The Fast Solution of Boundary Integral Equations

The Fast Solution of Boundary Integral Equations Sergej Rjasanow Universität des Saarlandes Olaf Steinbach Technische U...

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The Fast Solution of Boundary Integral Equations

Sergej Rjasanow Universität des Saarlandes Olaf Steinbach Technische Universität Graz

Sergej Rjasanow Fachrichtung 6.1 – Mathematik Universität des Saarlandes Postfach 151150 D-66041 Saarbrücken GERMANY Olaf Steinbach Institut für Numerische Mathematik Technische Universität Graz Steyrergasse 30 A-8010 Graz AUSTRIA Library of Congress Control Number: 2006927233 ISBN 978-0-387-34041-8 ISBN 0-387-34041-6

e-ISBN 978-0-387-34042-5 e-ISBN 0-387-34042-4

Printed on acid-free paper. c 2007 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 9 8 7 6 5 4 3 2 1 springer.com

Preface

Boundary Element Methods (BEM) play an important role in modern numerical computations in the applied and engineering sciences. Such algorithms are often more convenient than the traditional Finite Element Method (FEM), since the corresponding equations are formulated on the boundary, and, therefore, a signiﬁcant reduction of dimensionality takes place. Especially when the physical description of the problem leads to an unbounded domain, traditional methods like FEM become unalluring. A numerical procedure, called Boundary Element Methods (BEM), has been developed in the physics and engineering community since the 1950s. This method turns out to be a powerful tool for numerical studies of various physical phenomena. The most prominent examples of such phenomena are the potential equation (Laplace equation) in electromagnetism, gravitation theory, and in perfect ﬂuids. A further example leading to the Laplace equation is the steady state heat ﬂow. One of the most popular applications of the BEM is, however, the system of linear elastostatics which can be considered in both bounded and unbounded domains. A simple model for a ﬂuid ﬂow, the Stokes system, can also be solved by the use of the BEM. The most important examples for the Helmholtz equation are the acoustic scattering and the sound radiation. It has been known for a long time that boundary value problems for elliptic partial diﬀerential equations can be reformulated in terms of boundary integral equations. The trace of the solution on the boundary and its conormal derivative (Cauchy data) can be found by solving these equations numerically. The solution of the problem as well as its gradients or even high order derivatives are then given by the application of Green’s third formula (representation formula); this method based on Green’s formula is called the direct BEM approach. Another possibility is to use the property that single or double layer potentials solve the partial diﬀerential equation exactly for any given density function. Thus, this function can be used in order to fulﬁll the boundary conditions. The density function obtained this way has, in general,

VI

no physical meaning. Therefore, these boundary element methods are called indirect. When boundary integral equations are approximated and solved numerically, the study of stability and convergence is the most important issue. The most popular numerical methods are the Galerkin methods which perfectly ﬁt to the variational formulation of the boundary integral equations. The theoretical study of the Galerkin methods is now completed and provides a powerful theoretical background for BEM. Traditionally, however, the collocation methods were widely used, especially in the engineering community. These methods provide an easier practical implementation compared with the Galerkin methods. However, the stability and convergence theory for collocation methods is available only for two-dimensional problems. Furthermore, the error analysis of the collocation methods for three-dimensional problems, when assuming their stability, shows that the rate of convergence of the Galerkin methods is better, when assuming that the solution is smooth enough. In any case, a numerical procedure applied to the boundary integral equation leads to a linear system of algebraic equations. The matrix of this system is in general dense, i.e. almost all its entries are diﬀerent from zero, and, therefore, have to be stored in computer memory. It is clear that this is the main disadvantage of the BEM compared with FEM which leads to sparse matrices. This quadratic amount of computer memory sets very strong, unattractive bounds for the discretisation parameters and, often, force the user to switch to the out–of–core programming. However, so called fast BEM have been developed in the last two decades. The original methods are the Fast Multipole Method and the Panel Clustering; another example is the use of wavelets. Furthermore, the Adaptive Cross Approximation (ACA) was introduced and successfully applied to many practical problems in the last years. The purpose of this book is twofold. The ﬁrst goal is to give an exact mathematical description of various mathematical formulations and numerical methods for boundary integral equations in the three-dimensional case in an uniform and possibly compact form. The second goal is a systematic numerical treatment of a variety of boundary value problems for the Laplace equation, for the linear elastostatics system, and for the Helmholtz equation. This study will illustrate both the convergence of the Galerkin methods corresponding to the theory and the fast realisation of BEM based on the ACA method. We restrict our numerical tests to some more or less artiﬁcial surface examples. The simplest one is the surface of the unit sphere. Furthermore, two TEAM examples (Testing Electromagnetic Analysis Methods) will be considered besides some other non-trivial surfaces. This book is subdivided into four parts. Chapter 1 provides an overview of the direct and indirect reformulations of second order boundary value problems by using boundary integral equations, and it discusses the mapping properties of all boundary integral operators involved. From this, the unique solvability of the resulting boundary integral equations and the continuous depen-

VII

dence of the solution on the given boundary data can be deduced. Chapter 2 is concerned with boundary element methods, especially with the Galerkin method. The discrete version of the boundary integral equations from Chapter 1 and their variational formulations lead to systems of linear equations with diﬀerent matrices. The entries of these matrices are explicitly derived for all integral operators involved. Chapter 3 describes the Adaptive Cross Approximation of dense matrices and provides, in addition to the theory, some ﬁrst numerical examples. The largest part of the book, Chapter 4, contains some results of numerical experiments. First, the Laplace equation is considered, where we study Dirichlet, Neumann, and mixed boundary value problems as well as an inhomogeneous interface problem. Then, two mixed boundary value problems of linear elastostatics will be presented, and, ﬁnally, many examples for the Helmholtz equation are described. We consider again Dirichlet and Neumann, interior and exterior boundary value problems as well as multifrequency analysis. Many auxiliary results are collected in three appendices. The chapters are relatively independent of one another. Necessary notations and formulas are not only cross-referred to other chapters but usually repeated at the appropriate places. In 2003, Prof. Allan Jeﬀrey approached us with the idea to write a book about fast solutions of boundary integral equations. It has been delightful to write this book and we are also very thankful for his providing the opportunity to get this book published. We would like to thank our colleagues from the BEM community for many useful discussions and suggestions. We are grateful to our home institutions, the University of Saarland in Saarbr¨ ucken and the Technical University in Graz, for providing an excellent scientiﬁc environment and ﬁnancial funding to our research. We appreciate the help of J¨ urgen Rachor, who read the manuscript and made valuable comments and corrections. Furthermore, the authors would very much like to express their appreciation to Richard Grzibovski for his help in performing numerical tests.

Saarbr¨ ucken and Graz March 2007

Sergej Rjasanow Olaf Steinbach

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V 1

Boundary Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Laplace Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Interior Dirichlet Boundary Value Problem . . . . . . . . . . . 1.1.2 Interior Neumann Boundary Value Problem . . . . . . . . . . 1.1.3 Mixed Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . 1.1.4 Robin Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . 1.1.5 Exterior Dirichlet Boundary Value Problem . . . . . . . . . . . 1.1.6 Exterior Neumann Boundary Value Problem . . . . . . . . . . 1.1.7 Poisson Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.8 Interface Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Lam´e Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Dirichlet Boundary Value Problem . . . . . . . . . . . . . . . . . . 1.2.2 Neumann Boundary Value Problem . . . . . . . . . . . . . . . . . . 1.2.3 Mixed Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . 1.3 Stokes System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Interior Dirichlet Boundary Value Problem . . . . . . . . . . . 1.4.2 Interior Neumann Boundary Value Problem . . . . . . . . . . 1.4.3 Exterior Dirichlet Boundary Value Problem . . . . . . . . . . . 1.4.4 Exterior Neumann Boundary Value Problem . . . . . . . . . . 1.5 Bibliographic Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 10 13 17 19 21 22 24 26 27 35 36 37 40 44 49 50 52 54 56

2

Boundary Element Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Boundary Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Laplace Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Interior Dirichlet Boundary Value Problem . . . . . . . . . . . 2.3.2 Interior Neumann Boundary Value Problem . . . . . . . . . . 2.3.3 Mixed Boundary Value Problem . . . . . . . . . . . . . . . . . . . . .

59 59 61 65 65 72 77

X

Contents

2.3.4 Interface Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Lame Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Interior Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Interior Neumann Problem . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Exterior Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Exterior Neumann Problem . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Bibliographic Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84 87 91 91 93 97 98 99

3

Approximation of Boundary Element Matrices . . . . . . . . . . . . 101 3.1 Hierarchical Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.1.2 Hierarchical clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.2 Block Approximation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 3.2.1 Analytic Form of Adaptive Cross Approximation . . . . . . 112 3.2.2 Algebraic Form of Adaptive Cross Approximation . . . . . 119 3.3 Bibliographic Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4

Implementation and Numerical Examples . . . . . . . . . . . . . . . . . . 131 4.1 Geometry Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.1.1 Unit Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.1.2 TEAM Problem 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.1.3 TEAM Problem 24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.1.4 Relay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.1.5 Exhaust manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.2 Laplace Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.2.1 Analytical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.2.2 Discretisation, Approximation and Iterative Solution . . . 137 4.2.3 Generation of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.2.4 Interior Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.2.5 Interior Neumann Problem . . . . . . . . . . . . . . . . . . . . . . . . . 149 4.2.6 Interior Mixed Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 4.2.7 Inhomogeneous Interface Problem . . . . . . . . . . . . . . . . . . . 160 4.3 Linear Elastostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.3.1 Generation of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.3.2 Relay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 4.3.3 Foam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 4.4 Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 4.4.1 Analytical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 4.4.2 Discretisation, Approximation and Iterative Solution . . . 169 4.4.3 Generation of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 4.4.4 Interior Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.4.5 Interior Neumann Problem . . . . . . . . . . . . . . . . . . . . . . . . . 185 4.4.6 Exterior Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . . . 191 4.4.7 Exterior Neumann Problem . . . . . . . . . . . . . . . . . . . . . . . . . 196

Contents

XI

A

Mathematical Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 A.1 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 A.2 Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 A.2.1 Laplace Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 A.2.2 Lame System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 A.2.3 Stokes System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 A.2.4 Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 A.3 Mapping Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

B

Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 B.1 Variational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 B.2 Approximation Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

C

Numerical Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 C.1 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 C.2 Analytic Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 C.2.1 Single Layer Potential for the Laplace operator . . . . . . . . 246 C.2.2 Double Layer Potential for the Laplace operator . . . . . . . 249 C.2.3 Linear Elasticity Single Layer Potential . . . . . . . . . . . . . . 252 C.3 Iterative Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 C.3.1 Conjugate Gradient Method (CG) . . . . . . . . . . . . . . . . . . . 256 C.3.2 Generalised Minimal Residual Method (GMRES) . . . . . . 263

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

1 Boundary Integral Equations

The solutions of second order partial diﬀerential equations can be described by certain surface and volume potentials when a fundamental solution of the underlying partial diﬀerential equation is known. Although the existence of such a fundamental solution can be guaranteed for a wide class of partial diﬀerential operators, see for example [53], the explicite construction of fundamental solutions is a more diﬃcult task in the general case. Hence, we consider here partial diﬀerential operators with constant coeﬃcients only. In particular, we restrict ourselves to the Laplace operator, the Helmholtz operator, and the systems of linear elastostatics and of Stokes, which include the most important applications of boundary integral equations and boundary element methods. When using either a representation formula stemming from Green’s second formula or when considering indirect surface potential methods, one has to ﬁnd unknown density functions from the given boundary conditions. This is done by applying the corresponding trace operators to the surface and volume potentials yielding appropriate boundary integral equations to be solved. Depending on the given boundary conditions one can derive diﬀerent formulations of ﬁrst or second kind boundary integral equations. Although on the continuous level all boundary integral equations are equivalent to the original boundary value problem, and, therefore, to each other, they admit quite different properties when applying a numerical scheme to obtain an approximate solution. In this chapter we give an overview of direct and indirect reformulations of second order boundary value problems by using boundary integral equations and discuss the mapping properties of all boundary integral operators involved. From this we can deduce the unique solvability of the resulting boundary integral equations and the continuous dependence of the solution on the given boundary data.

2

1 Boundary Integral Equations

1.1 Laplace Equation The simplest example for a second order partial diﬀerential equation is the Laplace equation for a scalar function u : R3 → R satisfying −Δu(x) = −

3 ∂2 3 2 u(x) = 0 for x ∈ Ω ⊂ R . ∂x i i=1

(1.1)

This equation is used for the modelling of, for example, the stationary heat transfer, of electrostatic potentials, and of ideal ﬂuids. In (1.1), Ω ⊂ R3 is a bounded, multiply or simply connected domain with a Lipschitz boundary Γ = ∂Ω. Multiplying the partial diﬀerential equation (1.1) with a test function v, integrating over Ω, and applying integration by parts, this gives Green’s ﬁrst formula (−Δu(y)) v(y)dy = a(u, v) − γ1int u(y)γ0int v(y)dsy (1.2) Ω

Γ

with the symmetric bilinear form (∇u(y), ∇v(y))dy,

a(u, v) = Ω

with the interior trace operator γ0int v(y) =

lim

y∈Ω, y→y∈Γ

v( y) ,

and with the interior conormal derivative of u on Γ , γ1int u(y) = n(y), ∇yu( lim y) . y∈Ω, y→y∈Γ

Here, n(y) is the outer normal vector deﬁned for almost all y ∈ Γ . From Green’s ﬁrst formula (1.2) and by the use of the symmetry of the bilinear form a(·, ·), we deduce Green’s second formula − Δv(y) u(y)dy + γ1int v(y)γ0int u(y)dsy (1.3) Ω

=

Γ

− Δu(y) v(y)dy +

Ω

γ1int u(y)γ0int v(y)dsy .

Γ

Inserting v = v0 ≡ 1, we then obtain the compatibility condition − Δu(y) dy + γ1int u(y)dsy = 0 . Ω

Γ

(1.4)

1.1 Laplace Equation

3

Now, choosing in Green’s second formula (1.3) as a test function v a fundamental solution u∗ : R3 × R3 → R satisfying − Δy u∗ (x, y) u(y)dy = u(x) for x ∈ Ω , (1.5) Ω

the solution of the Laplace equation (1.1) is given by the representation formula ∗ int int u∗ (x, y)γ int u(y)ds u (x, y)γ1 u(y)dsy − γ1,y (1.6) u(x) = y 0 Γ

Γ

for x ∈ Ω. The fundamental solution of the Laplace equation is u∗ (x, y) =

1 1 4π |x − y|

for x, y ∈ R3 .

(1.7)

For a domain Ω with Lipschitz boundary Γ = ∂Ω, the solution (1.6) of the partial diﬀerential equation (1.1) has to be understood in a weak or distributional sense. For this, appropriate Sobolev spaces H α (Ω) and H β (Γ ) have to be introduced; see Appendix A.1. To derive suitable boundary integral equations from the representation formula (1.6), we ﬁrst have to investigate the surface potentials in (1.6) as well as their interior trace and conormal derivative. Single Layer Potential First we consider the single layer potential 1 w(y) dsy u∗ (x, y)w(y)dsy = (V w)(x) = 4π |x − y| Γ

for x ∈ Ω ,

Γ

which deﬁnes a continuous map from a given density function w on the boundary Γ to a harmonic function V w in the domain Ω. In particular, V : H −1/2 (Γ ) → H 1 (Ω) is continuous and V w ∈ H 1 (Ω) is a weak solution of the Laplace equation (1.1) for any w ∈ H −1/2 (Γ ). Using the mapping property of the interior trace operator γ0int : H 1 (Ω) → H 1/2 (Γ ) , we can deﬁne the corresponding boundary integral operator V = γ0int V with the following mapping properties, see for example [24, 71, 105].

4

1 Boundary Integral Equations

Lemma 1.1. The single layer potential operator V = γ0int V : H −1/2 (Γ ) → H 1/2 (Γ ) is bounded with V wH 1/2 (Γ ) ≤ cV2 wH −1/2 (Γ )

for all w ∈ H −1/2 (Γ ) ,

and H −1/2 (Γ )–elliptic, V w, w Γ ≥ cV1 w2H −1/2 (Γ )

for all w ∈ H −1/2 (Γ ).

Moreover, for w ∈ L∞ (Γ ) there holds the representation 1 w(y) ∗ dsy u (x, y)w(y)dsy = (V w)(x) = 4π |x − y| Γ

for x ∈ Γ

Γ

as a weakly singular surface integral. Double Layer Potential Next we consider the double layer potential (x − y, n(y)) int u∗ (x, y)v(y)ds = 1 γ1,y v(y)dsy (W v)(x) = y 4π |x − y|3 Γ

Γ

for x ∈ Ω, which again deﬁnes a continuous map from a given density function v on the boundary Γ to a harmonic function W v in the domain Ω. In particular, W : H 1/2 (Γ ) → H 1 (Ω) is continuous and W v ∈ H 1 (Ω) is a weak solution of the Laplace equation (1.1) for any v ∈ H 1/2 (Γ ). Applying the interior trace operator γ0int : H 1 (Ω) → H 1/2 (Γ ) , this deﬁnes an associated boundary integral operator [24, 71, 105]. Lemma 1.2. The boundary integral operator γ0int W : H 1/2 (Γ ) → H 1/2 (Γ ) is bounded with γ0int W vH 1/2 (Γ ) ≤ cW 2 vH 1/2 (Γ ) For v ∈ H 1/2 (Γ ) there holds the representation

for all v ∈ H 1/2 (Γ ).

1.1 Laplace Equation

γ0int (W v)(x) =

− 1 + σ(x) v(x) + (Kv)(x)

5

for x ∈ Γ ,

with the double layer potential operator int u∗ (x, y)v(y)ds γ1,y (Kv)(x) = lim y ε→0 y∈Γ :|y−x|≥ε

1 lim = 4π ε→0

(x − y, n(y)) v(y)dsy |x − y|3

y∈Γ :|y−x|≥ε

and

1 1 ε→0 4π ε2

dsy

σ(x) = lim

for x ∈ Γ.

y∈Ω:|y−x|=ε

Moreover, for v = v0 ≡ 1, we have for x ∈ Γ.

σ(x)v0 (x) + (Kv0 )(x) = 0

If x ∈ Γ is on a smooth part of the boundary Γ = ∂Ω, then we obtain σ(x) =

1 . 2

Otherwise, if x ∈ Γ is on an edge or in a corner point of the boundary Γ = ∂Ω, σ(x) is related to the interior angle of Ω in x ∈ Γ . However, without loss of generality, we assume σ(x) = 1/2 for almost all x ∈ Γ . By applying the interior trace operator γ0int to the representation formula (1.6), u(x) = (V γ1int u)(x) − (W γ0int u)(x) for x ∈ Ω , we obtain the boundary integral equation 1 γ0int u(x) = (V γ1int u)(x) + γ0int u(x) − (Kγ0int u)(x) 2

(1.8)

for almost all x ∈ Γ . In particular, this is a weakly singular boundary integral equation, 1 int u∗ (x, y)γ int u(y)ds u∗ (x, y)γ1int u(y)dsy = γ0int u(x) + γ1,y y 0 2 Γ

Γ

for x ∈ Γ , or, 1 1 int 1 1 (x − y, n(y)) int int γ1 u(y)dsy = γ0 u(x) + γ0 u(y)dsy . 4π |x − y| 2 4π |x − y|3 Γ

Γ

Instead of the interior trace operator γ0int , we may also apply the interior conormal derivative operator γ1int to the representation formula (1.6). To do

6

1 Boundary Integral Equations

so, we ﬁrst need to investigate the interior conormal derivatives of the single and double layer potentials V w and W v, which are both harmonic in Ω. Then, γ1int : H 1 (Ω, Δ) → H −1/2 (Γ ), where H 1 (Ω, Δ) =

−1 (Ω) . v ∈ H 1 (Ω) : Δv ∈ H

Adjoint Double Layer Potential Lemma 1.3. The boundary integral operator γ1int V : H −1/2 (Γ ) → H −1/2 (Γ ) is bounded with γ int V

γ1int V wH −1/2 (Γ ) ≤ c21

wH −1/2 (Γ )

for all w ∈ H −1/2 (Γ ).

For w ∈ H −1/2 (Γ ) there holds the representation γ1int (V w)(x) =

1 w(x) + (K w)(x) 2

in the sense of H −1/2 (Γ ), with the adjoint double layer potential operator int u∗ (x, y)w(y)ds (K w)(x) = lim γ1,x y ε→0 y∈Γ :|y−x|≥ε

1 lim = 4π ε→0

(y − x, n(x)) w(y)dsy . |x − y|3

y∈Γ :|y−x|≥ε

In particular, we have γ1int V w, v Γ =

1 1 w, v Γ + K w, v Γ = w, v Γ + w, Kv Γ 2 2

for all v ∈ H 1/2 (Γ ). Hypersingular Integral Operator In the same way as for the single layer potential V w, we now consider the interior conormal derivate of the double layer potential W v. Lemma 1.4. The operator D = −γ1int W : H 1/2 (Γ ) → H −1/2 (Γ )

1.1 Laplace Equation

7

is bounded with DvH −1/2 (Γ ) ≤ cD 2 vH 1/2 (Γ )

for all v ∈ H 1/2 (Γ ),

and H 1/2 (Γ )–semi–elliptic, 2 Dv, v Γ ≥ cD 1 |v|H 1/2 (Γ )

for all v ∈ H 1/2 (Γ ).

In particular, for v = v0 ≡ 1, we have (Dv0 )(x) = 0

for x ∈ Γ .

