,p (Ω) , 00

1,p

1−1/p,p

i.e., the continuity of TΓ : W∂Ω\Γ (Ω) → W00

(Γ ).

1−1/p,p W00 (Γ )

Conversely, let u ∈ be given. Then, by Definition 4.3.3, its extension u˜ by zero is an element of W 1−1/p,p (∂Ω) and can in turn be extended to an element E u˜ of W 1,p (Ω) due to Theorem 4.2.3. According to the construction, 1,p u˜ vanishes on ∂Ω \Γ , i.e., u˜ ∈ W∂Ω\Γ (Ω) and TΓ u˜ = u. Hence, EΓ can be defined as the composition of ˜· (extension by zero to ∂Ω) and E from Theorem 4.2.3. Then it is continuous: EΓ uW 1,p (Ω) = E u ˜ W 1,p (Ω) ≤ Cu ˜ W 1−1/p,p (∂Ω) = CuW 1−1/p,p (Γ ) . 00

 

A slight generalization of these concepts is discussed now. Consider & a domain Ω with three disjoint (connected) parts Γ1 , Γ2 , and Γ3 , such that ∂Ω = 3i=1 Γi . Then the space   1,p TΓ1 WΓ2 (Ω) 1−1/p,p

(Γ1 ) and W 1−1/p,p (Γ1 ) if and only if Γ1 and Γ3 is strictly between W00 have a common boundary. In that case, the extension by zero of a function in   1,p TΓ1 WΓ2 (Ω) is an element of W 1−1/p,p (Γ1 ∪ Γ2 ). The respective trace and extension operators remain continuous. A characterization of the Lions–Magenes spaces that does not consider extensions at all is proved next. It provides an equivalent norm and shows that functions in these spaces vanish at the boundary at a certain rate. A version for a domain Ω is followed by one for its boundary.

72

4 Traces

Proposition 4.3.5 Let Ω ⊂ Rd be a Lipschitz domain, 0 < s < 1, p > 1, and u ∈ W s,p (Ω). Furthermore, define  : Ω → R to be the distance function to the boundary: (x) := dist(x, ∂Ω). Then it holds s,p

u ∈ W00 (Ω) ⇐⇒ u/s ∈ Lp (Ω). Additionally, there exist positive constants C1 and C2 such that  1/p  p p C1 uW s,p (Ω) ≤ uW s,p (Ω) + u/s Lp (Ω) ≤ C2 uW s,p (Ω) . 00

00

Proof This proof can be found in [Tar07], Lemma 37.1. Again, denote the extension of u to all of Rd by u˜ such that uW s,p (Ω) = u ˜ W s,p (Rd ) . 00

“ ⇐ ”: The Lp (Rd )-norm of u˜ is finite, in fact it is equal to the Lp (Ω)-norm of u. The Sobolev–Slobodeckij semi-norm is split as follows: 

p

|u| ˜ W s,p (Rd ) =

|u(x) − u(y)|p Ω×Ω

|x − y|sp+d

 dx dy + 2

|u(x)|p Ω×(Rd \Ω)

|x − y|sp+d

p

dy dx. p

The first integral above is |u|W s,p (Ω) and the second can be written as uϕLp (Ω) where ϕ is defined for all x ∈ Ω such that  1 |ϕ(x)|p = dy. sp+d d |x − y| R \Ω Note that a ball around x with radius (x) is a subset of Ω, hence  |ϕ(x)|p ≤



1 Rd \B(x,(x))

|x − y|

sp+d

dy =

1 Rd \B(0,(x))

|z|sp+d

dz = ωd (x)−sp ,

where polar coordinates are used. A similar reasoning is employed in the proof p of Proposition 3.4.3 as well. The norm uϕLp (Ω) therefore can be bounded by p ωd u/s Lp (Ω) which is finite by assumption. Together, this shows that there is a constant C such that    p p p p uW s,p (Rd ) = u ˜ W s,p (Rd ) ≤ C uW s,p (Ω) + u/s Lp (Ω) , 00

i.e., setting C1 = C −1 suffices. “ ⇒ ”: Consider a localization (Vi , hi , φi ), i = 0, . . . , N, of Ω, see Definition 2.2.1, then it is   u/s p p = L (Ω)

p  N N   (φi u)(x) p i=0 (φi u)(x) p−1 dx ≤ (N +1) s (x) dx. s (x) Ω Ωi ∩Ω i=0

4.3 Characterization of the Kernel of the Trace Operator

73

The term for i = 0 in the above sum is bounded by the Lp (Ω)-norm of u, because φ0 has compact support in Ω. For i ≥ 1 one would like to straighten out the boundary. To find out how s in the denominator can be handled, for x = (x  , xd ) ∈ Ωi ∩ Ω, let p(x) = (y  , hi (y  )) ⊂ ∂Ω ∩ Ωi be a point such that (x) = dist(x, p(x)) = |x − p(x)|. Then it is xd − hi (x  ) ≤ xd − hi (y  ) + hi (y  ) − hi (x  ) ≤ xd − hi (y  ) + ∇hi ∞ x  − y 

(4.4)

≤ C(x), where C depends on the Lipschitz continuity of hi and on the space dimension d. Additionally, assuming the 1-norm on Rd for simplicity, with each y ∈ Rd+ and Rd− = Rd−1 × (−∞, 0) it is 

1 Rd−

|x − y|sp+d

 dx =

∞ yd

−sp−1 zd dzd

ωd−1 −sp = y sp d

 0

∞

 

1 Rd−1

|1 + |z ||sp+d





dz

r d−1 dr, (1 + r)sp+d

where at first the transformation zd = xd − yd , zi = (xi − yi )/(xd − yd ), i = 1, . . . , d − 1, is applied, while in the second equation polar coordinates are employed. Combining all the constants above into K yields  Rd−

−sp

|x − y|−sp−d dx = Kyd

.

(4.5)

With these results the relevant norm (i ≥ 1) can be bounded using the transformation (y  , yd ) = (x  , xd − hi (x  )): 

 (φi u)(x) p (φi u)(x) p (4.4) dx dx ≤ C sp s (x)  s Ωi ∩Ω Ωi ∩Ω |xd − hi (x )| p  (φi u)(y  , yd + hi (y  )) sp =C dy |yd |sp Rd+ p   (4.5) C sp (φi u)(y  , yd + hi (y  )) ≤ dx dy. K Rd+ Rd− |x − y|sp+d

74

4 Traces

The function vi : y → (φi u)(y  , yd + hi (y  )) in the numerator is in W s,p (Rd+ ) due to Propositions 3.2.1 and 3.2.3 and its extension by zero v˜ is even in W s,p (Rd ) because it is the transformed version of u. ˜ Hence, it is  (φi u)(x) p 1 p ˜ W s,p (Rd ) . s (x) dx ≤ CK v Ωi ∩Ω Combining all these results gives the bound u/s Lp (Ω) ≤ Cu ˜ W s,p (Rd ) for some 1/p p   constant C and hence the claim with C2 = (1 + C ) . s,p

The previous proposition shows the equivalence of two norms on W00 (Ω). On the boundary ∂Ω essentially the same result holds: Proposition 4.3.6 Let Ω ⊂ Rd be a Lipschitz domain, Γ ⊂ ∂Ω a part of its boundary which itself is Lipschitz continuous, s ∈ (0, 1), p > 1. Then it is s,p

u ∈ W00 (Γ ) ⇐⇒ u/s ∈ Lp (Γ ), where  is the distance from ∂Γ within Γ . An explicit proof is not given here. Just note that transforming to local coordinates in Rd−1 and extension by zero commute. Then, locally, Proposition 4.3.5 yields the desired result. Note that this representation is not affected by the existence of more than two boundary parts. Furthermore, it is completely local in the sense that no information of the rest ∂Ω \ Γ is needed.

4.4 Integration by Parts Having established a well defined notion of trace for Sobolev spaces, the important integration by parts formula can be extended to these spaces. Corollary 4.4.1 For all ϕ ∈ H 1 (Ω) and all v ∈ H 1 (Ω) it holds2 (v, ∇ϕ)0 + (∇ · v, ϕ)0 = (v · n, ϕ)0,∂Ω , where on the right-hand side both v and ϕ have to be understood in the sense of traces as in Theorem 4.1.1.  d Proof For all ϕ ∈ D(Ω) and v ∈ D(Ω) integration by parts yields  (v, ∇ϕ)L2 (Ω) + (∇ · v, ϕ)L2 (Ω) =

2 As

ϕ(x)(v · n)(x) dσx . ∂Ω

in Sect. 2.3 it is (·, ·)0 = (·, ·)L2 (Ω) and (·, ·)0,M = (·, ·)L2 (M) for sets M.

(4.6)

4.5 A Right Inverse of the Divergence Operator

75

Both sides are linear in v as well as ϕ and can be bounded by the product vH 1 (Ω) ϕH 1 (Ω) with a suitable constant. Due to Theorem 3.4.5 this equation holds also for ϕ ∈ H 1 (Ω), where on the right-hand side ϕ has to be understood in the sense of a trace. The same reasoning applies to v ∈ H 1 (Ω) where its trace is multiplied by the normal n which is in L∞ (∂Ω).   Remark 4.4.2 For p = 2 an integration by parts formula can be shown with the exact same steps. Let q be given such that p1 + q1 = 1. Then for all ϕ ∈ W 1,p (Ω) and all v ∈ W1,q (Ω) it is    v(x) · ∇ϕ(x) dx + (∇ · v)(x)ϕ(x) dx = ϕ(x)v · n(x) dσx . Ω

Ω

∂Ω

Remark 4.4.3 The integration by formula can be extended to hold also for  parts d  vector fields v ∈ H (div, Ω) = v ∈ L2 (Ω) ∇ · v ∈ L2 (Ω) . Such functions also admit a trace but only in normal  direction and ∗ the inner product on the boundary then turns into a duality product in H 1/2(∂Ω) ×H 1/2 (∂Ω). Special care then has to be taken if ϕ is in a space such as HΓ1 (Ω) or if the normal trace v · n vanishes on a part of the boundary. See [GR86], Theorems 2.2 and 2.3 and their corollaries in Chapter 1 as well as Proposition 3.58 in [DDE12] (also for p = 2).

4.5 A Right Inverse of the Divergence Operator In order to prove the existence and uniqueness of a solution to the Stokes equations in Sect. 5.3.2, a vector field u is used whose divergence is a given function f ∈ p 1,p Lp (Ω). It turns out that if f ∈ L0 (Ω), there is a function u ∈ W0 (Ω) such that p ∇ · u = f and uW 1,p (Ω) ≤ f Lp (Ω) . Note that L0 (Ω) is the subspace of Lp (Ω) with functions whose integral over Ω vanishes, see Eq. (2.9). The Sobolev space 1,p W0 (Ω) is the completion of smooth functions with compact support in W 1,p (Ω), see Definition 2.4.2 and Eq. (4.2). The typical proof of the existence of such a vector u for p = 2 uses results which are not (yet) shown in this monograph, namely higher regularity results and existence and uniqueness of solutions to Laplace problems. Such proofs can be found in [GR86], Lemma 3.2, and [Tem77], Lemma 2.4. There is however another approach which explicitly constructs the vector field u for a given f . Its proof is more involved and not presented in full detail here. Another advantage is that this works not only for p = 2. The following proposition shows that the divergence has a right inverse when restricted to appropriate spaces.

76

4 Traces

Proposition 4.5.1 Let Ω ⊂ Rd be a bounded Lipschitz domain, which is starp shaped3 with respect to a ball B ⊂ Ω, and p > 1. For all f ∈ L0 (Ω), there exists 1,p a u ∈ W0 (Ω) such that ∇·u=f

∇u(Lp (Ω))d×d ≤ f Lp (Ω) .

and

Proof Only a part of the proof is given here. The remaining details can be found in [DM01], see also [Rus13], Theorem 1 and the references therein. Let ω ∈ D(Ω)  be a test function whose support is contained in B and whose integral is one, Ω ω(x) dx = 1. Next, for any test function ϕ ∈ D(Ω) denote  ϕ :=

ϕ(x)ω(x) dx. Ω

Then for any x ∈ Ω it is  ϕ(x) − ϕ =

(ϕ(x) − ϕ(y))ω(y) dy Ω



 =

1

ω(y)

= 0

1



(x − y) · ∇ϕ(y + s(x − y)) ds dy

0

Ω





 z−x (x − z) · ∇ϕ(z) ω x + dz dt t d+1 t 1

Ω

G(z, x) · ∇ϕ(z) dz,

=− Ω

with 

1

G(z, x) := (z − x) 0

 z−x dt. ω x+ t t d+1 1

In the second equality above, the fundamental theorem of calculus is applied, in the third the transformations z = (1 − s)y + sx and t = 1 − s. The vector G(z, x) is bounded, because due to ω having compact support, the integrand is zero for t < t0 for some positive t0 which depends on |z − x| and the diameter of Ω. Hence, changing the order of integration is possible (Fubini’s theorem).

domain Ω ⊂ Rd is called star-shaped with respect to B ⊂ Ω, if for all x ∈ Ω and all y ∈ B the connecting line segment {z ∈ Rd | ∃t ∈ [0, 1] : z = y + t (x − y)} is a subset of Ω.

3A

4.5 A Right Inverse of the Divergence Operator

77

p

Since f ∈ L0 (Ω) by assumption, it is 

 f (x)ϕ(x) dx = Ω

f (x)(ϕ(x) − ϕ) dx Ω

  f (x)G(z, x) · ∇ϕ(z) dz dx

=− 

Ω

=−

Ω

u(z) · ∇ϕ(z) dz, Ω

where u is defined to be  u(z) :=

G(z, x)f (x) dx. Ω

According to the definition of the weak derivative, the above calculations show that u is the divergence of f , ∇ · u = f , see Definition 2.4.1. Additionally, G(z, x) = 0 for z ∈ ∂Ω, because y = x + (z − x)/t ∈ / B (otherwise it would be z = ty + (1 − t)x ∈ Ω because Ω is assumed to be star-shaped). Hence, it is u = 0 on ∂Ω. It remains to show that u ∈ W1,p (Ω) and the bound on its gradient. Note that due to Hölder’s inequality, Theorem 2.3.1, it is |u(z)| ≤ G(z, ·)(Lq (Ω))d f Lp (Ω) and therefore u(Lp (Ω))d ≤ Cf Lp (Ω) , i.e., u ∈ (Lp (Ω))p . In [DM01] it is shown also that u ∈ W1,p (Ω) and ∇u(Lp (Ω))d×d ≤ f Lp (Ω) as claimed.   The restriction on the domain to be star-shaped can be dropped to yield: Theorem 4.5.2 Let Ω ⊂ Rd be a bounded Lipschitz domain and p > 1. For all p 1,p f ∈ L0 (Ω), there exists a u ∈ W0 (Ω) such that ∇ ·u=f

and

∇u(Lp (Ω))d×d ≤ Cf Lp (Ω) ,

with a constant C independent of f . Proof Consider a covering of the domain Ω with subdomains Ωi ⊂ Ω, i = 1, . . . , N, which are star-shaped with respect to some open ball Bi ⊂ Ωi . Such a covering exists, because Ω is assumed  to be Lipschitz continuous. Additionally, let φi ∈ D(Ωi ) be a partition of unity, i.e., N i=1 φi = 1 on Ω. Due to Proposition 4.5.1 1,p there exist functions ui ∈ W0 (Ωi ) such that ∇ · ui = f φi in Lp (Ωi ) and through extension by zero also in Lp (Ω). Furthermore, it is ∇ui (Lp (Ω))d×d ≤  1,p (Ω), f φi Lp (Ω) . Then the combined function u = N i=1 ui is an element of W see Proposition 3.5.1. Furthermore, u vanishes on the boundary because all the ui

78

4 Traces 1,p

do, hence u ∈ W0 (Ω). Finally, it is ∇ ·u=

N 

∇ · ui = f

i=1

N 

φi = f

i=1

and ∇u(Lp (Ω))d×d ≤

N 

∇ui (Lp (Ω))d×d ≤

i=1

N 

f φi Lp (Ω) ≤ Cf Lp (Ω) .

 

i=1

This result can be extended to more general, so-called, John domains, see [Rus13], Theorem 2. Furthermore, the above Theorem 4.5.2 is equivalent to other well known results such as the Jaques-Louis Lions lemma, the Neças inequality, or a “coarse” version of de Rhams’s theorem, see [ACM14, ACM15] and the references therein.

4.6 Equivalent Norms on H 1 (Ω) In the analysis of partial differential equations it is sometimes necessary to define norms on (sub spaces of) H 1 (Ω) which are equivalent to the standard one. Socalled Poincaré type inequalities often provide the proof of this equivalence and some of its variants are presented here. The proof of the Poincaré type inequalities use the Rellich–Kondrachov theorem which is shown for C 1 domains in [Eva98], Theorem 1 in Section 5.7, and in [AF03], Theorem 6.3, for the case of a domain satisfying a cone condition, which Lipschitz domains do. See also a proof using extensions in [Tar07], Lemma 14.5. Theorem 4.6.1 (Rellich–Kondrachov) Let Ω be a Lipschitz domain in Rd , p ≥ 1. Then W 1,p (Ω) is compactly embedded in Lp (Ω). In other words, any uniformly bounded sequence in W 1,p (Ω) has a subsequence which converges in Lp (Ω). With the help of the Rellich–Kondrachov theorem it is possible to prove the Poincaré inequality. It can be found in many textbooks which treat Sobolev spaces, see for example [Tri92], Theorem 3 in Section 6.1.5, or Section 10 in [Tar07]. Theorem 4.6.2 (Poincaré Inequality) on Ω such that for all u ∈ H 1 (Ω) it is

There is a constant cP which only depends

u − u0 ≤ cP ∇u0 , where u is the mean value of u, u = |Ω|−1

 Ω

u(x) dx.

(4.7)

4.6 Equivalent Norms on H 1 (Ω)

79

Proof See § 5.8 in [Eva98]. Assume the inequality is not true, then there is a sequence uk ∈ H 1 (Ω) such that uk − uk 0 ≥ k∇uk 0 . Then define vk :=

uk − uk . uk − uk 0

This sequence has the properties vk 0 = 1, by construction vk = 0, and by assumption ∇vk 0 ≤ 1k , i.e., it is a uniformly bounded sequence in H 1 (Ω). Now the Rellich–Kondrachov theorem, Theorem 4.6.1, states there is a subsequence, still denoted by vk , which converges in L2 (Ω) to, say, v ∈ L2 (Ω). Due to continuity of the norm it is v0 = 1. Furthermore, for any φ ∈ D(Ω) it is  v Ω

∂φ = lim k→∞ ∂x i

 vk Ω

∂φ = − lim k→∞ ∂x i

 Ω

∂vk φ = 0, ∂x i

which means that v is in H 1 (Ω) and has vanishing weak derivatives, implying v is constant, v = |Ω|−1/2 . Since taking the mean value is continuous on H 1 (Ω) there is a contradiction 0 = lim vk = v = |Ω|−1/2 . k→∞

(4.8)  

Remark 4.6.3 In the proof of the previous theorem it is only used that taking the mean value, v → v is linear and continuous and that the constant function 1 is mapped to unity. In fact if one replaces the mean value by any such functional l the theorem is still valid, i.e., for a given linear and continuous l : H 1 (Ω) → R with l(1) = 1 it is u − l(u)0 ≤ cP ∇u0 .

(4.9)

However, cP might now depend on l. With the help of the Poincaré inequality one can easily prove that |||v||| := ∇v0 + |v|. is an equivalent norm on H 1 (Ω). Using the previous Remark 4.6.3, the mean value v can be replaced by l(v). Another common approach to define equivalent norms on H 1 (Ω) uses bilinear forms. For C 1 -domains a proof can also be found in [Tri92], Theorem 1, Paragraph 6.1.5.

80

4 Traces

Theorem 4.6.4 Let l : H 1 (Ω) → R be linear and continuous and let e : H 1 (Ω) × H 1 (Ω) → R be a continuous, coercive, and symmetric bilinear form with the additional property l(1)2 + e(1, 1) > 0. Then  |||v||| := ∇v20 + |l(v)|2 + e(v, v) defines an equivalent norm on H 1 (Ω). Proof To show that |||·||| in fact defines a norm, at first two properties of the bilinear form e are derived which are very similar to those of an inner product. Let v, w ∈ H 1 (Ω) and λ = e(v, v)/e(v, w). Then 0 ≤ e(v − λw, v − λw) = e(v, v) − 2λe(v, w) + λ2 e(w, w) = e(v, v) − 2e(v, v) + ⇒ e(v, w) ≤



 e(v, v) e(w, w),

e(v, v)2 e(w, w) e(v, w)2

which gives e(v + w, v + w) = e(v, v) + 2e(v, w) + e(w, w)   ≤ e(v, v) + 2 e(v, v) e(w, w) + e(w, w) 2   e(v, v) + e(w, w) . = Next, the properties of a norm are checked: Let again v, w ∈ H 1 (Ω) and η ∈ R be given, then |||v||| = 0 ⇐⇒ ∇v0 = 0 ⇐⇒ v(x) = c

and and

|l(v)|2 + e(v, v) = 0   |l(v)|2 + e(v, v) = c2 |l(1)|2 + e(1, 1) = 0

⇐⇒ v = 0, |||ηv||| = |η| |||v||| , |||v + w|||2 ≤ (∇v0 + ∇w0 )2 + (|l(v)| + |l(w)|)2  2  + e(v, v) + e(w, w) = |||v|||2 + |||w|||2     + 2 ∇v0 ∇w0 + |l(v)||l(w)| + e(v, v) e(w, w) ≤ |||v|||2 + |||w|||2 + 2 |||v||| |||w||| = (|||v||| + |||w|||)2 .

4.6 Equivalent Norms on H 1 (Ω)

81

The second estimate above uses the fact that for all nonnegative a1 , a2 , b1 , and b2 1/2  2 1/2  a2 + b22 it is a1 a2 + b1 b2 ≤ a12 + b12 and hence a1 a2 + b1 b2 + c1 c2 ≤ 1/2  2 1/2  2 2 2 2 2 for positive c1 and c2 . Due to the continuity of a 2 + b 2 + c2 a 1 + b 1 + c1 l and e the triple norm |||·||| is bounded by the standard norm on H 1 (Ω), i.e., for all v ∈ H 1 (Ω) it is 1/2 1/2   |||v||| ≤ ∇v20 + l2 v21 + ev21 v1 . ≤ 1 + l2 + e (4.10) Next, the opposite is shown, namely: There is a constant c > 0 such that for all u ∈ H 1 (Ω) it is u1 ≤ c |||u|||. To this end, it is enough to show u0 ≤ c |||u||| (for some other constant c). Assume the opposite is true, i.e., there exists a sequence 1 {uk }∞ k=1 ⊂ H (Ω) such that 1 = uk 0 > k |||uk |||

⇒

∇uk 0 ≤ |||uk |||
0 according to the assumption of the theorem, hence, there is a contradiction.   Remark 4.6.5 If l = 0 in the previous theorem, the norm also corresponds to an inner product which can be shown with the parallelogram law (2.2) and Eq. (2.3) |||v||| =



((v, v)),

with

((v, w)) := (∇v, ∇w)0 + e(v, w).

Note, that Theorem 4.6.4 is a generalization of the Poincaré inequality, Theorem 4.6.2, setting l to be the mean value and e to zero. Next a few examples are provided.

82

4 Traces

Example 4.6.6 On H 1 (Ω) the following norms are equivalent: v1 = v0 + ∇v0 ,  |||v||| = ∇v0 +

Ω0

ηv ,

where Ω0 ⊂ Ω has positive measure and η ∈ L2 (Ω0 ) with  Ω0 η(x) dx = 0,

 |||v||| = ∇v0 + ηv ,

where Γ ⊂ ∂Ω has positive measure and η ∈ L2 (Γ ) with  Γ η(x) dx = 0,

Γ

  |||v||| = ∇v20 +

1/2

Ω0

where Ω0 ⊂ Ω has positive measure and η ∈ L∞ (Ω0 ) is positive,

Γ

where Γ ⊂ ∂Ω has positive measure and η ∈ L∞ (Γ ) is positive.

ηv 2

 1/2  2 2 |||v||| = ∇v0 + ηv ,

,

Often one is interested in the subspace HΓ1 (Ω) where the last norm in the previous example reduces to ∇v0 which is therefore an equivalent norm on HΓ1 (Ω), in particular it is v0 ≤ CP ∇v0 ,

(4.12)

with the Poincaré constant CP ≥ 1. This equation is often referred to as the Poincaré inequality in the literature and proved without the use of the Rellich– Kondrachov theorem 4.6.1. Then the constant CP is given by s if Ω is contained in a d-dimensional cube of edge length s, see [Bra07a]. Remark 4.6.7 Due to the inequality of Korn (3.4) it is possible to define equivalent norms on H1 (Ω) similar to the ones above where the part ∇u0 is replaced by D(u)0 . In case the space is restricted on the boundary, another Poincaré type inequality holds: Kikuchi and Oden prove in [KO88] that there is a constant c > 0  such that for all v ∈ H1Γ (Ω) = v ∈ H1 (Ω) v = 0 on Γ it is v1 ≤ c D(v)0 .