Moreover, for continuous functions u, v ∈ H 1/2 (Γ ) ∩ C(Γ ) there holds the representation 1 (curlΓ u(y), curlΓ v(x)) dsy dsx (1.9) Du, v Γ = 4π |x − y| Γ Γ

where (x) curlΓ u(x) = n(x) × ∇x u

for x ∈ Γ

is the surface curl operator and u is some (locally deﬁned) extension of u into the neighbourhood of Γ . The boundary integral operator D = −γ1int W does not exhibit an explicit representation as a Cauchy singular surface integral, in particular, n(x), ∇x (W v)( lim x) = (Dv)(x) = −γ1int (W v)(x) = − x ∈Ω, x →x∈Γ (x − y, n(y))(x − y, n(x)) (n(x), n(y)) 1 lim 3 v(y)dsy − 4π ε→0 |x − y|5 |x − y|3 y∈Γ :|y−x|≥ε

does not exist. Therefore the boundary integral operator D is called hypersingular operator and it requires some appropriate regularisation procedure. Since u(x) = 1 for x ∈ Ω is a solution of the Laplace equation −Δu(x) = 0, the representation formula (1.6) reads for this special choice int u∗ ( x, y)dsy = 1 for x ∈ Ω. − γ1,y Γ

Thus, we have −∇x Γ

Then,

int u∗ ( γ1,y x, y)dsy = 0 for x ∈Ω.

8

1 Boundary Integral Equations

n(x), ∇x (W v)( (Dv)(x) = −γ1int (W v)(x) = − lim x) x ∈Ω, x →x∈Γ ⎛ ⎞ int u∗ ( ⎝n(x), ∇x γ1,y x, y)v(y)dsy ⎠ =− lim x ∈Ω, x →x∈Γ

⎛ =−

lim

x ∈Ω, x →x∈Γ

⎝n(x), ∇x

Γ

⎞ int u∗ ( γ1,y x, y) v(y) − v(x) dsy ⎠

Γ

exists as a Cauchy singular surface integral, (Dv)(x) = (x − y, n(y))(x − y, n(x)) (n(x), n(y)) 1 3 v(y) − v(x) dsy . − 4π |x − y|5 |x − y|3 Γ

However, using integration by parts as in the derivation of formula (1.9) of Lemma 1.4, the induced bilinear form of the hypersingular boundary integral operator can be transformed to a weakly singular bilinear form including some surface curl operators. Boundary Integral Equations Applying now the interior conormal derivative operator γ1int to the representation formula (1.6), u(x) = (V γ1int u)(x) − (W γ0int u)(x)

for x ∈ Ω,

this gives the boundary integral equation γ1int u(x) =

1 int γ u(x) + (K γ1int u)(x) + (Dγ0int u)(x) 2 1

(1.10)

in the sense of H −1/2 (Γ ). In particular, this is a hypersingular boundary integral equation, int int u∗ (x, y)γ int u(y)ds = 1 γ int u(x) − int u∗ (x, y)γ int u(y)ds , γ1,y γ1,x −γ1,x y y 0 1 2 1 Γ

Γ

or, (y − x, n(y))(y − x, n(x)) (n(x), n(y)) 3 × − |x − y|5 |x − y|3 Γ 1 int (y − x, n(x)) int int int γ0 u(y) − γ0 u(x) dsy = γ1 u(x) − γ1 u(y)dsy . 2 |x − y|3

1 4π

Γ

1.1 Laplace Equation

9

Combining (1.8) and (1.10), we can write both boundary integral equations by the use of the Calderon projector as 1 V γ0int u γ0int u 2I − K = . (1.11) 1 D γ1int u γ1int u 2I + K From the boundary integral equations of the Calderon projector (1.11) one can derive some important properties of boundary integral operators, i.e. 1 1 I −K I +K . V K = KV, V D = 2 2 Steklov–Poincar´ e Operator Since the single layer potential V is H −1/2 (Γ )–elliptic and therefore invertible, we obtain from the ﬁrst equation in (1.11) the Dirichlet to Neumann map 1 γ1int u(x) = V −1 I + K γ0int u(x) = (S int γ0int u)(x) for x ∈ Γ (1.12) 2 which deﬁnes the Steklov–Poincar´e operator S int : H 1/2 (Γ ) → H −1/2 (Γ ) associated to the partial diﬀerential equation (1.1). Inserting the Dirichlet to Neumann map (1.12) into the second equation of the Calderon projector (1.11), this gives γ1int u(x) = 1 1 D+ I + K V −1 I + K γ0int u(x) = (S int γ0int u)(x) 2 2 with the symmetric representation of the Steklov–Poincar´e operator 1 1 I + K V −1 I + K . S int = D + 2 2

(1.13)

Note that also the representation (1.12) of the Steklov–Poincar´e operator is symmetric. However, due to Lemma 1.1 and Lemma 1.4, we conclude from the symmetric representation (1.13) 1 1 S int v, v I +K v = Dv, v + V −1 I + K v, 2 2 Γ Γ Γ 2 ≥ Dv, v ≥ cD 1 |v|H 1/2 (Γ ) Γ

for all v ∈ H 1/2 (Γ ), and, therefore, S int is H 1/2 (Γ )–semi–elliptic. In particular, for v = v0 ≡ 1, we have (S int v0 )(x) = 0 for x ∈ Γ .

10

1 Boundary Integral Equations

The Steklov–Poincar´e operator S int deﬁnes the Dirichlet to Neumann map, int γ1 u = S int γ0int u, which is a relation of the Cauchy data associated to a solution of the homogeneous partial diﬀerential equation. This map will be used to handle more general, e.g. nonlinear, boundary conditions. Moreover, Steklov– Poincar´e operators play an important role in domain decomposition methods, e.g. when considering boundary value problems with piecewise constant coeﬃcients, or when considering the coupling of ﬁnite and boundary element methods, see, for example, [60, 107]. 1.1.1 Interior Dirichlet Boundary Value Problem We ﬁrst consider the interior Dirichlet boundary value problem for the Laplace equation, i.e., −Δu(x) = 0 for x ∈ Ω,

γ0int u(x) = g(x)

for x ∈ Γ.

(1.14)

Using the representation formula (1.6), the solution of the Dirichlet boundary value problem (1.14) is given by ∗ int u∗ (x, y)g(y)ds u (x, y)t(y)dsy − γ1,y for x ∈ Ω , u(x) = y Γ

Γ

where t = γ1int u is the unknown conormal derivative of u on Γ which has to be determined from some appropriate boundary integral equations. Direct Single Layer Potential Formulation Using the ﬁrst equation in the Calderon projector (1.11), we have to solve a ﬁrst kind boundary integral equation to ﬁnd t ∈ H −1/2 (Γ ), such that (V t)(x) = or, 1 4π

Γ

1 g(x) + (Kg)(x) 2

1 1 t(y) dsy = g(x) + |x − y| 2 4π

Γ

for x ∈ Γ,

(1.15)

(x − y, n(y)) g(y)dsy . |x − y|3

Using duality arguments, the boundary integral equation Vt =

1 g + Kg ∈ H 1/2 (Γ ) 2

corresponds to 1 = 0 = V t − g − Kg 1/2 2 H (Γ )

V t − 12 g − Kg, w Γ . wH −1/2 (Γ ) 0=w∈H −1/2 (Γ ) sup

1.1 Laplace Equation

11

Therefore, t ∈ H −1/2 (Γ ) is the solution of the variational problem 1 I + K g, w V t, w = for all w ∈ H −1/2 (Γ ), (1.16) 2 Γ Γ or, 1 t(y) dsy dsx = w(x) 4π |x − y| Γ Γ 1 1 (x − y, n(y)) w(x)g(x)dsx + w(x) g(y)dsy dsx . = 2 4π |x − y|3 Γ

Γ

Γ

Theorem 1.5. Let g ∈ H (Γ ) be given. Then there exists a unique solution t ∈ H −1/2 (Γ ) of the variational problem (1.16). Moreover, 1 gH 1/2 (Γ ) . tH −1/2 (Γ ) ≤ V 1 + cW 2 c1 1/2

Because the boundary integral equation (1.15) results from the representation formula (1.6) this approach is called direct. Since both the single and the double layer potentials are harmonic in Ω, the solution of the Dirichlet boundary value problem (1.14) can be represented also either by a single or by a double layer potential alone. Then the unknown density functions have no physical meaning in general. The resulting methods are called indirect and have a long history when solving boundary value problems for second order partial diﬀerential equations, see, e.g., [33]. Indirect Single Layer Potential Formulation Let us consider the indirect single layer potential approach w(y) 1 dsy for x ∈ Ω, u(x) = (V w)(x) = 4π |x − y| Γ

where we have to ﬁnd the unknown density function w ∈ H −1/2 (Γ ). Applying the interior trace operator γ0int , from the given Dirichlet boundary conditions, we then obtain the ﬁrst kind boundary integral equation u∗ (x, y)w(y)dsy = g(x) for x ∈ Γ , (1.17) (V w)(x) = Γ

which is equivalent to the variational problem V w, z Γ = g, z Γ or, 1 4π

z(x)

Γ

Γ

for all z ∈ H −1/2 (Γ ) ,

w(y) dsy dsx = |x − y|

z(x)g(x)dsx . Γ

(1.18)

12

1 Boundary Integral Equations

Theorem 1.6. Let g ∈ H 1/2 (Γ ) be given. Then there exists a unique solution w ∈ H −1/2 (Γ ) of the variational problem (1.18). Moreover, wH −1/2 (Γ ) ≤

1 gH 1/2 (Γ ) . cV1

Note that both boundary integral equations (1.15) and (1.17) are of the same structure, while they are diﬀerent in the deﬁnition of the right hand side. In fact, the boundary integral equation (1.15) of the direct approach involves the application of the double layer potential K to the given Dirichlet datum g, while the right hand side of the boundary integral equation (1.17) of the indirect approach is just the given Dirichlet datum g itself. Indirect Double Layer Potential Formulation Instead of the indirect single layer potential u = V w we now consider the indirect double layer potential approach (x − y, n(y)) 1 v(y)dsy for x ∈ Ω u(x) = −(W v)(x) = − 4π |x − y|3 Γ

which leads, by applying the interior trace operator γ0int and by the use of Lemma 1.2, to a second kind boundary integral equation to ﬁnd v ∈ H 1/2 (Γ ) such that 1 v(x) − (Kv)(x) = g(x) for x ∈ Γ, (1.19) 2 or, 1 1 (x − y, n(y)) v(x) − v(y)dsy = g(x) for x ∈ Γ. 2 4π |x − y|3 Γ

Since this boundary integral equation is formulated in H 1/2 (Γ ), the equivalent variational problem is to ﬁnd v ∈ H 1/2 (Γ ) such that 1 2

I − K v, w

=

g, w

Γ

Γ

for all w ∈ H −1/2 (Γ ).

The solution of the second kind boundary integral equation (1.19) is given by the Neumann series v(x) =

∞ 1 =0

2

I +K

g(x)

for x ∈ Γ.

(1.20)

The convergence of the Neumann series (1.20) and therefore the unique solvability of the boundary integral equation (1.19) can be established when using an appropriate norm in the Sobolev space H 1/2 (Γ ), see [108].

1.1 Laplace Equation

13

Theorem 1.7. Let g ∈ H 1/2 (Γ ) be given. Then there exists a unique solution v ∈ H 1/2 (Γ ) of the boundary integral equation (1.19). Moreover, vV −1 ≤

1 gV −1 1 − cK

where cK < 1 is the contraction rate, 1 I + K z −1 ≤ cK zV −1 2 V

for all z ∈ H 1/2 (Γ )

with respect to the norm induced by the inverse single layer potential, z2V −1 = V −1 z, z Γ

for all z ∈ H 1/2 (Γ ).

It seems to be a natural setting to consider the second kind boundary integral equation (1.19) in the trace space H 1/2 (Γ ), where Theorem 1.7 ensures the unique solvability. However, for practical reasons, the boundary integral equation (1.19) is often considered in L2 (Γ ). While it is known that the shifted double layer potential operator 1 I − K : L2 (Γ ) → L2 (Γ ) 2 is bounded, see [112], it is an open problem whether this operator is invertible in L2 (Γ ) or not for general Lipschitz boundaries Γ = ∂Ω. 1.1.2 Interior Neumann Boundary Value Problem For a simply connected domain Ω ⊂ R3 , we now consider the interior Neumann boundary value problem for the Laplace equation, −Δu(x) = 0 for x ∈ Ω,

γ1int u(x) = g(x)

for x ∈ Γ .

From (1.4), we have to assume the solvability condition g(y)dsy = 0.

(1.21)

(1.22)

Γ

Note that the solution of the Neumann boundary value problem (1.21) is only unique up to an additive constant. Using the representation formula (1.6), a solution of the Neumann boundary value problem (1.21) is given by int u∗ (x, y)γ int u(y)ds , x ∈ Ω. u∗ (x, y)g(y)dsy − γ1,y u(x) = y 0 Γ

Γ

Hence, we have to ﬁnd the yet unknown Dirichlet datum u ¯ = γ0int u on Γ .

14

1 Boundary Integral Equations

Direct Double Layer Potential Formulation Using the ﬁrst equation in (1.11), we have to solve a second kind boundary integral equation to ﬁnd u ¯ = γ0int u ∈ H 1/2 (Γ ) such that 1 u ¯(x) + (K u ¯)(x) = (V g)(x) 2

for x ∈ Γ,

(1.23)

or, 1 1 u ¯(x) + 2 4π

Γ

1 (x − y, n(y)) u ¯(y)dsy = |x − y|3 4π

Γ

g(y) dsy |x − y|

for x ∈ Γ .

As for the second kind boundary integral equation (1.19) for the Dirichlet boundary value problem (1.14), a solution of the second kind boundary integral equation (1.23) is given by the Neumann series u ¯(x) =

∞ 1 =0

2

I −K

(V g)(x)

for x ∈ Γ.

(1.24)

Since the given Neumann datum g ∈ H −1/2 (Γ ) has to satisfy the solvability condition (1.22), and since v0 ≡ 1 is the eigenfunction corresponding to the zero eigenvalue of 1/2 I + K, all members of the Neumann series (1.24), and, 1/2 therefore, u ¯ are in the subspace H∗ (Γ ) ⊂ H 1/2 (Γ ) deﬁned as follows

1/2 H∗ (Γ ) = v ∈ H 1/2 (Γ ) : V −1 v, 1 Γ = 0 . The general solution of the second kind boundary integral equation (1.23) is then given by u ¯α = u ¯ + c where c ∈ R is an arbitrary constant. To ﬁx the constant, we may require the scaling condition u ¯α (y)dsy = α , (1.25) ¯ uα , 1 Γ = Γ

where α ∈ R can be arbitrary, but prescribed. This ﬁnally leads to a variational problem to ﬁnd u ¯α ∈ H 1/2 (Γ ) such that 1 2

I +K u ¯α , w w, 1 + u ¯α , 1 Γ

Γ

Γ

=

V g, w Γ

+ α w, 1

(1.26) Γ

is satisﬁed for all w ∈ H −1/2 (Γ ). Note that the bilinear form of the extended variational problem (1.26) is regular due to the additional term ¯ uα , 1 Γ w, 1 Γ , which regularises the singular operator 1/2 I + K. Summarising the above, we obtain the following result:

1.1 Laplace Equation

15

Theorem 1.8. Let g ∈ H −1/2 (Γ ) and α ∈ R be given. Then there exists a unique solution u ¯α ∈ H 1/2 (Γ ) of the extended variational problem (1.26) satisfying ¯ uα H 1/2 (Γ ) ≤ c gH −1/2 (Γ ) + |α| . ¯α ∈ H 1/2 (Γ ) If g ∈ H −1/2 (Γ ) satisﬁes the solvability condition (1.22), then u is the unique solution of the boundary integral equation (1.23) satisfying the scaling condition (1.25). Direct Hypersingular Integral Operator Formulation When using the second equation in (1.11), we have to solve a ﬁrst kind boundary integral equation to ﬁnd u ¯ = γ0int u ∈ H 1/2 (Γ ) such that (D¯ u)(x) =

1 g(x) − (K g)(x) 2

for x ∈ Γ

(1.27)

is satisﬁed in a weak sense, in particular, in the sense of H −1/2 (Γ ). Since the hypersingular boundary integral operator D has a non–trivial kernel, we have to consider the equation (1.27) in suitable subspaces. For this we deﬁne

1/2 H∗∗ (Γ ) = v ∈ H 1/2 (Γ ) : v, 1 Γ = 0 . Then the variational problem of the boundary integral equation (1.27) reads 1/2 to ﬁnd u ¯ ∈ H∗∗ (Γ ) such that 1 D¯ u, v I − K g, v = (1.28) 2 Γ Γ 1/2

is satisﬁed for all v ∈ H∗∗ (Γ ). The general solution of the ﬁrst kind boundary ¯ + c where c ∈ R is a constant integral equation (1.27) is then given by u ¯α = u which can be determined by the scaling condition (1.25) afterwards. 1/2 Instead of solving the variational problem (1.28) in the subspace H∗∗ (Γ ) and ﬁnding the unique solution afterwards from the scaling condition (1.25), we can formulate an extended variational problem to ﬁnd u ¯α ∈ H 1/2 (Γ ) such that 1 D¯ uα , v I − K g, v v, 1 + u ¯α , 1 = + α v, 1 (1.29) 2 Γ Γ Γ Γ Γ is satisﬁed for all v ∈ H 1/2 (Γ ). Theorem 1.9. Let g ∈ H −1/2 (Γ ) and α ∈ R be given. Then there exists a unique solution u ¯α ∈ H 1/2 (Γ ) of the extended variational problem (1.29) satisfying ¯ uα H 1/2 (Γ ) ≤ c gH −1/2 (Γ ) + |α| . If g ∈ H −1/2 (Γ ) satisﬁes the solvability condition (1.22), then u ¯α is the unique solution of the hypersingular boundary integral equation (1.27) satisfying the scaling condition (1.25).

16

1 Boundary Integral Equations

Steklov–Poincar´ e Operator Formulation Instead of the hypersingular boundary integral equation (1.27) we may also consider a Steklov–Poincar´e operator equation to ﬁnd u ¯ = γ0int u ∈ H 1/2 (Γ ) such that ¯)(x) = g(x) for x ∈ Γ , (1.30) (S int u where the Steklov–Poincar´e operator S int is given either by the Dirichlet to Neumann map (1.12) or in the symmetric form (1.13). As for the hypersingular boundary integral equation (1.27), one can formulate an extended variational formulation to ﬁnd u ¯α ∈ H 1/2 (Γ ) such that ¯α , v Γ + ¯ uα , 1 Γ v, 1 Γ = g, v Γ + α v, 1 Γ S int u

(1.31)

is satisﬁed for all v ∈ H 1/2 (Γ ), and where α ∈ R is given by the scaling condition (1.25). Theorem 1.10. Let g ∈ H −1/2 (Γ ) a unique solution u ¯α ∈ H 1/2 (Γ ) of satisfying ¯ uα H 1/2 (Γ ) ≤ c

and α ∈ R be given. Then there exists the extended variational problem (1.31)

gH −1/2 (Γ ) + |α| .

If g ∈ H −1/2 (Γ ) satisﬁes the solvability condition (1.22), then u ¯α is the unique solution of the Steklov–Poincar´e operator equation (1.30) satisfying the scaling condition (1.25). Indirect Single and Double Layer Potential Formulations When using the indirect single layer potential ansatz u = V w in Ω, the application of the interior conormal derivative operator γ1int gives the second kind boundary integral equation 1 w(x) + (K w)(x) = g(x) 2

for x ∈ Γ.

(1.32)

As for the second kind boundary integral equation (1.23), the solution of the boundary integral equation (1.32) is given by the Neumann series w(x) =

∞ 1 =0

2

I − K

g(x)

for x ∈ Γ.

(1.33)

The convergence of the series (1.33) follows as in Theorem 1.7 due to the contraction estimate, see [108], 1 I − K w ≤ cK wV for all w ∈ H −1/2 (Γ ) : w, 1 Γ = 0 2 V with cK < 1.

1.1 Laplace Equation

17

The indirect double layer potential approach u = −W v in Ω leads, ﬁnally, to the hypersingular boundary integral equation (Dv)(x) = g(x)

for x ∈ Γ ,

which is of the same structure and hence can be handled like the hypersingular boundary integral equation (1.27); we skip the details. 1.1.3 Mixed Boundary Value Problem In most applications we have to deal with boundary value problems with boundary conditions of mixed type, e.g. with Dirichlet or Neumann boundary conditions on diﬀerent non–overlapping parts ΓD and ΓN of the boundary Γ = Γ D ∪ Γ N , respectively. Therefore, we now consider the mixed boundary value problem −Δu(x) = 0 for x ∈ Ω, γ0int u(x) = g(x)

for x ∈ ΓD ,

γ1int u(x) = f (x)

for x ∈ ΓN .