(4.13)

Chapter 5

Subproblems Individually

In this section the existence and uniqueness of solutions of the involved subproblems are proved. Besides the Laplace equation this includes the so-called saddle point theory which is an abstract framework and is then applied to the Stokes equations as well as to the coupled Stokes–Darcy problem in Chap. 6. In each case, let Ω ⊂ Rd be an open domain with Lipschitz boundary. As before, the exterior unit normal vector on the boundary ∂Ω is denoted by n, see Remark 2.5.1. First, the Laplace equation is introduced followed by the general formulation of saddle point problems. The last section in this chapter applies the abstract theory to the Stokes problem. Furthermore, for each of these problem classes, operators which map data on part of the boundary to solutions in the domain are introduced. These are useful to understand the Stokes–Darcy coupling in Chap. 6. Each equation is introduced in a similar manner. Starting from the partial differential equation a weak form is derived, for which existence and uniqueness of solutions is established. Finally, the solving can be viewed as an operator mapping data on a part Γ of the boundary to a solution inside the domain Ω. This data may be either of Dirichlet or Robin type resulting in operators KD and KR . Some interesting and helpful properties of both KD and KR are shown.

5.1 Laplace Equation The Laplace equation and its extensions model a wide variety of applications. In this work it is the pressure of a fluid in a porous medium. This is also why the solution is denoted by p instead of the much more common notation u.

© Springer Nature Switzerland AG 2019 U. Wilbrandt, Stokes–Darcy Equations, Advances in Mathematical Fluid Mechanics, https://doi.org/10.1007/978-3-030-02904-3_5

83

84

5 Subproblems Individually

sufficient conditions on b Proposition 5.2.6

existence and uniqueness for saddle point problems Theorem 5.2.2

sufficient conditions on a Proposition 5.2.5

existence and uniqueness for the Laplace equation Theorem 5.1.2

existence and uniqueness for the Stokes equation Theorem 5.3.5

Dirichlet operator Proposition 5.3.7

Dirichlet operator Proposition 5.1.3

Robin operator Proposition 5.3.8

Robin trace Definition 5.3.9

Robin operator Proposition 5.1.4

Robin trace Definition 5.1.5

Fig. 5.1 A graph showing the dependencies among all theorems, definitions, as well as propositions in this chapter. Note that implicit dependencies are not always shown

The Laplace problem is as follows: Find p : Ω → R such that − ∇ · (K∇p) = f

in Ω.

(5.1)

Here the source term f : Ω → R and the hydraulic conductivity tensor K : Ω → Rd×d are given data. Generally, it is assumed that K is symmetric and positive definite. Additionally, a solution p has to satisfy boundary conditions. Define two disjoint (relatively open) subsets ΓD and ΓR which in turn have a Lipschitz boundary, such that ΓD ∪ ΓR = ∂Ω. The boundary conditions read 

p = pD

on ΓD ,

K∇p · n + γp = gR

on ΓR .

(5.2)

Here pD : ΓD → R, gR : ΓR → R and γ : ΓR → R are given functions. The coefficient function γ is not negative, γ (x) ≥ 0 for x ∈ ΓR . The first boundary condition is of Dirichlet type, while the second is called Robin boundary condition. The case γ = 0 is referred to as of Neumann type. These different restrictions on the boundary are treated differently in the weak formulation. The Dirichlet condition is enforced in the definition of the solution space whereas the Robin (or Neumann respectively) condition becomes part of the formulation itself. Often the former are therefore called essential and the latter natural.

5.1 Laplace Equation

85

5.1.1 Weak Form Assume the above Eq. (5.1) admits a smooth solution p ∈ D(Ω) satisfying the boundary conditions (5.2). Next, multiply Eq. (5.1) with a test function q ∈ C ∞ (Ω) which vanishes in a neighborhood of the Dirichlet boundary part ΓD , q ∈ D(Ω ∪ ∂Ω \ ΓD ). Integrating this equation over the domain Ω and integration by parts lead to (K∇p, ∇q)0 − (K∇p · n, q)ΓR ,0 = (f, q)0 . The integral over the Robin boundary can be replaced using the second condition in (5.2) yielding a(p, q) = (q)

(5.3)

with the bilinear form a and linear form  defined by a(p, q) := (K∇p, ∇q)0 + (γp, q)0,ΓR , (q) := (f, q)0 + (gR , q)0,ΓR . Through approximation Eq. (5.3) can be meaningfully interpreted for p ∈ H 1 (Ω) and q ∈ HΓ1D (Ω), see Theorem 3.4.5 and Eq. (4.3) on page 70. Therefore, the bilinear form a maps from H 1 (Ω) × H 1 (Ω) to R and the data must fulfill the following minimal smoothness properties  ∗ f ∈ HΓ1D (Ω) ,  d×d K ∈ L∞ (Ω)

symmetric and positive definite,

pD ∈ H 1/2(ΓD ), γ ∈ L∞ (ΓR ),  ∗ 1/2 gR ∈ H00 (ΓR ) . Note that in case f and gR are not in more regular spaces (e.g. L2 (Ω) and L2 (ΓR )) the definition of the functional  must be understood as the respective duality pairings (q) = (f, q)

HΓ1 (Ω) D

∗

×HΓ1 (Ω) D

+ (gR , q)

1/2

H00 (ΓR )

∗

1/2

×H00 (ΓR )

.

86

5 Subproblems Individually

For the ease of presentation many authors still write it as an L2 inner product. Summarizing, the weak formulation of the Laplace equation (5.1) together with the boundary condition (5.2) is: Find p ∈ H 1 (Ω) such that p − pD ∈ HΓ1D (Ω) and for all q ∈ HΓ1D (Ω) it is a(p, q) = (q).

(5.4)

Remark 5.1.1 The expression p−pD is at first not reasonable, because p and pD do not belong to the same vector space. Instead, a continuous extension of pD according to Theorem 4.2.4 is used. It is called pD again to simplify notation, i.e., pD 1,Ω ≤ cEΓD pD 1/2,ΓD .

5.1.2 Existence and Uniqueness of a Solution The theorem of Lax–Milgram, Theorem 2.1.13, is the key to establish existence and uniqueness of a solution to the weak formulation (5.4) of the Laplace equation. The Hilbert space in that theorem is Q = HΓ1D (Ω). Since the solution p is in general not in Q (unless pD = 0), an equivalent problem is considered: As described in the previous remark, let pD denote not just the essential boundary data but also an extension of it into the domain Ω, pD ∈ H 1 (Ω) and let P = p − pD ∈ Q. Then consider the modified problem a(P , q) = (q) − a(pD , q).

(5.5)

Solving this problem is equivalent to the original weak formulation, p can be recovered as P + pD . This transformation is often used to justify an assumption of homogeneous Dirichlet data pD . In order to apply the Lax–Milgram theorem, Theorem 2.1.13, to the bilinear form a restricted to Q and the right-hand side L(q) = (q) + a(pD , q), one has to show continuity of a and L as well as the coercivity of a. This is done step-bystep in the following. Then, existence and uniqueness of the weak formulation (5.4) is established.

5.1.2.1 Continuity of a Let p, q ∈ Q be given, then, using the Hölder inequality, Theorem 2.3.1, to extract the L∞ -terms and a second time to split the products yields a(p, q) = (K∇p, ∇q)0 + (γp, q)0,ΓR ≤ K(L∞ (Ω))d×d ∇p · ∇qL1 (Ω) + γ L∞ (ΓR ) p qL1 (ΓR )

5.1 Laplace Equation

87

≤ K(L∞ (Ω))d×d p1 q1 + γ L∞ (ΓR ) p0,ΓR q0,ΓR ≤ ca p1 q1 . The constant ca depends on the hydraulic conductivity tensor K, the boundary data γ , and the domain Ω in form of the continuity constant of the trace operator mapping Q ⊂ H 1 (Ω) to L2 (ΓR ), Theorem 4.1.1.

5.1.2.2 Continuity of L Let q ∈ Q be given, then the continuity of the data f and gR as well as of the bilinear form a yields L(q) = (f, q)

HΓ1 (Ω)

∗

D

≤ f 

HΓ1 (Ω)

 ≤ f 

×HΓ1 (Ω)

+ (gR , q)

D

∗ q 1

+ gR 

D

HΓ1 (Ω)

∗

H00 (ΓR )

+ cTΓ gR 

D

1/2

1/2

1/2

H00 (ΓR )

∗

1/2

×H00 (ΓR )

∗ q 1/2 H00 (ΓR )

H00 (ΓR )

∗

− a(pD , q)

+ ca pD 1 q1

p  + ca D 1 q1

= cL q1 .

(5.6)

The constant cL depends on ca and additionally on f , gR , and the continuity 1/2 constant CTΓ of the trace operator mapping Q to H00 (ΓR ), see Theorem 4.3.4. 5.1.2.3 Coercivity of a Let q ∈ Q be given, using the lower bounds on the hydraulic conductivity tensor K and γ ≥ 0 leads to a(q, q) = (K∇q, ∇q)0 + (γ q, q)0,ΓR ≥ αK ∇q20 + inf γ (x) (q, q)0,ΓR x∈ΓR

≥ αa q21 . The constant αa depends on αK , the constant describing the positive definiteness of K, and on the inverse of the Poincaré constant CP from Eq. (4.12). Note that αa scales with K. If γ is not zero and q has a nonzero trace on the Robin boundary ΓR the constant αa could be chosen larger with an additional dependency on the boundary coefficient γ .

88

5 Subproblems Individually

5.1.2.4 Application of the Lax–Milgram Theorem According to the theorem of Lax–Milgram, Theorem 2.1.13, there exists a unique solution P ∈ Q to (5.5) and consequently a unique solution p = P + pD to the weak formulation (5.4) of the Laplace problem (5.1). For reference, this is repeated here in detail: Theorem 5.1.2 Let Ω ⊂ Rd be a Lipschitz domain together with two disjoint subsets ΓD and ΓR of the boundary ∂Ω, which are themselves Lipschitz, such that ΓD ∪ ΓR = ∂Ω. Furthermore, let  d×d K ∈ L∞ (Ω) ,

γ ∈ L∞ (ΓR ),

where K is symmetric and positive definite, γ nonnegative, and ∗  f ∈ HΓ1D (Ω) ,

 ∗ 1/2 gR ∈ H00 (ΓR ) .

pD ∈ H 1/2 (ΓD ),

Then there exists a unique solution p ∈ H 1 (Ω) such that TΓD p = pD , i.e., p = pD on ΓD , and for all q ∈ HΓ1D (Ω) it is a(p, q) = (q),

(5.4 revisited)

with a(p, q) := (K∇p, ∇q)0 + (γp, q)0,ΓR ,   ∗ + gR , TΓR q  (q) := (f, q) 1 1 HΓ (Ω) D

×HΓ (Ω)

1/2

H00 (ΓR )

D

∗

1/2

×H00 (ΓR )

.

An a priori estimate for the solution and right-hand side is (see also the end of the proof of the theorem of Lax–Milgram, Theorem 2.1.13) given for P = p − pD from Eq. (5.5) as P 1 ≤

cL , αa

cL ≤ ca P 1 . These bounds lead to p1 − pD 1 ≤ turn imply

cL αa

and cL ≤ ca p1 + ca pD 1 which in

cL c˜L p1 ≤ + pD 1 = + αa αa c˜L ≤ ca p1 ,



ca + 1 pD 1 , αa

(5.7) (5.8)

5.1 Laplace Equation

89

with c˜L = cL − ca pD 1 = f 

HΓ1 (Ω)

∗

+ cTΓ gR 

D

1/2

H00 (ΓR )

∗ ,

see also the definition of cL in Eq. (5.6). In the case of a small hydraulic conductivity K the constant on the right-hand side in the first estimate (5.7) becomes large.

5.1.3 View as an Operator on a Part of the Boundary The existence and uniqueness result for the Laplace problem, Theorem 5.1.2, and the bounds above allow the definition of particular extension operators. For example, setting all data to be zero except pD on the Dirichlet boundary and solving Eq. (5.4) yields a continuous extension operator, similar to the one defined in Theorem 4.2.4. Analogously, setting all data to zero except for the Robin data gR , i.e., solving a Neumann problem, leads to a continuous extension operator, similar to that defined in Theorem 4.3.4. Such operators are introduced in detail in the following, which facilitates the coupling of Stokes–Darcy equations in Chap. 6. Suppose the boundary ∂Ω is composed of three rather than two disjoint parts, ∂Ω = ΓD ∪ ΓR ∪ Γ . Then the weak solution of the Laplace equation with either Dirichlet or Robin data on Γ exists according to the previously introduced theory. Keeping other data fixed, one can define operators taking data on Γ and returning the weak solution. To make this more precise, define KD : H 1/2(Γ ) → H 1 (Ω), λ → KD (λ), where for all v ∈ QD = HΓ1D ∪Γ (Ω) the function φλ := KD (λ) solves (K∇φλ , ∇v)0 + (γ φλ , q)0,ΓR = (f, q)Q∗D ×QD + (gR , q)TΓ

R (QD )

∗ ×T (Q ) ΓR D

and has the correct traces TΓD φλ = pD and TΓ φλ = λ. Note that the space TΓR (QD ) 1/2 is H00 (ΓR ), see also Theorem 4.3.4. There are restrictions on the pre-image space of KD because the combined data on the entire Dirichlet boundary ΓD ∪ Γ has to be in H 1/2. This is made more precise in Proposition 5.1.3.

90

5 Subproblems Individually

Instead of Dirichlet data on Γ consider QR = HΓ1D (Ω) and Robin conditions to define a second operator KR : (TΓ QR )∗ → H 1 (Ω), μ → KR (μ), where for all q ∈ QR the function φμ := KR (μ) solves     K∇φμ , ∇q 0 + γ φμ , q 0,Γ

R ∪Γ

= (f, q)Q∗ ×QR + (gR , q)TΓ R

R

∗ (QR ) ×TΓR (QR )

+ (μ, q)(TΓ (QR ))∗ ×TΓ (QR )

and has the correct trace TΓD φλ = uD . Of special interest is the case where all data entering the right-hand side vanishes except the ones on Γ , that is f , pD , and gR . The coefficients K and γ remain as before. Then both KD and KR are linear operators which are continuous and bounded away from zero, which is shown in the following. Furthermore, they allow a deeper understanding of the Stokes–Darcy coupled problem in Chap. 6.

5.1.3.1 Dirichlet Operator on Γ Consider the case where Dirichlet data is prescribed on the boundary Γ . Since the data on the other Dirichlet boundary ΓD is now assumed to be zero, any function λ in the pre-image space of KD must be in H 1/2 (ΓD ∪ Γ ), when extended by zero outside of Γ . This imposes no further restriction if Γ and ΓD do not touch, i.e., λ ∈ H 1/2(Γ ). If however ΓD and Γ do touch then λ must be in a Lions–Magenes1/2 1/2 type space between H 1/2(Γ ) and H00 (Γ ). It is H00 (Γ ) if for example ΓR is   empty. In general, this space can be characterized as TΓ HΓ1D (Ω) and is denoted 1/2

by HΓD (Γ ) for brevity. 1/2

Proposition 5.1.3 Define the operator KD : HΓD (Γ ) → HΓ1D (Ω), where for all q ∈ HΓ1D ∪Γ the function φλ := KD (λ) solves (K∇φλ , ∇q)0 + (γ φλ , q)0,ΓR = 0

(5.9)

and TΓ φλ = λ. This operator is linear, continuous and positive, i.e., there exist constants cKD and αKD such that KD (λ)1 ≤ cKD λH 1/2 (Γ ) , ΓD

KD (λ)1 ≥ αKD λH 1/2 (Γ ) . ΓD

5.1 Laplace Equation

91

Proof The linearity of KD follows directly from the linearity of Eq. (5.9) and of the extension operator EΓ . Continuity is shown in the section on the existence and uniqueness of the Laplace weak form, Sect. 5.1.2, with a constant cKD depending on the continuity of the bilinear form, ca , and the inverse of its coercivity constant αa , in particular it is cKD = (ca /αa + 1)cEΓ , see Theorem 5.1.2 and the bound (5.7) where here it is c˜L = 0. The continuity of the trace operator TΓ , see Theorem 4.3.4 and the subsequent discussion, gives λH 1/2 (Γ ) = TΓ (KD (λ))H 1/2 (Γ ) ≤ cTΓ KD (λ)1 ΓD

ΓD

and therefore the positivity of KD with αKD = cT−1 . Γ

 

The operator KD defined above is therefore another linear and continuous right inverse of TΓ , see Theorem 4.3.4 and the subsequent discussion.

5.1.3.2 Robin Operator on Γ Consider the case where Robin data is prescribed on the boundary Γ. The pre-image  1/2 space of the operator KR consists of functionals on HΓD (Γ ) = TΓ HΓ1D (Ω) .  ∗ 1/2 Proposition 5.1.4 Define the operator KR : HΓD (Γ ) → HΓ1D (Ω), where for all q ∈ HΓ1D the function φμ := KR (μ) solves     K∇φμ , ∇q 0 + γ φμ , q 0,Γ

R ∪Γ

= (μ, q)

1/2 D

HΓ (Γ )

∗

1/2 D

×HΓ (Γ )

.

(5.10)

This operator is linear, continuous and positive, i.e., there exist constants cKR and αKR such that KR (μ)1 ≤ cKR μ

1/2 D

HΓ (Γ )

KR (μ)1 ≥ αKR μ

1/2 D

∗ ,

HΓ (Γ )

∗ .

Proof Because Eq. (5.10) is linear in φμ , so is the operator KR . As with KD before the continuity is shown in the section on existence and uniqueness of the Laplace weak form, Sect. 5.1.2, specifically in Theorem 5.1.2. The constant cKR depends on the inverse of the coercivity constant αa of the bilinear form a, in particular it is cKR = cTΓ /αa , see Eq. (5.7) where here it is pD = 0 and f = 0. Moreover, Eq. (5.8) yields αKR = cTΓ /ca .  

92

5 Subproblems Individually

Note that the Robin operator above is a continuous extension operator which maps μ to a function φμ = KR μ which, in a weak sense, satisfies K∇φμ ·n+γ φμ = μ on the boundary part Γ . In fact, this operator has a left inverse TΓR , i.e., it is TΓR φμ = μ: Definition 5.1.5 Let φ ∈ HΓ1D (Ω) be given such that for all q ∈ HΓ1D ∪Γ (Ω) it is (K∇φ, ∇q)0 + (γ φ, q)0,ΓR = 0.  ∗ 1/2 Then its Robin data TΓR φ ∈ HΓD (Γ ) on Γ is defined as 

TΓR φ, ξ



 ∗ 1/2 1/2 HΓ (Γ ) ×HΓ (Γ ) D

:= (K∇φ, ∇(EΓ ξ ))0 + (γ φ, EΓ ξ )0,ΓR ∪Γ ,

D

1/2

with the extension operator EΓ : HΓD (Γ ) → HΓ1D (Ω) defined in Theorem 4.3.4 or in its subsequent discussion, respectively. Due to the assumption on φ, the above definition does not depend on the particular extension operator EΓ . Additionally, TΓR is an extension of p → K∇p · n + γp on Γ . Furthermore, just as the trace operator TΓ is the left inverse of the Dirichlet operator KD , the left inverse TΓR of KR may be called a Robin trace operator.

5.1.3.3 A Connection of the Dirichlet and Robin Operator on Γ  ∗ 1/2 Suppose a μ ∈ HΓD (Γ ) is given. Then, for the function φμ = KR (μ) and all q ∈ HΓ1D ∪Γ it is     K∇φμ , ∇q 0 + γ φμ , q 0,Γ = 0. R

Thus the above Definition 5.1.5 applies and μ = TΓR φμ , i.e., TΓR ◦ KR is the identity  ∗ 1/2 on HΓD (Γ ) . At the same time the above equation means that the same φμ also is KD (TΓ φμ ). One way to understand this reasoning is that on the image of KR the operator KD ◦ TΓ is the identity. Combining these ideas leads to TΓR ◦ KD ◦ TΓ ◦ KR = id

 ∗ 1/2 in HΓD (Γ ) ,

TΓ ◦ KR ◦ TΓR ◦ KD = id

in HΓD (Γ )

1/2

where the second equation can be derived in a similar fashion.

5.2 Saddle Point Problems

93

5.2 Saddle Point Problems The special structure of Stokes (and other) equations, leads to an abstract formulation which requires a few tailored tools. The theory is commonly known as that of saddle point problems and well established. Classical literature includes [BBF13, BF91, GR86], see also [Joh16].

5.2.1 Notation and Formulation of the Abstract Problem Let V and Q be Hilbert spaces with inner products (·, ·)V and (·, ·)Q , norms ·V and ·Q , and duals V ∗ and Q∗ respectively. As before, the duality pairings between elements f ∈ V ∗ or g ∈ Q∗ and v ∈ V or q ∈ Q are denoted by (f, v)V ∗ ×V or (g, q)Q∗ ×Q respectively. Furthermore, let a(·, ·) : V × V → R and b(·, ·) : V × Q → R be two bilinear forms which are assumed to be continuous, i.e., there are positive constants ca and cb such that for all v, w ∈ V and q ∈ Q it holds a(v, w) ≤ ca vV wV ,

(5.11)

b(v, q) ≤ cb vV qQ .

(5.12)

Associated with these two bilinear forms define two linear operators A : V → V ∗ and B : V → Q∗ by

(Bv, q)Q∗ ×Q

(Av, w)V ∗ ×V := a(v, w)  ∗  = B q, v V ∗ ×V := b(v, q)

for all v, w ∈ V ,

(5.13)

for all v ∈ V and q ∈ Q.

(5.14)

Here B ∗ : Q → V ∗ is the dual operator of B, where the identification Q∗∗ = Q is used. The abstract problem to be solved is the following: Let f ∈ V ∗ and g ∈ Q∗ be given. Find u ∈ V and p ∈ Q such that for all v ∈ V and all q ∈ Q it holds 

a(u, v) + b(v, p) = (f, v)V ∗ ×V , b(u, q) = (g, q)Q∗ ×Q .

(5.15)

This can be rewritten as a system of equations in V ∗ and Q∗ using the operators A and B: Find u ∈ V and p ∈ Q such that 

Au + B ∗ p = f, Bu = g.

(5.16)

94

5 Subproblems Individually

Clearly, existence of a solution to the above set of equations can only be expected for g ∈ Im B. Further necessary assumptions to guarantee existence and uniqueness are discussed in the next part. Remark 5.2.1 If the bilinear form a(·, ·) is symmetric, Eq. (5.15) are optimality conditions of the saddle point problem inf sup

v∈V q∈Q

1 a(v, v) + b(v, q) − (f, v)V ∗ ×V − (g, q)Q∗ ×Q . 2

Even if a(·, ·) is not symmetric, problems of the form (5.15) or (5.16) are usually called saddle point problems in the literature, [BF91].

5.2.2 Existence and Uniqueness There are two essential conditions to prove existence and uniqueness of a solution of problem (5.16), one on each of the two operators A and B. In the literature one can find different assumptions which are equivalent or stronger than the ones needed. To clarify this, two propositions showing the relations between some popular formulations are given after the main theorem of this section. The proofs are mainly taken from [BF91] and [GR86]. Theorem 5.2.2 (Existence and Uniqueness) For any f ∈ V ∗ and g ∈ Im B ⊂ Q∗ problem (5.16) admits a solution (u, p) ∈ V × Q, where u is unique and p is unique up to an element of ker B ∗ , if (i) The restricted operator A0 : ker B → (ker B)∗ , defined for all v0 , w0 ∈ ker B ⊂ V by (A0 v0 , w0 )(ker B)∗ ×ker B = (Av0 , w0 )V ∗ ×V , is an isomorphism, (ii) The image of B is closed. Proof Assume (i) and (ii) are true and f ∈ V ∗ and g ∈ Im B ⊂ Q∗ are given. Since g ∈ Im B, there is a ug ∈ V such that Bug = g. Consider f − Aug as an element in (ker B)∗ . Then through (i) there exists u0 ∈ ker B such that A0 u0 = f − Aug . Denoting u = u0 + ug it holds (f − Au, v0 )(ker B)∗ ×ker B = 0 for all v0 ∈ ker B. That means f − Au ∈ (ker B)◦ . Since Im B is closed, the closed range Theorem 2.1.9 states (ker B)◦ = Im B ∗ . Therefore, there exists p ∈ Q, which is unique up to an element of ker B ∗ , such that B ∗ p = f − Au in V ∗ . The pair (u, p) solves problem (5.16). In order to prove uniqueness, i.e., that the pair (u, p) does not depend on the choice ug , it is shown that a solution (u, p) of the homogeneous problem 

Au + B ∗ p = 0, Bu = 0

5.2 Saddle Point Problems

95

must be trivial. Testing the first equation of this homogeneous system with v0 ∈ ker B, it follows (note u ∈ ker B and B ∗ p ∈ Im B ∗ = (ker B)◦ )   0 = (Au, v0 )V ∗ ×V + B ∗ p, v0 V ∗ ×V = (Au, v0 )(ker B)∗ ×ker B = (A0 u, v0 )(ker B)∗ ×ker B . The operator A0 is assumed to be invertible, and especially has trivial kernel so that u must vanish. Now it holds B ∗ p = 0 in V ∗ , i.e., p is zero up to an element of ker B ∗ . Therefore, Eq. (5.16) has a solution (u, p) which is unique up to an element of ker B ∗ for p.   Remark 5.2.3 From the last part of the proof one can see that if the operator B is assumed to be surjective, according to Proposition 2.1.8, its transposed B ∗ is injective, implying p = 0. In conclusion the solution of (5.16) is unique if, additionally, B is surjective. Remark 5.2.4 The conditions in the previous Theorem 5.2.2 are not just sufficient but also necessary, see for example the Remarks 4.1 in [GR86], Chapter 1. The first statement in the following proposition is exactly the first condition in the previous Theorem 5.2.2 showing existence and uniqueness. Proposition 5.2.5 Consider the following four statements. a) The operator A0 : ker B → (ker B)∗ , defined by (A0 v0 , w0 )ker B ∗ ×ker B = a(v0 , w0 ) for all v0 , w0 ∈ ker B ⊂ V , is an isomorphism. b) There exists α > 0 such that inf

sup

a(u0 , v0 ) ≥ α, u0 V v0 V

(5.17a)

inf

sup

a(u0 , v0 ) ≥ α. u0 V v0 V

(5.17b)

u0 ∈ker B v0 ∈ker B

v0 ∈ker B u0 ∈ker B

c) The operator A is coercive on ker B, i.e., there exists α > 0 such that a(v0 , v0 ) ≥ αv0 2V for all v0 ∈ ker B. d) The operator A is coercive on V , i.e., there exists α > 0 such that a(v, v) ≥ αv2V for all v ∈ V . Then the following implications hold a) ⇐⇒ b) ⇐ c) ⇐ d)

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5 Subproblems Individually

Proof “d) ⇒ c)”: This follows immediately because ker B ⊂ V . “c) ⇒ b)”: For all v0 ∈ ker B it is sup

u0 ∈ker B

a(u0 , v0 ) a(v0 , v0 ) ≥ ≥ α. u0 V v0 V v0 2V

This implies the inequality (5.17b). Analogously one can show (5.17a). “b) ⇒ a)”: Using the operator A0 inequality (5.17a) reads inf

sup

u0 ∈ker B v0 ∈ker B

(A0 u0 , v0 ) ≥ α, u0 V v0 V

which means A0 must have a trivial kernel, ker A0 = {0}, i.e., A0 is injective. Furthermore, it means inf

u0 ∈ker B

A0 u0 (ker B)∗ ≥ α. u0 V

(5.18)

Note that the norm of A0 u0 is the standard norm of the dual space (ker B)∗ , i.e., A0 u0 (ker B)∗ = supv0 ∈ker B (A0 u0 , v0 )/v0 ker B . According to Corollary 2.1.12 of the open mapping theorem, Theorem 2.1.10, the image of A0 is closed. Similarly, inequality (5.17b) shows that the transposed operator A∗0 : ker B → (ker B)∗ of A0 is as well injective, and therefore its image is dense, see Proposition 2.1.8. This proves the bijectivity and, together with (5.18), the continuous invertibility of A0 . ∗ “a) ⇒ b)”: Continuity of the operator A−1 0 : (ker B) → ker B ⊂ V means   ∗ ≤ there exists a constant 1/α ≥ 0 such that for all v0∗ ∈ (ker B)∗ it holds A−1 v 0 0 V   1  ∗ ∗ α v0 (ker B)∗ . Now let u0 ∈ ker B ⊂ V be given and denote v0 = A0 u0 , then it holds u0 V ≤ ⇒

1 1 A0 u0 (ker B)∗ = α α

sup

v0 ∈ker B

a(u0 , v0 ) ≥α v0 V u0 V

sup

v0 ∈ker B

a(u0 , v0 ) v0 V

for all u0 ∈ ker B.

That is equivalent to inequality (5.17a). Analogously one can show (5.17b).

 

Proposition 5.2.6 Consider the following nine statements. a) There exists a constant β > 0 such that inf sup

q∈Q v∈V

b(v, q) ≥ β. vV qQ

(5.19)

5.2 Saddle Point Problems

97

b) The operator B : (ker B)⊥ → Q∗ is an isomorphism and there exists a constant β > 0 such that for all v ∈ (ker B)⊥ it holds BvQ∗ ≥ βvV . c) The operator B ∗ : Q → (ker B)◦ is an isomorphism and there exists a constant β > 0 such that for all q ∈ Q it holds B ∗ qV ∗ ≥ βqQ . d) There exists a constant β > 0 such that for all q ∈ Q it holds b(v, q) ≥ βqQ/ ker B ∗ . vV

sup v∈V

(5.20)

e) There exists a constant β > 0 such that for all v ∈ V it holds sup q∈Q

f) g) h) i)

b(v, q) ≥ βvV / ker B . qQ

(5.21)

The image of B, Im B, is closed in Q∗ . The image of B ∗ , Im B ∗ , is closed in V ∗ . (ker B)◦ = Im B ∗ . (ker B ∗ )◦ = Im B.

The following implications hold a) ⇐⇒ b) ⇐⇒ c)

⇒

d) ⇐⇒ e) ⇐⇒ f ) ⇐⇒ g) ⇐⇒ h) ⇐⇒ i).

Proof “a) ⇒ c)”: Using the operator norm in V ∗ Eq. (5.19) reads  ∗  B q  ∗ ≥ βqQ V

(5.22)

for all q ∈ Q, i.e., B ∗ is injective. Furthermore, it implies the continuity of the inverse of B ∗ (defined on Im B ∗ ). Corollary 2.1.12 states that Im B ∗ is closed and Theorem 2.1.7 shows Im B ∗ = Im B ∗ = (ker B)◦ , i.e., B ∗ : Q → (ker B)◦ is an isomorphism. “c) ⇒ a)”: Rewriting the equation B ∗ qV ∗ ≥ βqQ gives sup v∈V

b(v, q) ≥β vV qQ

for all q ∈ Q. This implies (5.19). “b) ⇒ c)”: Due to Proposition 2.1.8 and B being surjective, B ∗ is injective. Furthermore, Im B is closed because B is bounded away from zero, see Corollary 2.1.12. Therefore, the closed range theorem, Theorem 2.1.9, implies Im B ∗ = Im B ∗ = (ker B)◦ , i.e., B ∗ is surjective. As a consequence of the open mapping theorem, Corollary 2.1.11, the inverse (B ∗ )−1 is continuous, i.e., for all q ∈ Q it holds B ∗ qV ∗ ≥ βqQ . Since B → B ∗ is isometric, the constant β is the same for b) and c). “c) ⇒ b)”: Interchanging B and B ∗ in “b) ⇒ c)” proves the claim.

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5 Subproblems Individually

“f) ⇐⇒ g) ⇐⇒ h) ⇐⇒ i)”: This is the closed range theorem, Theorem 2.1.9. “d) ⇒ g)”: Equation (5.20) implies B ∗ qV ∗ ≥ βqQ/ ker B ∗ for all q ∈ Q. Then Corollary 2.1.12 proves the claim, noting that the quotient space Q/ ker B ∗ is a Banach space. “g) ⇒ d)”: Corollary 2.1.12 shows B ∗ qV ∗ ≥ βqQ . Additionally, note that for all q ∈ Q it holds qQ ≥ qQ/ ker B ∗ . “e) ⇒ f)”: Equation (5.21) implies BvQ∗ ≥ βvV / ker B . Then copy the second part of the proof of Corollary 2.1.12 and note that the quotient space V / ker B is a Banach space. “f) ⇒ e)”: Corollary 2.1.12 shows BvQ∗ ≥ βvV . Additionally note that for all v ∈ V it holds vV ≥ vV / ker B . “a) ⇒ d)”: This follows directly, just note that qQ ≥ qQ/ ker B ∗ for all q ∈ Q.   Remark 5.2.7 If the operator B : V → Q∗ is surjective, then all of the statements in Proposition 5.2.6 are equivalent. One often chooses Q/ ker B ∗ instead of Q so that B is naturally surjective. Together with Remark 5.2.3 this means that any one of the statements a)–i) of the previous Proposition 5.2.6 implies unique solvability of the system (5.16) for u and p. Remark 5.2.8 The first statement in Proposition 5.2.6 is the celebrated inf–sup condition which one finds most often in the literature. It is also called Babuška– Brezzi, Ladyzhenskaya–Babuška–Brezzi or just LBB condition. In conclusion, existence and uniqueness is guaranteed if in each Propositions 5.2.5 and 5.2.6 any one of the conditions is satisfied. Using the constants α > 0 and β > 0 from these two propositions an estimate of the norm of the solution can be obtained: From the equation A0 u0 = f − Aug in the proof of the main Theorem 5.2.2 it follows u0 V ≤

   1 f V ∗ + ca ug V α

and therefore      1 uV ≤ ug V + f V ∗ + ca ug V . α   From Proposition 5.2.6, e), one can choose ug such that ug V ≤ 1 c 1 a f V ∗ + + 1 gQ∗ , α α β    1 ca ca  ca + 1 f V ∗ + 2 + 1 gQ∗ . ≤ β α β α

uV ≤ pQ/ ker B ∗

1 β gQ∗

to yield (5.23) (5.24)

The estimate on the pressure uses Proposition 5.2.6, c), together with B ∗ p = f − Au, i.e., pQ/ ker B ∗ ≤ β1 f − AuV ∗ ≤ β1 f V ∗ + cβa uV .

5.3 Application to the Stokes Problem

99

5.3 Application to the Stokes Problem The Stokes equations are a linear system describing the flow of a fluid. They can be deduced from the nonlinear Navier–Stokes equations when a slow and incompressible fluid is assumed. Then the convective term is small and can be neglected. The Stokes problem is given as follows: Find the velocity u : Ω → Rd and the pressure p : Ω → R such that 

−∇ · T(u, p) = f

in Ω,

∇ · u = 0 in Ω,

(5.25)

where T(u, p) = 2νD(u) − p id is the Cauchystress tensor,  id the identity tensor, ν > 0 the kinematic viscosity, and D(u) = 12 ∇u + ∇uT the symmetric part of ∇u also called the deformation tensor. The right-hand side vector field f : Ω → Rd is called source term. Additionally, a solution (u, p) has to satisfy boundary conditions. The focus here is on Dirichlet and Robin type boundary conditions similar to the case of the Laplace equation in Sect. 5.1. For this reason let ∂Ω be divided into two disjoint (relatively open) subsets ΓD and ΓR such that ΓD ∪ ΓR = ∂Ω. The boundary conditions then read  u = uD on ΓD , γ u + T(u, p) · n = g R

on ΓR .

Here uD : ΓD → Rd and g R : ΓR → Rd are given functions on the respective boundary parts. The function γ : ΓR → R is assumed to be not negative. In case γ = 0, Neumann boundary conditions are recovered. If furthermore also g R vanishes, the term outflow or do-nothing boundary condition is widely used. Remark 5.3.1 In the literature the Cauchy stress tensor T is often defined as T = ν∇u − p id. The resulting formulation (5.25) is equivalent for a smooth velocity solution u, which can be shown using the divergence constraint ∇ · u = 0. After discretization using, for example, the finite element method, this is no longer true. It is assumed that using the deformation tensor is the appropriate approach, see [LI06]. Remark 5.3.2 It is possible to vary boundary conditions not just on different parts of the boundary but also by components of the solution. For example one could want a no penetration boundary condition u · n = 0 and on the same boundary part a Neumann type condition in the tangential direction, τ · T(u, p) · n = 0, with τ being a tangential vector. For smooth boundaries this does not considerably complicate the analysis, but requires the introduction of much more notation.

100

5 Subproblems Individually

5.3.1 Weak Form In order to construct weak formulations of the Stokes problem assume (u, p) is a smooth solution to (5.25), e.g., u ∈ (D(Ω))d and p ∈ D(Ω). Then multiply the first equation of (5.25) with a test function v ∈ (C ∞ (Ω))d which vanishes in a neighborhood of the Dirichlet boundary part ΓD ,  d v ∈ D(Ω ∪ ∂Ω \ ΓD ) . The second equation of (5.25) is multiplied by a test function q ∈ D(Ω). Both equations are then integrated over the domain Ω and the first is integrated by parts yielding  (T(u, p), D(v))0 − (T(u, p) · n, v)0,ΓR = (f, v)0 , (∇ · u, q) = 0. Inserting the boundary conditions on ΓR this can be rewritten as  a(u, v) + b(v, p) = (v), b(u, q) = 0,

(5.26)

where the bilinear forms a(·, ·) and b(·, ·) and the right-hand side  are defined as a(u, v) = (2νD(u), D(v))0 + (γ u, v)0,ΓR , b(v, p) = −(∇ · v, p)0 ,   (v) = (f, v)0 + g R , v 0,Γ . R

As in the derivation of a weak form to the Laplace equation in Sect. 5.1, through approximation one can choose V := H1ΓD (Ω)

and

Q := L2 (Ω) as test spaces and Eq. (5.26) stay meaningful, see Theorem 3.4.5 and Eq. (4.3) for each velocity component and Theorem 3.4.5 also for the pressure. The data have to satisfy the following smoothness properties: f ∈ V ∗, uD ∈ H1/2 (ΓD ), γ ∈ L∞ (ΓR ),  ∗ 1/2 g R ∈ H00 (ΓR ) .

5.3 Application to the Stokes Problem

101

As in the case of the Laplace problem the definition of  involves duality pairings   (v) = (f, v)V ∗ ×V + g R , v 

1/2

H00 (ΓR )

∗

1/2

×H00 (ΓR )

The weak formulation of the Stokes problem (5.25) is: Find (u, p) ∈ H1 (Ω) × Q such that u − uD ∈ V and for all v ∈ V and all q ∈ Q it holds 

a(u, v) + b(v, p) = (v), b(u, q) = 0.

(5.27)

Remark 5.3.3 In the expression u − uD ∈ V the term uD has to be understood as a vector of extensions according to Theorem 4.2.4 for each of its components. See also Remark 5.1.1 in the case of the Laplace problem.

5.3.2 Existence and Uniqueness In order to put this formulation into the abstract setting for saddle point problems, define U = u − uD and consider 

a(U, v) + b(v, p) = (v) − a(uD , v) =: L(v), b(U, q) = −b(uD , q).

This way the bilinear form a(·, ·) acts on V ×V instead of H1 (Ω)×V . Furthermore, these equations are equivalent to the original weak formulation (5.27), the solution u can be recovered as U + uD . To prove existence and uniqueness of such solutions the coercivity of a(·, ·) and the inf–sup condition for b(·, ·) is shown and then Theorem 5.2.2 can be applied. The bilinear forms and the right-hand side are continuous with constants ca and cb respectively, which can be shown in the same way as in the Laplace case, see Sect. 5.1.2. Remark 5.3.4 The operator B of the abstract saddle point problem is the divergence: B : V → Q∗ ,

(Bu, q)Q∗ ×Q = −(∇ · u, q)0 .

The dual operator B ∗ : Q → V ∗ of B is related to the gradient. In case ΓD = ∂Ω, i.e., ΓR = ∅ and V = H10 (Ω), it is an extension of ∇ : Q ⊃ H 1 (Ω) → V ∗ ,

(∇q, v)V ∗ ×V = (∇q, v)0 ,

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5 Subproblems Individually

and integration by parts, see Corollary 4.4.1, results in 

B ∗ q, v



V ∗ ×V

= (∇q, v)0 = −(∇ · v, q)0 = (Bv, q)Q∗ ×Q ,

because v vanishes on the boundary. If ΓR = ∅, boundary terms may enter and B ∗ can not be interpreted as (an extension of) a gradient.

5.3.2.1 Coercivity of a Let v ∈ V be given, then using Eq. (4.13) on page 82, which is based on Korn’s inequality, Lemma 3.5.3, it is a(v, v) = 2ν(D(v), D(v))0 + (γ v, v)0,ΓR ≥ 2νcv1 + inf γ (x) v20,ΓR x∈ΓR

≥ αv1 . Therefore, a(·, ·) is coercive on V with a constant α, depending on the domain and ν, and according to Proposition 5.2.5 the first requirement of Theorem 5.2.2 is fulfilled. If γ is not zero and v has a nonzero trace on the Robin boundary ΓR the constant α could be chosen larger with an additional dependency on the boundary coefficient γ .

5.3.2.2 Inf–sup Condition for b The following proof of the inf–sup condition is from [QV99, Proposition 5.3.2]. Let  1 q(x) dx and the rest  q := q ∈ Q be given. Split q into a constant part q := |Ω| Ω q − q ∈ L20 (Ω) which has zero mean value over Ω. According to Theorem 4.5.2 choose v 0 ∈ H10 (Ω) such that ∇ · v 0 =  q and v 0 V ≤ C q Q to yield sup v∈V

  q 2Q q 2Q b(v, q) q , q)0 1 (∇ · v 0 , q)0 ( q Q , ≥ = + = ≥  vV v 0 V v 0 V v 0 V v 0 V C

 where it is used that ( q , q)0 = q Ω  q (x) dx = 0. In the case of Dirichlet conditions on the entire boundary, ΓD = ∂Ω, one typically chooses Q = L20 (Ω) and the inf– sup condition (5.19) is established by the above inequality with β = 1/C, where C is the continuity constant of the divergence operator. In case Q = L2 (Ω) it is q ˜ Q = qQ/ ker B ∗ , i.e., Proposition 5.2.6 (d) is fulfilled again with β = 1/C. If ΓD = ∂Ω the idea is to consider a larger Lipschitz domain Ω   Ω such that ∂Ω ∩ ∂Ω  = ΓD , i.e., the boundaries of the two domains share only the Dirichlet part of ∂Ω, while ΓR ∩∂Ω  = ∅, see Fig. 5.2 for an example. Let q  be the extension

5.3 Application to the Stokes Problem

103

Fig. 5.2 Sketch of Ω and Ω  inspired by Figure 5.3.2 in [QV99]

ΓR Ω

Ω

ΓD

of q by zero to Ω  , then the above reasoning applied to q  , the inclusion Ω ⊂ Ω  , and the triangle inequality yield   ∇ · v 0 , q  0,Ω  v 0 1,Ω 

≥

    1 1 1        q − − q ≥ q ≥ − q  q  q  , 0,Ω 0,Ω  0,Ω 0,Ω C C C

with v 0 ∈ H10 (Ω  ) and a possibly different constant C  . The norm of q  can be bounded using the Hölder inequality, Theorem 2.3.1, as follows  2   q 

0,Ω

=

|Ω| |Ω  |2

2

 Ω

q(x) dx

=

|Ω| |Ω  |2

2

 q(x) dx Ω

≤

|Ω|2 |Ω  |2

q20,Ω .

Combining these results and noting that v 0 1,Ω  ≥ v 0 1,Ω gives    ∇ · v 0 , q  0,Ω  (∇ · v 0 , q)0,Ω |Ω| 1 q0,Ω ≥ ≥  1− v 0 1,Ω v 0 1,Ω  |Ω  | C ' () * :=β

and therefore inf sup

q∈Q v∈V

b(v, q) ≥ β, vV qQ

i.e., the inf–sup condition (5.19).

5.3.2.3 Application of the Saddle Point Theory to the Stokes Problem Thanks to Propositions 5.2.5 and 5.2.6 together with Theorem 5.2.2 the weak formulation (5.27) of the Stokes Problem has a unique solution, at least up to an element of ker B ∗ in the pressure. In the case of Dirichlet conditions on the

104

5 Subproblems Individually

entire boundary, a function v ∈ V = H10 (Ω) has vanishing traces and therefore its  ∗ 1, v) ∗ divergence integrated over the domain Ω is also zero, = − ∇·v = (B V ×V Ω  − ∂Ω v · n = 0, see Corollary 4.4.1 with a constant ϕ = 1. Hence, the kernel of B ∗ consists of all constant functions and the pressure is only unique up to some constant. As in Remark 5.2.3 one can choose Q = L20 (Ω) (instead of Q = L2 (Ω)) which is isomorphic to L2 (Ω)/ ker B ∗ . Within this space the pressure is unique as well. For reference this result is repeated here in detail: Theorem 5.3.5 Let Ω ⊂ Rd be a Lipschitz domain together with two disjoint subsets ΓD and ΓR of the boundary ∂Ω which are themselves Lipschitz, such that Γ D ∪ Γ R = ∂Ω. Furthermore, let V := H1ΓD (Ω) and Q = L2 (Ω) (if ΓR = ∅, set Q = L20 (Ω)) and f ∈ V ∗,

uD ∈ H1/2(ΓD ),

 ∗ 1/2 gR ∈ H00 (ΓR ) ,

γ ∈ L∞ (ΓR ),

where γ is nonnegative. Then there exist unique u ∈ H1 (Ω) and q ∈ Q such that TΓD u = uD , i.e., u = uD on Γ , and for all v ∈ V and q ∈ Q it is 

a(u, v) + b(v, p) = (v),

(5.27)

b(u, q) = 0, with a(u, v) = (2νD(u), D(v))0 + (γ u, v)0,ΓR , b(v, p) = −(∇ · v, p)0 ,

  (v) = (f, v)V ∗ ×V + g R , v 

1/2

H00 (ΓR )

∗

1/2

×H00 (ΓR )

.

An a priori estimate on the solutions U = u − uD and p of the Stokes equations is then (see also Eq. (5.23) and (5.24) at the end of Sect. 5.2) 1 c 1 a LV ∗ + + 1 gQ , α α β    1 ca ca  ca + 1 LV ∗ + 2 + 1 gQ , ≤ β α β α

UV ≤ pQ/ ker B ∗ with

LV ∗ = fV ∗ + cTΓR gR 

1/2

H00 (ΓR )

gQ = cEΓD uD H 1/2 (ΓD ) .

∗

+ ca cEΓD uD H 1/2 (ΓD ) ,

5.3 Application to the Stokes Problem

105

The constants cTΓR and cEΓD are the continuity constants of the operators TΓR and EΓD , respectively, see also Theorems 4.2.4 and 4.3.4. An estimate for u then is 1 c 1 a LV ∗ + + 1 gQ + cEΓD uD H 1/2 (ΓD ) α α β    ca 1 c˜L + 1+ 1+ cEΓD uD H 1/2 (ΓD ) = α α β

u1 ≤

(5.28)

with c˜L = fV ∗ + cTΓR gR 

1/2

H00 (ΓR )

∗ .

In terms of c˜L the bound on the pressure can be expressed as   1  ca ca  ca 1 pQ/ ker B ∗ ≤ + 1 c˜L + +1 1+ cEΓD uD H 1/2 (ΓD ) . β α β α β (5.29) Note that the right-hand sides in both Eqs. (5.28) and (5.29) above include inverses of the viscosity ν, i.e., it is large for small ν. Remark 5.3.6 It is assumed that the divergence of u vanishes, i.e., the second equation in the Stokes problem (5.25) reads ∇ · u = 0. If a nonzero divergence is prescribed via ∇ · u = gdiv ∈ L2 (Ω), the norm of g in the above theorem changes to gQ = cEΓD uD H 1/2 (ΓD ) + gdiv Q . While in fluid dynamics the divergence is typically zero, this can be helpful to construct particular functions in H1 (Ω) with a prescribed divergence gdiv and a continuous dependence on gdiv where the boundary conditions and all data (including ν, γ , f) are free to choose. In Theorem 4.5.2 such a construction is available only in H10 (Ω) and gdiv ∈ L20 (Ω).

5.3.3 View as an Operator on a Part of the Boundary Very similar to the Laplace case one can view the above theorem as the definition of an operator which maps the data f, uD , and gR to a solution (u, p). And again setting all but one of these to zero results in a linear operator. Of special interest in the coupling of Stokes and Darcy equations is the case where data is only given at a particular part of the boundary. For that purpose let ∂Ω be split into three disjoint parts ∂Ω = ΓD ∪ ΓR ∪ Γ . For simplicity assume that ΓR = ∅, otherwise the adjustments mentioned above apply. Then the operators introduced below are well defined according to Theorem 5.3.5. The presentation is not as detailed as in the case of the Laplace equation in Sect. 5.1.3.

106

5 Subproblems Individually

5.3.3.1 Dirichlet Operator on Γ   1/2 Denote the space TΓ H1ΓD (Ω) as HΓD (Γ ). 1/2

Proposition 5.3.7 Define the operator KD : HΓD (Γ ) → H1ΓD (Ω) × L2 (Ω), where for all v ∈ H1ΓD ∪Γ (Ω) and all q ∈ L2 (Ω) the pair (u, p) = KD (λ) solves 

(2νD(u), D(v))0 + (γ u, v)0,ΓR − (∇ · v, p)0 = 0,

(5.30)

(∇ · u, q)0 = 0

and TΓ u = λ. This operator is linear, continuous and positive, i.e., there exist constants cKD > 0 and αKD > 0 such that KD (λ)H1 (Ω)×L2 (Ω) ≤ cKD λH1/2 (Γ ) , ΓD

KD (λ)H1 (Ω)×L2 (Ω) ≥ αKD λH1/2 (Γ ) . ΓD

Proof The linearity of KD follows directly from the linearity of Eq. (5.30) and of the extension operator EΓ . Continuity is shown in Sect. 5.3.2 with a constant depending on the inverse of the coercivity constant αa , the continuity constant of ca of the bilinear form a, and the inverse of the inf–sup constant β. In particular, Eqs. (5.28) and (5.29) apply with c˜L = 0. The continuity of the trace operator TΓ yields λH1/2 (Γ ) = TΓ (KD (λ))H1/2 (Γ ) ≤ cTΓ KD (λ)H1 (Ω)×L2(Ω) ΓD

ΓD

and therefore the positivity of KD with α = cT−1 . Γ

 

The operator KD restricted to its first component (the velocity) is another 1/2 linear continuous right inverse of TΓ on HΓD (Γ ), see also Theorem 4.3.4 and the subsequent discussion.

5.3.3.2 Robin Operator on Γ Consider now the case where only Robin data is prescribed on Γ . Then define the Robin operator for the Stokes equations as follows:  ∗ 1/2 Proposition 5.3.8 Define the operators KR : HΓD (Γ ) → H1ΓD (Ω) × L2 (Ω), where for all v ∈ H1ΓD (Ω) and all q ∈ L2 (Ω) the pair (u, p) = KR (μ) solves ⎧ ⎨ (2νD(u), D(v))0 + (γ u, v)0,ΓR ∪Γ − (∇ · v, p)0 = (μ, v)H1/2 (Γ )∗ ×H1/2 (Γ ) , ⎩

ΓD

ΓD

(∇ · u, q)0 = 0. (5.31)

5.3 Application to the Stokes Problem

107

This operator is linear, continuous and positive, i.e., there exist constants cKR > 0 and αKR > 0 such that KR (μ)H1 (Ω)×L2 (Ω) ≤ cKR μ

1/2 D

HΓ (Γ )

KR (μ)H1 (Ω)×L2 (Ω) ≥ αKR μ

1/2 D

∗ ,

HΓ (Γ )

∗ .