(1.34)

Note that for simplicity the domain Ω is supposed to be simply connected. The solution of the mixed boundary value problem (1.34) is then given by the representation formula ∗ int u (x, y)γ1 u(y)dsy + u∗ (x, y)f (y)dsy u(x) = − ΓD

ΓD

int u∗ (x, y)g(y)ds − γ1,y y

ΓN

int u∗ (x, y)γ int u(y)ds γ1,y y 0

for x ∈ Ω ,

ΓN

where we have to ﬁnd the yet unknown Cauchy data γ0int u on ΓN and γ1int u on ΓD . As we have seen in the two previous subsections on the Dirichlet and on the Neumann problem, there exist diﬀerent approaches leading to diﬀerent boundary integral equations to ﬁnd the unknown Cauchy data. However, we consider here only two direct methods, which seem to be the most convenient approaches to solve mixed boundary value problems by boundary element −1/2 (ΓD ) can 1/2 (ΓN ) and H methods. The deﬁnition of the Sobolev spaces H be seen in Appendix A.1. Symmetric Formulation of Boundary Integral Equations The symmetric formulation (cf. [103]) is based on the use of the ﬁrst kind boundary integral equation (1.15) to ﬁnd the unknown Neumann datum γ1int u on the Dirichlet part ΓD , while the hypersingular boundary integral equation (1.27) is used to ﬁnd the unknown Dirichlet datum γ0int u on the Neumann part ΓN :

18

1 Boundary Integral Equations

1 g(x) + (Kγ0int u)(x) for x ∈ ΓD , 2 1 (Dγ0int u)(x) = f (x) − (K γ1int u)(x) for x ∈ ΓN . 2 1/2 −1/2 Let g ∈ H (Γ ) and f ∈ H (Γ ) be some arbitrary, but ﬁxed extensions of the given boundary data g ∈ H 1/2 (ΓD ) and f ∈ H −1/2 (ΓN ), respectively. Then, we have to ﬁnd (V γ1int u)(x) =

1/2 (ΓN ), u = γ0int u − g ∈ H

−1/2 (ΓD ) t = γ1int u − f ∈ H

satisfying the system of boundary integral equations 1 g(x) + (K g )(x) − (V f)(x) for x ∈ ΓD , 2 1 t )(x) + (D u )(x) = f (x) − (K f)(x) − (D (K g )(x) for x ∈ ΓN . 2 (V t )(x) − (K u )(x) =

The associated variational problem is to ﬁnd 1/2 (ΓN ) −1/2 (ΓD ) × H ( t, u ) ∈ H such that

a( t, u ; w, v) = F (w, v) −1/2

is satisﬁed for all (w, v) ∈ H

1/2

(ΓD ) × H

(1.35)

(ΓN ) with

t, v ΓN + D , w ΓD + K u, v ΓN , a( t, u ; w, v) = V t, w ΓD − K u 1 1 g + K g − V f, w f − K f − D F (w, v) = + . g, v 2 2 ΓD ΓN Since the bilinear form a(·, · ; ·, ·) is skew–symmetric, i.e. a(w, v; w, v) = V w, w ΓD + Dv, v ΓN , the unique solvability of the variational problem (1.35) follows from the mapping properties of the single layer potential V and of the hypersingular integral operator D. Theorem 1.11. Let g ∈ H 1/2 (ΓD ) and f ∈ H −1/2 (ΓN ) be given. Then there −1/2 (ΓD ) × H 1/2 (ΓN ) of the variational exists a unique solution ( t, u ) ∈ H problem (1.35) satisfying 2 2 u2H t2H −1/2 (Γ ) + 1/2 (Γ ) ≤ c gH 1/2 (ΓD ) + f H −1/2 (ΓN ) . D

N

1.1 Laplace Equation

19

Steklov–Poincar´ e Operator Formulation Instead of using the weakly singular boundary integral equation (1.15) on ΓD and the hypersingular boundary integral equation (1.27) on ΓN , we may also use the Dirichlet to Neumann map (1.12) to derive a second boundary integral approach to ﬁnd the yet unknown Cauchy data. Then we have to 1/2 (ΓN ) such that solve an operator equation to ﬁnd u ∈H (S int u )(x) = f (x) − (S int g)(x)

for x ∈ ΓN ,

(1.36)

where the Steklov–Poincar´e operator S int : H 1/2 (Γ ) → H −1/2 (Γ ) is either given by the representation (1.12) or by the symmetric version (1.13). Although both representations are equivalent in the continuous case, they exhibit diﬀerent stability properties when applying some numerical approximation schemes. Theorem 1.12. Let g ∈ H 1/2 (ΓD ) and f ∈ H −1/2 (ΓN ) be given. Then there 1/2 (ΓN ) of the Steklov–Poincar´e operator equaexists a unique solution u ∈H tion (1.36) satisfying uH 1/2 (ΓN ) ≤ c g||H 1/2 (ΓD ) + f H −1/2 (ΓN ) . When the Dirichlet datum γ0int u = u + g is known on the whole boundary Γ , we can ﬁnd the complete Neumann datum γ1int u by solving the corresponding Dirichlet boundary value problem afterwards. 1.1.4 Robin Boundary Value Problem Besides of standard Dirichlet or Neumann boundary conditions also linear or nonlinear boundary conditions of Robin type have to be included, as for example in radiosity transfer problems. Linear Robin Boundary Conditions Hence we now consider the Robin boundary value problem −Δu(x) = 0 for x ∈ Ω, γ1int u(x) + κ(x)γ0int u(x) = g(x) for x ∈ Γ,

(1.37)

where κ ∈ L∞ (Γ ) is strictly positive with κ(x) ≥ κ0 > 0 for x ∈ Γ . Using the Dirichlet to Neumann map γ1int u = S int γ0int u on Γ with the Steklov–Poincar´e operator S int : H 1/2 (Γ ) → H −1/2 (Γ ) either deﬁned by (1.12) or by (1.13), we can ﬁnd the unknown Dirichlet datum γ0int u ∈ H 1/2 (Γ ) by solving the boundary integral equation (S int γ0int u)(x) + κ(x)γ0int u(x) = g(x)

for x ∈ Γ .

20

1 Boundary Integral Equations

Since κ is assumed to be strictly positive, the additive term regularises the H 1/2 (Γ )–semi–elliptic Steklov–Poincar´e operator S int yielding the unique solvability of the equivalent variational problem to ﬁnd u ¯ ∈ H 1/2 (Γ ) such that ¯, v Γ + κ¯ u, v Γ = g, v Γ S int u

for all v ∈ H 1/2 (Γ ).

(1.38)

Theorem 1.13. Let g ∈ H −1/2 (Γ ) and κ ∈ L∞ (Γ ) with κ(x) ≥ κ0 > 0 for x ∈ Γ be given. Then there exists a unique solution of the variational problem (1.38). Moreover, ¯ uH 1/2 (Γ ) ≤ c gH −1/2 (Γ ) . When the complete Dirichlet datum u ¯ = γ0int u ∈ H 1/2 (Γ ) is known we can ﬁnd the Neumann datum γ1int u by solving the corresponding Dirichlet boundary value problem. Nonlinear Robin Boundary Conditions Instead of linear Robin boundary conditions in the boundary value problem (1.37), we may also consider a boundary value problem with nonlinear Robin boundary conditions, −Δu(x) = 0 for x ∈ Ω, γ1int u(x) + f (γ0int u, x) = g(x) for x ∈ Γ , m where f (·, ·) is nonlinear in the ﬁrst argument, for example f (u, x) = u(x) with m ∈ N, typical choices are m = 3 or m = 4. Using again the Dirichlet to Neumann map γ1int u = S int γ0int u on Γ , we can ﬁnd the unknown Dirichlet datum u ¯ = γ0int u ∈ H 1/2 (Γ ) by solving the nonlinear boundary integral equation ¯)(x) + f (¯ u, x) = g(x) (S int u

for x ∈ Γ .

The equivalent variational problem is to ﬁnd u ¯ ∈ H 1/2 (Γ ) such that ¯, v Γ + f (¯ u, ·), v Γ = g, v Γ S int u

for all v ∈ H 1/2 (Γ ).

(1.39)

The unique solvability of the nonlinear variational problem (1.39) follows from appropriate assumptions on the nonlinear function f , see, e.g., [32, 95]. Theorem 1.14. Let g ∈ H −1/2 (Γ ) be given and let f be strongly monotone satisfying ≥ c u − v2L2 (Γ ) for all u, v ∈ L2 (Γ ). f (u, ·) − f (v, ·), u − v Γ

Then there exists a unique solution u ¯ ∈ H 1/2 (Γ ) of the nonlinear variational problem (1.39) satisfying ¯ uH 1/2 (Γ ) ≤ c gH −1/2 (Γ ) .

1.1 Laplace Equation

21

1.1.5 Exterior Dirichlet Boundary Value Problem One of the main advantages in using boundary element methods for the approximate solution of boundary value problems is their applicability to problems in exterior unbounded domains. As a ﬁrst model problem we consider the exterior Dirichlet boundary value problem −Δu(x) = 0

γ0ext u(x) = g(x)

for x ∈ Ω e = R3 \Ω,

with the radiation condition

|u(x) − u0 | = O

1 |x|

for x ∈ Γ

(1.40)

as |x| → ∞ ,

(1.41)

where u0 ∈ R is given. We denote by γ0ext u(x) =

lim

x ∈Ω e , x →x∈Γ

u( x)

the exterior trace of u on Γ and by γ1ext u(x) =

lim

x ∈Ω e , x →x∈Γ

n(x), ∇x u( x)

the exterior conormal derivative of u on Γ . Note that the outer normal vector n(x) is still deﬁned with respect to the interior domain Ω. For a ﬁxed y0 ∈ Ω and R > 2 diam Ω, let BR (y0 ) be a ball of radius R with centre in y0 and including Ω. The solution of the boundary value problem (1.40) is then given by the representation formula, see (1.6), for x ∈ BR (y0 )\Ω ∗ ext ext u∗ (x, y)g(y)ds u(x) = − u (x, y)γ1 u(y)dsy + γ1,y y Γ

Γ

u∗ (x, y)γ1int u(y)dsy −

+ ∂BR (y0 )

int u∗ (x, y)γ int u(y)ds . γ1,y y 0

∂BR (y0 )

Taking the limit R → ∞ and incorporating the radiation condition (1.41), this gives the representation formula in the exterior domain Ω e ∗ ext ext u∗ (x, y)g(y)ds u(x) = u0 − u (x, y)γ1 u(y)dsy + γ1,y (1.42) y Γ

Γ

for x ∈ Ω e . To ﬁnd the yet unknown Neumann datum t = γ1ext u ∈ H −1/2 (Γ ), we apply the exterior trace operator γ0ext to obtain the boundary integral equation 1 (1.43) (V t)(x) = − g(x) + (Kg)(x) + u0 for x ∈ Γ. 2

22

1 Boundary Integral Equations

As for the direct and the indirect approach for the interior Dirichlet boundary value problem, we can conclude the unique solvability of the ﬁrst kind boundary integral equation (1.43) from Lemma 1.1. We then obtain the Dirichlet to Neumann map 1 γ1ext u(x) = V −1 − I + K γ0ext u(x) + (V −1 u0 )(x) 2

for x ∈ Γ

(1.44)

associated to the exterior Dirichlet boundary value problem (1.40). Applying the exterior conormal derivative to the representation formula (1.42), and inserting the Dirichlet to Neumann map (1.44), this gives 1 γ1ext u(x) = I − K γ1ext u(x) − (Dγ0ext u)(x) 2 1 1 −1 ext −1 I −K V − I + K γ0 u(x) + (V u0 )(x) − (Dγ0ext u)(x) = 2 2 1 I − K (V −1 u0 )(x) = −(S ext γ0ext u)(x) + (1.45) 2 with the Steklov–Poincar´e operator (cf. (1.13)) 1 1 S ext = D + − I + K V −1 − I + K : H 1/2 (Γ ) → H −1/2 (Γ ) (1.46) 2 2 associated to the exterior boundary value problem (1.40). 1.1.6 Exterior Neumann Boundary Value Problem Instead of the exterior Dirichlet boundary value problem (1.40), we now consider the exterior Neumann boundary value problem −Δu(x) = 0 for x ∈ Ω e ,

γ1ext u(x) = g(x)

for x ∈ Γ

(1.47)

with the radiation condition (1.41) |u(x) − u0 | = O

1 |x|

as |x| → ∞ ,

where u0 ∈ R is given. Note that, due to the radiation condition, we have unique solvability of the exterior Neumann boundary value problem (1.47). As for the exterior Dirichlet boundary value problem, the solution of the exterior Neumann boundary value problem is given by the representation formula (1.42) ∗ ext u∗ (x, y)γ ext u(y)ds . (1.48) u(x) = u0 − u (x, y)g(y)dsy + γ1,y y 0 Γ

Γ

1.1 Laplace Equation

23

for x ∈ Ω e . To ﬁnd the yet unknown Dirichlet datum u ¯ = γ0ext ∈ H 1/2 (Γ ), ext we apply the exterior trace operator γ0 to obtain the boundary integral equation 1 u ¯(x) − (K u ¯)(x) = u0 − (V g)(x) for x ∈ Γ. (1.49) 2 As for the indirect double layer potential formulation for the interior Dirichlet boundary value problem, the solution of the boundary integral equation (1.49) is given by the Neumann series u ¯(x) = u0 +

∞ 1 =0

2

I +K

(V g)(x)

for x ∈ Γ ,

(1.50)

where we have used (1/2 I + K)u0 = 0. The convergence of the Neumann series (1.50) in H 1/2 (Γ ) follows from Theorem 1.7. Note that the boundary integral equation (1.49), and, therefore, the exterior Neumann boundary value problem (1.47) with the radiation condition (1.41) is uniquely solvable for any given g ∈ H −1/2 (Γ ). When applying the exterior conormal derivative γ1ext to the representation formula (1.48), this gives the hypersingular boundary integral equation to ﬁnd u ¯ = γ0ext u ∈ H 1/2 (Γ ) satisfying 1 (D¯ u)(x) = − g(x) − (K g)(x) 2

for x ∈ Γ.

(1.51)

The boundary integral equation (1.51) is equivalent to the variational problem to ﬁnd u ¯ ∈ H 1/2 (Γ ) such that 1 I +K v D¯ u, v = − g, (1.52) 2 Γ Γ is satisﬁed for all v ∈ H 1/2 (Γ ). Using the test function v = v0 ≡ 1, this gives the trivial equality 1 I + K v0 = u ¯, Dv0 = − g, = 0. D¯ u, v0 2 Γ Γ Γ This shows that the variational problem (1.52) has to be considered in a subspace of H 1/2 (Γ ) which is orthogonal to constants. In particular, the solution of the variational problem (1.52) is only unique up to a constant. Since the hypersingular boundary integral operator D : H 1/2 (Γ ) → H −1/2 (Γ ) is only H 1/2 (Γ )–semi elliptic, see Lemma 1.4, a suitable regularisation of the hypersingular boundary integral operator has to be introduced. As in (1.29), we obtain an extended variational problem to ﬁnd u¯ ∈ H 1/2 (Γ ) such that 1 D¯ u, v I +K v v, 1 + u ¯, 1 = − g, (1.53) 2 Γ Γ Γ Γ is satisﬁed for all v ∈ H 1/2 (Γ ). The extended variational problem (1.53) is uniquely solvable yielding a solution u ¯ ∈ H 1/2 (Γ ) satisfying the orthogonality

24

1 Boundary Integral Equations

¯ u, 1 Γ = 0. Since u(x) = 1 for x ∈ Ω e is a solution of the Laplace equation −Δu(x) = 0 with the radiation condition (1.41) for u0 = 1, the representation formula (1.48) reads ext u∗ (x, y)ds u(x) = u0 + γ1,y y Γ

implying

ext u∗ (x, y)ds = 0 for x ∈ Ω e . γ1,y y

Γ

This shows that the scaling condition for the solution u ¯ of the extended variational problem (1.53) can be chosen in an arbitrary way, the representation formula (1.48) describes the correct solution for any scaling parameter. 1.1.7 Poisson Problem Instead of the homogeneous Laplace equation (1.1), we now consider an inhomogeneous Poisson equation with some given right hand side. The Dirichlet boundary value problem for the Poisson equation reads −Δu(x) = f (x)

for x ∈ Ω,

γ0int u(x) = g(x)

for x ∈ Γ.

(1.54)

From Green’s second formula (1.3), we then obtain the representation formula int u∗ (x, y)g(y)ds + u(x) = u∗ (x, y)t(y)dsy − γ1,y u∗ (x, y)f (y)dy y Γ

Γ

Ω

for x ∈ Ω, where t = γ1int u is the yet unknown Neumann datum. As for the interior Dirichlet boundary value problem (1.14), we have to solve a ﬁrst kind boundary integral equation to ﬁnd t ∈ H −1/2 (Γ ) such that (V t)(x) =

1 g(x) + (Kg)(x) − (N0 f )(x) 2

where (N0 f )(x) =

u∗ (x, y)f (y)dy

for x ∈ Γ ,

(1.55)

for x ∈ Γ

Ω

is the Newton potential entering the right hand side. Hence, the unique solvability of the boundary integral equation (1.55) follows, as in Theorem 1.5, for the ﬁrst kind boundary integral equation (1.15), which is associated to the Dirichlet boundary value problem (1.14). The drawback in considering the boundary integral equation (1.55) is the evaluation of the Newton potential N0 f . Besides a direct computation there exist several approaches leading to more eﬃcient methods.

1.1 Laplace Equation

25

Particular Solution Approach Let up be a particular solution of the Poisson equation in (1.54) satisfying −Δup (x) = f (x)

for x ∈ Ω.

Then, instead of (1.54), we consider a Dirichlet boundary value problem for the Laplace operator, −Δu0 (x) = 0 for x ∈ Ω,

γ0int u0 (x) = g(x) − γ0int up (x)

for x ∈ Γ.

The solution u of (1.54) is then given by u0 + up . The unknown Neumann datum t0 = γ1int u0 is the unique solution of the boundary integral equation 1 g(x) − γ0int up (x) + K g − γ0int up (x) for x ∈ Γ. (V t0 )(x) = 2 On the other hand we have t0 = γ1int u0 = γ1int u − up = t − γ1int up (x). Hence, we obtain (V t)(x) =

1 1 g(x) + (Kg)(x) − γ0int up (x) − (Kγ0int up )(x) + (V γ1int up )(x) 2 2

for x ∈ Γ , and, therefore, (N0 f )(x) =

1 int γ up (x) + (Kγ0int up )(x) − (V γ1int up (x)) 2 0

for x ∈ Γ.

Thus, we can evaluate a Newton potential N0 f by the use of the surface potentials, when a particular solution up of the Poisson equation is known. Integration by Parts In several applications the given function f in (1.54) satisﬁes a certain homogeneous partial diﬀerential equation. For simplicity, we assume that −Δf (x) = 0 Using u∗ (x, y) =

for x ∈ Ω.

1 1 1 = Δy |x − y| 4π |x − y| 8π

we obtain from the Green’s second formula (1.3) 1 ∗ u (x, y)f (y)dy = f (y)Δy |x − y|dy 8π Ω Ω 1 int |x − y|γ int f (y)ds − 1 int |x − y|γ int f (y)dy. γ1,y γ0,y = y 0 1 8π 8π Γ

Γ

26

1 Boundary Integral Equations

1.1.8 Interface Problem In addition to interior and exterior boundary value problems, we may also consider an interface problem, i.e., −αi Δui (x) = f (x)

for x ∈ Ω,

−αe Δue (x) = 0 for x ∈ Ω e ,

(1.56)

with transmission conditions describing the continuity of the potential and of the ﬂux, respectively, γ0int ui (x) = γ0ext ue (x) ,

αi γ1int ui (x) = αe γ1ext ue (x)

for x ∈ Γ ,

and with the radiation condition for a given u0 ∈ R, ue (x) − u0 = O 1 as |x| → ∞. |x|

(1.57)

(1.58)

The solution of the above interface problem is given by the representation formula int u∗ (x, y)γ int u (y)ds ui (x) = u∗ (x, y)γ1int ui (x)dsx − γ1,y i y 0 Γ

1 + αi

Γ ∗

u (x, y)f (y)dy Ω

for x ∈ Ω and

ue (x) = u0 −

u∗ (x, y)γ1ext ue (x)dsx +

Γ

ext u∗ (x, y)γ ext u (y)ds γ1,y e y 0

Γ

int/ext

int/ext

for x ∈ Ω e . To ﬁnd the unknown Cauchy data γ0 u and γ1 u, which are linked via the transmission conditions (1.57), we have to solve appropriate boundary integral equations on the interface boundary Γ . Using the Dirichlet to Neumann map associated to the interior Dirichlet boundary value problem (1.54), in particular, solving the boundary integral equation (1.55), (V γ1int ui )(x) =

1 int 1 γ ui (x) + (Kγ0int ui )(x) − (N0 f )(x) 2 0 αi

for x ∈ Γ,

we obtain γ1int ui (x) = V −1

1 I + K γ0int ui (x) − V −1 (N0 f )(x) 2 αi

1

for x ∈ Γ.

Let us assume that there is given a particular solution up satisfying −Δup (x) = f (x) for x ∈ Ω .

1.2 Lam´e Equations

27

Hence, we obtain (N0 f )(x) =

1 int γ up (x) + (Kγ0int up )(x) − (V γ1int up )(x) 2 0

for x ∈ Γ ,

and, therefore, γ1int ui (x) = 1 1 1 1 V −1 I + K γ0int ui (x) + γ1int up (x) − V −1 I + K γ0int up (x) = 2 αi αi 2 1 int 1 int int int int (S γ0 ui )(x) + γ1 up (x) − (S γ0 up )(x) αi αi for x ∈ Γ with the Steklov–Poincar´e operator S int . Correspondingly, the Dirichlet to Neumann map (1.45) associated to the exterior Dirichlet boundary value problem (1.40) gives γ1ext ue (x) = −(S ext γ0ext ue )(x) +

1 2

I − K (V −1 u0 )(x)

for x ∈ Γ.