Proof Since Eq. (5.31) are linear, so is KR . As before, continuity is shown in Sect. 5.3.2 with a constant depending on the inverse of the coercivity constant αa , the continuity constant of ca of the bilinear form a, and the inverse of the inf–sup constant β. More precisely, here Eqs. (5.28) and (5.29) reduce to  cT 1  ca u1 ≤ Γ μ 1/2 ∗ p0 ≤ + 1 cTΓ μ 1/2 ∗ . and HΓ (Γ ) HΓ (Γ ) α β α D D 1/2

To show positivity, for all λ ∈ HΓD (Γ ) denote (uλ , pλ ) = KD (λ) and (uμ , pμ ) =  ∗ 1/2 KR (μ) for all μ ∈ HΓD (Γ ) , then it is (μ, λ)

1/2 D

HΓ (Γ )

∗

1/2 D

×HΓ (Γ )

= (μ, TΓ uλ )

1/2 D

HΓ (Γ )

∗

1/2 D

×HΓ (Γ )

= a(uμ , uλ )   ≤ ca cKD uμ 1 λH1/2 (Γ ) ΓD

≤ ca cKD KR (μ)H1 (Ω)×L2 (Ω) λH1/2 (Γ ) ΓD

and hence (μ, λ) μ

1/2 D

HΓ (Γ )

∗

=

sup 1/2 D

λ∈HΓ (Γ )

1/2 D

HΓ (Γ )

∗

1/2 D

×HΓ (Γ )

λH1/2 (Γ ) ΓD

≤ ca cKD KR (μ)H1 (Ω)×L2 (Ω) , i.e., αKR = 1/(ca cKD ).

 

Just as the corresponding operator in the Laplace case, the first  Sect. 5.1.3, ∗ 1/2 component of the Robin operator KR is an extension from HΓD (Γ ) to H1ΓD (Ω) and a left inverse TΓR can be defined: Definition 5.3.9 Let (u, p) ∈ H1ΓD × L2 (Ω) be given such that for all (v, q) ∈ H1ΓD ∪Γ × L2 (Ω) it is 

(2νD(u), D(v))0 + (γ u, v)0,ΓR − (∇ · v, p)0 = 0, (∇ · u, q)0 = 0.

108

5 Subproblems Individually

 ∗ 1/2 Then its Robin data TΓR (u, p) ∈ HΓD (Γ ) on Γ is defined to be   TΓR (u, p), ξ 

1/2 D

HΓ (Γ )

∗

1/2 D

×HΓ (Γ )

:= (2νD(u), D(EΓ ξ ))0 + (γ u, EΓ ξ )0,ΓR ∪Γ − (∇ · (EΓ ξ ), p)0 , 1/2

where EΓ ξ is an extension of (each component of) ξ ∈ HΓD (Γ ), see Theorem 4.3.4 and in its subsequent discussion. The above definition is well posed in the sense that it does not depend on the particular extension EΓ used. Additionally, it is a generalization of the operator (u, p) → T(u, p) · n + γ u on the Robin boundary ΓR .

5.3.3.3 A Connection of the Dirichlet and Robin Operator on Γ With the same reasoning as in the Laplace case the following identities hold: TΓR ◦ KD ◦ TΓ ◦ KR = id

 ∗ 1/2 in HΓD (Γ ) ,

TΓ ◦ KR ◦ TΓR ◦ KD = id

in HΓD (Γ ).

1/2

In fact, the notation in this section is chosen such that these equations are exactly the same as in the Laplace case.

Chapter 6

Stokes–Darcy Equations

6.1 The Setting Let Ω ⊂ Rd be a Lipschitz domain split into two disjoint nonempty subdomains Ωp and Ωf which are Lipschitz, too. The index p refers to the Darcy subdomain where a porous medium is modeled, while the index f refers to the free flow domain with a Stokes model. The intersection ΓI := Ωp ∩Ωf is called interface in the following. The boundary of the domain Ω is split into four relatively open parts Γf,N ⊂ ∂Ωf , Γf,D ⊂ ∂Ωf , Γp,N ⊂ ∂Ωp , and Γp,D ⊂ ∂Ωp where Dirichlet (essential) and Neumann (natural) boundary conditions, respectively, are prescribed. Further, let n be the unit normal vector pointing outward of Ω on ∂Ω and pointing outward of Ωf on the interface ΓI . Figure 6.2 shows such a setting as an example. Consider a Darcy problem in Ωp and a Stokes problem in Ωf , −∇ · T(uf , pf ) = ff

in Ωf ,

(6.1a)

∇ · uf = 0

in Ωf ,

(6.1b)

up + K∇ϕp = 0

in Ωp ,

(6.1c)

in Ωp ,

(6.1d)

∇ · up = fp

together with boundary conditions on the outer boundary ∂Ω uf = ub on Γf,D ,

(6.2a)

T(uf , pf ) · n = gf on Γf,N ,

(6.2b)

up · n = gp on Γp,N,

(6.2c)

ϕp = ϕb on Γp,D.

(6.2d)

© Springer Nature Switzerland AG 2019 U. Wilbrandt, Stokes–Darcy Equations, Advances in Mathematical Fluid Mechanics, https://doi.org/10.1007/978-3-030-02904-3_6

109

110

6 Stokes–Darcy Equations trace opearators Tf and Tp Definition 6.2.3

trace spaces Λf and Λp Definition 6.4.1

Neumann trace TpN Definition 6.4.5

Neumann trace TfN Definition 6.4.6

solution operators Ki,j Definition 6.4.4

interface operators Hi,j Definition 6.4.9

Steklov–Poincaré equations Definition 6.4.12

fixed point equations Definition 6.4.11

equivalence of formulations, part 2 Corollary 6.4.14

affine spaces Definition 6.4.2

equivalence of formulations, part 1 Corollary 6.4.13

Neumann–Neumann weak formulation Definition 6.2.1

Robin–Robin weak formulation Definition 6.6.3

existence and uniqueness Theorem 6.3.1

equivalence of weak forms Theorem 6.6.4

μ-fixed point eq. ⇐⇒ weak form Theorem 6.4.16

λ-fixed point eq. ⇐⇒ weak form Theorem 6.4.15

R Robin traces Ti,γ Definition 6.6.7

interface operators Li,j Definition 6.5.1

R extended Robin traces Ti,γ Definition 6.6.8

characterization of Li,j Lemma 6.5.2

solution operators i,j Definition 6.5.3

Symmetry and positivity of Li,j Lemma 6.5.4

Steklov–Poincaré equations reformulated Definition 6.5.5

existence and uniqueness Theorem 6.5.6

equivalence of fixed point eqs. Corollary 6.6.11

γ

solution operators Ki i Definition 6.6.6

fixed point eq. ⇐⇒ weak form part 1, Theorem 6.6.12

γi ,γj

interface operators Hi Definition 6.6.9

fixed point equations Definition 6.6.10

fixed point eq. ⇐⇒ weak form part 2, Theorem 6.6.13

Fig. 6.1 A graph showing the dependencies among all theorems, definitions, corollaries, as well as propositions in this chapter. Rounded corners indicate items related to the Robin–Robin approach from Sect. 6.6. Note that implicit dependencies are not always shown

6.1 The Setting

111

Fig. 6.2 Sketch of the general setting

Ωf

ΓI

Ωp

Here uf : Ωf → Rd and up : Ωp → Rd are vector-valued functions, called the Stokes velocity and Darcy velocity (also hydraulic discharge), respectively. The other two unknowns pf : Ωf → R and ϕp : Ωp → R are scalar functions which are called Stokes and Darcy pressure, respectively. Often ϕp is also referred to as hydraulic head. The vector-valued function ff : Ωf → Rd and the scalar fp : Ωp → R are the right-hand sides and K is the hydraulic conductivity tensor, which is assumed to be symmetric and positive definite. Furthermore, as in the previous chapter, T(u, p) = 2νD(u)− p id is the  Cauchy stress tensor, ν > 0 the kinematic viscosity, and D(u) = 12 ∇u + ∇uT the symmetric part of ∇u or the deformation tensor. The functions prescribing the Dirichlet and Neumann boundary conditions are given. Other types such as Robin or periodic boundary conditions are possible with the usual modifications. To complete the description of the general setting these equations are equipped with conditions on the interface ΓI : uf · n = up · n −n · T(uf , pf ) · n = gϕp uf · τ i + ατ i · T(uf , pf ) · n = 0

on ΓI ,

(6.3a)

on ΓI ,

(6.3b)

on ΓI , i = 1, . . . , d − 1.

(6.3c)

Here (τ i )i=1,...d−1 is an orthonormal basis of the tangential space on ΓI and g the gravitational acceleration which is omitted, i.e., assumed to be one in the following. The first interface condition (6.3a) describes the mass conservation and the second equation (6.3b) represents the balance of momentum. The third interface condition (6.3c) is called Beavers–Joseph–Saffman1 condition, based on

1 Sometimes

also called Beavers–Joseph–Saffman–Jones condition.

112

6 Stokes–Darcy Equations

experimental findings, and necessary for the problem to be well posed. Note that it is a Robin type condition for the tangential components of the Stokes velocity.  The parameter α is set to be α = α0 (τi · K · τi )/(νg), where α0 is a constant depending on the porous medium. See the original works [Jon73, Saf71] as well as [JM00]. A common simplification for the two equations in the porous medium Ωp is to use a Laplace instead of a Darcy model, which is derived from taking the divergence of Eq. (6.1c) and inserting Eq. (6.1d). In this monograph, only this simplification is analyzed, i.e., Eqs. (6.1c) and (6.1d) are replaced by   − ∇ · K∇ϕp = fp

in Ωp .

(6.4)

In the boundary as well as the interface conditions the normal flux up · n is replaced by −K∇ϕp · n to yield the following coupled system: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

−∇ · T(uf , pf ) = ff ∇ · uf = 0  −∇ · K∇ϕp = fp 

in Ωf , in Ωf , in Ωp ,

uf = ub

on Γf,D ,

T(uf , pf ) · n = gf

on Γf,N ,

−K∇ϕp · n = gp

on Γp,N ,

ϕp = ϕb

on Γp,D ,

uf · n = −K∇ϕp · n −n · T(uf , pf ) · n = ϕp uf · τ i + ατ i · T(uf , pf ) · n = 0

(6.5)

on ΓI , on ΓI , on ΓI , i = 1, . . . , d − 1.

This model is analyzed in detail in this chapter. Before weak formulations together with existence and uniqueness results are shown, some possible extensions to this model are mentioned.

6.1.1 Possible Extensions of the Model The coupled system (6.5) consists of rather simple models but already proves to be numerically challenging. Various extensions and modifications are studied in the literature. In the free flow domain, the Stokes equations describe a slow incompressible flow. This model is often replaced by the nonlinear Navier–Stokes equations, see for example Section 6 in [DQ09] or the shallow water equations, [DMQ02]. The mixed Darcy equations (6.1c) and (6.1d) can be used instead of the simpler Laplace

6.2 Weak Formulation

113

formulation (6.4) used in this work, see [LSY02, MMS15, GS07]. The porous medium is modeled as fully saturated in the equations above, a two-phase approach is considered for example in [RGM15]. The Brinkman model combines both Stokes and Darcy and can therefore be used in either subdomain (e.g., [Ang11]) or as a unified approach, where coefficients in the partial differential equations determine the subdomains and no particular interface conditions are needed, see for example [BC09]. The system (6.5) presented here is stationary, for the time dependent case, see [DZ11, CGHW14, RM14]. The interface conditions are in fact a simplification of the more difficult Beavers–Joseph condition, see for example [JM00, CGHW10]. Furthermore, the Stokes–Darcy system can be extended by a transport equation, [VY09].

6.2 Weak Formulation The weak formulation is derived as in Sects. 5.1.1 and 5.3.1: multiplication with a test function (denoted by v, q, and ψ) and integration over the respective subdomain, followed by an integration by parts and insertion of boundary and interface conditions. The involved spaces are V f = H1Γf,D (Ωf ), Qf = L2 (Ωf ),

(6.6)

Qp = HΓ1p,D (Ωp ), together with the standard norms: vV f := v1,Ωf ,

pQf := p0,Ωf ,

ψQp := ψ1,Ωp .

The term (T(uf , pf ) · n, v)0,ΓI which enters the equations after integration by parts in the Stokes subdomain, is decomposed in normal and tangential parts as follows (T(uf , pf ) · n, v)0,ΓI = (n · T(uf , pf ) · n, v · n)0,ΓI +

d−1 

(τ i · T(uf , pf ) · n, v · τ i )0,ΓI ,

i=1

d−1 where the identity v = (v · n)n + i=1 (v · τ i )τ i is employed. In the two terms above the conservation of momentum (6.3b) and the Beavers–Joseph–Saffman condition(6.3c) are inserted. The mass conservation (6.3a) replaces K∇ϕp · n in  the term K∇ϕp · n, ψ 0,Γ which enters due to integration by parts in Ωp . This I

114

6 Stokes–Darcy Equations

leads to the equations d−1 1 (u · τ i , v · τ i )0,ΓI α i=1   −(∇ · v, p)0,Ωf + ϕp , v · n 0,Γ = (ff , v)0 + (gf , v)0,Γf,N ,

(2νD(u), D(v))0,Ωf +

I

(K∇ϕ, ∇ψ)0,Ωp

(∇ · uf , q)0,Ωf = 0,     − (uf · n, ψ)0,ΓI = fp , ψ 0 − gp , ψ 0,Γ

p,N

.

The terms on the right-hand sides can be extended to respective duals as before in Sects. 5.1.1 and 5.3.1. The following definition additionally groups the terms above and serves as a summary and reference. Since in each subdomain a Neumanntype problem is posed, this approach is referred to as the Neumann–Neumann weak formulation. Another suggestion with Robin type problems is introduced in Sect. 6.6. Definition 6.2.1 (Neumann–Neumann Weak Formulation) With the spaces d×d  ∞ positive definite, as well from Eq. (6.6), let ν > 0, α > 0, K ∈ L (Ωp ) as ff ∈ V ∗f ,

∗  gf ∈ TΓf,N (V f ) ,

fp ∈ Q∗p ,

 ∗  gp ∈ TΓp,N Qp

and ub ∈ H1/2(Γf,D ),

ϕb ∈ H 1/2 (Γp,D),

be given. Define the bilinear and linear forms af : H1 (Ωf ) × H1 (Ωf ) → R, bf : H1 (Ωf ) × L2 (Ωf ) → R, ap : H 1 (Ωp ) × H 1 (Ωp ) → R, f : V f → R, and p : Qp → R, as af (u, v) = (2νD(u), D(v))0,Ωf

d−1 1 + (u · τ i , v · τ i )0,ΓI , α i=1

bf (v, p) = −(∇ · v, p)0,Ωf , ap (ϕ, ψ) = (K∇ϕ, ∇ψ)0,Ωp , f (v) = (ff , v)V ∗f ×V f + (gf , v)

∗  , TΓf,N (V f ) × TΓf,N (V f )

    p (ψ) = fp , ψ Q∗ ×Qp − gp , ψ  p

∗  . TΓp,N (Qp ) × TΓp,N (Qp )

Then the weak formulation of the Stokes–Darcy coupled problem is: Find (uf , pf , ϕp ) ∈ H1 (Ωf ) × L2 (Ωf ) × H 1 (Ωp ), such that for all v ∈ V f , q ∈ Qf ,

6.2 Weak Formulation

115

and all ψ ∈ Qp it is   ⎧ af (uf , v) + bf (v, pf ) + ϕp , v · n 0,Γ = f (v) ⎪ I ⎪ ⎨ bf (uf , q) = 0 ⎪ ⎪   ⎩ ap ϕp , ψ − (uf · n, ψ)0,ΓI = p (ψ),

(6.7)

together with uf = ub on Γf,D and ϕp = ϕb on Γp,D . The bilinear and linear forms are essentially the ones introduced in the sections on the respective subproblems, see Sects. 5.1 and 5.3. The main difference is that the boundary integral related to the Robin condition on the interface ΓI in the definition of af only includes tangential components. Remark 6.2.2 Let Γ be a Lipschitz continuous part of the boundary of a domain Ω. A vector-valued function v ∈ H1/2 (Γ ) is defined such that each component is in H 1/2(Γ ). However, this does not imply that also v · n ∈ H 1/2(Γ ), since n may have jumps. In fact, considering a piecewise smooth part Γ , then the normal component is in H 1/2 only piecewise, and the same is valid for the tangential ones. However all components are in L2 (Γ ), so that all integrals over ΓI above are well defined. As a part of a Lipschitz boundary with a localization as in Definition 2.2.1, the interface ΓI is represented as a set of Lipschitz continuous functions hi which, due to Rademacher’s theorem ([Eva98], Theorem 6 in Section 5.8.3), are differentiable almost everywhere. To avoid technical difficulties, it is assumed that the hi are not differentiable at only finitely many points (and lines for d = 3). In practice, especially in geometries which are resolved by a finite element grid, the interface is often piecewise smooth so that this assumption is no restriction. The (normal) traces of functions in H1 (Ωf ) and H 1 (Ωp ) onto the interface Γ play a central role in the following analysis and are therefore introduced additionally in a separate definition. For notational consistency, the normal trace of a vector field in the Stokes subdomain is defined on the entire solution space, including the pressure space. Definition 6.2.3 (Trace Operators) Let Tf : (H1 (Ωf ) × L2 (Ωf )) → L2 (ΓI ), (v, q) → v · n, and Tp : H 1 (Ωp ) → H 1/2(ΓI ) be the trace operators onto the interface. According to Theorems 4.1.1 and 4.2.4 they are continuous. The image space of Tf is in fact piecewise H 1/2 on the interface, see the previous Remark 6.2.2. The continuity constants of the two trace operators are denoted by cTf and cTp , respectively. Note that the continuity of Tf holds even on the piecewise H 1/2 space on the interface ΓI , because it is a composition of local (piecewise) traces. As before, the explicit statement of the trace operators is sometimes omitted for better readability, for example v · n on ΓI rather than Tf (v, q) in the weak formulation (6.7). The trace spaces are studied in more detail in Sect. 6.4.1.

116

6 Stokes–Darcy Equations

6.3 Existence and Uniqueness The theory of the Neumann–Neumann formulation has been studied in the literature, see for example [DQ09] and the references therein. To put the weak formulation (6.7) of the coupled Stokes–Darcy equations (6.5) into the setting of abstract saddle point theory, see Sect. 5.2, for all w = (w, ω) ∈ H1 (Ωf ) × H 1 (Ωp ), v = (v, ψ) ∈ V f × Qp and all q ∈ Qf define A(w, v) = af (w, v) + ap (ω, ψ) − (w · n, ψ)0,ΓI + (ω, v · n)0,ΓI , B(w, q) = bf (w, q),

  F (v) = f (v) + p (ψ) − A (EΓf,D ub , EΓp,D ϕb ), v , G(q) = −bf (EΓf,D ub , q).

The operators EΓf,D : H1/2 (Γf,D ) → H1 (Ωf ) and EΓp,D : H 1/2(Γp,D ) → H 1 (Ωp ) extend Dirichlet boundary data into the respective subdomain, see Theorem 4.2.4. Then the weak formulation (6.7) can be rewritten to: Find u = (u0f , ϕp0 ) ∈ V f × Qp and pf ∈ Qf such that for all v = (v, ψ) ∈ V f × Qp and q ∈ Qf it holds 

A(u, v) + B(v, pf ) = F (v) B(u, q) = G(q).

(6.8)

The solution can be recovered via uf = u0f + EΓf,D ub and ϕp = ϕp0 + EΓp,D ϕb . In order to apply Theorem 5.2.2 together with Propositions 5.2.5 and 5.2.6 to these bilinear forms, the following properties are proved in the next three subsections. 1. continuity of A, B, F, and G with constants  cA , cB , cF , and cG ,respectively 2. coercivity of A on v ∈ V f × Qp B v, q = 0 for all q ∈ Qf with constant αA 3. inf–sup condition for B with constant β. Once these properties are proved, the existence and uniqueness can be established: Theorem 6.3.1 (Existence and Uniqueness) The Neumann–Neumann weak formulation (6.8) of the Stokes–Darcy problem admits a unique solution (u, pf ), u = (u0f , ϕp0 ) ∈ V f × Qp , pf ∈ Qf . Moreover, the following a priori estimates hold:    1 √ α + 2cA  0 0  cG , 2cF +  uf , ϕp  ≤ W αA β  2cA √ 1 2cA (α + 2cA ) pf Qf ≤ 1+ cG . 2cF + β α αβ

6.3 Existence and Uniqueness

117

6.3.1 Continuity of A, B, F , and G The bilinear forms are continuous with constants caf = 2ν + α1 (d − 1)cT2f , cbf = 1, and cap = K(L∞ (Ω))d×d , respectively, see Sects. 5.3 and 5.1. Let w = (w, ω), v = (v, ψ) ∈ W = V f × Qp and q ∈ Qf be given. Then A(w, v) = af (w, v) + ap (ω, ψ) − (w · n, ψ)ΓI + (ω, v · n)ΓI ≤ caf wVf vVf + cap ωQp ψQp + cTf cTp wV f ψQp + cTf cTp vV f ωQp   cA  wV f + ωQp vV f + ψQp ≤ 2  1/2  1/2 v2V f + ψ2Qp ≤ cA w2V f + ω2Qp     = cA w W v W , where   cA = 2 max caf , cap , cTf cTp and the continuity constants of the respective traces onto the interface are cTf and cTp , see Definition 6.2.3. This proves the continuity of A. For B it holds   B(w, q) = bf (w, q) ≤ cbf wV f qQf ≤ cbf wW qQf , which means cB = cbf = 1. The linear forms F and G are continuous due to       F (v) ≤ cf vV f + cp ψQp + cA (EΓf,D ub , EΓp,D ϕb )W v W = cF v W , G(q) ≤ cEΓf,D ub H 1/2 (Γf,D ) qQf = cG qQf , with  1/2  2 2 2 2 cF = max cf , cp , cA cEΓ ub H 1/2 (Γ ) + cEΓ ϕb H 1/2 (Γ ) , f,D p,D p,D f,D 

cG = cEΓf,D ub H 1/2 (Γf,D ) ,

118

6 Stokes–Darcy Equations

the continuity constants of the right-hand sides cf = ff V ∗f + gf 

∗ , TΓf,N (V f )

    cp = fp Q∗ + gp  p

TΓp,N (Qp )

∗ ,

and of the extension operators EΓf,D and EΓp,D .

6.3.2 Coercivity of A Let v = (v, ψ) ∈ W = V f ×Qp be given. The two interface terms in A(v, v) cancel and it is  2 A(v, v) = af (v, v) + ap (ψ, ψ) ≥ αaf v2V f + αap ψ2Qp = αA v W ,   with αA = min αaf , αap and the coercivity constants αaf and αap of the two subproblems, see also Sects. 5.1.2 and 5.3.2.

6.3.3 Inf–sup Condition The bilinear form bf is inf–sup stable, i.e., there is a constant β > 0 such that for all q ∈ Qf there exists a w ∈ V f such that bf (w, q) ≥ βqQf /M , wV f with the set M := {q ∈ Qf | bf (v, q) = 0 for all v ∈ V f }, see Sect. 5.3.2.2. Choosing w = (w, 0) for a given q yields B(w, q)   ≥ βqQf /M . w  W

  Note that the set M equals q ∈ Qf B(v, q) = 0 for all v ∈ W which implies that the same β can be used as the inf–sup constant for B.

6.4 Weak Formulation Rewritten

119

6.4 Weak Formulation Rewritten While in Sect. 6.3 the weak formulation (6.7) is rewritten as a saddle point problem where the bilinear form A included both af and ap , i.e., parts from both subdomains, instead, in this section a different but equivalent formulation is introduced which acts on suitable spaces on the interface ΓI .

6.4.1 Spaces On the interface the trace spaces play a central role in the analysis. Definition 6.4.1 (Trace Spaces) The restrictions of the test spaces onto the interface via the trace operators from Definition 6.2.3 are denoted by Λp := Tp (Qp ), Λf := Tf (V f , Qf ). 1/2

While the first space (Λp ) is a subspace of H 1/2(ΓI ) containing H00 (ΓI ), the second (Λf ) is a subspace of L2 (ΓI ) and only included in H 1/2 (ΓI ) if Γ is smooth, see Remark 6.2.2. The norms can therefore be the natural ones or, equivalently, μΛp := inf ψQp , ψ∈Qp Tp ψ=μ

μΛf :=

inf

(v,q)∈V f ×Qf Tf (v,q)=μ

vV f .