Inserting the transmission conditions (1.57), u ¯ = γ0ext ue (x) = γ0int ui (x) ,

αi γ1int ui (x) = αe γ1ext ue (x)

for x ∈ Γ,

we obtain a coupled Steklov–Poincar´e operator equation to ﬁnd u ¯ ∈ H 1/2 (Γ ) such that ¯)(x) + αe (S ext u ¯)(x) = αi (S int u (S int γ0int up )(x) − γ1int up (x) + αe

1 2

I − K (V −1 u0 )(x)

is satisﬁed for x ∈ Γ . This is equivalent to a variational problem to ﬁnd u ¯ ∈ H 1/2 (Γ ) such that (αi S int + αe S ext )¯ u, v = (1.59) Γ 1 S int γ0int up − γ1int up + αe I − K V −1 u0 , v 2 Γ is satisﬁed for all v ∈ H 1/2 (Γ ). The unique solvability of (1.59) ﬁnally follows from the ellipticity estimates for the interior and exterior Steklov–Poincar´e operators S int and S ext .

1.2 Lam´ e Equations In linear isotropic elastostatics the displacement ﬁeld u of an elastic body occupying some reference conﬁguration Ω ⊂ R3 satisﬁes the equilibrium equations

28

1 Boundary Integral Equations

−

3 ∂ σij (u, x) = 0 for x ∈ Ω, i = 1, 2, 3 , ∂xj j=1

(1.60)

where σ ∈ R3×3 denotes the stress tensor. For a homogeneous isotropic material, the linear stress–strain relation is given by Hooke’s law Eν E δij eij (u, x) ekk (u, x) + (1 + ν)(1 − 2ν) 1+ν 3

σij (u, x) =

k=1

for i, j = 1, 2, 3. Here, E > 0 is the Young modulus, and ν ∈ (0, 1/2) denotes the Poisson ratio. The strain tensor e is deﬁned as follows, ∂ 1 ∂ uj (x) + ui (x) for i, j = 1, 2, 3 . eij (u, x) = 2 ∂xi ∂xj Inserting the strain and stress tensors, we obtain from (1.60) the Navier system −μΔu(x) − (λ + μ)grad div u(x) = 0 for x ∈ Ω with the Lam´e constants λ =

Eν , (1 + ν)(1 − 2ν)

μ =

E . 2(1 + ν)

Multiplying the equilibrium equations (1.60) with some test function vi , integrating over Ω, applying integration by parts, and taking the sum over i = 1, 2, 3, this gives the ﬁrst Betti formula 3 ∂ − γ1int u(y), γ0int v(y) dsy (1.61) σij (u, y)vi (y)dy = a(u, v) − ∂yj i,j=1 Ω

Γ

with the symmetric bilinear form a(u, v) =

3

σij (u, y)eij (v, y)dy

Ω i,j=1

= 2μ

3

eij (u, y)eij (v, y)dy + λ

Ω i,j=1

div u(y) div v(y)dy Ω

and with the boundary stress operator (γ1int u)i (y) =

3

σij (u, y)nj (y)

j=1

and for i = 1, 2, 3, which can be written as

for y ∈ Γ ,

1.2 Lam´e Equations

(γ1int u)(y) = λ div u(y) n(y) + 2μ

∂ u(y) + μ n(y) × curl u(y) ∂n(y)

29

for y ∈ Γ .

From (1.61) and using the symmetry of the bilinear form a(·, ·), we can deduce the second Betti formula −

3 ∂ γ1int v(y), γ0int u(y) dsy σij (v, y)ui (y)dy + ∂yj i,j=1

Ω

Γ

3 ∂ γ1int u(y), γ0int v(y) dsy . σij (u, x)vi (x)dy + = − ∂yj i,j=1 Ω

Let

(1.62)

Γ

⎧⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎫ 0 x3 ⎬ 0 0 −x2 ⎨ 1 R = span ⎝ 0 ⎠ , ⎝ 1 ⎠ , ⎝ 0 ⎠ , ⎝ x1 ⎠ , ⎝ −x3 ⎠ , ⎝ 0 ⎠ ⎩ ⎭ 0 x2 −x1 0 0 1

(1.63)

be the space of the rigid body motions which are solutions of the homogeneous Neumann boundary value problem −

3 ∂ σij (v, x) = 0 for x ∈ Ω , ∂xj j=1

(γ1int v)i (x) = 0 for x ∈ Γ ,

for i = 1, 2, 3, and v ∈ R. Then there holds 3 ∂ − γ1int u(y), γ0int v(y) dsy = 0 σij (u, y)vi (y)dy + ∂yj i,j=1 Ω

Γ

for i = 1, 2, 3, and for all v ∈ R. Choosing in (1.62) as a test function v a fundamental solution U ∗ (x, y) having the property 3 ∂ σij (U ∗ (x, y), y)ui (y)dy = u (x), − ∂y j i,j=1

(1.64)

Ω

the displacement ﬁeld u satisfying the equilibrium equations (1.60) is given by the Somigliana identity int U ∗ (x, y), γ int u(y) ds (1.65) γ1,y u (x) = γ1int U ∗ (x, y), u(y) dsy − y 0 Γ

Γ

for x ∈ Ω and = 1, 2, 3. The fundamental solution of linear elastostatics is given by the Kelvin tensor

30

1 Boundary Integral Equations ∗ Uk (x, y) =

1 1 1+ν 8π E 1 − ν

δk (xk − yk )(x − y ) (3 − 4ν) + |x − y| |x − y|3

(1.66)

for x, y ∈ R3 and k, = 1, 2, 3. Note that the fundamental solution is deﬁned even in the incompressible case ν = 1/2. The mapping properties of all boundary potentials and the related boundary integral operators follow as in the case of the Laplace operator. Single Layer Potential The single layer potential of linear elastostatics is given as 3 1 1 1+ν Lame (3 − 4ν)(V wk )(x) + w)k (x) = (Vk w )(x) , (V 2E 1−ν =1

where (V wk )(x) =

1 4π

Γ

wk (y) dsy |x − y|

is the single layer potential of the Laplace operator, and 1 (xk − yk )(x − y ) w (y)dsy (Vk w )(x) = 4π |x − y|3 Γ 1 ∂ 1 dsy = w (y)(xk − yk ) 4π ∂y |x − y| Γ

for k, = 1, 2, 3. The single layer potential V Lame deﬁnes a continuous map from a given vector function w on the boundary Γ to a vector ﬁeld V Lame w which is a solution of the homogeneous equilibrium equations (1.60). In particular, V Lame : [H −1/2 (Γ )]3 → [H 1 (Ω)]3 is continuous. Using the mapping property of the interior trace operator γ0int : H 1 (Ω) → H 1/2 (Γ ) for (V Lame w) , = 1, 2, 3, this deﬁnes a continuous boundary integral operator V Lame = γ0int V Lame . Lemma 1.15. The single layer potential operator V Lame : [H −1/2 (Γ )]3 → [H 1/2 (Γ )]3 is bounded with V Lame w[H 1/2 (Γ )]3 ≤ cV2 w[H −1/2 (Γ )]3

for all w ∈ [H −1/2 (Γ )]3

1.2 Lam´e Equations

31

and, if ν ∈ (0, 1/2), [H −1/2 (Γ )]3 –elliptic, V Lame w, w Γ ≥ cV1 w2[H −1/2 (Γ )]3

for all w ∈ [H −1/2 (Γ )]3 ,

where the duality pairing ·, · is now deﬁned as follows u, v = (u(y), v(y))dsy . Γ

Moreover, for w ∈ [L∞ (Γ )]3 there holds the representation 3 1 1 1+ν Lame (3 − 4ν)(V wk )(x) + w)k (x) = (Vk w )(x) , (V 2E 1−ν =1

where

1 (V wk )(x) = 4π

Γ

wk (y) dsy |x − y|

is the single layer potential of the Laplace operator, and 1 ∂ 1 dsy w (y)(xk − yk ) (Vk w )(x) = 4π ∂y |x − y| Γ

for k, = 1, 2, 3, all deﬁned as weakly singular surface integrals. Note that the single layer potential V Lame of linear elastostatics can be written as (V Lame w)k (x) = 3 1 1+ν 1 1 1+ν (V wk )(x) + (1 − 2ν)(V wk )(x) , (Vk w )(x) + 2E 1−ν E 1−ν =1

where the ﬁrst part corresponds to the single layer potential V Stokes of the Stokes problem (see Section 1.3). From V Stokes n = 0, we then obtain (cf. [106]) 3 1 1+ν (1 − 2ν) (V nk , nk ) , (V Lame n, n) = E 1−ν k=1

behaves like O(1 − 2ν) for ν → 1/2. showing that the ellipticity constant In particular, we have lim cV1 (ν) = 0. cV1

ν→1/2

32

1 Boundary Integral Equations

Double Layer Potential The double layer potential of linear elastostatics is Lame int U ∗ (x, y), v(y)) ds v) (x) = (γ1,y (W y Γ

for = 1, 2, 3. The double layer potential W Lame deﬁnes a continuous map from a given vector function v on the boundary Γ to a vector ﬁeld W Lame v which is a solution of the homogeneous equilibrium equations (1.60). In particular, W : [H 1/2 (Γ )]3 → [H 1 (Ω)]3 is continuous. Using the mapping property of the interior trace operator γ0int : H 1 (Ω) → H 1/2 (Γ ) applied to the components (W v) , = 1, 2, 3, this deﬁnes an associated boundary integral operator. Lemma 1.16. The boundary integral operator γ0int W Lame v : [H 1/2 (Γ )]3 → [H 1/2 (Γ )]3 is bounded with γ0int W Lame v[H 1/2 (Γ )]3 ≤ cW 2 v[H 1/2 (Γ )]3

for all v ∈ [H 1/2 (Γ )]3 .

For continuous v there holds the representation γ0int (W Lame v)(x) =

1 v + (K Lame v)(x) 2

for x ∈ Γ with the double layer potential operator (K Lame v)(x) = (Kv)(x) − (V M (∂, n)v)(x) +

E (V Lame M (∂, n)v)(x) , 1+ν

where K and V are the double and single layer potential for the Laplace operator, and V Lame is the single layer potential of linear elasticity, respectively. In addition, we have used the matrix surface curl operator given by Mij (∂y , n(y)) = nj (y)

∂ ∂ − ni (y) ∂yi ∂yj

for i, j = 1, 2, 3. Moreover, we have 1 I + K Lame v(x) = 0 2

for all v ∈ R ,

where R is the space of the rigid body motions.

(1.67)

1.2 Lam´e Equations

33

By applying the interior trace operator γ0int to the representation formula (1.65), we obtain the ﬁrst boundary integral equation 1 γ0int u(x) = (V Lame γ1int u)(x) + γ0int u(x) − (K Lame γ0int u)(x) 2

(1.68)

for x ∈ Γ . Instead of the interior trace operator γ0int , we may also apply the interior boundary stress operator γ1int to the representation formula (1.65). To do so, we ﬁrst need to investigate the application of the boundary stress operator to the single and double layer potentials V Lame w and W Lame v which are both solutions of the homogeneous equilibrium equations (1.60). Adjoint Double Layer Potential Lemma 1.17. The boundary integral operator γ1int V Lame : [H −1/2 (Γ )]3 → [H −1/2 (Γ )]3 is bounded with γ int V

γ1int V Lame w[H −1/2 (Γ )]3 ≤ c21

w[H −1/2 (Γ )]3

for all w ∈ [H −1/2 (Γ )]3 .

For w ∈ [H −1/2 (Γ )]3 there holds the representation (γ1int V Lame w)(x) =

1 w(x) + K Lame w (x) 2

in the sense of [H −1/2 (Γ )]3 . In particular, for v ∈ [H 1/2 (Γ )]3 we have γ1int V Lame w, v Γ =

1 w, v Γ + w, K Lame v Γ . 2

Hypersingular Integral Operator In the same way as for the single layer potential V Lame w, we now consider the application of the boundary stress operator γ1int to the double layer potential W Lame v. Lemma 1.18. The boundary integral operator DLame = −γ1int W Lame : [H 1/2 (Γ )]3 → [H −1/2 (Γ )]3 is bounded with DLame v[H −1/2 (Γ )]3 ≤ cD 2 v[H 1/2 (Γ )]3 1/2

and HR (Γ )–elliptic,

for all v ∈ [H 1/2 (Γ )]3

34

1 Boundary Integral Equations 2 DLame v, v Γ ≥ cD 1 v[H 1/2 (Γ )]3

1/2

for all v ∈ HR (Γ ) ,

1/2

where HR (Γ ) is the space of all vector functions which are orthogonal to the space R of rigid body motions. In particular, there holds (DLame v)(x) = 0

for all v ∈ R.

Moreover, for continuous vector functions u, v ∈ [H 1/2 (Γ ) ∩ C(Γ )]3 , there holds the representation DLame u, v Γ = 3 ∂ μ 1 ∂ u(y), v(x) dsy dsx + 4π |x − y| ∂Sk (y) ∂Sk (x) k=1 Γ Γ (M (∂x , n(x))v(x)) × Γ Γ

I μ − 4μ2 U ∗ (x, y) M (∂y , n(y))u(y)dsy dsx + 2π |x − y| 3 1 μ Mki (∂y , n(y))vj (y)dsy dsx Mkj (∂x , n(x))vi (x) 4π |x − y| Γ Γ

i,j,k=1

with the surface curl operator M (∂, n) as deﬁned in (1.67) and ∂ = M32 (∂x , n(x)), ∂S1 (x) ∂ = M13 (∂x , n(x)), ∂S2 (x) ∂ = M21 (∂x , n(x)). ∂S3 (x) Boundary Integral Equations Applying the interior boundary stress operator γ1int to the Somigliana identity (1.65), u(x) = (V Lame γ1int u)(x) − (W Lame γ0int u)(x)

for x ∈ Ω,

this gives a second boundary integral equation 1 γ1int u(x) = γ1int u(x) + (K Lame ) γ1int u (x) + (DLame γ0int u)(x) 2

(1.69)

for x ∈ Γ . As in (1.11), we can write the boundary integral equations (1.68) and (1.69) by the use of the Calderon projector as

1.2 Lam´e Equations

γ0int u

γ1int u

1 2I

− K Lame DLame

V Lame Lame 1 2I + K

γ0int u γ1int u

35

.

(1.70)

Since the single layer potential V Lame is [H −1/2 (Γ )]3 –elliptic and therefore invertible, we obtain from the ﬁrst equation in (1.70) the Dirichlet to Neumann map (1.71) γ1int u(x) = (S Lame γ0int u)(x) for x ∈ Γ with the Steklov–Poincar´e operator −1 1 S Lame = V Lame I + K Lame 2 −1 1 1 Lame I + K Lame I + K Lame . V Lame =D + 2 2 Note that it holds (S Lame γ0int v)(x) = 0 for all v ∈ R. 1.2.1 Dirichlet Boundary Value Problem When considering the Dirichlet boundary value problem of linear elastostatics, −

3 ∂ σij (u, x) = 0 ∂x j j=1

γ0int u(x) = g(x)

for x ∈ Ω, i = 1, 2, 3, for x ∈ Γ,

the displacement ﬁeld u can be described by the Somigliana identity int U ∗ (x, y), g(y) ds U ∗k (x, y), t(y) dsy − γ1,y uk (x) = y k Γ

Γ

for x ∈ Ω and k = 1, 2, 3, where the boundary stress t = γ1int u has to be determined from some appropriate boundary integral equation. Using the ﬁrst equation in the Calderon projector (1.70), we have to solve a ﬁrst kind boundary integral equation to ﬁnd t ∈ [H −1/2 (Γ )]3 such that (V Lame t)(x) =

1 g(x) + (K Lame g)(x) 2

for x ∈ Γ.

This boundary integral equation corresponds to ﬁnding the solution t of the variational problem 1 I + K Lame g, w = (1.72) V Lame t, w 2 Γ Γ in [H −1/2 (Γ )]3 for all test functions w ∈ [H −1/2 (Γ )]3 . Since the single layer potential V Lame is [H −1/2 (Γ )]3 –elliptic, the unique solvability of the variational problem (1.72) follows due to the Lax–Milgram theorem.

36

1 Boundary Integral Equations

1.2.2 Neumann Boundary Value Problem For a simply connected domain Ω ⊂ R3 , we now consider the Neumann boundary value problem −

3 ∂ σij (u, x) = 0 ∂x j j=1

for x ∈ Ω, i = 1, 2, 3,

γ1int u(x) = g(x)

(1.73)

for x ∈ Γ

where we have to assume the solvability conditions g(y), γ0int v(y) dsy = 0 for all v ∈ R.

(1.74)

Γ

Note that the solution of the Neumann boundary value problem (1.73) is only unique up to the rigid body motions v ∈ R. Using the Somigliana identity (1.65), a solution of the Neumann boundary value problem (1.73) is given by the representation formula int U ∗ (x, y), γ int u(y) ds U ∗ (x, y), g(y) dsy − γ1,y u (x) = y 0 Γ

Γ

for x ∈ Ω and = 1, 2, 3. Hence, we have to ﬁnd the yet unknown Dirichlet datum u ¯ = γ0int u on Γ . When using the second equation in the Calderon projector (1.70), we have to solve a ﬁrst kind boundary integral equation to ﬁnd u¯ ∈ [H 1/2 (Γ )]3 such that 1 Lame Lame (D (1.75) u ¯)(x) = g(x) − K g (x) for x ∈ Γ 2 is satisﬁed in a weak sense, in particular, in the sense of [H −1/2 (Γ )]3 . Since the hypersingular boundary integral operator DLame has the non–trivial kernel of the rigid body motions, we have to consider the boundary integral equation (1.75) in suitable subspaces. To this end, we deﬁne

1/2 HR (Γ ) = u ∈ [H 1/2 (Γ )]3 : u, v Γ = 0 for all v ∈ R . Then the variational problem of the boundary integral equation (1.75) is to 1/2 ﬁnd u ¯ ∈ HR (Γ ) such that

DLame u ¯, v

= Γ

1/2

1 2

I − K Lame g, v

(1.76)

Γ

is satisﬁed for all v ∈ HR (Γ ). The general solution of the hypersingular boundary integral equation (1.75) is then given by

1.2 Lam´e Equations

u ¯α (x) = u ¯(x) +

6

37

ck v k (x) ,

k=1

where the vectors v k , k = 1, . . . , 6 build a basis in the space of rigid body motions (cf. (1.63)). To ﬁx the constants ck , we may require the scaling conditions u ¯(y), γ0int v k (y) dsy = αk (1.77) Γ

for k = 1, . . . , 6, where the αk ∈ R can be arbitrary, but prescribed. 1/2 Instead of solving the variational problem (1.76) in the subspace HR (Γ ) and ﬁnding the unique solution afterwards from the scaling conditions (1.77), one can formulate an extended variational problem to ﬁnd u ¯α ∈ [H 1/2 (Γ )]3 such that

¯α , v DLame u

1 2

+ Γ

6

u ¯α , v k

k=1

Γ

v, v k

=

(1.78)

Γ

6 I − K Lame g, v + αk v, v k Γ

Γ

k=1

is satisﬁed for all v ∈ [H 1/2 (Γ )]3 . The extended variational problem (1.78) is uniquely solvable for any given g ∈ [H −1/2 (Γ )]3 . If g satisﬁes the solvability conditions (1.74), then u ¯α is the unique solution of the hypersingular boundary integral equation (1.75) satisfying the scaling conditions (1.77). 1.2.3 Mixed Boundary Value Problem Let Ω ⊂ R3 be simply connected. Then we consider the mixed boundary value problem 3 ∂ σij (u, x) = 0 for x ∈ Ω , − ∂xj j=1 3

γ0int ui (x) = gi (x)

for x ∈ ΓD,i ,

σij (u, x)nj (x) = fi (x)

for x ∈ ΓN,i ,

(1.79)

j=1

and for i = 1, 2, 3. We assume that Γ = Γ N,i ∪ Γ D,i ,

ΓN,i ∩ ΓD,i = ∅ ,

meas ΓD,i > 0

for i = 1, 2, 3 is satisﬁed. Using the Somigliana identity (1.65), the solution of the mixed boundary value problem (1.79) is given by the representation formula

38

1 Boundary Integral Equations

u (x) =

3 i=1Γ

3 i=1Γ

∗ fi (x)Ui (x, y)dsy −

3 i=1Γ

N,i

∗ (γ1int u)i (x)Ui (x, y)dsy −

3 i=1Γ

D,i

int U ∗ ) (x, y)ds + gi (y)(γ1,y y i

D,i

int U ∗ ) (x, y)ds γ0int ui (y)(γ1,y y i

N,i

for x ∈ Ω and for = 1, 2, 3. Hence, we have to ﬁnd the yet unknown Cauchy data (γ1int u)i on ΓD,i and γ0int ui on ΓN,i . The symmetric formulation of boundary integral equations is based on the use of the ﬁrst kind boundary integral equation (1.68) for those components, where the boundary displacement γ0int ui = gi is given, while the hypersingular boundary integral equation (1.69) is used when the boundary stress (γ1int u)i = fi is prescribed. Let gi ∈ H 1/2 (Γ ) and fi ∈ H −1/2 (Γ ) be some arbitrary but ﬁxed extensions of the given boundary data gi ∈ H 1/2 (ΓD,i ) and fi ∈ H −1/2 (ΓN,i ), respectively. Then, we have to ﬁnd 1/2 (ΓN,i ), u i = γ0int ui − gi ∈ H

−1/2 (ΓD,i ) ti = (γ1int u)i − fi ∈ H

satisfying a system of boundary integral equations, t)i (x) − (K Lame u )i (x) = (V Lame 1 gi (x) + (K Lame g)i (x) − (V Lame f)i (x) 2 for x ∈ ΓD,i , and t (x) = K Lame i 1 Lame fi (x) − K f (x) − (DLame g)i (x) 2 i

)i (x) + (DLame u

for x ∈ ΓN,i , and for i = 1, 2, 3. The associated variational problem is to ﬁnd ) ∈ ( t, u

3

−1/2 (ΓD,i ) × H

i=1

such that

3

1/2 (ΓN,i ) H

i=1

a( t, u ; w, v) = F (w, v)

is satisﬁed for all (w, v) ∈

3 i=1

with the bilinear form

−1/2 (ΓD,i ) × H

3 i=1

1/2 (ΓN,i ) H

(1.80)

1.2 Lam´e Equations

a( t, u ; w, v) = 3 (V Lame t )i , wi

ΓD,i

i=1 3 ti , (K Lame v )i i=1

−

(K Lame u )i , wi

i=1

+ ΓN,i

3

3

(DLame u )i , vi

39

+ ΓD,i

i=1

ΓN,i

and with the linear form F (w, v) = 3 1 Lame Lame gi , wi + + (K g)i , wi − (V f )i , wi 2 ΓD,i ΓD,i ΓD,i i=1 3 1 fi , vi . − fi , (K Lame v)i − (DLame g)i , vi 2 ΓN,i ΓN,i ΓN,i i=1 Since the bilinear form a(·, · ; ·, ·) is skew–symmetric, the unique solvability of the variational problem (1.80) follows from the mapping properties of all boundary integral operators involved. In the mixed boundary value problem (1.79), diﬀerent boundary conditions in the cartesian coordinate system are prescribed. In many practical applications, however, boundary conditions are given with respect to some diﬀerent orthogonal coordinate system. As an example, we consider the mixed boundary value problem −

3 ∂ σij (u, x) = 0 ∂x j j=1

for x ∈ Ω ,

i = 1, 2, 3

(γ0int u(x), n(x)) = g(x) for x ∈ Γ, γ1int u(x) − (γ1int u(x), n(x))n(x) = 0

(1.81)

for x ∈ Γ.

An elastic body, which is modelled by the mixed boundary value problem (1.81), can slide in tangential direction while in the normal direction a displacement is given. Note that the boundary value problem (1.81) may arise when considering a linearisation of nonlinear contact (Signorini) boundary conditions. Using the Dirichlet to Neumann map (1.71), it remains to ﬁnd the boundary displacements γ0int u and the boundary stresses γ1int u satisfying γ1int u(x) = (S Lame γ0int u(x)

for x ∈ Γ ,

as well as the boundary conditions (γ0int u(x), n(x)) = g(x) ,

γ1int u(x) − (γ1int u(x), n(x))n(x) = 0 for x ∈ Γ.

40

1 Boundary Integral Equations

Using γ0int u(x) = g(x)n(x) + uT (x) , we have to ﬁnd a tangential displacement ﬁeld 1/2 uT ∈ HT (Γ ) = v ∈ [H 1/2 (Γ )]3 : (v(x), n(x)) = 0

for x ∈ Γ

as a solution of the boundary integral equation (S Lame gn)(x) + (S Lame uT )(x) − (γ1int u(x), n(x))n(x) = 0 for x ∈ Γ. 1/2

For a test function v T ∈ HT (Γ ), we then obtain the variational problem S Lame uT , v T Γ = − S Lame gn, v T Γ , which is uniquely solvable due to the mapping properties of the Steklov– Poincar´e operator. Note that one may also consider mixed boundary value problems with sliding boundary conditions only on a part ΓS , but standard Dirichlet or Neumann boundary conditions elsewhere. However, to ensure uniqueness, one needs to assume Dirichlet boundary conditions somewhere for each component.

1.3 Stokes System The Stokes problem is to ﬁnd a velocity ﬁeld u ∈ R3 and a pressure p such that −Δu(x) + ∇p(x) = 0, div u(x) = 0 for x ∈ Ω (1.82) is satisﬁed, where is the viscosity of the ﬂuid. Note that the Stokes system (1.82) also arises in the limiting case when considering the Navier system −μΔu(x) − (λ + μ)grad div u(x) = 0 for x ∈ Ω for incompressible materials. Introducing the pressure p(x) = −(λ + μ)div u(x)

for x ∈ Ω ,

we get −μΔu(x) + ∇p(x) = 0 for x ∈ Ω , as well as div u(x) = −

2 1 p(x) = − (1 + ν)(1 − 2ν)p(x) = 0 λ+μ E

in the incompressible case ν = 1/2.

1.3 Stokes System

41

Using integration by parts, we obtain from the second equation in (1.82) the compatibility condition 0 = div u(y) dy = (u(y), n(y)) dsy . (1.83) Ω

Γ

Green’s ﬁrst formula for the Stokes system (1.82) reads a(u, v) =

3 ∂ −Δui (y) + p(y) vi (y)dy ∂yi i=1

Ω

3

p(y)div v(y)dy +

+ Ω

Γ

(1.84)

ti (u(y), p(y))vi (y)dsy

i=1

with the symmetric bilinear form a(u, v) = 2

3

eij (u, y)eij (v, y)dy −

Ω i,j=1

div u(y) div v(y) dy Ω

and with the associated boundary stress ti (u(y), p(y)) = −p(y)ni (y) + 2

3

eij (u, y)nj (y),

y ∈ Γ, i = 1, 2, 3.

j=1

From Green’s ﬁrst formula (1.84), we now derive Green’s second formula which reads for the solution (u, p) of (1.82) as 3 Ω i=1

∂ −Δvi (y) + q(y) ui (y)dy − p(y)div v(y)dy ∂yi

=

3 Γ

Ω

ti (u(y), p(y))vi (y)dsy −

i=1

3 Γ

ti (v(y), q(y))ui (y)dsy .

i=1

Choosing as test functions a pair of fundamental solutions U ∗ (x, y) and q∗ (x, y), i.e. satisfying 3

∗ −ΔUi (x, y) +

Ω i=1

∂ ∗ q (x, y) ui (y)dy = u (x), ∂yi

div U ∗ (x, y) = 0,

we obtain a representation formula for x ∈ Ω u (x) =

3 Γ k=1

∗ tk (u, p)Uk (x, y)dsy −

3 Γ k=1

tk (U ∗ (x, y), q∗ )uk (y)dsy

42

1 Boundary Integral Equations

for = 1, 2, 3. The fundamental solution of the Stokes system is given by δk (xk − yk )(x − y ) 1 1 ∗ + (1.85) (x, y) = Uk 8π |x − y| |x − y|3 for k, = 1, 2, 3, and q∗ (x, y) =

1 y − x 4π |x − y|3

for = 1, 2, 3. Note that the fundamental solution (1.85) coincides with the Kelvin tensor (1.66) for E 1 ν= , = . 2 3 Hence, we can deﬁne and analyse all the boundary integral operators and related boundary integral equations as for the system of linear elastostatics. The only exception is the Dirichlet boundary value problem of the Stokes system which requires a special treatment of the associated single layer potential. As in linear elastostatics the single layer potential of the Stokes system is given by 3 1 Stokes (V wk )(x) + w)k = (Vk w )(x) (V 2 =1

for k = 1, 2, 3. As before, V Stokes : [H −1/2 (Γ )]3 → [H 1 (Ω)]3 deﬁnes a continuous map. Combining this with the mapping properties of the interior trace operator γ0int : H 1 (Ω) → H 1/2 (Γ ) , we can deﬁne the continuous boundary integral operator V Stokes = γ0int V Stokes : [H −1/2 (Γ )]3 → [H 1/2 (Γ )]3 allowing the representation (V Stokes w)

k

11 = 2

(V wk )(x) +

3

(Vk w )(x)

for x ∈ Γ ,

=1

and for k = 1, 2, 3 as a weakly singular surface integral; see also Lemma 1.15. When considering the interior Dirichlet boundary value problem for the Stokes system −Δu(x) + ∇p(x) = 0 div u(x) = 0 γ0int u(x) = g(x)

for x ∈ Ω, for x ∈ Ω, for x ∈ Γ ,

(1.86)

1.3 Stokes System

and using (1.83), we ﬁrst have to assume the solvability condition (g(y), n(y)) dsy = 0.

43

(1.87)

Γ

On the other hand, it is obvious, that the pressure p satisfying the ﬁrst equation in (1.86) is only unique up to an additive constant. In particular, the homogeneous Dirichlet boundary value problem −Δu(x) + ∇p(x) = 0,

div u(x) = 0 for x ∈ Ω, u(x) = 0 for x ∈ Γ

has the non–trivial pair of solutions u∗ (x) = 0 and p∗ (x) = −1 for x ∈ Ω. The ﬁrst kind boundary integral equation of the direct approach for the Dirichlet boundary value problem (1.86) is (V Stokes t)(x) =

1 g(x) + (K Stokes g)(x) 2

for x ∈ Γ.

(1.88)

For the homogeneous Dirichlet boundary value problem with g = 0, we therefore obtain (V Stokes t∗ )(x) = 0 for x ∈ Γ with

t∗ (u∗ (x), p∗ (x)) = −p∗ (x)n(x) = n(x)

for x ∈ Γ.

Thus, t∗ = n is an eigenfunction of the single layer potential V Stokes yielding a zero eigenvalue. Therefore, we conclude that the boundary integral equation (1.88) is only solvable modulo t∗ , and we have to consider the boundary integral equation (1.88) in an appropriate factor space [90]. Hence, we deﬁne 3 −1/2 (Γ ) = w ∈ [H −1/2 (Γ )]3 : w, n V = V wk , nk Γ = 0 , H∗ k=1

where

V : H −1/2 (Γ ) → H 1/2 (Γ )

is the single layer potential of the Laplace operator. Considering the boundary −1/2 (Γ ), this can be rewritten as an extended integral equation (1.88) in H∗ variational problem to ﬁnd t ∈ [H −1/2 (Γ )]3 such that

V Stokes t, w

w, n + t, n

Γ

V

V

=

1 2

I + K Stokes g, w

(1.89)

Γ

is satisﬁed for all w ∈ [H −1/2 (Γ )]3 . Note that there exists a unique solution t ∈ [H −1/2 (Γ )]3 of the extended variational problem (1.89) for any given Dirichlet datum g ∈ [H 1/2 (Γ )]3 . If g satisﬁes the solvability condition (1.87), −1/2

we then obtain t ∈ H∗

(Γ ).

44

1 Boundary Integral Equations

1.4 Helmholtz Equation Let U : R+ × Ω → R be a scalar function which satisﬁes the wave equation 1 ∂2 U (t, x) = ΔU (t, x) c2 ∂t2

for t > 0 , x ∈ Ω .

(1.90)

The equation (1.90) is valid for the wave propagation in a homogeneous, isotrop, friction-free medium having the constant speed of sound c. The most important examples are the acoustic scattering and the sound radiation. The time harmonic acoustic waves are of the form U (t, x) = Re u(x)e−ı ωt , (1.91) where ı is the imaginary unit. In (1.91), u : Ω → C is a scalar, complex valued function and ω > 0 denotes the frequency. Inserting (1.91) into the wave equation (1.90), we obtain the reduced wave equation or the Helmholtz equation (1.92) −Δu(x) − κ2 u(x) = 0 for x ∈ Ω , where κ = ω/c > 0 is the wave number. First we consider the Helmholtz equation (1.92) in a bounded domain Ω ⊂ R3 . Multiplying this equation (1.92) with a test function v, integrating over Ω, and applying integration by parts, this gives Green’s ﬁrst formula (−Δu(y) − κ2 u(y))v(y)dy = a(u, v) − γ1int u(y)γ0int v(y)dsy (1.93) Ω

Γ

with the symmetric bilinear form 2 ∇u(y), ∇v(y) dy − κ u(y)v(y)dy. a(u, v) = Ω

Ω

From Green’s formula (1.93) and by the use of the symmetry of the bilinear form a(· , ·), we deduce Green’s second formula, 2 (−Δu(y) − κ u(y))v(y)dy + γ1int u(y)γ0int v(y)dsy = Ω

Γ

(−Δv(y) − κ v(y))u(y)dy + 2

Ω

γ1int v(y)γ0int u(y)dsy .

Γ

Now, choosing as a test function v a fundamental solution u∗κ : R3 × R3 → C satisfying − Δu∗κ (x, y) − κ2 u∗κ (x, y) u(y)dy = u(x) for x ∈ Ω, (1.94) Ω

1.4 Helmholtz Equation

45

the solution of the Helmholtz equation (1.92) is given by the representation formula int u∗ (x, y)γ int u(y)ds u∗κ (x, y)γ1int u(y)dsy − γ1,y (1.95) u(x) = y κ 0 Γ

Γ

for x ∈ Ω. The fundamental solution of the Helmholtz equation (1.92) is u∗κ (x, y) =

1 eı κ|x−y| 4π |x − y|

for x, y ∈ R3 .

(1.96)

As for the Laplace operator, we consider the single layer potential ı κ|x−y| 1 e w(y)dsy for x ∈ Ω u∗κ (x, y)w(y)dsy = (Vκ w)(x) = 4π |x − y| Γ

Γ

which deﬁnes a continuous map from a given density function w on the boundary Γ to a function Vκ w, which satisﬁes the partial diﬀerential equation (1.92) in Ω. In particular, Vκ : H −1/2 (Γ ) → H 1 (Ω) is continuous and Vκ w ∈ H 1 (Ω) is a weak solution of the Helmholtz equation (1.92) for any w ∈ H −1/2 (Γ ). Using the mapping properties of the interior trace operators γ0int : H 1 (Ω) → H 1/2 (Γ ) and

γ1int : H 1 (Ω, Δ + κ2 ) → H −1/2 (Γ ) ,

we can deﬁne corresponding boundary integral operators, e.g. the single layer potential operator Vκ = γ0 Vκ : H −1/2 (Γ ) → H 1/2 (Γ ) , as follows:

(Vκ w)(x) =

u∗κ (x, y)w(y)dsy =

1 4π

Γ

Γ

eı κ|x−y| w(y)dsy |x − y|

for x ∈ Γ .

Its conormal derivative is 1 I + Kκ : H −1/2 (Γ ) → H −1/2 (Γ ) 2 with the adjoint double layer potential operator int u∗ (x, y)w(y)ds (Kκ w)(x) = lim γ1,x y κ γ1int Vκ =

ε→0 y∈Γ :|y−x|≥ε

1 = lim ε→0 4π

y∈Γ :|y−x|≥ε

∇x

eı κ|x−y| , n(x) w(y)dsy . |x − y|

46

1 Boundary Integral Equations

Note that H 1 (Ω, Δ + κ2 ) =

−1 (Ω) . v ∈ H 1 (Ω) : Δv + κ2 v ∈ H

Since the density functions of the boundary integral operators introduced above may be complex valued, we consider v(x)w(x)dsx v, w Γ = Γ

as an appropriate duality pairing for v ∈ H 1/2 (Γ ) and w ∈ H −1/2 (Γ ). Then the single layer potential operator is complex symmetric, i.e. the following property holds for w, z ∈ H −1/2 (Γ ) Vκ w, z Γ =

1 4π

Γ Γ

1 = 4π

Γ

eı κ|x−y| w(y)dsy z(x)dsx |x − y|

−ı κ|x−y| e z(x)dsx dsy = w, V−κ z Γ . w(y) |x − y| Γ

If Γ is a Lipschitz boundary, the operator Vκ − V0 : H −1/2 (Γ ) → H 1/2 (Γ ) is compact. Since the single layer potential V0 of the Laplace operator is H −1/2 (Γ )–elliptic (see Lemma 1.1), the single layer potential Vκ is coercive, arding’s inequality i.e. with the compact operator C = V0 − Vκ , the G˚ (Vk + C)w, w Γ = V0 w, w Γ ≥ cV1 0 w2H −1/2 (Γ )

(1.97)

is satisﬁed for all w ∈ H −1/2 (Γ ). Next we consider the double layer potential 1 eı κ|x−y| int ∗ , n(y) v(y)dsy ∇y γ1,y uκ (x, y)v(y)dsy = (Wκ v)(x) = 4π |x − y| Γ

Γ

for x ∈ Ω, which again deﬁnes a continuous map from a given density function v on the boundary Γ to a function Wκ v satisfying the Helmholtz equation (1.92). In particular, Wκ : H 1/2 (Γ ) → H 1 (Ω) is continuous and Wκ v ∈ H 1 (Ω) is a weak solution of the Helmholtz equation (1.92) for any v ∈ H 1/2 (Γ ). Using the mapping properties of the interior trace operator γ0int : H 1 (Ω) → H 1/2 (Γ )

1.4 Helmholtz Equation

and

47

γ1int : H 1 (Ω, Δ + κ2 ) → H −1/2 (Γ ) ,

we can deﬁne corresponding boundary integral operators, i.e. the trace 1 γ0int W = − I + Kκ 2 with the double layer potential operator eı κ|x−y| 1 (Kκ v)(x) = lim , n(y) v(y)dsy ∇y ε→0 4π |x − y|

for x ∈ Γ.

y∈Γ :|y−x|≥ε

As for the single layer potential, we have w Γ Kκ v, w Γ = v, K−κ

for all v ∈ H 1/2 (Γ ) and w ∈ H −1/2 (Γ ). The conormal derivative of the double layer potential deﬁnes the hypersingular boundary integral operator Dκ = −γ1int Wκ : H 1/2 (Γ ) → H −1/2 (Γ ) . For a Lipschitz boundary Γ , the operator Dκ − D0 : H 1/2 (Γ ) → H −1/2 (Γ ) is compact. Since the regularised hypersingular boundary integral operator D0 + I of the Laplace operator is H 1/2 (Γ )–elliptic, and since the embedding H 1/2 (Γ ) → H −1/2 (Γ ) is compact, the hypersingular boundary integral operator Dκ is coercive, i.e. with the compact operator C = Dκ − D0 − I, the G˚ arding’s inequality

2 0 (Dκ + C)v, v Γ = (D0 + I)v, v Γ ≥ cD 1 vH 1/2 (Γ )

(1.98)

is satisﬁed for all v ∈ H 1/2 (Γ ). As for the bilinear form for the hypersingular boundary integral operator for the Laplace equation (see (1.9)), there holds an analogue result for the Helmholtz equation, see [78]: ı κ|x−y| 1 e curlΓ u(y), curlΓ v(x) dsy dsx (Dκ u)(x)v(x)dsx = 4π |x − y| Γ

κ2 − 4π

Γ Γ

Γ Γ ı κ|x−y|

e u(y)v(x) n(x), n(y) dsy dsx . |x − y|

(1.99)

In addition to the interior boundary value problem for the Helmholtz equation (1.92), we also consider the exterior boundary value problem

48

1 Boundary Integral Equations

−Δu(x) − κ2 u(x) = 0 for x ∈ Ω e = R3 \Ω ,

(1.100)

where we have to add the Sommerfeld radiation condition x 1 = O , ∇u(x) − ı κu(x) as |x| → ∞. |x| |x|2

(1.101)

For a ﬁxed y0 ∈ Ω and R > 2 diam Ω, let BR (y0 ) be a ball of radius R with centre y0 and including Ω. Let u be a solution of the exterior boundary value problem for the Helmholtz equation (1.100) satisfying the radiation condition (1.101). Considering Green’s ﬁrst formula (1.93) with respect to the bounded domain ΩR = BR (y0 )\Ω and choosing v = u as test function, we obtain |∇u(y)|2 dy − k 2 |u(y)|2 = γ1int u(y)γ0int u(y)dsy ΩR

=

ΩR

∂ΩR

γ1int u(y)γ0int u(y)dsy −

γ1ext u(y)γ0ext u(y)dsy ,

Γ

∂BR (y0 )

when taking into account the opposite direction of the normal vector n(x) for x ∈ Γ . Since the left hand side of the above equation is real, we conclude γ1int u(y)γ0int u(y)dsy = Im γ1ext u(y)γ0ext u(y)dsy . Im Γ

∂BR (y0 )

By the use of this property, the Sommerfeld radiation condition (1.101) implies 2 int 0 = lim γ1 u(y) − ı κγ0int u(y) dsy R→∞ ∂BR (y0 )

⎡

⎢ = lim ⎣

R→∞

|γ1int u(y)|2 dsy + κ2

∂BR (y0 )

−2κ Im

⎢ = lim ⎣ R→∞

⎤

⎥ γ1int u(y)γ0int u(y)dsy ⎦

∂BR (y0 )

|γ1int u(y)|2 dsy + κ2

∂BR (y0 )

|γ0int u(y)|2 dsy

∂BR (y0 )

−2κ Im Γ

and, therefore,

|γ0int u(y)|2 dsy

∂BR (y0 )

⎡

⎤

γ1ext u(y)γ0ext u(y)dsy ⎦

1.4 Helmholtz Equation

49

γ1ext u(y)γ0ext u(y)dsy

2κ Im Γ

⎡

⎢ = lim ⎣

R→∞

|γ1int u(y)|2 dsy + κ2

∂BR (y0 )

In particular, this gives

⎤ ⎥ |γ0int u(y)|2 dsy ⎦ ≥ 0.