1/2

In case ΓI is smooth, both of these two spaces therefore contain H00 (Γ ) and can but need not be equal, depending on where the Dirichlet boundary conditions are prescribed. If Γp,D ∩ ΓI = Γf,D ∩ ΓI then Λp = Λf holds true. A longer discussion on this issue follows in Sect. 6.6.2. The solution to the coupled problem, Eq. (6.5), in general has nonzero Dirichlet boundary data, it is sought in affine spaces. Consequently, the corresponding restrictions to the interface (through Tf and Tp ) do not necessarily belong to Λp or Λf . Definition 6.4.2 (Affine Spaces) The solution spaces for the Stokes velocity and the Darcy pressure are affine spaces

 p = ϕ ∈ H 1 (Ωp ) TΓp,D ϕ = ϕb , Q

  f = v ∈ H1 (Ωf ) TΓf,D v = ub . V

120

6 Stokes–Darcy Equations

Additionally their restrictions onto the interface ΓI via the trace operators from Definition 6.2.3 are also affine and denoted by   p , p := Tp Q Λ    f × L2 (Ωf ) . f := Tf V Λ In fact, these trace spaces could be defined with the help of two extensions EΓp,D ϕb ∈ H 1 (Ωp ) and EΓf,D ub ∈ H1 (Ωf ) of the Dirichlet boundary data f = p = Tp (EΓp,D ϕb ) + Λp and Λ in the respective subdomains. Then it is Λ  Tf (EΓf,D ub , 0)+Λf . The affine space Λp is different from the linear space Λp if, and only if, the boundary data cannot be extended by 0 to a function in H 1/2 (ΓI ∪ Γp,D ). This occurs if the Dirichlet boundary Γp,D touches the interface ΓI and nonzero data f and Λf is prescribed in the vicinity of Γp,D ∩ ΓI = ∅. Similar ideas apply to Λ where also the normal has to be taken into account. Remark 6.4.3 In general, the restriction spaces on the interface are not equal, however at least for a smooth interface ΓI the following inclusions hold, i, j ∈ {p, f},  ∗  ∗ 1/2 1/2 H00 (ΓI ) ⊂ Λi ⊂ H 1/2 (ΓI ) ⊂ L2 (ΓI ) ⊂ H 1/2(ΓI ) ⊂ Λ∗j ⊂ H00 (ΓI ) , where the dual spaces are denoted by the superscript ∗ . Furthermore, even if the interface is only Lipschitz continuous the affine spaces are subspaces of L2 (ΓI ) implying p ⊂ Λ∗f Λ

and

f ⊂ Λ∗p . Λ

The case Λp = Λf is discussed in more detail in Sect. 6.6.2.

6.4.2 Revisiting Operators on the Boundary In Chap. 5 the Laplace as well as the Stokes problem is introduced and analyzed. At the end of Sects. 5.1.3 and 5.3.3 these subproblems are reinterpreted as operators which take data on a particular boundary part and return the solution in the respective subdomain. In this section these ideas are applied to the Stokes–Darcy case. First, only Dirichlet (index D ) and Neumann (index N ) conditions on the interface ΓI are of interest. Definition 6.4.4 (Neumann and Dirichlet Operators) With the spaces from p , Kp,D : Λ p , Kf,N : Λ∗ → p → Q Definitions 6.4.1 and 6.4.2 let Kp,N : Λ∗p → Q f  f × Qf be operators which map given Neumann (N)  f × Qf , and Kf,D : Λ f → V V or Dirichlet (D) data on the interface ΓI to solutions of the subproblems. In strong

6.4 Weak Formulation Rewritten

121

form these are described by: Kp,N (μ) = ϕ solves ⎧ ⎪ ⎪ −∇ · (K∇ϕ) = fp ⎪ ⎪ ⎪ ⎨ −K∇ϕ · n = gp

in Ωp ,

⎪ ⎪ ⎪ ⎪ ⎪ ⎩

on Γp,D ,

ϕ = ϕb −K∇ϕ · n = μ

Kf,N (μ) = (u, p) solves ⎧ −∇ · T(u, p) = ff ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∇·u=0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ T(u, p) · n = gf

on Γp,N ,

⎪ ⎪ u = ub ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u · τ i + ατ i · T(u, p) · n = 0 ⎪ ⎪ ⎪ ⎪ ⎩ −n · T(u, p) · n = μ

on ΓI .

Kp,D (λ) = ϕ solves ⎧ −∇ · (K∇ϕ) = fp ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −K∇ϕ · n = gp

in Ωp ,

⎪ ⎪ ⎪ ⎪ ⎪ ⎩

ϕ = ϕb

on Γp,D,

ϕ=λ

on ΓI .

on Γp,N,

Kf,D (λ) = (u, p) solves ⎧ −∇ · T(u, p) = ff ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∇ ·u =0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ T(u, p) · n = gf ⎪ u = ub ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u · τ i + ατ i · T(u, p) · n = 0 ⎪ ⎪ ⎪ ⎪ ⎩ u·n =λ

in Ωf , in Ωf , on Γf,N , on Γf,D , on ΓI , on ΓI .

in Ωf , in Ωf , on Γf,N , on Γf,D , on ΓI , on ΓI .

Transformed to the weak formulations, the above operators are characterized by • Kp,N (μ) = ϕ such that TΓp,D ϕ = ϕb and for all ψ ∈ Qp it is   ap (ϕ, ψ) = p (ψ) + μ, Tp ψ Λ∗ ×Λp , p

• Kp,D (λ) = ϕ such that TΓp,D ϕ = ϕb , Tp ϕ = λ, and for all ψ ∈ Qp with Tp ψ = 0 (i.e., ψ ∈ HΓ1p,D ∪ΓI (Ωp )) it is ap (ϕ, ψ) = p (ψ), • Kf,N (μ) = (u, p) such that TΓf,D u = ub and for all (v, q) ∈ V f × Qf it is 

af (u, v) + bf (v, p) = f (v) − (μ, Tf v)Λ∗f ×Λf bf (u, q) = 0,

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6 Stokes–Darcy Equations

• Kf,D (λ) = (u, p) such that TΓf,D u = ub , Tf (u, p) = λ, and for all (v, q) ∈ V f × Qf with Tf (v, q) = 0 it is 

af (u, v) + bf (v, p) = f (v) bf (u, q) = 0.

Note that the ranges of the respective Neumann and Dirichlet operators are the same, i.e., Im Ki,N = Im Ki,D , i ∈ {p, f}. For example the trace (using Tp or Tf ) of a solution to a Neumann problem can be used as the argument to Ki,D . The uniqueness of the solutions of these equations assures that the result is that exact same solution from before. Instead of recovering the Dirichlet data on the interface ΓI via the two trace operators Tp and Tf , it is possible to recover the Neumann data as well. A similar construction as in Definitions 5.1.5 and 5.3.9 is made individually for each subproblem. Definition 6.4.5 Let φ ∈ H 1 (Ωp ) be given such that for all ψ ∈ Qp ∩ ker Tp = HΓ1p,D ∪ΓI (Ω) it is ap (φ, ψ) = p (ψ). Then its Neumann data TpN φ ∈ Λ∗p on ΓI is defined as 

TpN φ, ξ

 Λ∗p ×Λp

  := ap φ, ψξ − p (ψξ )

with an extension ψξ ∈ Qp defined in Theorem 4.3.4 or in its subsequent discussion, respectively. Next, Neumann data for the Stokes subdomain is introduced. Definition 6.4.6 Let (u, p) ∈ H1 (Ωf ) × L2 (Ωf ) be given such that for all (v, q) ∈ (V f ∩ ker Tf ) × Qp it is 

af (u, v) + bf (v, p) = f (v), bf (u, q) = 0.

Then its Neumann data TfN (u, p) ∈ Λ∗f on ΓI is defined as   TfN (u, p), ξ

Λ∗f ×Λf

  := f (v ξ ) − af u, v ξ − b(v ξ , p),

with an extension v ξ ∈ V f defined in Theorem 4.3.4 or in its subsequent discussion, respectively. Both of the above definitions do not depend on the particular choice of the extension, because any two extensions differ by an element in Qp ∩ ker Tp or

6.4 Weak Formulation Rewritten

123

(V f × Qf ) ∩ ker Tf , respectively, for which the defining expression is zero by assumption. Furthermore, this means that the Neumann data can be defined in a meaningful way only if the domain of definition is suitably restricted. In fact, the assumptions on φ and (u, p) above are satisfied for any Darcy or Stokes solution p and including functions from the image spaces Im(Kp,N ) = Im(Kp,D ) ⊂ Q  Im(Kf,N ) = Im(Kf,D ) ⊂ V f × Qf , respectively. These Neumann data operators TiN are left inverses to the respective Neumann operators Ki,N in the subdomains, very similar to the trace and the respective Dirichlet operator. Additionally, on the image spaces Im(Ki,j ) they are also right inverses, i.e., for i ∈ {p, f} it is Ti ◦ Ki,D = id

i , in Λ

(6.9a)

Ki,D ◦ Ti = id

in Im(Ki,N ) = Im(Ki,D ),

(6.9b)

TiN ◦ Ki,N = id

in Λ∗i ,

(6.9c)

Ki,N ◦ TiN = id

in Im(Ki,N ) = Im(Ki,D ).

(6.9d)

In case the data (f and p ) is zero, taking the Neumann data is a linear operation. Remark 6.4.7 The operators TiN are indeed an extension of the Neumann data: For smooth functions φ and (u, p) integration by parts shows TpN φ = −K∇φ · n and TfN (u, p) = −n · T(u, p) · n on ΓI . While TpN ϕ has the usual sign, because n is pointing into the Darcy subdomain Ωp , the Neumann data in the Stokes part has a minus sign. This matches the definition of Kf,N (μ) where also it is −n·T(u, p)·n = μ. The reason for this notational choice is mainly the conservation of momentum (6.3b). Remark 6.4.8 The above identities (6.9) are an elaborated way of stating that it is possible to recover • the data on the interface with which a subproblem is solved, and • a solution by extracting its Dirichlet/Neumann data and solving a suitable Dirichlet/Neumann problem with it.

6.4.3 Operators Acting Only on the Interface The weak form (6.8) of the Stokes–Darcy coupled problem is rewritten in terms of operators acting solely on spaces on the interface ΓI . Definition 6.4.9 (Operators on the Interface) With the solution operators Ki,j from Definition 6.4.4 and the traces from Definitions 6.2.3, 6.4.5 and 6.4.6 define p , Hp,N : Λ∗p → Λ

Hp,N = Tp ◦ Kp,N ,

(6.10a)

p → Λ∗p Hp,D : Λ

Hp,D = TpN ◦ Kp,D ,

(6.10b)

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6 Stokes–Darcy Equations

f Hf,N : Λ∗f → Λ

Hf,N = Tf ◦ Kf,N ,

(6.10c)

f → Λ∗f Hf,D : Λ

Hf,D = TfN ◦ Kf,D ,

(6.10d)

essentially mapping Dirichlet to Neumann data (Hi,D ) or vice versa (Hi,N ). As discussed in Sects. 5.1.3.3 and 5.3.3.3, and also from Eqs. (6.9), these operators are related to each other through the identities Hp,N ◦ Hp,D = id

p , in Λ

(6.11a)

Hp,D ◦ Hp,N = id

in

Λ∗p ,

(6.11b)

Hf,N ◦ Hf,D = id

f , in Λ

(6.11c)

Hf,D ◦ Hf,N = id

in Λ∗f .

(6.11d)

i to Λ∗ (Hi,D ) Remark 6.4.10 These operators are therefore bijective maps from Λ i ∗  and Λi to Λi (Hi,N ), respectively.

6.4.4 Equations on the Interface The solutions of the following equations on the interface are related to the solution of the Stokes–Darcy coupled system which is made precise in Sect. 6.4.5. Here only the equations and a few reformulations are presented. Definition 6.4.11 (Fixed Point Equations) Using the operators on the interface from Definition 6.4.9 define the following two fixed point equations   Hf,N Hp,N(λ) = λ,   Hp,N Hf,N (μ) = μ.

(6.12) (6.13)

Using the invertibility properties (6.11a)–(6.11d), the solution to the fixed point equation (6.12) also solves   Hp,D Hf,D (λ) = λ

(6.14)

and similarly the solution to (6.13) also solves   Hf,D Hp,D (μ) = μ.

(6.15)

Note that the nested operators on the left-hand sides of (6.14) and (6.15) are not p , since the range of the inner operator is larger f and μ ∈ Λ defined for all λ ∈ Λ than the domain of definition of the outer one. These two equations are therefore

6.4 Weak Formulation Rewritten

125

inappropriate in an iterative scheme trying to find the fixed point, i.e., a Dirichlet– Dirichlet method is not studied here. Therefore, from now on, the focus is on the f plays the role other two formulations (6.12) and (6.13). Note that in (6.12) λ ∈ Λ  of the normal velocity on the interface while μ ∈ Λp in (6.13) is the pressure (or normal stress). Another possible formulation is introduced in the following definition: Definition 6.4.12 (Steklov–Poincaré Equations) Using the operators on the interface from Definition 6.4.9 define the following Steklov–Poincarè equations Hp,N(λ) − Hf,D (λ) = 0,

(6.16)

Hf,N (μ) − Hp,D (μ) = 0.

(6.17)

The first equation (6.16) is derived from Eq. (6.12) through multiplying with Hf,D and Eq. (6.11d). Similarly, Eqs. (6.13) and (6.11b) yield (6.17). Already from here one can see that the operators solving Stokes and Darcy systems are not a composition in the Steklov–Poincarè equation, while they are in the fixed point formulation. This means evaluating the left-hand side for a given λ or μ can be done in parallel only for the former. Note that in other publications, e.g. [DQ09], some of the above operators are defined with an additional minus sign so that the Steklov–Poincaré equations are sums rather than differences. The operators involved in the Steklov–Poincaré equations (6.16) and (6.17) have different ranges and domains of definition. An iterative scheme therefore needs to apply another operator, e.g., a preconditioner, which corrects this drawback. Solving a Neumann problem in one of the subdomains (depending on which of the two Steklov–Poincaré equations is considered) has the desired property. The mentioned formulations are summarized in the following: Corollary 6.4.13 (Equivalence of Formulations, Part 1) Steklov–Poincaré equations are equivalent:   f Hf,N Hp,N(λ) = λ ∈ Λ   p Hp,N Hf,N (μ) = μ ∈ Λ

The fixed point and

⇐⇒

p , Hp,N (λ) − Hf,D (λ) = 0 ∈ Λ

(6.18)

⇐⇒

f . Hf,N (μ) − Hp,D (μ) = 0 ∈ Λ

(6.19)

Additionally, with a solution of either one of the fixed point equations from Definition 6.4.11, one can find a solution to the respective other using the identities (6.11): Corollary 6.4.14 (Equivalence of Formulations, Part 2) If λ solves (6.12), μ := Hp,N(λ) solves (6.13). Conversely, if μ solves (6.13), λ := Hf,N (μ) solves (6.12). Combining this last result with Corollary 6.4.13, means that when λ is the solution of the fixed point equation (6.12), then μ := Hp,N(λ) equals Hf,D (λ). Similarly, it holds λ := Hf,N (μ) = Hp,D (μ) for the solution μ of the second fixed point equation (6.13).

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6 Stokes–Darcy Equations

6.4.5 Equivalence to the Weak Formulation In this subsection the connection between the weak solution of the coupled system, Definition 6.2.1, and the interface equations, Definitions 6.4.11 and 6.4.12, is established. Theorem 6.4.15 (Equivalence of Fixed Point Equation and Weak Formulation, f solves the fixed point equation (6.12), then Part 1) If λ ∈ Λ ϕp := Kp,N (λ)

and

(uf , pf ) := Kf,N (Hp,N(λ))

p × V  f × Qf solves (6.7), then λ = solve (6.7). Conversely if (ϕp , uf , pf ) ∈ Q  Tf (uf , pf ) ∈ Λf solves (6.12). f solve (6.12). By definition (ϕp , uf , pf ) satisfy the correct Proof First let λ ∈ Λ boundary data away from the interface and, using Eq. (6.12), uf · n = Tf (uf , pf ) = Hf,N (Hp,N (λ)) = λ on ΓI . Let v ∈ V f , q ∈ Qf , and ψ ∈ Qp be given. Then, using Hp,N(λ) = Tp (ϕ), the definitions of Kf,N and Kp,N read   af (uf , v) + bf (v, pf ) = f (v) − ϕp , v · n 0,Γ , I

bf (uf , q) = 0,   ap ϕp , ψ = p (ψ) + (λ, ψ)Λ∗p ×Λp = p (ψ) + (uf · n, ψ)0,ΓI , which is precisely Eq. (6.7). Conversely, let (ϕp , uf , pf ) solve (6.7) and λ := Tf (uf , pf ) = uf · n. Then by definition of Kp,N and Kf,N it is ϕp = Kp,N (λ) and (uf , pf ) = Kf,N (Tp (ϕp )). Hence, λ = Tf (Kf,N (Tp (ϕp ))) = Hf,N (Tp (ϕp )) = Hf,N (Tp (Kp,N (λ))) = Hf,N (Hp,N (λ)),  

i.e., Eq. (6.12) holds.

Theorem 6.4.16 (Equivalence of Fixed Point Equation and Weak Formulation, p solves the interface equation (6.13), then Part 2) If μ ∈ Λ (uf , pf ) := Kf,N (μ)

and

ϕp := Kp,N (Hf,N (μ))

p × V  f × Qf solves (6.7), then μ = solve (6.7). Conversely if (ϕp , uf , pf ) ∈ Q p solves (6.13). Tp (ϕp ) ∈ Λ Proof Combining Corollary 6.4.14 and Theorem 6.4.15 shows the claim. Alternatively, a very similar proof as in Theorem 6.4.15 can be used directly.  

6.5 Linear Operators on the Interface

127

Remark 6.4.17 Using the identities (6.9) and (6.11), in Theorem 6.4.15 it also is (uf , pf ) = Kf,D (λ) and ϕp = Kp,D (Hf,D (λ)). Similarly in Theorem 6.4.16 it is ϕp = Kp,D (μ) and (uf , pf ) = Kf,D (Hp,D(μ)). Once either one of the fixed point equations (6.12) and (6.13) or one of the Steklov–Poincaré equations (6.16) and (6.17) is solved, Corollaries 6.4.13 and 6.4.14 together with the previous theorems provide a solution to the coupled weak formulation (6.7). Thus, it is possible to approach the coupled Stokes–Darcy problem through these interface equations and develop algorithms for them.

6.4.6 Decoupling the Equations Further In this subsection another view on the presented equations is developed. While in the weak formulation (6.7) the solution variables are u, p, and ϕ, in the interface equations these are λ and μ. Combining these to one formulation where all the above variables are the solutions at once formally decouples the problems in the subdomains further. For all v ∈ V f , q ∈ Qf , ψ ∈ Qp , λf ∈ Λf , and λp ∈ Λp it is af (uf , v) + bf (v, pf ) + (μ, v · n)0,ΓI = f (v), 

λ, λp

 0,ΓI

bf (uf , q) = 0,   − uf · n, λp 0,Γ = 0, I

(6.20)

ap (ϕp , ψ) − (λ, ψ)0,ΓI = p (ψ),   (μ, λf )0,ΓI − ϕp , λf 0,Γ = 0. I

These equations are further decoupled in the sense that the equations in the subdomains do not directly include the solutions from the other subdomain.

6.5 Linear Operators on the Interface The operators Hi,j , i ∈ {f, p}, j ∈ {N, D}, have some interesting properties which p , V f , Λ p , are inherited by the respective subproblems. However the affine spaces Q  and Λf are not suitable to define linear operators solving certain subproblems. For i be in the domain of definition of Hi,D . In fact, any i ∈ {f, p}, let λ0,i = Hi,N (0) ∈ Λ  λ0,i ∈ Λi can be chosen here. The following definition introduces the linear parts of the interface operators Hi,j from Definition 6.4.9. While the Neumann operators Hi,N are affine linear this is not possible for the Dirichlet operators Hi,D , because they are defined on an affine space. However also in that case similar linear operators exist:

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6 Stokes–Darcy Equations

Definition 6.5.1 (Linear Operators on the Interface) i , define from Definition 6.4.9 and λ0,i = Hi,N (0) ∈ Λ

Using the operators Hi,j

Lp,N : Λ∗p → Λp ,

Lp,N (μ) = Hp,N (μ) − λ0,p ,

Lf,N : Λ∗f → Λf ,

Lf,N (μ) = Hf,N (μ) − λ0,f .

Lp,D : Λp → Λ∗p ,

Lp,D (λ) = Hp,D (λ + λ0,p ) − Hp,D (λ0,p ),

Lf,D : Λf → Λ∗f ,

Lf,D (λ) = Hf,D (λ + λ0,f ) − Hf,D (λ0,f ).

The claim that these operators are indeed linear is shown together with other properties next.

6.5.1 Linearity, Invertibility, Symmetry, and Positivity The operators Li,j from Definition 6.5.1 in fact represent Neumann/Dirichlet data of a solution in the respective subdomain which has zero data everywhere except on the interface ΓI : Lemma 6.5.2 (Characterization of Li,j ) The operators Li,j can be characterized as follows: • for all μ ∈ Λ∗p there exists a unique function ϕμ ∈ Qp such that Lp,N (μ) = Tp ϕμ and it holds for all ψ ∈ Qp ap (ϕμ , ψ) = (μ, ψ)Λp ∗ ×Λp , • for all λ ∈ Λp there exists a unique function ϕλ ∈ Qp such that Lp,D (λ) = TpN ϕλ , Tp ϕλ = λ, and for all ψ ∈ Qp with Tp ψ = 0 it is ap (ϕλ , ψ) = 0, • for all μ ∈ Λ∗f there exists a unique pair (uμ , pμ ) ∈ V f ×Qf such that Lf,N (μ) = Tf (uμ , pμ ) and for all (v, q) ∈ V f × Qf it is    af uμ , v + bf (v, pμ ) = −(μ, v · n)Λf ∗ ×Λf , bf (uμ , q) = 0, • for all λ ∈ Λf there exists a unique pair (uλ , pλ ) ∈ V f × Qf such that Lf,D (λ) = TfN (uλ , pλ ), Tf (uλ , pλ ) = λ, and for all (v, q) ∈ V f × Qf with Tf (v, q) = 0 it is 

af (uλ , v) + bf (v, pλ ) = 0, bf (uλ , q) = 0.

6.5 Linear Operators on the Interface

129

Proof First, let j = N, i.e., consider the case Li,N (·) = Ti (Ki,N (·)) − Ti (Ki,N (0)). Let ϕμ := Kp,N (μ) − Kp,N (0) and (uμ , pμ ) := Kf,N (μ) − Kf,N (0), so that Tp ϕμ = Lp,N (μ) and Tf (uμ , pμ ) = Lf,N (μ), because the traces Tp and Tf are linear. Furthermore, for all ψ ∈ Qp and all (v, q) ∈ V f × Qf , the definition of Ki,N leads to       ap ϕμ , ψ = ap Kp,N (μ), ψ − ap Kp,N (0), ψ     = p (ψ) + μ, Tp ψ Λ ∗ ×Λ − p (ψ) = μ, Tp ψ Λ p

p

p

∗ ×Λ p

,

  af uμ , v + bf (v, pμ ) = f (v) − (μ, v · n)Λf ∗ ×Λf − f (v) = −(μ, v · n)Λf ∗ ×Λf , bf (uμ , p) = 0, as claimed. For j = D, i.e., the Dirichlet operators, and λ ∈ Λi , let ϕλ := Kp,D (λ+λp,0 )−Kp,D (λp , 0) and (uλ , pλ ) := Kf,D (λ+λf,0 )−Kf,D (λf,0 ). Then, the identities (6.9) and the linearity of the traces Tp and Tf yield Tp ϕλ = λ+λp,0 −λp,0 = λ and Tf (uλ , pλ ) = λ + λf,0 − λf,0 = λ. Additionally, for all ψ ∈ Qp with Tp ψ = 0 and all (v, q) ∈ V f × Qf with Tf (v, q) = 0, the definition of Ki,D gives     ap (ϕλ , ψ) = ap Kp,D (λ + λp,0 ), ψ − ap Kp,D (λp,0 ), ψ = p (ψ) − p (ψ) = 0, af (uλ , v) + bf (v, pλ ) = f (v) − f (v) = 0, bf (uλ , q) = 0. Finally TpN ϕλ = Lp,D (λ) and TfN (uλ , pλ ) = Lf,D (λ) because taking the Neumann data is a linear operation in case the data away from the interface is zero.   Compare the previous characterization of the operators Li,j with the definition of Ki,j . The only difference is that all data (fp , gp , ϕb , ff , gf , ub ) vanishes for Li,j . This is the same setting as in Propositions 5.1.3, 5.1.4, 5.3.7, and 5.3.8, where instead of Robin here Neumann conditions are considered. Hence, these operators are linear and therefore called the linear parts of Hi,j in the following. Definition 6.5.3 (Solution Operators Li,j ) Lemma 6.5.2 assures that the operators

are well defined.

Lp,N : Λ∗p → Qp ,

μ → ϕμ ,

Lp,D : Λp → Qp ,

λ → ϕλ ,

Lf,N : Λ∗f → V f × Qf ,

μ → (uμ , pμ ), and

Lf,D : Λf → V f × Qf ,

λ → (uλ , pλ ),

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6 Stokes–Darcy Equations

As in Sects. 5.1.3 and 5.3.3 these operators are linear and continuous where the continuity constants cLi,j reduce to cTi /αai for j = N and cai cEi /αai for j = D. Furthermore, similar identities as in Eqs. (6.9) hold, namely Ti ◦ Li,D = id

in Λi ,

(6.21a)

Li,D ◦ Ti = id

in Im(Li,N ) = Im(Li,D ),

(6.21b)

TiN ◦ Li,N = id

in Λ∗i ,

(6.21c)

Li,N ◦ TiN = id

in Im(Li,N ) = Im(Li,D ).

(6.21d)

i ) and the image spaces are Here the linear spaces Λi replace the affine ones (Λ contained in the respective test spaces, namely Im(Lp,N) = Im(Lp,D ) ⊂ Qp and Im(Lf,N ) = Im(Lf,D ) ⊂ V f × Qf . Furthermore, the Li,j are to the operators Li,j what Ki,j are to Hi,j , i.e., the operators Li,j could as well be defined analogously to Hi,j , see Eqs. (6.10), Lp,N = Tp ◦ Lp,N,

Lp,D = TpN ◦ Lp,D,

Lf,N = Tf ◦ Lf,N ,

Lf,D = TfN ◦ Lf,D .

(6.22)

Following Eqs. (6.21), identities similar to those in (6.11) hold Lp,N ◦ Lp,D = id

in Λp ,

(6.23a)

Lp,D ◦ Lp,N = id

in

Λ∗p ,

(6.23b)

Lf,N ◦ Lf,D = id

in Λf ,

(6.23c)

Lf,D ◦ Lf,N = id

in Λ∗f .

(6.23d)

Besides linearity also symmetry and positivity of the two subproblems are inherited by the linear parts Li,j introduced above: Lemma 6.5.4 (Symmetry and Positivity of Li,j ) The linear operators Li,j , i ∈ {f, p} and j ∈ {N, D}, are symmetric and positive/negative definite, i.e., it is 

   ξ, Li,N (μ) Λ∗ ×Λ = μ, Li,N (ξ ) Λ∗ ×Λ , i i i i     Li,D (λ), ξ Λ∗ ×Λ = Li,D (ξ ), λ Λ∗ ×Λ , i

i

i

i

and there are positive constants αLi,j > 0 such that   ξ, Lp,N (ξ ) Λp ∗ ×Λp ≥ αLp,N ξ 2Λ∗p ,   Lp,D (λ), λ Λp ∗ ×Λp ≥ αLp,N λ2Λp .



 ξ, −Lf,N (ξ ) Λ ∗ ×Λ ≥ αLf,N ξ 2Λ∗ , f f f   2 −Lf,D (λ), λ Λ ∗ ×Λ ≥ αLf,N λΛf . f

f

6.5 Linear Operators on the Interface

131

Proof Using Eqs. (6.21) and (6.22) and the definitions of TiN , it is     ξ, Lp,N (μ) Λp ∗ ×Λp = TpN (Lp,N (ξ )), Tp (Lp,N (μ))

Λp ∗ ×Λp



ξ, −Lf,N (μ)

 Λp ∗ ×Λp

= ap (Lp,N(ξ ), Lp,N (μ)),   = − TfN (Lf,N (ξ )), Tf (Lf,N (μ))

Λp ∗ ×Λp

= af (uξ , uμ ), where Lf,N (k) = (uk , pk ), k ∈ {ξ, μ} (the term bf (uμ , pξ ) vanishes). Hence, the symmetry of ai carries over to Lp,N and −Lf,N . Furthermore, with the same reasoning also the positivity is inherited by the respective bilinear form ai and Li,N . For the Dirichlet operators it is     Lp,D (λ), ξ Λp ∗ ×Λp = TpN (Lp,D (λ)), Tp (Lp,D (ξ )) Λp ∗ ×Λp   −Lf,D (λ), ξ Λp ∗ ×Λp

= ap (Lp,D(λ), Lp,D (ξ )),   = − TfN (Lf,D (λ)), Tf (Lf,D (ξ )) Λp ∗ ×Λp = af (uλ , uξ ),

where this time it is Lf,D (k) = (uk , pk ), k ∈ {ξ, μ}. Hence, also in this case the symmetry and positivity of ai carries over to Lp,D and Lf,N .   In summary, the four operators Lp,D , −Lf,D , Lp,N , and −Lf,N are linear, symmetric, and positive. This furthermore implies that for all appropriate λ, ξ it is     Lf,D (λ) − Lp,N (λ), ξ Λ ∗ ×Λ = Lf,D (ξ ) − Lp,N (ξ ), λ Λ ∗ ×Λ , f f f f     Lf,N (λ) − Lp,D (λ), ξ Λp ∗ ×Λp = Lf,N (ξ ) − Lp,D (ξ ), λ Λp ∗ ×Λp ,        Lf,N Lp,N (λ) , ξ Λp ∩Λ = λ, Lp,N Lf,N (ξ ) Λp ∩Λ . f

f

The last equation can be interesting because it states that the operators in the fixed point equations (6.12) and (6.13) below are in fact transposed to each other up to some constant terms. Iterative schemes requiring the transposed operator can therefore be exploited as well.

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6 Stokes–Darcy Equations

6.5.2 Equations on the Interface Revisited The aim of this subsection is to write the interface equations of Sect. 6.4.4 in terms of the linear operators Li,j introduced in Sect. 6.5. The fixed point equation (6.12) f which together with the definitions reads Hf,N (Hp,N( λ)) =  λ with  λ = λ+λ0,f ∈ Λ of Li,N leads to   Lf,N Lp,N ( λ) + λ0,p + λ0,f =  λ.   Denoting χf = Lf,N (λ0,p ) + Lf,N Lp,N (λ0,f ) ∈ Λf this can be reformulated as an equation in Λf :   λ − Lf,N Lp,N (λ) = χf .

(6.24)

This is not a fixed point but a linear equation with a nonzero right-hand side. Similarly the other fixed point equation (6.13) can be rewritten as   μ − Lp,N Lf,N (μ) = χp ,

(6.25)

  with right-hand side χp = Lp,N (λ0,f ) + Lp,N Lf,N (λ0,p ) ∈ Λp . Combining Eqs. (6.24) and (6.25) with the inverse identities (6.23) leads to Steklov–Poincaré type equations: Definition 6.5.5 (Steklov–Poincaré Equations) With the linear operators Li,j from Definition 6.5.1, define the following two Steklov–Poincaré type equations p Lf,D (λ) − Lp,N (λ) = Lf,D (χf ) = λ0,p + Lp,N (λ0,f ) ∈ Λ

(6.26)

f . Lp,D (μ) − Lf,N (μ) = Lp,D (χp ) = λ0,f + Lf,N (λ0,p ) ∈ Λ

(6.27)

The linear operators are positive/negative definite and therefore these equations have a unique solution: Theorem 6.5.6 (Existence and Uniqueness, Steklov–Poincaré Equations) Both of the two Steklov–Poincaré equations (6.26) and (6.27) admit unique solutions λ ∈ Λf and μ ∈ Λp . Proof The operators Lp,N − Lf,D and Lp,D − Lf,N both are positive definite (and symmetric) according to Lemma 6.5.4. Therefore, the bilinear and continuous forms ef : Λf × Λf → R, and ep : Λp × Λp → R,

  ef (λ, η) := Lp,N (λ) − Lf,D (λ), η Λ ∗ ×Λ , f f   ep (μ, η) := Lp,D (μ) − Lf,N (μ), η Λp ∗ ×Λp ,

are both coercive. Furthermore, the right-hand sides λ0,p + Lp,N (λ0,f ) and λ0,f + Lf,N (λ0,p ) both define linear continuous functions on the trace spaces

6.6 Robin–Robin

133

  Λf and Λp respectively, namely ηf → λ0,p + Lp,N (λ0,f ), ηf 0,Γ and I   ηp → λ0,f + Lf,N (λ0,p ), ηp 0,Γ . Then, using the theorem of Lax–Milgram, I Theorem 2.1.13, there exist unique solutions λ ∈ Λf and μ ∈ Λp such that for all ηf ∈ Λf and ηp ∈ Λp it is   ef (λ, ηf ) = λ0,p + Lp,N (λ0,f ), ηf 0,Γ , I   ep (μ, ηp ) = λ0,f + Lf,N (λ0,p ), ηp 0,Γ . I

With the positive constants αLi,j from Lemma 6.5.4 together with cf  λ0,p + Lp,N (λ0,f ) and cp = λ0,f + Lf,N (λ0,p )0,Γ it is 0,Γ I

λΛf ≤



=

I

cf

min αLp,N , αLf,D



and

μΛp ≤



cp

min αLp,D , αLf,N

.  

Note that if λ ∈ Λf and μ ∈ Λp are the unique solutions to the two Steklov– Poincaré equations (6.26) and (6.27), then they also solve the linear equations (6.24) and (6.25). Furthermore, according to the construction the sums  λ = λ + λ0,f and  μ = μ+λ0,p solve the respective fixed point and Steklov–Poincaré equations (6.18) and (6.19). Therefore, with Theorems 6.4.15 and 6.4.16 the above Theorem 6.5.6 provides an alternative proof for the existence and uniqueness of the Stokes–Darcy coupled problem (6.7). In contrast to the proof given in Sect. 6.3, where a bilinear form A is defined which includes terms from both subdomains, here only properties of the respective subproblems are used. In Sect. 7.3 the expected convergence behavior of several algorithms are studied. Some minor modifications there also show the existence of solutions to the Stokes–Darcy coupled problem.

6.6 Robin–Robin In Sect. 6.2 the interface conditions (6.3a) and (6.3b) are inserted directly to get a weak formulation where in each subdomain a Neumann-type problem is solved, see Eq. (6.7). In this section the two conditions are replaced by suitable linear combinations. Let γf ≥ 0 and γp > 0 be two constants. Together with the Beavers– Joseph–Saffman condition (6.3c) the following interface conditions are imposed: γf uf · n + n · T(uf , pf ) · n = γf up · n − ϕp

on ΓI ,

(6.28a)

−γp up · n − ϕp = −γp uf · n + n · T(uf , pf ) · n on ΓI .

(6.28b)

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6 Stokes–Darcy Equations

Remark 6.6.1 (Motivation) In Chap. 7 it is shown that iterative methods based on the Neumann–Neumann weak formulation, Definition 6.2.1, are slow or diverging whenever the viscosity ν and hydraulic conductivity K are small. The two modified interface conditions (6.28) above lead to Robin problems in each subdomain and iterative algorithms based on this approach converge reasonably fast even for small ν and K. The overall problem to be solved is defined by Eqs. (6.1a), (6.1b), and (6.4), together with the boundary and interface conditions (6.2), (6.3c), and (6.28), where up · n is replaced by −K∇ϕp · n: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

−∇ · T(uf , pf ) = ff ∇ · uf = 0   −∇ · K∇ϕp = fp

in Ωf , in Ωf , in Ωp ,

uf = ub

on Γf,D ,

T(uf , pf ) · n = gf

on Γf,N ,

⎪ ⎪ −K∇ϕp · n = gp ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕp = ϕb ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ γf uf · n + n · T(uf , pf ) · n = −γf K∇ϕp · n − ϕp ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ γp K∇ϕp · n − ϕp = −γp uf · n + n · T(uf , pf ) · n ⎪ ⎪ ⎪ ⎩ uf · τ i + ατ i · T(uf , pf ) · n = 0

on Γp,N, on Γp,D, on ΓI , on ΓI , on ΓI , (6.29)

with i = 1, . . . , d − 1.

6.6.1 Weak Form The same steps as in the Neumann–Neumann case in Sect. 6.2 are done here, namely multiplication with a suitable test function, integration by parts, followed by insertion of boundary data and interface conditions. The involved spaces do not change, i.e., it remains V f = H1Γf,D (Ωf ), Qf = L2 (Ωf ), Qp = HΓ1p,D (Ωp ),

6.6 Robin–Robin

135

together with the standard norms: vV f := v1,Ωf ,

pQf := p0,Ωf ,

ψQp := ψ1,Ωp .

The weak formulation is then: Find (uf , pf , ϕp ) ∈ H1 (Ωf ) × L2 (Ωf ) × H 1 (Ωp ), such that for all v ∈ V f , q ∈ Qf and all ψ ∈ Qp it is     ⎧ R af (uf , v) + bf (v, pf ) + ϕp , v · n 0,Γ + γf K∇ϕp · n, v · n 0,Γ = f (v) ⎪ ⎪ I I ⎪ ⎪ ⎨ bf (uf , q) = 0, ⎪ ⎪   ⎪ 1 ⎪ ⎩ apR ϕp , ψ − (uf · n, ψ)0,ΓI + (n · T(uf , pf ) · n, ψ)0,ΓI = p (ψ), γp (6.30) and uf = ub on Γf,D as well as ϕp = ϕb on Γp,D . The bilinear forms afR : H1 (Ωf ) × H1 (Ωf ) → R, bf : H1 (Ωf ) × L2 (Ωf ) → R, and apR : H 1 (Ωp ) × H 1 (Ωp ) → R are defined as afR (u, v) = (2νD(u), D(v))0,Ωf +

d−1 1 (u · τ i , v · τ i )0,ΓI + γf (u · n, v · n)0,ΓI α i=1

= af (u, v) + γf (u · n, v · n)0,ΓI , bf (v, p) = −(∇ · v, p)0,Ωf , apR (ϕ, ψ) = (K∇ϕ, ∇ψ)0,Ωp + = ap (ϕ, ψ) +

1 (ϕ, ψ)0,ΓI γp

1 (ϕ, ψ)0,ΓI , γp

while the right-hand sides f ∈ V ∗f and p ∈ Q∗p are unchanged: f (v) = (ff , v)0 + (gf , v)0,Γf,N ,     p (ψ) = fp , ψ 0 − gp , ψ 0,Γ . p,N

In Eq. (6.30) the terms involving γf and γp only make sense if the Robin data ϕp + γf K∇ϕp ·n and −uf ·n+γp −1 n·T(uf , pf )·n are in L2 (ΓI ) which is in general not true. Note that the Neumann data TiN extends the notion of K∇ϕp ·n and  n·T(uf , pf )·n  to be in Λ∗i , see Definitions 6.4.5 and 6.4.6. However, the terms K∇ϕp · n, v · n 0,Γ I and (n · T(uf , pf ) · n, ψ)0,ΓI pair objects in Λ∗p × Λf and Λ∗f × Λp , respectively which can only be meaningful if Λp = Λf ,

(6.31)

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6 Stokes–Darcy Equations

which is therefore now implicitly assumed. Then the weak form can be reformulated to be ⎧ R a (uf , v) + bf (v, pf ) + cf (ϕp , v) = f (v) + f,p (v), ⎪ ⎪ ⎨ f bf (uf , q) = 0, (6.32) ⎪ ⎪   ⎩ apR ϕp , ψ + cp ((uf , pf ), ψ) = p (ψ) + p,f (ψ), with the coupling terms and additional right-hand sides cf (ϕ, v) = (ϕ, v · n)0,ΓI − γf ap (ϕ, ψv ), cp ((u, p), ψ) = −(u · n, ψ)0,ΓI +

1 1 af (u, v ψ ) + bf (v ψ , p), γp γp

f,p (v) = −γf p (ψv ), p,f (ψ) =

1 f (v ψ ), γp

where v ψ is an extension of ψ to V f (with v ψ · n = ψ) and ψv extends v · n to Qp . Note that these definitions do not depend on the particular choice of extensions as long as the same ones are used for both cf and f,p as well as for cp and p,f , compare with Definitions 5.1.5 and 5.3.9. Remark 6.6.2 Another way to obtain the Robin–Robin weak form (6.32) is to  add multiples of the terms (u · n, v · n)0,ΓI + K∇ϕp · n, v · n 0,Γ to the first and I (ϕ, ψ)0,ΓI +(n · T(uf , pf ) · n, ψ)0,ΓI to the last equation in the Neumann–Neumann weak formulation (6.7). These terms are consistent in the sense that they vanish for the (unique) solution of the Neumann–Neumann problem. This leads to a strategy for the proof of the equivalence to the Neumann–Neumann formulation in Eq. (6.6.3).

6.6.2 Remarks on the Condition Λp = Λf Starting with the formulation (6.29) where the interface conditions are of Robin type, a meaningful weak formulation is derived under the condition Λp = Λf , i.e., Eq. (6.31). There are two common situations where this condition is violated. One is concerned with the boundary at the interface ΓI . Note that Λp is the subspace of functions λ ∈ H 1/2(ΓI ) which can be extended by zero to H 1/2(ΓI ∪Γp,D ), so λ must vanish at ΓI ∩ Γp,D fast enough, see Theorem 4.3.4, the subsequent discussion, and Proposition 4.3.6. Similar statements are true for Λf . This means, condition (6.31) can only hold if the essential boundary parts in the respective subdomains touch the interface at the same points, i.e., if Γp,D ∩ ΓI = Γf,D ∩ ΓI . A special situation is one

6.6 Robin–Robin

137

Fig. 6.3 Sketch of two examples where one subdomain is completely enclosed by the other so that the interface ΓI has no boundary

subdomain being completely inside the other, for example a porous obstacle in a free flow or a crack inside a porous medium, see Fig. 6.3. In this case the boundary of ΓI is empty and therefore ΓI ∩ Γi,D = ∅, i ∈ {p, f}. Hence, the first common situation where the condition Λp = Λf is violated does not apply here. Some analysis is devoted to this special case in [GOS11a, GOS11b]. The other possible situation where the condition Λp = Λf is violated is the presence of a nonsmooth interface. While the trace of a function v ∈ H1 (Ωf ) is in the vector-valued space H1/2(ΓI ), its normal component v · n on the interface need not be in H 1/2(ΓI ) because the normal might have jumps, see also Remark 6.2.2. Then Λf contains H 1/2(ΓI ) but is not equal, while Λp is a subspace. In the left picture of Fig. 6.3 the interface is only Lipschitz continuous and therefore Λp = Λf , while in the right picture the interface is smooth and hence Λp = Λf . In many applications the interface is modeled to be Lipschitz continuous, for example piecewise linear. Also the condition Γp,D ∩ ΓI = Γf,D ∩ ΓI is not in general satisfied such that it is quite common that the interface spaces do not coincide. The Robin–Robin formulation can then be viewed as the Neumann– Neumann formulation (6.7) but with additional terms, which have to be suitably adapted whenever the extensions ψv ∈ Qp of v · n ∈ Λf and v ψ ∈ V f of ψ ∈ Λp do not exist. If the interface is piecewise smooth, with only finitely many kinks, then in some sense (Λp ∪ Λf ) \ (Λp ∩ Λf ) is small and these extensions mostly do exist. The above considerations therefore lead to a definition of a Robin–Robin weak formulation even if Λp = Λf . However in such a case the conditions on the interface are fulfilled only in some weaker sense. To this end, define Pi : Λj → Λi , i = j , such that Pi (ξ ) = ξ for all ξ ∈ Λj ∩ Λi and Pi (ξ ) = 0 if ξ ∈ Λj \ Λi , i.e., Pi = χΛi id, where χM is the indicator function of the set M and id the identity. The functions Pi have the property Pi ◦ Pi = Pi but are no projections, because the spaces Λi on the interface ΓI are not closed subspaces. Also both Pi are not continuous, however, they are bounded. With the help of the functions Pi , the bilinear forms afR and apR , the coupling terms cf and cf , as well as additional

138

6 Stokes–Darcy Equations

right-hand sides f,p and p,f , which are used in the weak formulation (6.32), are extended: Definition 6.6.3 (Robin–Robin Weak Formulation) Let γf ≥ 0, γp > 0, and all the data as well as (bi)linear forms from Definition 6.2.1 be given. Furthermore, let Pi = χΛi id : Λj → Λi , j = i. Define the forms afR : H 1 (Ωf ) × H 1 (Ωf ) → R, apR : H 1 (Ωp ) × H 1 (Ωp ) → R, cf : H 1 (Ωp ) × H 1 (Ωf ) → R, cp : H 1 (Ωf ) × H 1 (Ωp ) → R, f,p : H 1 (Ωf ) → R, and p,f : H 1 (Ωp ) → R as   afR (u, v) = af (u, v) + γf u · n, Pp (v · n) 0,Γ , I

1 apR (ϕ, ψ) = ap (ϕ, ψ) + (ϕ, Pf (ψ))0,ΓI , γp   cf (ϕ, v) = (ϕ, v · n)0,ΓI − γf ap ϕ, ψPp (v·n) , cp ((u, p), ψ) = −(u · n, ψ)0,ΓI +   f,p (v) = −γf p ψPp (v·n) , p,f (ψ) =

  1  1  af u, v Pf (ψ) + bf v Pf (ψ) , p , γp γp

(6.33)

 1  f v Pf (ψ) . γp

Then the Robin–Robin weak formulation of the Stokes–Darcy coupled problem is: Find (uf , pf , ϕp ) ∈ H1 (Ωf ) × L2 (Ωf ) × H 1 (Ωp ), such that for all v ∈ V f , q ∈ Qf , and all ψ ∈ Qp it is ⎧ R a (uf , v) + bf (v, pf ) + cf (ϕp , v) = f (v) + f,p (v) ⎪ ⎪ ⎨ f bf (uf , q) = 0 ⎪ ⎪   ⎩ apR ϕp , ψ + cp ((uf , pf ), ψ) = p (ψ) + p,f (ψ),

(6.34)

together with uf = ub on Γf,D and ϕp = ϕb on Γp,D . Essentially the above extensions (6.34) can be summarized as follows: Whenever a trace of a test function (v or ψ) is not in Λf ∩ Λp it is set to zero in some terms and Eq. (6.34) reduce to the Neumann–Neumann system (6.7). Note that the forms above indeed are not (bi)linear, because the functions Pp and Pf are not. However, if Λf = Λp this extended weak formulation reduces to that in Eq. (6.32) and all the defined forms are (bi)linear.

6.6 Robin–Robin

139

6.6.3 Equivalence to the Neumann–Neumann Formulation It is shown in Sect. 6.3 that the Neumann–Neumann weak formulation (6.7) admits a unique solution. Furthermore, the difference to the Robin–Robin weak formulation is the addition of terms which are consistent. This is made more precise in the following theorem: Theorem 6.6.4 (Equivalence of Weak Forms) Let (uf , pf , ϕp ) be the unique solution to the Neumann–Neumann weak formulation (6.7), Definition 6.2.1. Then this solution also solves the Robin–Robin formulation (6.34), Definition 6.6.3. Conversely, a solution of (6.34) also solves (6.7). Proof Let (v, q, ψ) ∈ V f × Qf × Qp . If the function Pp (Tf (v, q)) = Pp (v · n) onto Λp vanishes, the first equation in (6.34) reduces to that of (6.7), similarly for the last equation if Pf (Tp ψ) = 0. Now assume Pp (Tf (v, q)) = 0 and denote its extension to Qp by ψv = ψPp (v·n) . Then the additional terms in the first equation of (6.34) are γf (uf · n, ψv )0,ΓI − γf ap (ϕ, ψv ) = γf p (ψv ). This identity is the last equation in (6.7) with ψ = ψv and therefore holds for all such v. Now consider the case where Pf (Tp ψ) = 0. Then the additional terms in the last equation in the Robin–Robin formulation (6.34) read  1 1 1 1 ϕp , ψ 0,Γ + af (uf , v ψ ) + bf (v ψ , pf ) = f (v ψ ), I γp γp γp γp where v ψ denotes the extension of Pf (Tp ψ) to V f , i.e., v ψ · n = Pf (Tp ψ). Therefore, this identity holds because of the first equation in the Neumann– Neumann formulation (6.7). Now let (uf , pf , ϕp ) be a solution to the Robin–Robin formulation (6.34). As a reminder the relevant two equations of (6.34) are rewritten here:     af (uf , v) + bf (v, pf ) + γf uf · n, Pp (v · n) 0,Γ + ϕp , v · n 0,Γ I

I

− γf ap (ϕp , ψv ) = f (v) − γf p (ψv ),    1 ϕp , Pf (ψ) 0,Γ − (uf · n, ψ)0,ΓI a p ϕp , ψ + I γp +

1 1 1 af (uf , v ψ ) + bf (v ψ , pf ) = p (ψ) + f (v ψ ). γp γp γp

The extensions ψv ∈ Qp and v ψ ∈ V f of Pp (v ·n) and Pf (ψ) on the interface ΓI can be inserted into these two equations. In particular, the second equation with ψ = ψv and multiplied by γf can be added to the first, and the first equation with v = v ψ

140

6 Stokes–Darcy Equations

multiplied by − γ1p can be added to the second, yielding     γf γf γf af uf , v + v ψv + bf v + v ψv , pf + ϕp , v + v ψv · n γp γp γp 0,ΓI  γf = f v + v ψv , γp    γf γf γf ap ϕp , ψ + ψv ψ − uf · n, ψ + ψv ψ = p ψ + ψv ψ . γp γp γp 0,ΓI These two equations are the first and last equation in the Neumann–Neumann weak formulation (6.7) with v + γγpf v ψv and ψ + γγpf ψvψ instead of v and ψ. The function v ψv ∈ V f has the trace Tf (v ψv , ·) = Pf (Pp (Tf (v, ·))) and similarly it is Tp (ψv ψ ) = Pp (Pf (Tp ψ)). Finally, any v˜ ∈ V f and ψ˜ ∈ Qp can be written as v˜ = v + γγpf v ψv and −1  ψ˜ = ψ + γγpf ψvψ with v ∈ V f and ψ ∈ Qp . Indeed, let α = 1 + γf /γp , define ψ := ψ˜ − γf ψv , and note that γp

α ψ˜

   γf ˜ Pf Tp ψ = Pf Tp ψ − Tp ψv αψ˜ γp  γf ˜ = Pf Tp ψ˜ − α Pp (Pf (Tp ψ)) γp  γf = Pf 1 − α Tp ψ˜ γp   = Pf Tp (α ψ˜ ) , i.e., v ψ = v α ψ˜ and therefore ψv ψ = ψvαψ˜ and hence ψ˜ = ψ + γγpf ψv ψ . Here it is used that Tp ψv η = Pf (Tp η) for all η ∈ Qp . Using the same ideas lead to a construction of v for a given v˜ . Thus, it is shown that the Neumann–Neumann formulation (6.7) holds if the Robin–Robin one does.   Remark 6.6.5 The previous result, Theorem 6.6.4, shows existence of a solution to the Robin–Robin formulation (6.34), Definition 6.6.3. Furthermore, its second part assures that this solution is unique. In other words, unique solvability of the Neumann–Neumann problem (6.7) carries over to (6.34).