∂BR (y0 )

lim

R→∞ ∂BR (y0 )

and, therefore,

|u(y)|2 dsy = O(1) ,

|u(x)| = O

1 |x|

as |x| → ∞.

(1.102)

For the bounded domain ΩR , we can apply the representation formula (1.95) to obtain ∗ int int u∗ (x, y)γ int u(y)ds u(x) = − uκ (x, y)γ1 u(y)dsy + γ1,y y 0 κ Γ

+

Γ

u∗κ (x, y)γ1int u(y)dsy

∂BR (y0 )

−

int u∗ (x, y)γ int u(y)ds γ1,y y κ 0

∂BR (y0 )

for x ∈ ΩR . Taking the limit R → ∞ and incorporating the radiation conditions (1.101) and (1.102), this gives the representation formula in the exterior domain Ω e , i.e. for x ∈ Ω e ∗ int int u∗ (x, y)γ int u(y)ds . u(x) = − uκ (x, y)γ1 u(y)dsy + γ1,y (1.103) y κ 0 Γ

Γ

1.4.1 Interior Dirichlet Boundary Value Problem We ﬁrst consider the interior Dirichlet boundary value problem for the Helmholtz equation, i.e. −Δu(x) − κ2 u(x) = 0 for x ∈ Ω,

γ0int u(x) = g(x)

for x ∈ Γ .(1.104)

Using the representation formula (1.95), the solution of the above Dirichlet boundary value problem is given by int u∗ (x, y)g(y)ds u(x) = u∗κ (x, y)t(y)dsy − γ1,y for x ∈ Ω, y Γ

Γ

where t = γ1int u is the unknown conormal derivative of u on Γ which has to be determined from some appropriate boundary integral equation.

50

1 Boundary Integral Equations

Applying the interior trace operator γ0int to the representation formula, this gives a boundary integral equation to ﬁnd t ∈ H −1/2 (Γ ) such that (Vκ t)(x) =

1 g(x) + (Kκ g)(x) 2

for x ∈ Γ.

(1.105)

Note that t ∈ H −1/2 (Γ ) is the solution of the variational problem

Vκ t, w

=

1

Γ

2

I + Kκ g, w

Γ

for all w ∈ H −1/2 (Γ ).

(1.106)

When applying the interior normal derivative γ1int to the representation formula, this gives a second kind boundary integral equation to ﬁnd t ∈ H −1/2 (Γ ) such that 1 t(x) − (Kκ t)(x) = (Dκ g)(x) for x ∈ Γ. (1.107) 2 To investigate the unique solvability of the variational problem (1.106), and, therefore, of the boundary integral equation (1.105) as well as of the boundary integral equation (1.107), we ﬁrst consider the Dirichlet eigenvalue problem for the Laplace operator, −Δu(x) = λu(x)

for x ∈ Ω,

γ0int u(x) = 0

for x ∈ Γ.

(1.108)

Let λ ∈ R+ be a certain eigenvalue, and let uλ be the corresponding eigenfunction. Since the eigenvalue problem (1.108) can be seen as the Helmholtz equation with the wave number κ satisfying κ2 = λ, we obtain for the conormal derivative tλ = γ1int uλ the boundary integral equations (Vκ tλ )(x) =

1 int γ uλ (x) + (Kκ γ0int uλ )(x) = 0 for x ∈ Γ 2 0

and

1 tλ (x) − (Kκ tλ )(x) = (Dκ γ0int uλ )(x) = 0 for x ∈ Γ. 2 Thus, the boundary integral operators Vκ and 1/2 I − Kκ are singular, and, therefore, not invertible, if κ2 = λ is an eigenvalue of the Dirichlet eigenvalue problem (1.108). On the other hand, if κ2 is not an eigenvalue of the Dirichlet eigenvalue problem (1.108), the single layer potential Vκ is injective and hence, since Vκ is coercive, also invertible. This shows the unique solvability of the variational problem (1.106) and of the boundary integral equation (1.105) in this case. Note that also in this case the second kind boundary integral equation (1.107) is uniquely solvable. 1.4.2 Interior Neumann Boundary Value Problem Next we consider the interior Neumann boundary value problem for the Helmholtz equation, i.e.

1.4 Helmholtz Equation

γ1int u(x) = g(x)

−Δu(x) − κ2 u(x) = 0 for x ∈ Ω,

51

for x ∈ Γ .(1.109)

From the representation formula (1.95), we can obtain the solution of the above boundary value problem as int u∗ (x, y)γ int u(y)ds u∗κ (x, y)g(y)dsy − γ1,y for x ∈ Ω. u(x) = y κ 0 Γ

Γ

Applying the interior trace operator γ0int to the above representation formula, this gives a ﬁrst boundary integral equation to ﬁnd u ¯ = γ0int u ∈ H 1/2 (Γ ) such that 1 u ¯(x) + (Kκ u ¯)(x) = (Vκ g)(x) for x ∈ Γ. (1.110) 2 When applying the conormal derivative operator γ1int to the above representation formula, this gives a second boundary integral equation, ¯)(x) = (Dκ u

1 g(x) − (Kκ g)(x) 2

for x ∈ Γ.

(1.111)

Hence, u ¯ ∈ H 1/2 (Γ ) is a solution of the variational problem

Dκ u ¯, v

=

1 2

Γ

I − Kκ g, v

Γ

for all v ∈ H 1/2 (Γ ).

(1.112)

To investigate the unique solvability of the variational problem (1.112), and, therefore, of the boundary integral equations (1.110) and (1.111), we now consider the Neumann eigenvalue problem for the Laplace operator, −Δu(x) = μu(x)

for x ∈ Ω,

γ1int u(x) = 0

for x ∈ Γ.

(1.113)

Let μ ∈ R+ be a certain eigenvalue, and let uμ be the corresponding eigenfunction. Since the eigenvalue problem (1.113) can be seen as the Helmholtz equation with the wave number κ satisfying κ2 = μ, we then obtain the boundary integral equations (Dκ uμ )(x) = and

1 int γ uμ (x) − (Kκ γ1int uμ )(x) = 0 for x ∈ Γ 2 1

1 uμ (x) + (Kκ uμ )(x) = (Vκ γ1int uμ )(x) = 0 for x ∈ Γ . 2 Thus, the boundary integral operators Dκ and 1/2 I + Kκ are singular and, therefore, not invertible if κ2 = μ is an eigenvalue of the Neumann eigenvalue problem (1.113). On the other hand, if κ2 is not an eigenvalue for the Neumann eigenvalue problem (1.113), the hypersingular boundary integral operator Dκ is injective and coercive, and, therefore, invertible.

52

1 Boundary Integral Equations

1.4.3 Exterior Dirichlet Boundary Value Problem The exterior Dirichlet boundary value problem for the Helmholtz equation reads −Δu(x) − κ2 u(x) = 0 for x ∈ Ω e ,

γ0ext u(x) = g(x)

for x ∈ Γ, (1.114)

where, in addition, we have to require the Sommerfeld radiation condition (1.101), x 1 = O , ∇u(x) − ı κu(x) as |x| → ∞. |x| |x|2 Note that the exterior Dirichlet boundary value problem is uniquely solvable due to the radiation condition. The solution of the above problem is given by the representation formula (1.103), ∗ ext ext u∗ (x, y)g(y)ds for x ∈ Ω e . u(x) = − uκ (x, y)γ1 u(y)dsy + γ1,y y κ Γ

Γ

To ﬁnd the yet unknown Neumann datum t = γ1ext u, we consider the boundary integral equation which results from the representation formula when applying the exterior trace operator γ0ext , 1 (Vκ t)(x) = − g(x) + (Kκ g)(x) 2

for x ∈ Γ.

(1.115)

This boundary integral equation is equivalent to a variational problem to ﬁnd t ∈ H −1/2 (Γ ) such that 1 Vκ t, w = − I + Kκ g, w for all w ∈ H −1/2 (Γ ) . (1.116) 2 Γ Γ Since the single layer potential Vκ of the exterior Dirichlet boundary value problem coincides with the single layer potential of the interior Dirichlet boundary value problem, Vκ is not invertible when κ2 = λ is an eigenvalue of the Dirichlet eigenvalue problem (1.108). However, we have 1 1 − I + Kκ g, tλ I − K−κ tλ = − g, = 0, 2 2 Γ Γ and, therefore,

1 − I + Kκ g ∈ Im Vκ . 2 In fact, the variational problem (1.116) of the direct approach is solvable, but the solution is not unique. As for the Neumann problem (1.21) for the Laplace equation, we can use a stabilised variational formulation to obtain a unique solution t ∈ H −1/2 (Γ ) satisfying some prescribed side condition, e.g., V0 t, tλ Γ = 0 ,

1.4 Helmholtz Equation

53

where V0 : H −1/2 (Γ ) → H 1/2 (Γ ) is the single layer potential operator of the Laplace equation. Instead of the variational problem (1.116), we then have to ﬁnd the function t ∈ H −1/2 (Γ ) as the unique solution of the stabilised variational problem 1 V0 w, tλ + V0 t, tλ = − I + Kκ g, w Vκ t, w 2 Γ Γ Γ Γ for all w ∈ H −1/2 (Γ ). Since this formulation requires the a priori knowledge of the eigensolution tλ , this approach does not seem to be applicable in general. If κ2 is not an eigenvalue of the interior Dirichlet eigenvalue problem (1.108), then the unique solvability of the boundary integral equation (1.115) follows, since Vκ is coercive and injective. Instead of a direct approach, we may also consider an indirect single layer potential approach, u∗κ (x, y)w(y)dsy for x ∈ Ω e . u(x) = (Vκ w)(x) = Γ

Then, applying the exterior trace operator, this leads to a boundary integral equation to ﬁnd w ∈ H −1/2 (Γ ) such that (Vκ w)(x) = g(x)

for x ∈ Γ.

(1.117)

Again, we have unique solvability of the boundary integral equation (1.117) only for those wave numbers κ2 , which are not eigenvalues of the interior Dirichlet eigenvalue problem (1.108). When using an indirect double layer potential approach, ext u∗ (x, y)v(y)ds γ1,y for x ∈ Ω e , u(x) = (Wκ v)(x) = y κ Γ

this leads to a boundary integral equation to ﬁnd v ∈ H 1/2 (Γ ) such that 1 v(x) + (Kκ v)(x) = g(x) 2

for x ∈ Γ.

(1.118)

The boundary integral operator 1/2 I + Kκ is singular, and, therefore, not invertible when κ2 is an eigenvalue of the interior Neumann boundary value problem (1.113). If κ2 is not an eigenvalue of the interior Neumann eigenvalue problem (1.113), the unique solvability of the boundary integral equation (1.118) follows. Although the exterior Dirichlet boundary value problem for the Helmholtz equation is uniquely solvable, the related boundary integral equations may not be solvable, in particular, when κ2 either coincides with an eigenvalue of the interior Dirichlet boundary value problem or with an eigenvalue of the interior Neumann boundary value problem. However, in any case at least one of the

54

1 Boundary Integral Equations

boundary integral equations (1.117) or (1.118) is uniquely solvable since κ2 can not be an eigenvalue of both the interior Dirichlet and the interior Neumann boundary value problem. Thus, we may combine both, the indirect single and double layer potential formulations to derive a boundary integral equation, which is uniquely solvable for arbitrary wave numbers. This leads to the well known Brakhage–Werner formulation (see [13]) u(x) = (Wκ w)(x) + ı η(Vκ w)(x)

for x ∈ Ω e ,

which leads to the boundary integral equation 1 w(x) + (Kκ w)(x) + ı η(Vκ w)(x) = g(x) 2

for x ∈ Γ .

(1.119)

Here, η ∈ R is some real parameter. Note that this equation is usually considered in the L2 (Γ ) sense. The numerical analysis to investigate the unique solvability of the combined boundary integral equation (1.119) is based on the coercivity of the underlying boundary integral operator, and, therefore, on some compactness argument. In general, this may require more regularity assumptions for the boundary surface under consideration (cf. [23]). Instead of considering the boundary integral equation (1.119) in L2 (Γ ), one may formulate some modiﬁed boundary integral equations to be considered in the energy spaces H 1/2 (Γ ) or H −1/2 (Γ ), see [17, 18]. 1.4.4 Exterior Neumann Boundary Value Problem Finally, we consider the exterior Neumann boundary value problem −Δu(x) − κ2 u(x) = 0 for x ∈ Ω e ,

γ1int u(x) = g(x)

for x ∈ Γ, (1.120)

where we have to require the Sommerfeld radiation condition (1.101), x 1 = O , ∇u(x) − ı κu(x) as |x| → ∞ . |x| |x|2 Due to the radiation condition, the exterior Neumann boundary value problem is uniquely solvable. The solution of the above boundary value problem is given by the representation formula (1.103), ext u∗ (x, y)γ ext u(y)ds for x ∈ Ω e . u(x) = − u∗κ (x, y)g(y)dsy + γ1,y y κ 0 Γ

Γ

To ﬁnd the yet unknown Dirichlet datum u = γ0ext u, we consider the boundary integral equation which results from the representation formula when applying the exterior conormal derivative γ1ext , 1 (Dκ u ¯)(x) = − g(x) − (Kκ g)(x) 2

for x ∈ Γ .

(1.121)

1.4 Helmholtz Equation

55

This hypersingular boundary integral equation is equivalent to the variational problem to ﬁnd u ¯ ∈ H 1/2 (Γ ) such that 1 I + Kκ g, v Dκ u ¯, v = − for all v ∈ H 1/2 (Γ ) . (1.122) 2 Γ Γ Since the hypersingular boundary integral operator Dκ of the exterior Neumann boundary value problem coincides with the operator which is related to the interior Neumann boundary value problem, Dκ is not invertible when κ2 = μ is an eigenvalue of the interior Neumann eigenvalue problem (1.113). However, we have 1 1 I + Kκ g, uλ I + K−κ uλ = g, = 0, 2 2 Γ Γ and, therefore,

1

I + Kκ g ∈ Im Dκ .

2 In fact, the variational problem (1.122) of the direct approach is solvable, but the solution is not unique. Again, one can formulate a suitable stabilised variational problem; we skip the details. If κ2 is not an eigenvalue of the Neumann eigenvalue problem (1.113), then the unique solvability of the variational problem (1.122) follows, since Dκ is coercive and injective. When applying the exterior trace operator γ0ext to the representation formula, this gives a second kind boundary integral equation to be solved, 1 − u(x) + (Kκ u)(x) = (Vκ g)(x) 2

for x ∈ Γ.

(1.123)

If κ2 = μ is an eigenvalue of the Dirichlet eigenvalue problem (1.108), the operator 1 I − K−κ : H −1/2 (Γ ) → H −1/2 (Γ ) 2 is singular and, therefore, not invertible. Then, the adjoint operator 1 I − Kκ : H 1/2 (Γ ) → H 1/2 (Γ ) 2 is also not invertible. As for the exterior Dirichlet boundary value problem, one may formulate a combined boundary integral equation in L2 (Γ ), i.e., a linear combination of the boundary integral equations (1.121) and (1.123) gives (cf. [19]) 1 1 g(x) + (Kκ g)(x) − u(x) + (Kκ u)(x) + ı η(Dκ u)(x) = (Vκ g)(x) − ıη 2 2 for x ∈ Γ , which is uniquely solvable due to the coercivity of the underlying boundary integral operators when assuming suﬃcient smoothness of the boundary Γ .

56

1 Boundary Integral Equations

1.5 Bibliographic Remarks The history of using surface potentials to describe solutions of partial diﬀerential equations goes back to the middle of the 19th century. Already C. F. Gauß [33, 34] proposed to solve the Dirichlet boundary value problem for the Laplace equation in a suﬃciently smoothly bounded domain by using an indirect double layer potential. To ﬁnd the yet unknown density function, a second kind boundary integral equation has to be solved. C. Neumann [80] applied a series representation to construct this solution, and he showed the convergence, i.e. a contraction property, when the domain is convex. These results were then extended by several authors, see also the discussion in [108], where the solvability of second kind boundary integral equations was considered for domains with non–smooth boundaries. This proof is based on diﬀerent representations of the boundary integral operators which follow from the Calderon projection property. In particular, the symmetry of the double layer potential, with respect to an inner product induced by the single layer potential, was already observed for a simple model problem by J. Plemelj [88]. A diﬀerent view on the historical development of those results was given recently in [25]. For a general review on the history of boundary integral and boundary element methods, see, for example, [22]. For a long time, direct and indirect boundary integral formulations have been a standard approach to describe solutions of partial diﬀerential equations in mathematical physics, see, for example, [29, 55, 59, 61, 62, 70, 72, 74, 91]. While second kind boundary integral equation methods [6] resulting from an indirect approach have a long tradition in both the analysis and numerical treatment of boundary value problems [81, 82], direct formulations and ﬁrst kind boundary integral equation methods became more popular in the last decades. This is mainly due to the rigorous mathematical analysis of boundary integral formulations and related numerical approximation schemes, which is available for ﬁrst kind equations in the setting of energy spaces. First results were obtained simultaneously by J. C. N´ed´elec and J. Planchard [79] and by G. C. Hsiao and W. L. Wendland [57]. More general results on the mapping properties of boundary integral operators in Sobolev spaces were later given by M. Costabel and W. L. Wendland [24, 26], see also the monograph [71] by W. McLean. While for boundary value problems with pure Dirichlet or pure Neumann boundary conditions one may use either ﬁrst or second kind boundary integral equations, the situation becomes more complicated when considering boundary value problems with mixed boundary conditions. Direct formulations, which are based on the weakly singular boundary integral equation only, then lead to systems combining boundary integral operators of both the ﬁrst and the second kind, see, e.g., [56]. Today, the symmetric formulation of boundary integral equations [103] seems to be more popular, see also [109, 114].

1.5 Bibliographic Remarks

57

Alternative representations of boundary integral operators are important for both analytical and numerical considerations. In particular, by using integration by parts, the bilinear form of the hypersingular boundary integral operator, which is the conormal derivative of the double layer potential, can be transformed into a linear combination of weakly singular forms, see [77] for the Laplace and for the Helmholtz operator. In fact, this also remains true for the system of linear elastostatics [47]. Moreover, also the double layer potential of linear elastostatics, which is deﬁned as a Cauchy singular integral operator, can be written as a combination of weakly singular boundary integral operators [62]. The use of boundary integral equation methods to describe solutions of boundary value problems is essentially based on the knowledge of a fundamental solution of the underlying partial diﬀerential operator. In fact, a fundamental solution is a solution of the partial diﬀerential equation with a Dirac impulse as the right hand side. While the existence of such a fundamental solution can be ensured for a quite large class of partial diﬀerential operators, in particular for partial diﬀerential operators with constant coeﬃcients [30, 53, 73], the explicite construction can be a complicated task in general, see for example [66, 85, 86]. For more general partial diﬀerential operators, i.e. with variable coeﬃcients, the concept of a parametrix, also known as a Levi function [73], was introduced by D. Hilbert [52]. A Levi function is a solution of the partial diﬀerential equation where the right hand side is given by a Dirac impulse and some more regular remainder. For example, such an approach was used in [89] to model shells by using a boundary–domain integral method.

2 Boundary Element Methods

The numerical approximation of boundary integral equations leads to boundary element methods in general. Since already the formulation of boundary integral equations is not unique, the choice of an appropriate discretisation scheme gives even more variety. The most common approximation methods are the Collocation scheme and the Galerkin method. In this chapter we ﬁrst introduce boundary element spaces of piecewise constant piecewise linear basis functions. Then we describe some discretisation methods for diﬀerent boundary integral formulations and we discuss the corresponding error estimates.

2.1 Boundary Elements Let Γ = ∂Ω be the boundary of a Lipschitz domain Ω ⊂ R3 . For N ∈ N, we consider a sequence of boundary element meshes ΓN =

N '

τ .

(2.1)

=1

In the most simple case, we assume that Γ is piecewise polyhedral and that each boundary element mesh (2.1) consists of N plane triangular boundary elements τ with mid points x∗ . Using the reference element ) ( τ = ξ ∈ R2 : 0 < ξ1 < 1, 0 < ξ2 < 1 − ξ1 , the boundary element τ = χ (τ ) with nodes xi for i = 1, 2, 3 can be described via the parametrisation x(ξ) = χ (ξ) = x1 + ξ1 (x2 − x1 ) + ξ2 (x3 − x1 ) ∈ τ

for ξ ∈ τ.

For the area Δ of the boundary element τ , we then obtain * 1* dsx = EG − F 2 dξ = EG − F 2 , Δ = 2 τ

τ

60

2 Boundary Element Methods

where 2 3 ∂ xi (ξ) = |x2 − x1 |2 , E= ∂ξ 1 i=1 2 3 ∂ xi (ξ) = |x3 − x1 |2 , G= ∂ξ 2 i=1 F =

3 ∂ ∂ xi (ξ) xi (ξ) = (x2 − x1 , x3 − x1 ) . ∂ξ1 ∂ξ2 i=1

Using Δ , we deﬁne the local mesh size of the boundary element τ as * h = Δ for = 1, . . . , N implying the global mesh sizes h = hmax = max h , 1≤≤N

hmin = min h . 1≤≤N

(2.2)

The sequence of boundary element meshes (2.1) is called globally quasi uniform if the mesh ratio hmax ≤ cG hmin is uniformly bounded by a constant cG which is independent of N ∈ N. Finally, we introduce the element diameter d = sup |x − y| . x,y∈τ

We assume that all boundary elements τ are uniformly shape regular, i.e., there exists a global constant cB independent of N such that d ≤ c B h With

for all = 1, . . . , N.

⎞ x2 ,1 − x1 ,1 x3 ,1 − x1 ,1 J = ⎝ x2 ,2 − x1 ,2 x3 ,2 − x1 ,2 ⎠ ∈ R3×2 x2 ,3 − x1 ,3 x3 ,3 − x1 ,3 ⎛

and using the parametrisation τ = χ (τ ), a function v deﬁned on τ can be interpreted as a function v with respect to the reference element τ , v(x) = v(x1 + J ξ) = v (ξ)

for ξ ∈ τ,

x = χ (ξ).

Vice versa, a function v deﬁned in the parameter domain τ implies a function v on the boundary element τ , v(ξ) = v(x1 + J ξ) = v (x)

for ξ ∈ τ,

x = χ (ξ).

Hence, we can deﬁne boundary element basis functions on τ by deﬁning associated shape functions on the reference element τ .

2.2 Basis Functions

61

2.2 Basis Functions Piecewise Constant Basis Functions The piecewise constant shape function ψ 0 (ξ) = 1 for ξ ∈ τ implies the piecewise constant basis functions on Γ 1 for x ∈ τ , ψ (x) = 0 elsewhere

(2.3)

for = 1, . . . , N , and, therefore, the global trial space N Sh0 (Γ ) = span ψ , =1

dim Sh0 (Γ ) = N.

Note that any wh ∈ Sh0 (Γ ) can be written as wh =

N

w ψ ∈ Sh0 (Γ ),

w ∈ R for = 1, . . . , N.

=1

Moreover, a function wh ∈ Sh0 (Γ ) can be identiﬁed with the vector w ∈ RN deﬁned by the components w for = 1, . . . , N . In what follows, we will consider the approximation property of the trial space Sh0 (Γ ) ⊂ L2 (Γ ). For this, we introduce the L2 projection of a given function w ∈ L2 (Γ ), N w ψ ∈ Sh0 (Γ ), Qh w = =1

which minimises the error w − Qh w in the L2 (Γ )–norm, 2 2 w(x) − w Qh w = arg min w − w = arg min (x) dsx . h h L (Γ ) 2 0 0 wh ∈Sh (Γ )

wh ∈Sh (Γ )

Γ

Note that Qh w is the unique solution of the variational problem (Qh w)(x)ψk (x)dsx = w(x)ψk (x)dsx for k = 1, . . . , N, Γ

Γ

or, N =1

w

ψ (x)ψk (x)dsx =

Γ

w(x)ψk (x)dsx Γ

for k = 1, . . . , N.

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2 Boundary Element Methods

Due to

ψ (x)ψk (x)dsx =

Δ 0

Γ

we obtain w =

1 Δ

for k = , , for k =

w(x)dsx

for = 1, . . . , N.

τ

From this explicit representation of w , one can prove the error estimate, see Appendix B.2, w − Qh w2L2 (Γ ) ≤ c

N

2 2s 2 h2s s (Γ ) |w|H s (τ ) ≤ c h |w|Hpw

(2.4)

=1 s (Γ ) and s ∈ (0, 1]. The semi-norm for a suﬃciently regular function w ∈ Hpw in (2.4) is deﬁned as |w(x) − w(y)|2 2 dsx dsy for s ∈ (0, 1) |w|H s (τ ) = |x − y|2+2s τ τ

and

|∇ξ w(χ (ξ))|2 dξ

|w|2H 1 (τ ) =

for s = 1.

τ

From the above variational formulation, we conclude the Galerkin orthogonality w(x) − (Qh w)(x) vh (x)dsx = 0 for all vh ∈ Sh0 (Γ ) , Γ

and, therefore, the trivial error estimate w − Qh wL2 (Γ ) ≤ wL2 (Γ ) . Using a duality argument, we further obtain s (Γ ) w − Qh wH σ (Γ ) ≤ c hs−σ |w|Hpw

for σ ∈ [−1, 0] and s ∈ [0, 1]. Summarising the above, we obtain the following approximation property in Sh0 (Γ ). s Theorem 2.1. Let w ∈ Hpw (Γ ) for some s ∈ [0, 1]. Then there holds

inf

0 (Γ ) wh ∈Sh

s (Γ ) w − wh H σ (Γ ) ≤ c hs−σ |w|Hpw

(2.5)

for all σ ∈ [−1, 0]. Moreover, the approximation property (2.5) remains valid for all σ ≤ s ≤ 1 with σ < 1/2.

2.2 Basis Functions

63

Piecewise Linear Discontinuous Basis Functions With respect to the reference element τ , we may also deﬁne local polynomial shape functions of higher order. In particular, we introduce the linear shape functions ψ11 (ξ) = 1 − ξ1 − ξ2 ,

ψ21 (ξ) = ξ1 ,

ψ31 (ξ) = ξ2

for ξ ∈ τ.

(2.6)

These shape functions imply globally discontinuous piecewise linear basis functions for x = χ (ξ) ∈ τ , ψi1 (ξ) ψ,i (x) = 0 elsewhere for = 1, . . . , N , i = 1, 2, 3, and, therefore, the global trial space

N , Sh1,−1 (Γ ) = span ψ,1 (x), ψ,2 (x), ψ,3 (x) =1

dim Sh1,−1 (Γ ) = 3N.

Any function wh ∈ Sh1,−1 (Γ ) can be written as wh =

3 N

w,i ψ,i ∈ Sh1,−1 (Γ ) .

=1 i=1

Moreover, a function wh ∈ Sh1,−1 (Γ ) can be identiﬁed with the vector w ∈ R3N which is deﬁned by the coeﬃcients w,i for i = 1, 2, 3 and = 1, . . . , N . As for piecewise constant basis functions, we may also deﬁne the corresponding L2 projection Qh w ∈ Sh1,−1 (Γ ) ⊂ L2 (Γ ), Qh w =

3 N

w,i ψ,i ∈ Sh1,−1 (Γ ),

=1 i=1

as the unique solution of the variational problem (Qh w)(x)ψk,j (x)dsx = w(x)ψk,j (x)dsx , j = 1, 2, 3 , k = 1, . . . , N Γ

Γ

satisfying the error estimate w − Qh w2L2 (Γ ) ≤ c

N

h4 |w|2H 2 (τ ) ≤ c h4 |w|2Hpw 2 (Γ )

=1 2 when assuming w ∈ Hpw (Γ ). Combining this with the trivial error estimate

w − Qh wL2 (Γ ) ≤ wL2 (Γ ) , and using an interpolation argument, the ﬁnal error estimate

64

2 Boundary Element Methods s (Γ ) w − Qh wL2 (Γ ) ≤ c hs |w|Hpw

s follows when assuming w ∈ Hpw (Γ ) for some s ∈ [0, 2]. Using again a duality argument, we ﬁnally obtain s (Γ ) w − Qh wH σ (Γ ) ≤ c hs−σ |w|Hpw

(2.7)

for σ ∈ [−2, 0] and s ∈ [0, 2]. Summarising the above, we obtain the approximation property in Sh1,−1 (Γ ). s (Γ ) for some s ∈ [0, 2]. Then there holds Theorem 2.2. Let w ∈ Hpw

inf

1,−1 wh ∈Sh (Γ )

s (Γ ) w − wh H σ (Γ ) ≤ c hs−σ |w|Hpw

(2.8)

for all σ ∈ [−2, 0]. Moreover, the approximation property (2.8) remains valid for all σ ≤ s ≤ 2 with σ < 1/2. Piecewise Linear Continuous Basis Functions Up to now, we have considered only globally discontinuous basis functions which do not require any admissibility condition of the triangulation (2.1). But such a condition is needed to deﬁne globally continuous basis functions. Let {xj }M j=1 be the set of all nodes of the triangulation (2.1). A boundary element mesh consisting of plane triangular elements is called admissible, if the intersection of two neighboured elements τ¯ and τ¯k is just one common edge or one common node. Then I(j) is the index set of all boundary elements τ containing the node xj while J() is the three–dimensional index set of the nodes deﬁning the triangular element τ . For j = 1, . . . , M , one can deﬁne globally continuous piecewise linear basis functions ϕj with ⎧ 1 for x = xj , ⎪ ⎨ 0 for x = xi = xj , ϕj (x) = ⎪ ⎩ piecewise linear elsewhere. Note that the restrictions of ϕj onto a boundary element τk for k ∈ I(j) can be represented by the linear shape functions ψj1k , ϕj (x) = ψj1k (ξ)

for x = χk (ξ) ∈ τk .

(2.9)

The basis functions ϕj are used to deﬁne the trial space M Sh1 (Γ ) = span ϕj , j=1

dim Sh1 (Γ ) = M.

The piecewise linear continuous L2 projection Qh w ∈ Sh1 (Γ ) is then deﬁned as the unique solution of the variational problem

2.3 Laplace Equation

Qh w(x)ϕj (x)dsx = Γ

Due to

65

w(x)ϕj (x)dsx

for j = 1, . . . , M.

Γ

Sh1 (Γ )

⊂

Sh1,−1 (Γ )

we immediately ﬁnd the error estimate

s (Γ ) w − Qh wH σ (Γ ) ≤ c hs−σ |w|Hpw

s when assuming w ∈ Hpw (Γ ), σ ∈ [−2, 0], s ∈ [0, 2]. 1 Deﬁning Ph u ∈ Sh (Γ ) as the unique solution of the variational problem

Ph w, vh H 1 (Γ ) = w, vh Γ

for all vh ∈ Sh1 (Γ )

we can show the error estimate s (Γ ) w − Ph wH σ (Γ ) ≤ c hs−σ |w|Hpw

s when assuming w ∈ Hpw (Γ ) and σ ∈ (0, 1], s ∈ [1, 2]. Hence, we have the following result. s (Γ ) for some s ∈ [1, 2]. Then there holds Theorem 2.3. Let v ∈ Hpw

inf

1 (Γ ) vh ∈Sh

s (Γ ) v − vh H σ (Γ ) ≤ c hs−σ |v|Hpw

(2.10)

for all σ ∈ [−2, 1]. Moreover, the approximation property (2.10) remains valid for all σ ≤ s ≤ 2 with σ < 3/2.

2.3 Laplace Equation 2.3.1 Interior Dirichlet Boundary Value Problem The solution of the interior Dirichlet boundary value problem (cf. (1.14)) γ0int u(x) = g(x)

−Δu(x) = 0 for x ∈ Ω,

for x ∈ Γ,

is given by the representation formula (cf. (1.6)) int u∗ (x, y)g(y)ds u∗ (x, y)t(y)dsy − γ1,y u(x) = y Γ

for x ∈ Ω,

Γ

where the unknown Neumann datum t = γ1int u ∈ H −1/2 (Γ ) is the unique solution of the boundary integral equation (cf. (1.15)) 1 1 1 1 (x − y, n(y)) t(y)dsy = g(x) + g(y)dsy for x ∈ Γ. 4π |x − y| 2 4π |x − y|3 Γ

Γ

Replacing t ∈ H −1/2 (Γ ) by a piecewise constant approximation th =

N

t ψ ∈ Sh0 (Γ ),

(2.11)

=1

we have to ﬁnd the unknown coeﬃcient vector t ∈ RN from some appropriate system of linear equations.

66

2 Boundary Element Methods

Collocation Method Inserting (2.11) into the boundary integral equation (1.15), and choosing the boundary element mid points x∗k as collocation nodes, we have to solve the collocation equations 1 1 1 (x∗k − y, n(y)) 1 ∗ t g(x (y)ds = ) + g(y)dsy (2.12) h y k ∗ 4π |xk − y| 2 4π |x∗k − y|3 Γ

Γ

for k = 1, . . . , N , or using the deﬁnition (2.3) of the piecewise constant basis functions ψ , N =1

t

1 4π

τ

1 1 1 dsy = g(x∗k ) + |x∗k − y| 2 4π

Γ

(x∗k − y, n(y)) g(y)dsy |x∗k − y|3

for k = 1, . . . , N . With 1 Vh [k, ] = 4π

τ

1 dsy |x∗k − y|

for k, = 1, . . . , N , and fk =

1 1 g(x∗k ) + 2 4π

Γ

(x∗k − y, n(y)) g(y)dsy |x∗k − y|3

for k = 1, . . . , N , this results in a linear system of equations, Vh t = f . The stiﬀness matrix Vh of the collocation method is in general non–symmetric and the stability of the collocation scheme (2.12) and therefore the invertibility of the stiﬀness matrix Vh is still an open problem when Γ is the boundary of a general Lipschitz domain Ω ⊂ R3 . When assuming the stability of the collocation scheme (2.12), the quasi optimal error estimate, i.e., Cea’s lemma, t − th H −1/2 (Γ ) ≤ c

inf

0 (Γ ) wh ∈Sh

t − wh H −1/2 (Γ )

follows. Combining this with the approximation property (2.5) for σ = −1/2, we get the error estimate s (Γ ) , t − th H −1/2 (Γ ) ≤ c hs+1/2 |t|Hpw

s (Γ ) for some s ∈ [0, 1]. Applying the Aubin–Nitsche when assuming t ∈ Hpw trick (for σ < −1/2) and an inverse inequality argument (for σ ∈ (−1/2, 0]), we also obtain the error estimate

2.3 Laplace Equation s (Γ ) , t − th H σ (Γ ) ≤ c hs−σ |t|Hpw

67

(2.13)

s when assuming t ∈ Hpw (Γ ) for some s ∈ [0, 1], and σ ∈ [−1, 0]. Note that the lower bound σ ≥ −1 is due to the collocation approach, independently of the degree of the used polynomial basis functions. Inserting the computed solution th into the representation formula (1.6), this gives an approximate representation formula int u∗ (x, y)g(y)ds u (x) = u∗ (x, y)th (y)dsy − γ1,y y Γ

Γ

for x ∈ Ω, describing an approximate solution of the Dirichlet boundary value problem (1.14). Note that u is harmonic, satisfying the Laplace equation, but the Dirichlet boundary conditions are satisﬁed only approximately. For an arbitrary x ∈ Ω, the error is given by u∗ (x, y) t(y) − th (y) dsy . u(x) − u (x) = Γ

Using a duality argument, the error estimate |u(x) − u (x)| ≤ u∗ (x, ·)H −σ (Γ ) t − th H σ (Γ ) for some σ ∈ R follows. Combining this with the error estimate (2.13) for the minimal possible value σ = −1, we obtain the pointwise error estimate s (Γ ) , |u(x) − u (x)| ≤ c hs+1 u∗ (x, ·)H 1 (Γ ) |t|Hpw

s (Γ ) for some s ∈ [0, 1]. Hence, if t is suﬃciently when assuming t ∈ Hpw 1 smooth, i.e. t ∈ Hpw (Γ ), we obtain as the optimal order of convergence for s=1 1 (Γ ) . (2.14) |u(x) − u (x)| ≤ c h2 u∗ (x, ·)H 1 (Γ ) |t|Hpw

Note that the error estimate (2.14) involves the position of the observation point x ∈ Ω. In particular, the error estimate (2.14) does not hold in the limiting case x ∈ Γ . Galerkin Method The boundary integral equation (cf. (1.15)) 1 1 1 (x − y, n(y)) 1 t(y)dsy = g(x) + g(y)dsy 4π |x − y| 2 4π |x − y|3 Γ

Γ

is equivalent to the variational problem (1.16),

for x ∈ Γ

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2 Boundary Element Methods

V t, w

=

1 2

Γ

I + K g, w

for all w ∈ H −1/2 (Γ ),

Γ

and to the minimisation problem F (t) =

min

w∈H −1/2 (Γ )

F (w)

with

1 1 V w, w I + K g, w . − 2 2 Γ Γ Using a sequence of ﬁnite dimensional subspaces Sh0 (Γ ) spanned by piecewise constant basis functions, associated approximate solutions F (w) =

th =

N

t ψ ∈ Sh0 (Γ )

=1

are obtained from the minimisation problem F (th ) =

min

0 (Γ ) wh ∈Sh

F (wh ).

The solution th ∈ Sh0 (Γ ) of the above minimisation problem is deﬁned via the Galerkin equations 1 I + K g, ψk = for k = 1, . . . , N. (2.15) V th , ψk 2 Γ Γ With (2.11) and by using the deﬁnition (2.3) of the piecewise constant basis functions ψ , this is equivalent to N =1

1 t 4π

τk

1 dsy dsx = |x − y| τ 1 (x − y, n(y)) 1 g(x)dsx + g(y)dsy dsx 2 4π |x − y|3 τk

for k = 1, . . . , N . With 1 Vh [k, ] = 4π

τk Γ

τk τ

1 dsy dsx |x − y|

for k, = 1, . . . , N , and 1 1 (x − y, n(y)) fk = g(x)dsx + g(y)dsy dsx 2 4π |x − y|3 τk

τk Γ

for k = 1, . . . , N , we ﬁnd the coeﬃcient vector t ∈ RN as the unique solution of the linear system

2.3 Laplace Equation

Vh t = f .

69

(2.16)

The Galerkin stiﬀness matrix Vh is symmetric and positive deﬁnite. Therefore, one may use a conjugate gradient scheme for an iterative solution of the linear system (2.16). Since the spectral condition number of Vh behaves like O(h−1 ), i.e., 1 λmax (Vh ) ≤ c , κ2 (Vh ) = Vh 2 Vh−1 2 = λmin (Vh ) h an appropriate preconditioning is sometimes needed. Moreover, since the stiﬀness matrix Vh is dense, fast boundary element methods are required to construct more eﬃcient algorithms, see Chapter 3. From the H −1/2 (Γ )–ellipticity and the boundedness of the single layer potential V : H −1/2 (Γ ) → H 1/2 (Γ ) , see Lemma 1.1, we conclude the unique solvability of the Galerkin variational problem (2.15), or, correspondingly, of the linear system (2.16), as well as the quasi optimal error estimate, i.e. Cea’s lemma, t − th H −1/2 (Γ ) ≤

cV2 cV1

inf

0 (Γ ) wh ∈Sh

t − wh H −1/2 (Γ ) .

Combining this with the approximation property (2.5) for σ = −1/2, we get 1

s (Γ ) , t − th H −1/2 (Γ ) ≤ c hs+ 2 |t|Hpw

s (Γ ) and s ∈ [0, 1]. Applying the Aubin–Nitsche trick when assuming t ∈ Hpw (for σ < −1/2) and an inverse inequality argument (for σ ∈ (−1/2, 0]), we also obtain the error estimate s (Γ ) , t − th H σ (Γ ) ≤ c hs−σ |t|Hpw

(2.17)

s when assuming t ∈ Hpw (Γ ) for some s ∈ [0, 1] and σ ∈ [−2, 0]. Inserting the computed Galerkin solution th ∈ Sh0 (Γ ) into the representation formula (1.6), this gives an approximate representation formula int u∗ (x, y)g(y)ds u∗ (x, y)th (y)dsy − γ1,y for x ∈ Ω , (2.18) u (x) = y Γ

Γ

describing an approximate solution of the Dirichlet boundary value problem (1.14). Note that u is harmonic satisfying the Laplace equation, but the Dirichlet boundary conditions are satisﬁed only approximately. For an arbitrary x ∈ Ω, the error is given by u∗ (x, y) t(y) − th (y) dsy . u(x) − u (x) = Γ

70

2 Boundary Element Methods

Using a duality argument, the error estimate |u(x) − u (x)| ≤ u∗ (x, ·)H −σ (Γ ) t − th H σ (Γ ) for some σ ∈ R follows. Combining this with the error estimate (2.17) for the minimal value σ = −2, we obtain the pointwise error estimate s (Γ ) , |u(x) − u (x)| ≤ c hs+2 u∗ (x, ·)H 2 (Γ ) |t|Hpw

s 1 (Γ ) for some s ∈ [0, 1]. Hence, if t ∈ Hpw (Γ ) is when assuming t ∈ Hpw suﬃciently smooth, we obtain the optimal order of convergence for s = 1, 1 (Γ ) . |u(x) − u (x)| ≤ c h3 u∗ (x, ·)H 2 (Γ ) |t|Hpw

(2.19)

Note that the error estimate (2.19) involves the position of the observation point x ∈ Ω again. In particular, the error estimate (2.19) does not hold in the limiting case x ∈ Γ . The computation of the right hand side f in the linear system (2.16) requires the evaluation of the integrals 1 1 (x − y, n(y)) g(x)dsx + g(y)dsy dsx fk = 2 4π |x − y|3 τk

τk Γ

for k = 1, . . . , N . An approximation of the given Dirichlet datum g ∈ H 1/2 (Γ ) by a globally continuous and piecewise linear function gh =

M

gj ϕj ∈ Sh1 (Γ )

j=1

can be obtained either by interpolation, gh =

M

g(xj ) ϕj ,

(2.20)

j=1

or by the L2 projection, gh =

M

gj ϕj ,

j=1

where the coeﬃcients gj , j = 1, . . . , M satisfy M j=1

This leads to

gj ϕj , ϕi L2 (Γ ) = g, ϕi L2 (Γ )

for i = 1, . . . , M .

(2.21)

2.3 Laplace Equation

1 gj fk = 2 j=1 M

=

M

gj

j=1

ϕj (x)dsx +

M

gj

j=1

τk

1 4π

1 Mh [k, j] + Kh [k, j] 2

with the matrix entries ϕj (x)dsx , Mh [k, j] =

Kh [k, j] =

τk Γ

1 4π

τk

71

(x − y, n(y)) ϕj (y)dsy dsx |x − y|3

τk Γ

(x − y, n(y)) ϕj (y)dsy dsx |x − y|3

for j = 1, . . . , M and k = 1, . . . , N . Instead of the linear system (2.16), we then have to solve a linear system with a perturbed right hand side f, yielding a perturbed solution vector t, i.e., we have to solve the linear system t = Vh

1 2

Mh + Kh g .