6.6 Robin–Robin

141

6.6.4 Subdomain Operators The Neumann–Neumann formulation (6.7) could be rewritten using operators on the interface, see Definitions 6.4.11 and 6.4.12 in Sect. 6.4.4. Analogously, such operators solving Robin instead of Dirichlet or Neumann problems are introduced. γ

Definition 6.6.6 (Solution Operators Ki i ) With the spaces from Definitions 6.4.1 γp and 6.4.2, define for γp > 0 and γf ≥ 0 the Robin solution operators Kp : Λ∗p → γ  f × Qf which map given Robin data μ on the interface ΓI to p and K f : Λ∗ → V Q f f a Darcy and Stokes solution, respectively: γp

Kp (μ) = ϕ solves ⎧ −∇ · (K∇ϕ) = fp ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ −K∇ϕ · n = gp

in Ωp ,

⎪ ⎪ ⎪ ⎪ ⎪ ⎩

on Γp,D ,

ϕ = ϕb −γp K∇ϕ · n + ϕ = μ

on Γp,N ,

on ΓI .

γ

Kf f (μ) = (u, p) solves ⎧ −∇ · T(u, p) = ff ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∇·u=0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ T(u, p) · n = fN f ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

u = fD f

in Ωf , in Ωf , on Γf,N , on Γf,D ,

u · τ + ατ · T(u, p) · n = 0

on ΓI ,

−γf u · n − n · T(u, p) · n = μ

on ΓI .

The respective weak formulations are obtained as before, see Sects. 5.1.1 and 5.3.1. In particular for all ψ ∈ Qp and all (v, q) ∈ V f × Qf it is apR (ϕ, ψ) = p (ψ) +

 1 μ, Tp ψ Λp ∗ ×Λp γp

and afR (u, v) + bf (v, p) = f (v) − (μ, Tf v)Λf ∗ ×Λf , bf (u, q) = 0.

142

6 Stokes–Darcy Equations

Note that in Sect. 6.6.2 functions Pp and Pf are introduced which slightly modified the bilinear forms apR and afR whenever Tp ψ ∈ / Λf and Tf (v, ·) ∈ / Λp respectively. γ Without this modification the above operators Ki i are of the type introduced in the γ Sects. 5.1.3 and 5.3.3. With it however, the Robin operators Ki i have the form of the Neumann operators Ki,N whenever Pf (Tp ψ) = 0 and Pp (Tf (v, ·)) = 0. Despite this difficulty these operators are still referred to as of Robin type. Furthermore, the γ image spaces of Ki i coincide with those of the respective Neumann and Dirichlet γ operators, i.e., Im Ki,N = Im Ki,D = Im Ki for all γ > 0. Also, it is Kf0 = Kf,N γ while the other operators Ki,j can not be identified with the Ki with a suitable γ . It is possible to recover Robin data similar to the Neumann case in Definitions 6.4.5 and 6.4.6: Definition 6.6.7 Let ϕ ∈ H 1 (Ωp ) be given such that for all ψ ∈ Qp ∩ ker Tp = HΓ1p,D up ΓI (Ωp ), it is2 ap (ϕ, ψ) = p (ψ). Furthermore, let (u, p) ∈ H 1 (Ωf ) × L2 (Ωf ) be given such that for all (v, q) ∈ (V f × Qf ) ∩ ker Tf it is3  af (u, v) + bf (v, p) = f (v), bf (u, q) = 0. R : H 1 (Ω ) → Λ∗ and T R : H1 (Ω ) × L2 (Ω ) → Λ∗ is Then the Robin data Tp,γ p f f p f,γ f defined as     R Tp,γ ϕ, λ = γ p (ψλ ) − ap (ϕ, ψλ ) + (ϕ, λ)0,ΓI , ∗ Λp ×Λp

  R Tf,γ (u, p), λ

Λf ∗ ×Λf

= f (v λ ) − af (u, v λ ) − bf (v λ , p) + γ (u · n, λ)0,ΓI , (6.35)

Compare this also with Definitions 5.1.5 and 5.3.9. This is equivalent to adding the Neumann and Dirichlet data with appropriate scaling, i.e., R Tp,γ = −γ TpN + Tp

R and Tf,γ = −TfN + γ Tf .

Identities similar to the ones for the operators Ki,N , i ∈ {p, f} in Eqs. (6.9) hold whenever Λf = Λp : γ

R ◦ Ki i = id Ti,−γ i γ

R Ki i ◦ Ti,−γ = id i

2 With 3 With

this ψ it is ap (ϕ, ψ) = apR (ϕ, ψ). this v it is afR (u, v) = af (u, v).

in Λ∗i , γ

in Im Ki,N = Im Ki,D = Im Ki i .

6.6 Robin–Robin

143

If however the interface spaces Λf and Λp do not coincide, such identities do not hold because terms involving id −Pi enter the equation. In Sect. 6.6.6 the operators R are concatenated with K γ , j ∈ {p, f}, j = i, which requires that the image Ti,γ j R maps to Λ∗ and T R space of one is in the preimage space of the other, i.e., that Tp,γ f f,γ to Λ∗p . The following extension is suitable: R from Definition 6.6.7 are extended Definition 6.6.8 The Robin data operators Ti,γ 2 R : H 1 (Ω ) → Λ∗ via the functions Pi : L (ΓI ) → Λi from Definition 6.6.3 to Tp,γ p f R 1 2 ∗ and Tf,γ : H (Ωf ) × L (Ωf ) → Λp as follows

  R Tp,γ (ϕ), η   R Tf,γ (u, p), η

Λf ∗ ×Λf

Λp ∗ ×Λp

  = γ p (ψPp (η) ) − ap (ϕ, ψPp (η) ) + (ϕ, η)0,ΓI ,

(6.36a)

= f (v Pf (η) ) − af (u, v Pf (η) ) − bf (v Pf (η) , p) + γ (u · n, λ)0,ΓI .

(6.36b)

This means η is replaced by zero whenever the earlier definitions are not applicable; in that case the Robin data operators above reduce to the trace operator Tp and γ Tf , respectively.

6.6.5 Operators Acting Only on the Interface In Sect. 6.4.3 the operators Hi,j are defined which map Dirichlet/Neumann data to Neumann/Dirichlet data solving one of the two subproblems. Here, similarly, γ ,γ operators Hi 1 2 which solve Robin problems using γ1 and return other Robin data (using γ2 ) are introduced. With their help the Robin–Robin weak formulation (6.34) can be rewritten. Definition 6.6.9 (Robin Operators on the Interface) Using the Robin solution and data operators, Definitions 6.6.6 and 6.6.7, define for all γp > 0 and γf ≥ 0 γp ,γf

R = Tp,γ ◦ Kp , f

γ ,γp

R = Tf,γ ◦ Kf f . p

Hp

Hf f

γp

γ

Note that the conditions on γp and γf can be slightly weakened without sacrificing γ the well-posedness of Ki i . For negative values of γi the respective bilinear form aiR is still positive definite if |γi | is small compared to K or ν respectively, see also the proofs of the coercivity of ai in Sects. 5.1.2 and 5.3.2. This is however a strong limitation, because small values of K and ν are of interest. So even though the following inverse identities hold formally, they have little practical relevance: γ ,−γ2

Hi 1

γ ,−γ1

◦ Hi 2

= id

in Λ∗i .

(6.37)

144

6 Stokes–Darcy Equations

6.6.6 Weak Formulation Rewritten Analogously to the fixed point equations (6.12) and (6.13) and Definition 6.4.11, for the Neumann–Neumann problem consider the following Definition 6.6.10 (Robin Fixed Point Equations) Using the Robin operators on the interface from Definition 6.6.9, define the two fixed point equations  γ ,γ  p Hp f (λ) = λ,  γ ,γ  γp ,γ p Hp f Hf f (μ) = μ. γ ,γp

Hf f

(6.38) (6.39)

According to the extensions of the Robin data operators in Definition 6.6.8, these fixed point equations are well defined. Corollary 6.4.14 shows that the fixed point equations (6.12) and (6.13) are equivalent. Its counterpart with respect to the equations above is the following. Corollary 6.6.11 (Equivalence of Fixed Point Equations) If λ is a solution γp ,γ of (6.38), then μ = Hp f (λ) solves (6.39). Conversely if μ solves (6.39), then γf ,γp λ = Hf (μ) solves (6.38). γp ,γf

Proof Apply Hp

γ ,γp

to (6.38) and Hf f

 

to (6.39).

The fixed point formulations furthermore allow for a connection to the coupled Robin–Robin problem (6.34). The following theorem is an analogue to Theorem 6.4.15. Theorem 6.6.12 Let λ solve the fixed point equation (6.38). Then  γ ,γ  γ p (u, p) := Kf f Hp f (λ)

and

γp

ϕ := Kp (λ)

R (u, p) solves (6.38). solve (6.34). If conversely (u, p, ϕ) solve (6.34), then λ = Tf,γ p   γ ,γ γf ,γp p f Proof Let λ solve Eq. (6.38), i.e., λ = Hf Hp (λ) . By definition the solution (u, p, ϕ) satisfies the correct Dirichlet data away from the interface, u = ub on Γf,D and ϕ = ϕb on Γp,D . Now let v ∈ V f , q ∈ Qf , and ψ ∈ Qp be given test functions γp ,γ γ and μ = Hp f (λ). Then according to the definition of Kf f it is

afR (u, v) + bf (v, p) = f (v) − (μ, Tf v)Λf ∗ ×Λf , bf (u, q) = 0.

6.6 Robin–Robin

145

R yields The definition of Tp,γ f

  R (ϕ), v · n (μ, v · n)Λf ∗ ×Λf = Tp,γ f

Λp ∗ ×Λp

  = γf p (ψPp (v·n) ) − ap (ϕ, ψPp (v·n) ) + (ϕ, v · n)0,ΓI = cf (ϕ, v) − f,p (v) with the function Pp introduced in Sect. 6.6.2, and hence the first equation in (6.34). γp Similarly, the definition of Kp gives apR (ϕ, ψ) = p (ψ) +

1 (λ, ψ)Λp ∗ ×Λp . γp γ ,γp

Using the fixed point equation (6.38), it is λ = Hf f therefore

R (u, p), and (μ), i.e., λ = Tf,γ p

 1 1 R Tf,γp (u, p), ψ (λ, ψ)Λp ∗ ×Λp = Λp ∗ ×Λp γp γp        1 = f v Pf (ψ) − af u, v Pf (ψ) − bf v Pf (ψ) , p γp + (u · n, ψ)0,ΓI = p,f (ψ) − cp ((u, p), ψ),  

which is the last equation in (6.34).

With a very similar proof one can also show that a solution of the other fixed point equation (6.39) on the interface leads to a solution of the weak formulation (6.34). However, Corollary 6.6.11 together with the previous result, Theorem 6.6.12, provide an easier proof of the following analogue to Theorem 6.4.16: Theorem 6.6.13 Let μ solve the interface condition (6.39). Then γ

(u, p) := Kf f (μ)

and

γp

ϕ := Kp

 γ ,γ  p Hf f (μ)

R (ϕ) solves (6.39). solve (6.34). If conversely (u, p, ϕ) solve (6.34), then μ = Tp,γ f γ ,γp

Proof With μ solving (6.39), also λ = Hf f

γ Kf f (μ)

(μ) solves  γ (6.38),  see Corolγf p ,γf = Kf Hp (λ) and ϕ =

lary 6.6.11. Then note that (u, p) =   γ ,γ γ γ Kp p Hf f p (μ) = Kp p (λ), so that (u, p, ϕ) solves the Robin–Robin weak formulation (6.34), according to Theorem 6.6.12. If conversely (u, p, ϕ) solves (6.34),

146

6 Stokes–Darcy Equations

R (u, p) solves (6.38), and hence then the same theorem assures that λ = Tf,γ p γ ,γ

R (ϕ) solves (6.39), see Corollary 6.6.11. μ = Hp p f (λ) = Tp,γ f

 

Remark 6.6.14 Together with the identities (6.37) it can be seen that there are equations of Steklov–Poincaré type which are equivalent to the fixed point problems (6.38) and (6.39): 

 (λ) = λ  γ ,γ  γp ,γ p Hp f Hf f (μ) = μ γ ,γp

Hf f

γp ,γf

Hp

−γp ,−γf

γp ,γf

(λ) − Hf

γ ,γp

(μ) − Hp

⇐⇒

Hp

⇐⇒

Hf f

(λ) = 0,

−γf ,−γp

(μ) = 0.

Note that while these equations are formally correct, they suffer the same deficiencies as the identities (6.37). This is why the Steklov–Poincaré equations with Robin operators are not further studied. Remark 6.6.15 Similar to Sect. 6.5 one can show that the Robin operators above are affine linear and derive a linear equation on the interface similar to (6.24) and (6.25). No particular problems arise in this approach but for ease of notation and presentation it is avoided here.

6.6.7 Decoupling the Equations Further In this subsection another view on the Robin–Robin weak formulation (6.34) is developed, very similar to the decoupling in Sect. 6.4.6. For all v ∈ V f , q ∈ Qf , ψ ∈ Qp , ξf ∈ Λf , and ξp ∈ Λp it is afR (uf , v) + bf (v, pf ) + (μ, v · n)Λf ∗ ×Λf = f (v), 

λ, ξp





Λp ∗ ×Λp

− γp uf · n, ξp

bf (uf , q) = 0,

 0,ΓI

+af (uf , v Pf (ξp ) ) + bf (v Pf (ξp ) , pf ) − f (v Pf (ξp ) ) = 0, apR (ϕp , ψ) −

1 (λ, ψ)Λp ∗ ×Λp = p (ψ), γp

  (μ, ξf )Λf ∗ ×Λf − ϕp , ξf 0,ΓI + γf ap (ϕp , ψPp (ξf ) ) − γf p (ψPp (ξf ) ) = 0,

(6.40) where v λp and ψλf are extensions of λp and λf to V f and Qp such that v λp · n = λp and Tp ψλf = ψλf Γ = λf . The second and fourth equations above are essentially I the application of cp and cf from Eq. (6.33).

6.6 Robin–Robin

147

6.6.8 Alternative Robin–Robin Method In [DQV07] and [CGHW11] an alternative weak formulation is introduced which solves Robin problems in each subdomain, see also the C-RR method in [CJW14]. It is assumed that Λp = Λf . Here a solution consists of uf , pf , and ϕp together with two variables on the interface ηf , ηp ∈ Λp = Λf . Then for each v ∈ V f , q ∈ Qf and ψ ∈ Qp it shall hold afR (uf , v) + bf (v, pf ) − (ηf , v · n)0,ΓI = f (v), b(uf , q) = 0, ηp = duf · n + cηf , apR (ϕp , ψ)

 1 − ηp , ψ 0,Γ = p (ψ), I γp

(6.41)

ηf = bϕp + aηp , where a, b, c, and d are constants which are chosen such that the solution to the Neumann–Neumann formulation (6.7) also solves the above.4 In contrast to Eqs. (6.20) and (6.40) the intermediate variables on the interface now are directly coupled. Subtracting the equations in (6.7) from the first and third equation in (6.41) yields   γf (uf · n, v · n)0,ΓI − (ηf , v · n)0,ΓI − ϕp , v · n 0,Γ = 0, I

ηp = duf · n + cηf ,   1 1 ϕp , ψ 0,Γ − ηp , ψ 0,Γ + (uf · n, ψ)0,ΓI = 0, I I γp γp ηf = bϕp + aηp . The first and third equations only include inner products on the interface ΓI , hence can be simplified to be equations solely in Λf = Λp : ηf = γf uf · n − ϕp , ηp = duf · n + cηf , ηp = γp uf · n + ϕp , ηf = bϕp + aηp .

4 The

signs and positions of ηf and ηp are chosen such that the resulting system matches that in [DQV07] and [CGHW11].

148

6 Stokes–Darcy Equations

These equations can be solved for a, b, c, and d yielding a=

γf , γp

b = −1 − a,

c = −1,

d = γf + γp ,

(6.42)

see also Lemma 2.2 in [CGHW11]. In comparison to the Robin–Robin weak formulation (6.40) the interface variables in Eq. (6.41) are of higher regularity, namely in Λf = Λp instead of its dual. However, ηf and ηp are directly coupled to each other while this is not the case for λ and μ in Eq. (6.40). Both approaches share the advantage that each interface variable only depends on one of the two subdomain solutions directly. Remark 6.6.16 The notation ηf and ηp is chosen to match that in the literature [DQV07, CGHW11, CJW14]. These interface variables play a similar role to λ and μ in the previous Sect. 6.6.7, but in general are not equal.

6.7 The Finite Element Method for the Stokes–Darcy Problem A general introduction to the finite element method is not given here, instead it is referred to the classic literature, e.g., [Bra07b, BS08, BF91, Cia02]. This is also where relevant notions such as grid, edge, face, vertex, and finite element space, as well as properties of the linear systems obtained through the finite element method for the subproblems can be found. Consider one of the weak formulations of the coupled Stokes–Darcy problem (6.7) or (6.34). Furthermore, let {Th } be a regular family of grids for the entire domain Ω such that the interface ΓI does not intersect with the interior of any grid cell, i.e., it is the union of a number of edges/faces of the grid. This leads to subgrids Th,f and Th,p in each subdomain Ωf and Ωp which match at the interface in the sense that each edge/face F ⊂ ΓI of Th,f is exactly one edge/face of Th,p . In general it is possible to use non-matching grids, but for simplicity here this is not pursued. Each of the spaces H 1 (Ωf ), L2 (Ωf ), and H 1 (Ωp ) in which the solution is sought is approximated by suitable conforming finite element spaces V h,f , Qh,f , and Qh,p leading to a linear system of equations of the following form: 

S C1 C2 D

 (uh,f , ph,f ) = b. ϕh,p

(6.43)

Here S is a saddle point matrix and D is the discretization of the bilinear form ap or apR , respectively. The matrices C1 and C2 represent the coupling terms. In the case of the Neumann–Neumann weak form (6.7) it is C1 = −C2 while in the case of the Robin–Robin form (6.34) no such relation holds. To be precise, let {v i }, {qi }, and {ψi } be a finite element basis of V h,f , Qh,f , and Qh,p , respectively. Then the system

6.7 The Finite Element Method for the Stokes–Darcy Problem

149

to be solved is ⎞ ⎛ ⎞ ⎞⎛ b1 uh,f A B  Cf ⎝ B 0 0 ⎠ ⎝ ph,f ⎠ = ⎝ 0 ⎠ . Cp,1 Cp,2 D ϕh,p b2 ⎛

The matrices and right-hand side have the following entries for the two considered weak formulations: Robin–Robin (6.34)

Neumann–Neumann (6.7)

Aij = afR (v j , v i ),

Aij = af (v j , v i ),

Bij = bf (v j , qi ),   (Cf )ij = ψj , v i · n 0,Γ

Bij = bf (v j , qi ),   (Cf )ij = ψj , v i · n 0,Γ ,

I

I

  (Cp,1 )ij = − ψi , v j · n 0,Γ , I

− γf ap (ψj , ψPp (vi ·n) ),   (Cp,1 )ij = − ψi , v j · n 0,Γ I

1 + af (v j , v Pf (ψi ) ), γp (Cp,2 )ij = 0, Dij = ap (ψj , ψj ),

(Cp,2 )ij =

1 bf (v Pf (ψi ) , qj ), γp

Dij = apR (ψj , ψj ),

(b1 )i = f (v i ),

(b1 )i = f (v i ) + f,p (v i ),

(b2 )i = f (ψi ),

(b2 )i = f (ψi ) + p,f (ψi ).

Specific algorithms to solve these type of linear systems are discussed in Chap. 7. In the Sects. 6.4.4 and 6.6.6 several equations on the interface ΓI are introduced which are equivalent to the Neumann–Neumann and Robin–Robin weak formulations (6.7) and (6.34), respectively, namely the fixed point equations (6.12), (6.13), (6.38), and (6.39) as well as the Steklov–Poincaré equations (6.16) and (6.17), see also Theorems 6.4.15, 6.4.16, 6.6.12, and 6.6.13. Additionally, in Sects. 6.4.6 and 6.6.7 the respective linear system is rewritten with the help of the interface variables λ and μ. This can be discretized in a straightforward way leading to a system of the form ⎛

⎞ ⎛ ⎞⎛ ⎞ S Ef (b1 , 0) (uh,f , ph,f ) ⎜Rf − id ⎟ ⎜ 0 ⎟ ⎟⎜ λh ⎜ ⎟=⎜ ⎟⎜ ⎟, ⎝ ⎠ ⎝ ϕh,p ⎠ ⎝ b2 ⎠ Ep D Rp − id μh 0

(6.44)

150

6 Stokes–Darcy Equations

where Ef Rp = C1 and Ep Rf = C2 , see also Section 2 in [CJW14]. For the Neumann–Neumann problem the spaces for the interface variables are the restrictions of the respective finite element spaces in the subdomains. Hence, μh is continuous because only conforming finite elements are used for ϕh,p and λh is in general only continuous if the interface ΓI does not contain jumps in the normal, i.e., is flat, compare with Remark 6.2.2. Therefore, one of the discretized interface spaces might have to be discontinuous. Implementing the decoupled system (6.44) rather than the coupled one (6.43) may be helpful because the solver for one subdomain does not need any information on the other subdomain which allows greater modularization. Furthermore, the discretization of the Steklov–Poincaré operators is just the application of Ei , i ∈ {f, p}, followed by a solving step (inverting S or D, respectively) and an application of the restriction Ri . Also the alternative Robin–Robin system (6.41) can be written in the form of Eq. (6.44). Here the variables on the interface, ηf and ηp , are continuous whenever the finite element spaces for the Stokes velocity and the pressure in the Darcy subdomain are discretized with continuous, i.e., conforming, finite elements.5 This is why this approach is denoted C-RR (continuous Robin–Robin) in [CJW14]. Note that in the C-RR method, due to the direct coupling of the interface variables, the system to be solved cannot be written in the form of Eq. (6.43).

6.7.1 The D-RR Method In [CJW14], another method is proposed and called D-RR. It directly discretizes the preliminary weak form (6.30), i.e., derivatives of functions in the subdomain are evaluated at the interface. The resulting linear system of equations has the same form as the other approaches, and the matrices A, B, and D have the same entries. The coupling terms however differ, for the D-RR method these are     (Cf )ij = ψj , v i · n 0,Γ + γf K∇ψj · n, v i · n 0,Γ , I

I

   1 n · 2νD(v j ) · n, ψi 0,Γ , (Cp,1 )ij = − ψi , v j · n 0,Γ + I I γp (Cp,2 )ij = −

 1 qj , ψi 0,Γ , I γp

(b1 )i = f (v i ), (b2 )i = f (ψi ).

5 For

this approach it is assumed that Λf = Λp , see Sect. 6.6.8.

6.7 The Finite Element Method for the Stokes–Darcy Problem

151

In other words, compared with the Neumann–Neumann discretization, the terms     γf uh,f · n, v i · n 0,Γ + γf K∇ϕh,p · n, v i · n 0,Γ I

I

and   1 1 ϕh,p , ψi 0,Γ + n · T(uh,f , ph,f ) · n, ψi 0,Γ I I γp γp are added to the system, i.e., the two interface equations (6.3a) and (6.3b) tested with the appropriate traces of the respective test functions. In this sense, the DRR method is a conforming discretization. However, these additions involve lower order terms such that the accuracy of the resulting discrete solution might be worse compared with the Neumann–Neumann one. On the other hand, the implementation cost is comparable and much smaller than that of the Robin–Robin approach. Since the used finite element functions are piecewise smooth this is possible on each cell separately yielding a discontinuous interface function even for straight interfaces, hence the name discontinuous Robin–Robin (D-RR). This approach can also be viewed as an approximation to the discretization of Eq. (6.34), after R and T R , see their definitions (6.35), are extensions of all the Robin traces Tp,γ f,γ suitable combinations of derivatives and values, compare also with Definitions 5.1.5 and 5.3.9.

Chapter 7

Algorithms

The Neumann–Neumann as well as the Robin–Robin systems (6.7) and (6.34) along with their decoupled variants (6.20) and (6.40) can be solved iteratively. The first two can be represented as systems 

S C1 C2 D

  b (uf , pf ) = 1 . ϕp b2

(7.1)

Compare this with the discrete system (6.43) where for simplicity the same symbols in the matrix and right-hand side are used.1 The decoupled approaches (6.20) and (6.40) lead to systems of the form ⎞⎛ ⎞ ⎛ ⎞ (uf , pf ) b1 S Ef ⎜Rf − id ⎟⎜ λ ⎟ ⎜0⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ϕp ⎠ = ⎝ b 2 ⎠ , Ep D Rp − id μ 0 ⎛

(7.2)

with C1 = Ef Rp and C2 = Ep Rf in correspondence to the discrete version in Eq. (6.44). In the case of the Neumann–Neumann approach the interface   operators are recovered as Hf,N (μ) = Rf S −1 (b1 − Ef μ) and Hp,N (λ) =    Rp D −1 b2 − Ep (λ) , while in the Robin–Robin case Hf,N and Hp,N have to be γ ,γp γp ,γ replaced with Hf f and Hp f , respectively. Note that the entire system (including S, D, b1 , b2 , Ef , Ep , Rf , Rp , λ, and μ) differs compared with the Neumann– Neumann case, but the structure of the equations remains.

1 There

is a slight notational inconsistency here. For simplicity, the vector (b1 , 0) in Eqs. (6.43) and (6.44) is denoted only b1 here.

© Springer Nature Switzerland AG 2019 U. Wilbrandt, Stokes–Darcy Equations, Advances in Mathematical Fluid Mechanics, https://doi.org/10.1007/978-3-030-02904-3_7

153

154

7 Algorithms

equivalence of all Neumann–Neumann algorithms, Proposition 7.2.5 Richardson iteration for λ Algorithm 7.8 Proposition 7.2.3 fixed point algorithm for λ Algorithm 7.4

simple iterative algorithm Algorithm 7.1

Proposition 7.2.1

Proposition 7.1.1

decoupled iterative algorithm Algorithm 7.2

fixed point algorithm for μ Algorithm 7.5 Proposition 7.2.4

Richardson iteration for μ Algorithm 7.9

Fig. 7.1 A graph showing the relationships among all algorithms and propositions related to the Neumann–Neumann approach. The expected convergence behavior is shown in Proposition 7.3.1

equivalence of all Robin–Robin algorithms, Proposition 7.2.6 iterative algorithm for the modified Robin–Robin method Algorithm 7.3

fixed point algorithm for λ Algorithm 7.6

simple iterative algorithm Algorithm 7.1

Proposition 7.2.2 fixed point algorithm for μ Algorithm 7.7

Proposition 7.1.1 decoupled iterative algorithm Algorithm 7.2

Fig. 7.2 A graph showing the relationships among all algorithms and propositions related to the Robin–Robin approaches. The expected convergence behavior is shown in Proposition 7.3.3 and in 7.3.2 for Algorithm 7.3

7.1 Classical Iterative Subdomain Methods In this section several algorithms are presented which solve the Stokes–Darcy problem (7.1) and (7.2). In the finite element setting it is possible to set up the entire matrix at once. However, this typically requires a lot of coding and direct access to the underlying finite element spaces and matrices. Instead, all the presented

7.1 Classical Iterative Subdomain Methods

155

Algorithm 7.1 choose ϕp0 for k = 0, 1, . . . do ufk+1 , pfk+1 = S −1 b1 − C1 ϕpk ϕpk+1 = D −1 b2 − C2 ufk+1 , pfk+1 end for A simple iterative subdomain algorithm to solve the discrete Stokes–Darcy problem (7.1)

algorithms are based on solving the individual subproblems separately. The coupled system does not need to exist in memory at all. This enables the use of possibly highly tailored codes for each subproblem which only need to be able to solve Dirichlet, Neumann, and/or Robin problems and return data on the interface. In principle, the two subproblems could be solved using different codes. A first rather simple approach to the Stokes–Darcy problem (7.1) is shown in Algorithm 7.1. Here each iteration consists of one solving step in each subdomain. Using the k-th iterate instead of the (k + 1)-th in the second step yields a version where each solving step can be done in parallel. The main advantage of Algorithm 7.1 is its simplicity and straightforward implementation. To better understand it, a reformulation is meaningful. Let s k be the pair (ukf , pfk ). Then Algorithm 7.1 leads to

s k+1 ϕpk+1



 S −1 (b1 − C1 ϕpk ) = D −1 (b2 − C2 s k+1 ) 

S −1 (b1 − C1 ϕpk ) = D −1 b2 − D −1 C2 S −1 (b1 − C1 ϕpk )   b1 − C1 ϕpk 0 S −1 = −D −1 C2 S −1 D −1 b2 /  k 0  s 0 b1 0 C1 S −1 − = −1 −1 −1 ϕpk b2 0 0 −D C2 S D  −1 /  k 0 s S 0 b1 0 C1 = − ϕpk b2 0 0 C2 D

  −1 /  k 0 sk s b1 S 0 S C1 = + . − ϕpk ϕpk b2 C2 D C2 D

156

7 Algorithms

This is the form a block-wise Gauss–Seidel2 iteration has. Furthermore, a damping can be added via a weighting with the previous iterate:

s k+1 ϕpk+1





 sk S −1 (b1 − C1 ϕpk ) = (1 − ω) k + ω ϕp D −1 (b2 − ωC2 s k+1 ) 

k − C ϕk ) Ss S −1 (b1 + 1−ω 1 p ω =ω k k+1 ) D −1 (b2 + 1−ω ω Dϕp − ωC2 s

 k k S −1 (b1 + 1−ω ω Ss − C1 ϕp )   =ω 1−ω k −1 k k D −1 b2 + 1−ω ω Dϕp − ωC2 S (b1 + ω Ss − C1 ϕp )   k − C ϕk b1 + 1−ω Ss 0 S −1 1 p ω =ω k −D −1 ωC2 S −1 D −1 b2 + 1−ω ω Dϕp   −1 /  k k 0 1 s s S 0 b1 S 0 S C1 =ω + − k ϕpk ϕ ωC2 D ωC b2 C2 D D ω 2 p

  −1 /  k 0 s sk S 0 b1 S C1 +ω . − = k ϕp ϕpk b2 C2 D ωC2 D

This is then the block-wise SOR method. Also SSOR can be done in a blockwise manner. In essence, Algorithm 7.1 is a fixed point (Richardson) iteration preconditioned with a block-wise Gauss–Seidel method. It is shown that the decoupled system (7.2) is equivalent to the smaller coupled problem (7.1), see Eqs. (6.7) and (6.20) in the Neumann–Neumann case and Eqs. (6.34) and (6.40) for the Robin–Robin approach. A straightforward recursion to solve (7.2) leads to Algorithm 7.2, which therefore produces the same iterates as Algorithm 7.1, if the initial guess is chosen appropriately: Proposition 7.1.1 Let ϕp0 be given. Then the iterates (ukf , pfk ) and ϕpk , k ∈ N, obtained from Algorithm 7.1, initialized with ϕp0 , coincide with those from Algorithm 7.2, initialized with μ0 = Rp ϕp0 . If in contrast μ0 is given, Algorithm 7.2 yields the same iterates as Algorithm 7.1 with ϕp0 = Ep μ0 (or any other extension). The alternative Robin–Robin method (C-RR) is only representable in the decoupled way, see also Sect. 6.6.8, leading to Algorithm 7.3. Here the updates of the interface variables are modified in comparison with Algorithm 7.2. Both algorithms correspond to blockwise Gauss–Seidel iterations and can be turned into Jacobi type if the indices k+1 are replaced by k in the right-hand sides. 2 See for example [Saa03] for an introduction to classical iterative solvers such as Gauss–Seidel, Jacobi, SOR (successive overrelaxation), and SSOR (symmetric SOR) as well as for Krylov solvers, such as cg (conjugate gradients) and gmres (generalized minimal residual).

7.2 Algorithms for Interface Equations

157

Algorithm 7.3

Algorithm 7.2 choose μ0 for k = 0, 1, . . . do ufk+1 , pfk+1 = S −1 b1 − Ef μk λk+1 = Rf ufk+1 , pfk+1

ϕpk+1 = D −1 b2 − Ep λk+1 μk+1

=

Rp ϕpk+1

end for

choose ηf0 for k = 0, 1, . . . do ufk+1 , pfk+1 = S −1 b1 − Ef ηfk ηpk+1 = (γf + γp )uk+1 · f D −1

ϕpk+1

=

ηfk+1

=− 1+

− ηfk

b2 − Ep ηpk+1 γf γp

ϕpk+1 +

γf k+1 γp η f

end for

Simple iterative algorithms to solve the decoupled Stokes–Darcy problem (7.2) (left) and its variant for the C-RR method (right)

The main difference between Algorithms 7.1 and 7.2 is that the latter stores the intermediate results λk+1 and μk+1 which are only implicitly given in the former. This idea can be extended to store only one solution variable: If one is not directly interested in, say (uf , pf ), one can rewrite Algorithm 7.1 as a recursion for ϕpk+1 :    ϕpk+1 = D −1 b2 − C2 S −1 b1 − C1 ϕpk .

(7.3)

The same strategy in Algorithm 7.2 leads to    ϕpk+1 = D −1 b2 − Ep Rf S −1 b1 − Ef Rp ϕpk .

(7.4)

Using the identities Ef Rp = C1 and Ep Rf = C2 as in Sect. 6.7, this is exactly the recursion in Eq. (7.3), and is the claim of Proposition 7.1.1. The additional coupling between the interface variables in the C-RR method, Algorithm 7.3, does not allow such a simple representation.

7.2 Algorithms for Interface Equations The simplest algorithm to solve the fixed point equations (6.12) and (6.13) is to define the next iterate by the left-hand side of these equations, see Algorithms 7.4 and 7.5. As before, a damping strategy can be added via a weighting with the previous iterate and is left out here only for brevity. Since the fixed point equations (6.12) and (6.13) are equivalent, Corollary 6.4.14, so are the above algorithms: Proposition 7.2.1 Let λk , k ∈ N, be the iterates of Algorithm 7.4 initialized with λ0 . Then μk = Hp,N (λk ) are the iterates of Algorithm 7.5 initialized with μ0 = Hp,N(λ0 ). Conversely, if μk , k ∈ N, are the iterates of Algorithm 7.5 initialized

158

7 Algorithms

Algorithm 7.4 λ0

choose for k = 0, 1, . . . do λk+1 = Hf,N Hp,N (λk ) end for

Algorithm 7.5 choose μ0 for k = 0, 1, . . . do μk+1 = Hp,N Hf,N (μk ) end for

Simple fixed point algorithms to solve Eqs. (6.12) (left) and (6.13) (right)

Algorithm 7.6 λ0

choose for k = 0, 1, . . . do γ ,γ γ ,γ λk+1 = Hf f p Hp p f (λk ) end for

Algorithm 7.7 choose μ0 for k = 0, 1, . . . do γ ,γ γ ,γ μk+1 = Hp p f Hf f p (μk ) end for

Simple fixed point algorithms to solve Eqs. (6.38) (left) and (6.39) (right)

with μ0 , then λk = Hf,N (μk ) are the iterates of Algorithm 7.4 initialized with λ0 = Hf,N (μ0 ). The algorithms above can as well be applied to solve the fixed point equations (6.38) and (6.39) for the Robin–Robin problem, see Algorithms 7.6 and 7.7. Similar to Proposition 7.2.1 these two are equivalent: Proposition 7.2.2 Let λk , k ∈ N, be the iterates of Algorithm 7.6 initialized with γp ,γ λ0 . Then μk = Hp f (λk ) are the iterates of Algorithm 7.7 initialized with μ0 = γp ,γ Hp f (λ0 ). Conversely, if μk , k ∈ N, are the iterates of Algorithm 7.7 initialized γ ,γp with μ0 , then λk = Hf f (μk ) are the iterates of Algorithm 7.6 initialized with γ ,γp λ0 = Hf f (μ0 ). A first algorithm for the Steklov–Poincaré equations (6.16) could consist of a recursion of the form   λk+1 = λk − ω Hf,D (λk ) − Hp,N(λk ) , f (as required by Hf,D ) it would be with a damping ω > 0. However with λk ∈ Λ ∗ k+1 λ ∈ Λf , because the image space of Hf,D is Λ∗f and includes that of Hp,N , see also Eqs. (6.10). Therefore, to allow a meaningful definition of λk+2 in the next f . This leads to step, it is necessary to insert an operator P which maps Λ∗f to Λ the preconditioned Richardson iteration of Algorithm 7.8 with the preconditioner P . A candidate for P is Hf,N as it has the desired domain of definition and image space. However it is not linear, especially it does not map zero to zero, which would imply that the iteration would not stay at the sought solution. The linear part of Hf,N , Lf,N = Hf,N − λ0,f = Hf,N − Hf,N (0), does not suffer from this drawback and is the

7.2 Algorithms for Interface Equations

159

Algorithm 7.8 choose λ0 for k = 0, 1, . . . do λk+1 = λk − ωP Hf,D (λk ) − Hp,N (λk ) end for Preconditioned Richardson iteration for the Steklov–Poincaré equation (6.16)

canonical preconditioner P , see also Definition 6.5.1. With this choice the recursion in Algorithm 7.8 is   λk+1 = λk − ωLf,N Hf,D (λk ) − Hp,N(λk )      = λk − ω Lf,N Hf,D (λk ) − Lf,N Hp,N (λk )      = λk − ω Hf,N Hf,D (λk ) − Hf,N Hp,N(λk )   = (1 − ω)λk + ωHf,N Hp,N (λk ) , where the linearity of Lf,N , Definition 6.5.1, and the identity (6.11c) are used. This is exactly Algorithm 7.4 for the fixed point equation (6.12) with a damping ω which is stated in the following proposition. Proposition 7.2.3 Let λ0 be fixed. Then Algorithm 7.8 with P = Lf,N yields the same iterates as Algorithm 7.4 (or rather a damped version of it, if ω = 1). Similarly, the preconditioned Richardson iteration for the Steklov–Poincaré equation (6.17) leads to Algorithm 7.9. Here the canonical preconditioner P = Lp,N is appropriate and leads to the same iterates as a damped version of Algorithm 7.5. Proposition 7.2.4 Let μ0 be fixed. Then Algorithm 7.9 with P = Lp,N yields same iterates as Algorithm 7.5 (or rather a damped version of it, if ω = 1). In contrast to both Robin–Robin approaches the Neumann–Neumann method allows (useful) Steklov–Poincaré equations (6.16) and (6.17), see also Remark 6.6.14. Therefore, also other Krylov type iterative methods, such as

Algorithm 7.9 choose μ0 for k = 0, 1, . . . do μk+1 = μk − ωP Hp,D (μk ) − Hf,N (μk ) end for Preconditioned Richardson iteration for the Steklov–Poincaré equations (6.17)

160

7 Algorithms

conjugate gradients (cg) or generalized minimal residuals (gmres), can be applied and are converging much faster.

7.2.1 Connection to the Classical Iterative Subdomain Methods It is noted at the beginning of this chapter that in the case of the Neumann–Neumann coupled system (6.7), are represented as Hf,N (μ) =  the operators on the   interface   Rf S −1 (b1 − Ef μ) and Hp,N (λ) = Rp D −1 b2 − Ep (λ) , while in the case of   γ ,γ the Robin–Robin coupled system it is3 Hf f p (μ) = Rf S −1 (b1 − Ef μ) and    γ ,γ Hp p f (λ) = Rp D −1 b2 − Ep (λ) , such that the recursions in the Algorithms 7.4 and 7.6 turn into    λk+1 = Rf S −1 b1 − Ef Rp D −1 b2 − Ep (λk ) which is exactly what Algorithm 7.2 does. The same approach can be done for the Algorithms 7.5 and 7.7 which solve for the fixed points μ:    μk+1 = Rp D −1 b2 − Ep Rf S −1 b1 − Ef μk .

(7.5)

In fact, applying Rp to Eq. (7.4) leads to the above Eq. (7.5). Hence, assuming suitable initial conditions, all the fixed point Algorithms 7.4–7.7 are equivalent to the simple iterative Algorithms 7.1 and 7.2. Furthermore, for the Neumann–Neumann problem these are also equivalent to the preconditioned Richardson iterations with preconditioners P = Lf,N and P = Lp,N , respectively. This is summarized in the following two propositions. Proposition 7.2.5 Considering the Neumann–Neumann approach, the Algorithms 7.1, 7.2, 7.4, 7.5, 7.8, and 7.9 are equivalent. Proposition 7.2.6 For the Robin–Robin approach, Algorithms 7.1, 7.2, 7.6, and 7.7 are equivalent. In conclusion, only three distinct algorithms are introduced in this chapter, namely the ones based on the Neumann–Neumann and Robin–Robin formulation as well as the modified Robin–Robin method (C-RR), Algorithm 7.3, which behaves differently compared with the first two approaches.

3 Reminder:

The operators S, D, b1 , b2 , Ef , Ep , Rf , and Rp differ compared with the Neumann– Neumann case.

7.3 Convergence Behavior

161

7.3 Convergence Behavior In this section the convergence behavior of three distinct algorithms is analyzed with an emphasis on its dependence on the kinematic viscosity ν and the hydraulic conductivity tensor K. For simplicity, K is assumed to be a positive constant instead of a symmetric positive definite tensor. The general case however poses no further difficulties, other than notation. First, the Neumann–Neumann approach is studied and it turns out that the iteration diverges whenever ν and K are small. The alternative Robin–Robin method (C-RR) is studied next. Its convergence is slow for small ν and K. Finally, the Robin–Robin approach is analyzed and its convergence is somewhat independent of ν and K. To keep the analysis as similar as possible among the three methods, the focus is on the decoupled Algorithms 7.2 and 7.3. For the C-RR method this is done in Section 3.2 in [CGHW11] and the proofs here use similar ideas and notation.

7.3.1 Neumann–Neumann Consider Algorithm 7.2 solving the Neumann–Neumann problem (6.20). Writing each step in weak form leads to • Find (uk+1 , pfk+1 ) such that for all v ∈ V f and q ∈ Qf it is f   af (uk+1 , v) + bf (v, pfk+1 ) = f (v) − μk , v · n f

0,ΓI

,

, q) = 0, bf (uk+1 f • set λk+1 = uk+1 · n, f • find ϕpk+1 such that for all ψ ∈ Qp it is   ap (ϕpk+1 , ψ) = p (ψ) + λk+1 , ψ

0,ΓI

,

• set μk+1 = ϕpk+1 Γ . I

Next, define the errors of the iterates: ελk = λ − λk ,

εμk = μ − μk ,

eϕk = ϕp − ϕpk ,

euk = uf − ukf ,

epk = pf − pfk .

162

7 Algorithms

Since the constituting equations are linear the errors solve related equations, namely for all v ∈ V f , q ∈ Qf , and ψ ∈ Qp it is   af (euk+1 , v) + bf (v, epk+1 ) = − εμk , v · n

0,ΓI

(7.6)

,

bf (euk+1 , q) = 0,

(7.7)

ελk+1 = euk+1 · n,   ap (eϕk+1 , ψ) = ελk+1 , ψ

(7.8)

εμk+1 = eϕk+1 Γ .

0,ΓI

(7.9)

,

(7.10)

I

Using the notation introduced in Sect. 6.5, this means (euk+1 , epk+1 ) = Lf,N (εμk ) and eϕk+1 = Lp,N (ελk+1 ) as well as ελk+1 = Tf (euk+1 , epk+1 ) = Lf,N (εμk ) and εμk+1 = Tp (eϕk+1 ) = Lp,N (ελk+1 ). Combining these equations leads to         k+1      =  Tf ◦ Lf,N ◦ Tp ◦ Lp,N ελk  ≤ cNN ελk  , ελ  0,ΓI 0,ΓI 0,ΓI              k+1  =  Tp ◦ Lp,N ◦ Tf ◦ Lf,N εμk  ≤ cNN εμk  , εμ  0,ΓI

0,ΓI

0,ΓI

with cNN :=

cT2f cT2p αaf αap

.

Here it is used that the continuity constants of the solution operators Li,N , i ∈ {f, p}, are cTi /αai , see the discussion after Definition 6.5.3. Similar results are obtained for the errors in the solution variables euk+1 and eϕk+1 to give    k+1  eu 

1,Ωf

    ≤ cNN euk 

1,Ωf

,

   k+1  eϕ 

1,Ωp

    ≤ cNN eϕk 

1,Ωp

.

The following proposition summarizes this result. Proposition 7.3.1 The reduction of the error in each step of the Algorithms using the Neumann–Neumann approach, see Proposition 7.2.5, is cNN =

cT2f cT2p αaf αap

.

The coercivity constants αaf and αap scale like ν and K, respectively, see Sect. 6.3.2 as well as Sects. 5.1.2 and 5.3.2. Therefore, cNN → ∞ for ν → 0 and K → 0 such that the Neumann–Neumann algorithms are only expected to converge

7.3 Convergence Behavior

163

for larger ν and K and diverge otherwise. This can be compensated with a damping ω. However, the algorithm then only converges if the damping is very small, which makes it slow.

7.3.2 Alternative Robin–Robin (C-RR) Consider Algorithm 7.3 which solves the alternative Robin–Robin method introduced in Sect. 6.6.8. The proof presented here is essentially taken from [CGHW11], Section 3.2. There, a slightly modified version is presented which corresponds to a Jacobi type method whereas here a Gauss–Seidel type iteration is analyzed. As in the Neumann–Neumann case each step in the algorithm is written using the weak formulations: γ

, pfk+1 ) = Kf f (−ηf ), i.e., such that for all v ∈ V f and q ∈ Qf it is • Find (uk+1 f   k+1 k , v) + b (v, p ) =  (v) + η , v · n , afR (uk+1 f f f f f 0,ΓI

, q) bf (uk+1 f

= 0,

• set ηpk+1 = cηfk + duk+1 · n, f γp k+1 k+1 • find ϕp = Kp (γp ηp ) such that for all ψ ∈ Qp it is   apR (ϕpk+1 , ψ) = p (ψ) + ηpk+1 , ψ

0,ΓI

,

• set ηfk+1 = aηpk+1 + bϕpk+1 . The errors of the iterates εpk = ηp − ηpk ,

εfk = ηf − ηfk ,

eϕk = ϕp − ϕpk ,

euk = uf − ukf ,

epk = pf − pfk ,

solve the following equations4 for all v ∈ V f , q ∈ Qf , and ψ ∈ Qp : afR (euk+1 , v) + bf (v, epk+1 )     = af (euk+1 , v) + bf (v, epk+1 ) + γf euk+1 · n, v · n = εfk , v · n

,

(7.11)

bf (euk+1 , q) = 0,   1  k+1  eϕ , ψ apR (eϕk+1 , ψ) = ap (eϕk+1 , ψ) + = εpk+1 , ψ . 0,ΓI 0,ΓI γp

(7.12)

0,ΓI

0,ΓI

γ

(7.13)

that these equations are the linear parts of the Robin solution operators Ki , i ∈ {f, p}, introduced in Sect. 6.6.4, similar to Li,N and the respective Neumann operators Ki,N in Sect. 6.5.

4 Note

164

7 Algorithms

Additionally, the interface errors satisfy εpk+1 = ηp − ηpk+1 = γp uf · n + ϕp − cηfk − duk+1 ·n f = (d − γf )uf · n + ϕp − cηfk − duk+1 ·n f

(d = γf + γp )

= cεfk + deuk+1 · n,

(c = −1, ηf = γf uf · n−ϕp)

εfk+1 = ηf − ηfk+1 = γf uf · n − ϕp − aηpk+1 − bϕpk+1 = aγpuf · n + (a + b)ϕp −aηpk+1 −bϕpk+1

(−1 = a + b, γf = aγp )

= aεpk+1 + beϕk+1 .

(ηp = γp uf · n + ϕp )

Assume the algorithm is initialized with a function on the interface which is in L2 (ΓI ), then all the iterates and errors on the interface are in L2 (ΓI ) as well. Then the norm of εpk+1 is    k+1 2 εp 

0,ΓI

 2   = c2 εfk 

0,ΓI

2    + d 2 euk+1 · n

0,ΓI

  + 2cd εfk , euk+1 · n

0,ΓI

.

The last term also appears in Eq. (7.11) with v = euk+1 : 

εfk , euk+1

 0,ΓI

2    = af (euk+1 , euk+1 ) + γf euk+1 · n

0,ΓI

(7.14)

,

where also the incompressibility condition, Eq. (7.12), is taken into account. Combining the last two equations gives    k+1 2 εp 

0,ΓI

 2   = c2 εfk 

0,ΓI

 2   + (d 2 + 2cdγf )euk+1 · n + 2cdaf (euk+1 , euk+1 ) . 0,ΓI ' () * =:Ip

(7.15) Similarly, the norm of εfk+1 is given as    k+1 2 εf 

0,ΓI

 2   = a 2 εpk 

0,ΓI

 2   + b2 eϕk+1 

0,ΓI

  + 2ab εpk , eϕk+1

0,ΓI

and inserting ψ = eϕk+1 in Eq. (7.13) yields   εpk , eϕk+1

0,ΓI

 2   = γp ap (eϕk+1 , eϕk+1 ) + eϕk+1 

0,ΓI

.

7.3 Convergence Behavior

165

Again, combining these two equations leads to    k+1 2 εf 

0,ΓI

 2   = a 2 εpk 

0,ΓI

2    + (b 2 + 2ab)eϕk+1  + 2abγpap (eϕk+1 , eϕk+1 ) . 0,ΓI () * ' =:If

(7.16) In [CGHW11] it is shown that for γf = γp these errors converge to zero, however no particular convergence speed is shown. In case the parameters 0 < γf < γp furthermore satisfy γp − γf
0 and af (euk+1 , euk+1 ) ≥ 0. Next, consider Eq. (7.16). Note that using b = −1 − a and a = γf /γp it is b2 + 2ab = b(a − 1) = 1 − a 2 > 0 and hence    k+1 2 eϕ 

0,ΓI

 2   ≤ CT2p eϕk+1 

1,Ωp

≤

CT2p αap

ap (eϕk+1 , eϕk+1 )

(7.19)

166

7 Algorithms

implies

If ≤ b

CT2p αap

 (b + 2a) + 2aγp ap (eϕk+1 , eϕk+1 )

CT2p  γf





− 1 + 2γf ap (eϕk+1 , eϕk+1 ) αap γp

2   CTp 1 1 = −bγf − 2 ap (eϕk+1 , eϕk+1 ) − αap γf γp

=b

≤ 0, where the last inequality holds because −γf b > 0, the term involving ap is positive, and the second condition in (7.17) implies that the factor in parentheses above is negative. Therefore, it is shown that Ip ≤ 0 and If ≤ 0 so that Eqs. (7.15) and (7.16) simplify to    k+1  εp 

0,ΓI

    ≤ εfk 

0,ΓI

and

   k+1  εf 

0,ΓI

≤

 γf   k εp  0,ΓI γp

which in turn gives    k+1  εp 

0,ΓI

≤

 γf   k−1  εp  0,ΓI γp

and

   k+1  εf 

0,ΓI

≤

 γf   k−1  εf  , 0,ΓI γp

i.e., an expected error reduction by γf /γp during every second iteration step. Hence, it is desirable to choose γp much larger than γf , however, the conditions (7.17) imply that γp ≈ γf at least for small viscosity ν and conductivity K, which determine the coercivity constants αaf and αap . A fast convergence can therefore only be expected for larger ν and K. Unlike in the Neumann–Neumann case, a divergence does not occur in a C-RR iteration as long as the Robin parameters γp and γf are suitably chosen. The following proposition serves as a summary of the expected convergence properties of the modified Robin–Robin method. Proposition 7.3.2 Solving the alternative Robin–Robin method via the Algorithm 7.3 reduces the error at every second step with a factor γf /γp whenever the following restrictions hold γp − γf