(2.22)

For the perturbed boundary element solution th ∈ Sh0 (Γ ), the error estimate t − th H σ (Γ ) ≤ c1 t − th H σ (Γ ) + c2 g − gh H σ+1 (Γ ) follows with σ ∈ [−2, 0], when the L2 projection (2.21) is used to deﬁne gh ∈ Sh1 (Γ ). Note that σ ∈ [−1, 0] in the case of the interpolation (2.20). s+1 s (Γ ) and g ∈ Hpw (Γ ) for some s ∈ [0, 1], we then obtain Assuming t ∈ Hpw the error estimate s (Γ ) + c2 |g| s+1 t − th H σ (Γ ) ≤ hs−σ c1 |t|Hpw Hpw (Γ ) . For the approximate representation formula int u∗ (x, y)g (y)ds u (x) = u∗ (x, y)th (y)dsy − γ1,y h y Γ

for x ∈ Ω ,

Γ

we then obtain the optimal error estimate |u(x) − u (x)| ≤ c(x, t, g) h3 ,

(2.23)

1 (Γ ) and when using the L2 projection (2.21) and when assuming t ∈ Hpw 2 g ∈ Hpw (Γ ). When using the interpolation (2.20) instead, the error estimate

|u(x) − u (x)| ≤ c(x, t, g) h2 follows.

72

2 Boundary Element Methods

2.3.2 Interior Neumann Boundary Value Problem Let Ω ⊂ R3 be a simply connected domain. The solution of the interior Neumann boundary value problem (cf. (1.21)) −Δu(x) = 0

γ1int u(x) = g(x)

for x ∈ Ω,

for x ∈ Γ,

is given by the representation formula (cf. (1.6)) ∗ int u∗ (x, y)¯ u (x, y)g(y)dsy − γ1,y u(y)dsy u(x) = Γ

for x ∈ Ω,

Γ

where the unknown Dirichlet datum u ¯ = γ0int u ∈ H 1/2 (Γ ) is a solution of the hypersingular boundary integral equation (cf. (1.27)) 1 int u∗ (x, y)¯ int u∗ (x, y)g(y)ds u(y)dsy = g(x) − γ1,x −γ1int γ1,y y 2 Γ

Γ

for x ∈ Γ . Since the hypersingular boundary integral operator D has a non– trivial kernel, we consider the extended variational problem (cf. (1.29)) to ﬁnd u ¯α ∈ H 1/2 (Γ ) such that v, 1 + u ¯α , 1 D¯ uα , v Γ

Γ

=

1

Γ

2

I − K g, v + α v, 1 Γ

Γ

is satisﬁed for all v ∈ H 1/2 (Γ ). Note that from the solvability condition (1.22), we reproduce the scaling condition (1.25). Since the bilinear form of this variational problem is strictly positive, the variational problem is equivalent to the minimisation problem F (¯ uα ) =

min

v∈H 1/2 (Γ )

F (v)

with F (v) =

2 1 1 Dv, v I − K g, v − + v, 1 − α v, 1 . 2 2 Γ Γ Γ Γ

Using a sequence of ﬁnite dimensional subspaces Sh1 (Γ ) ⊂ H 1/2 (Γ ) spanned by piecewise linear and continuous basis functions, an associated approximate function M u ¯α,j ϕj ∈ Sh1 (Γ ) (2.24) u ¯α,h = j=1

is obtained from the minimisation problem F (¯ uα,h ) =

min

1 (Γ ) vh ∈Sh

F (vh ).

2.3 Laplace Equation

73

The solution u ¯α,h ∈ Sh1 (Γ ) of the above minimisation problem is then deﬁned via the Galerkin equations D¯ uα,h , ϕi ϕi , 1 + u ¯α,h , 1 = Γ Γ Γ 1 I − K g, ϕi + α ϕi , 1 (2.25) 2 Γ Γ for i = 1, . . . , M . Using (2.24), this becomes M j=1

Dϕj , ϕi

u ¯α,j

1 2

ϕi , 1 = + ϕj , 1

Γ

Γ

I − K g, ϕi

Γ

Γ

+ α ϕi , 1

Γ

for i = 1, . . . , M . With Dh [i, j] = Dϕj , ϕi Γ

1 = 4π

ai = ϕi , 1 Γ =

Γ Γ

curlΓ ϕj (y) , curlΓ ϕi (x) dsx dsy , |x − y|

ϕi (x)dsx , Γ

fi =

1

1 = 2

2

Γ

g(x)ϕi (x)dsx − Γ

1 = 2

I − K g, ϕi

Γ

ϕi (x)

Γ

1 g(x)ϕi (x)dsx − 4π

int u∗ (x, y)g(y)ds ds γ1,x y x

Γ

ϕi (x)

Γ

Γ

(y − x, n(x)) g(y)dsy dsx |x − y|3

for i, j = 1, . . . , M , we ﬁnd the coeﬃcient vector u ¯ α ∈ RM as the unique solution of the linear system ¯ α = f + α a. Dh + a a u (2.26) The extended stiﬀness matrix Dh +a a is symmetric and positive deﬁnite. Therefore, one may use a conjugate gradient scheme for an iterative solution of the linear system (2.26). However, due to the estimate for the spectral condition number 1 κ2 (Dh + a a ) ≤ c , h an appropriate preconditioning is sometimes needed.

74

2 Boundary Element Methods

Note, that instead of a direct evaluation of the hypersingular boundary integral operator D, we apply integration by parts to obtain the representation (1.9) in Lemma 1.4, where i (x) curlΓ ϕi (x) = n(x) × ∇x ϕ

for x ∈ Γ

is the surface curl operator, and ϕ i is some locally deﬁned extension of ϕi into the neighbourhood of Γ . Since ϕi is linear on every boundary element τk , and deﬁning the extension ϕ i to be constant along n(x), we obtain curlΓ ϕi to be a piecewise constant vector function. Hence, we get Dh [i, j] =

(2.27) 1 1 dsx dsy . curlΓ ϕi|τk , curlΓ ϕj|τ 4π |x − y|

τk ∈ supp ϕi τ ∈ supp ϕj

τk τ

Thus, the entries of the stiﬀness matrix Dh of the hypersingular boundary integral operator D are linear combinations of the entries Vh [k, ] of the single layer potential matrix Vh . Hence, we can write ⎛ ⎞ Vh 0 0 Dh = T ⎝ 0 Vh 0 ⎠ T 0 0 Vh with some sparse transformation matrix T ∈ RM ×3N . From the H 1/2 (Γ )–ellipticity of the extended bilinear form, i.e., 2 Dv, v Γ + v, 1 2Γ ≥ cD 1 vH 1/2 (Γ )

for all v ∈ H 1/2 (Γ ),

we conclude the unique solvability of the variational problem (2.25), or correspondingly, of the linear system (2.26). Furthermore, the quasi optimal error estimate, i.e., Cea’s lemma, ¯α,h H 1/2 (Γ ) ≤ c ¯ uα − u

inf

1 (Γ ) vh ∈Sh

¯ uα − vh H 1/2 (Γ )

holds. Combining this with the approximation property (2.10) for σ = 1/2, we get s (Γ ) , ¯ uα − u ¯α,h H 1/2 (Γ ) ≤ c hs−1/2 |¯ uα |Hpw (2.28) s (Γ ) for some s ∈ [1/2, 2]. Applying the Aubin– when assuming u ¯α ∈ Hpw Nitsche trick, we also obtain the error estimate s (Γ ) , ¯α,h H σ (Γ ) ≤ c hs−σ |¯ uα |Hpw ¯ uα − u

s when assuming u ¯α ∈ Hpw (Γ ) for some s ∈ [1/2, 2] and σ ∈ [−1, 1/2]. Inserting the computed Galerkin solution u ¯α,h ∈ Sh1 (Γ ) into the representation formula (1.6), this gives an approximate representation formula

2.3 Laplace Equation

u (x) =

u∗ (x, y)g(y)dsy −

Γ

int u∗ (x, y)¯ γ1,y uα,h (y)dsy

75

for x ∈ Ω

Γ

describing an approximate solution of the Neumann boundary value problem (1.21). For an arbitrary x ∈ Ω, the error is given by int u∗ (x, y) u γ1,y ¯α,h (y) − u ¯α (y) dsy . u(x) − u (x) = Γ

Using a duality argument, the error estimate int ∗ u (x, ·) |u(x) − u (x)| ≤ γ1,y

H −σ (Γ )

¯ uα − u ¯α,h H σ (Γ )

for some σ ∈ R follows. Combining this with the error estimate (2.28) for the minimal value σ = −1, we obtain the pointwise error estimate int ∗ s (Γ ) , u (x, ·) 1 |¯ uα |Hpw |u(x) − u (x)| ≤ c hs+1 γ1,y H (Γ )

s 2 (Γ ) for some s ∈ [1/2, 2]. Hence, if u ¯α ∈ Hpw (Γ ) is when assuming u ¯α ∈ Hpw suﬃciently smooth, we obtain the optimal order of convergence for s = 2,

int u∗ (x, ·) 1 |¯ 2 (Γ ) . |u(x) − u (x)| ≤ c h3 γ1,y H (Γ ) uα |Hpw

(2.29)

Again, the error estimate (2.29) involves the position of the observation point x ∈ Ω, and, therefore, it is not valid in the limiting case x ∈ Γ . As in the boundary element method for the Dirichlet boundary value problem, we may also approximate the given Neumann datum g ∈ H −1/2 (Γ ) ﬁrst. If gh ∈ Sh0 (Γ ) is deﬁned by the L2 projection, i.e. if it is the unique solution of the variational problem gh (x)ψk (x) dsx = g(x)ψk (x) dsx for k = 1, . . . , N , Γ

Γ

then the error estimate s (Γ ) g − gh H σ (Γ ) ≤ c hs−σ |g|Hpw

s holds, when assuming g ∈ Hpw (Γ ) for some s ∈ [0, 1] and σ ∈ [−1, 0]. Hence, 1 (Γ ), we get the optimal error estimate if g is suﬃciently smooth, i.e., g ∈ Hpw 1 (Γ ) . g − gh H −1 (Γ ) ≤ c h2 |g|Hpw

Then,

(2.30)

76

2 Boundary Element Methods

fi =

1 2

1 = 2 =

=

I − K gh , ϕi

N

g

=1

ϕi (x)dsx −

g τ

N =1

τ

=1

N

Γ

ϕi (x)dsx −

g

=1

N 1

2

N =1

g

ϕi (x) Γ

1 g 4π

τ

ϕi (x) Γ

1 Mh [, i] − Kh [, i] . 2

int u∗ (x, y)ds ds γ1,x y x

τ

(y − x, n(x)) dsy dsx |x − y|3

Instead of the linear system (2.26), we now have to solve a linear system with a perturbed right hand side f yielding a perturbed solution vector u α , i.e., we have to solve the linear system 1 Mh − Kh g + αa . Dh + a a u α = (2.31) 2 For the associated boundary element solution u α,h ∈ Sh1 (Γ ), the error estimate ¯ uα − u α,h H 1/2 (Γ ) ≤ ¯ uα − u ¯α,h H 1/2 (Γ ) + c g − gh H −1/2 (Γ ) 2 (Γ ) + |g|H 1 (Γ ) uα |Hpw , ≤ c h3/2 |¯ pw 2 1 holds, when assuming u ¯ ∈ Hpw (Γ ) and g ∈ Hpw (Γ ). Applying the Aubin– Nitsche trick to obtain an error estimate in lower order Sobolev spaces, the restriction due to the error estimate (2.30) has to be considered. Hence, we obtain the error estimate

¯ u−u h H σ (Γ ) ≤ c1 ¯ u−u ¯h H σ (Γ ) + c2 g − gh H σ−1 (Γ ) 2 (Γ ) + |g|H 1 (Γ ) u|Hpw , ≤ c h2−σ |¯ pw 2 1 when assuming u ¯α ∈ Hpw (Γ ), g ∈ Hpw (Γ ), and σ ≥ 0. Therefore, the optimal error estimate reads 2 (Γ ) + |g|H 1 (Γ ) uα |Hpw . (2.32) α,h L2 (Γ ) ≤ c h2 |¯ ¯ uα − u pw

For the approximate representation formula int ∗ int u∗ (x, y) γ0,y u (x, y)gh (y)dsy − γ1,y uα,h (y)dsy u (x) = Γ

(2.33)

Γ

for x ∈ Ω, we then obtain the best possible error estimate |u(x) − u (x)| ≤ c(x, g, u ¯ α ) h2 , 2 1 when assuming u ¯α ∈ Hpw (Γ ) and g ∈ Hpw (Γ ).

(2.34)

2.3 Laplace Equation

77

2.3.3 Mixed Boundary Value Problem The solution of the mixed boundary value problem (cf. (1.34)) −Δu(x) = 0

for x ∈ Ω,

γ0int u(x) = g(x) γ int u(x) = f (x)

for x ∈ ΓD ,

1

for x ∈ ΓN

is given by the representation formula u∗ (x, y)γ1int u(y)dsy + u∗ (x, y)f (y)dsy u(x) = ΓD

−

ΓN

int u∗ (x, y)g(y)ds − γ1,y y

ΓD

int u∗ (x, y)γ int u(y)ds γ1,y y 0

ΓN

for x ∈ Ω, where we have to ﬁnd the yet unknown Cauchy data γ0int u on ΓN and γ1int u on ΓD . Let g ∈ H 1/2 (Γ ) and f ∈ H −1/2 (Γ ) be some arbitrary, but ﬁxed extensions of the given boundary data g ∈ H 1/2 (ΓD ) and f ∈ H −1/2 (ΓN ), respectively. The new Cauchy data 1/2 (ΓN ) u = γ0int u − g ∈ H and

−1/2 (Γ ) t = γ1int u − f ∈ H

are the unique solutions of the variational problem (cf. (1.35)) a( t, u ; w, v) = F (w, v) −1/2 (ΓD ) with the bilinear form 1/2 (ΓN ) and w ∈ H for all v ∈ H 1 1 t(y)dsy dsx w(x) a(t, u ; w, v) = 4π |x − y| ΓD ΓD 1 (x − y, n(y)) − w(x) u (y)dsy dsx 4π |x − y|3 ΓD ΓN 1 (y − x, n(x)) t(y)dsy dsx + v(x) 4π |x − y|3 ΓN ΓD curlΓ v(x) , curlΓ u (y) 1 dsy dsx + 4π |x − y| Γ Γ

and with the linear form

78

2 Boundary Element Methods

(x − y, n(y)) g(y)dsy dsx |x − y|3 ΓD ΓD Γ 1 1 1 − w(x) v(x)f (x)dsx f (y)dsy dsx + 4π |x − y| 2 ΓD Γ ΓN 1 (y − x, n(x)) − v(x) f (y)dsy dsx 4π |x − y| ΓN Γ curlΓ v(x) , curlΓ g(y) 1 dsy dsx . − 4π |x − y|

F (w, v) =

1 2

w(x)g(x)dsx +

1 4π

w(x)

ΓN Γ

To be able to deﬁne approximate solutions of the above variational problem, we ﬁrst deﬁne suitable trial spaces, ND −1/2 (ΓD ) = span ψ , Sh0 (ΓD ) = Sh0 (Γ ) ∩ H =1

MN 1/2 (ΓN ) = span ϕj Sh1 (ΓN ) = Sh1 (Γ ) ∩ H . j=1

The Galerkin formulation of the variational problem (1.35) is to ﬁnd th ∈ Sh0 (ΓD ) and u h ∈ Sh1 (ΓN ) such that

h ; wh , vh ) = F (wh , vh ) a( th , u

(2.35)

is satisﬁed for all wh ∈ and vh ∈ This formulation is equivalent to a linear system of equations g Vh −Kh t = (2.36) f Kh Dh u Sh0 (ΓD )

Sh1 (ΓN ).

with the following blocks: Vh ∈ RND ×ND ,

Kh ∈ RND ×MN ,

Dh ∈ RMN ×MN .

The matrix entries of these blocks are deﬁned by 1 1 dsy dsx , Vh [k, ] = 4π |x − y| τk τ 1 (x − y, n(y)) Kh [k, j] = ϕj (y)dsy dsx , 4π |x − y|3 τk Γ 1 (curlΓ ϕj (y), curlΓ ϕi (x)) Dh [i, j] = dsy dsx 4π |x − y| Γ Γ

2.3 Laplace Equation

79

for all k, = 1, . . . , ND and i, j = 1, . . . , MN . The components of the right hand side, g ∈ RND and f ∈ RMN , are given by gk =

1 2

g(x)dsx + τk

1 4π

τ

Γ

(x − y, n(y)) g(y)dsy dsx |x − y|3

k 1 1 f (y)dsy dsx , − 4π |x − y| τk Γ 1 1 (y − x, n(x)) fi = ϕi (x)f (x)dsx − ϕi (x) f (y)dsy dsx 2 4π |x − y| ΓN ΓN Γ 1 (curlΓ g(y), curlΓ ϕi (x)) dsy dsx − 4π |x − y|

ΓN Γ

for all k = 1, . . . , ND and i = 1, . . . , MN . Since the trial spaces Sh0 (ΓD ) ⊂ Sh0 (Γ ) and Sh1 (ΓN ) ⊂ Sh1 (Γ ) are subspaces of the trial spaces already used for the Dirichlet and for the Neumann boundary value problems, the blocks of the matrix in (2.36) are submatrices of the stiﬀness matrices already used in (2.22) and in (2.31), respectively. In particular, the evaluation of the discrete hypersingular integral operator Dh can be reduced to the evaluation of some linear combinations of the matrix entries of the discrete single layer potential Vh . Since the stiﬀness matrix in (2.36) is positive deﬁnite but block skew symmetric, we have to apply a generalised Krylov subspace method such as the Generalised Minimal Residual Method (GMRES) (see Appendix C.3) to solve (2.36) by an iterative method. Here we will describe two alternative approaches to apply the conjugate gradient scheme to solve (2.36). Since the discrete single layer potential Vh is symmetric and positive deﬁnite and hence invertible, we can solve the ﬁrst equation in (2.36) to ﬁnd . t = Vh−1 g + Kh u Inserting this into the second equation in (2.36), this gives the Schur complement system = f − Kh Vh−1 g (2.37) Sh u with the symmetric and positive deﬁnite Schur complement matrix Sh = Dh + Kh Vh−1 Kh . Therefore, we can apply a conjugate gradient scheme to solve (2.37), where we eventually need a suitable preconditioning matrix for Sh . Note that the matrix by vector multiplication with the Schur complement matrix Sh involves one application of the inverse single layer potential matrix Vh . This can be realised either by a direct inversion, if the dimension ND is small, or

80

2 Boundary Element Methods

by the application of an inner conjugate gradient scheme. Again, a suitable preconditioning matrix is eventually needed, which is spectrally equivalent to Vh . Following [14], we can also apply a suitable transformation to (2.36) to obtain a linear system with a symmetric, positive deﬁnite matrix. In particular, the transformed matrix Vh −Kh Vh CV−1 − I 0 = Kh Dh −Kh CV−1 I Vh CV−1 Vh − Vh (I − Vh CV−1 )Kh Kh (I − CV−1 Vh ) Dh + Kh CV−1 Kh

is symmetric and positive deﬁnite. Hence, instead of (2.36), we now solve the transformed linear system Vh CV−1 Vh − Vh (I − Vh CV−1 )Kh t = (2.38) −1 −1 u Kh (I − CV Vh ) Dh + Kh CV Kh g Vh CV−1 − I 0 −1 f −Kh CV I by a preconditioned conjugate gradient scheme. In the above, CV is a suitable preconditioning matrix, which is spectrally equivalent to the discrete single layer potential Vh , i.e., c1 (CV w, w) ≤ (Vh w, w) ≤ c2 (CV w, w)

for all w ∈ RND .

To ensure that (2.38) is equivalent to (2.36), we have to require the invertibility of Vh CV−1 − I = (Vh − CV )CV−1 . Due to ((Vh − CV )w, w) ≥ (c1 − 1)(CV w, w)

for all w ∈ RND ,

a suﬃcient condition is c1 > 1, which ensures the positive deﬁniteness of Vh − CV , and, therefore, its invertibility. A suitable preconditioning matrix for (2.38) is V h − CV 0 , CM = 0 CS where CS is a preconditioning matrix for the Schur complement Sh . 1/2 (ΓN )–ellipticity of the underlying bilinear form −1/2 (ΓD )×H From the H a(·, · ; ·, ·), we conclude the unique solvability of the Galerkin variational problem (2.35), and, therefore, of the linear system (2.36). In particular, we obtain the quasi optimal error estimate

2.3 Laplace Equation

t− th 2H −1/2 (Γ ) + u−u h 2H 1/2 (Γ ) ≤ c inf t − wh 2H −1/2 (Γ ) + 0 wh ∈Sh (ΓD )

81

inf

1 (Γ ) vh ∈Sh N

u−

vh 2H 1/2 (Γ )

from Cea’s lemma. Using the approximation property (2.5) for σ = −1/2 as well as the approximation property (2.10) for σ = 1/2, this gives s1 s2 u−u h 2H 1/2 (Γ ) ≤ c1 h2s1 +1 | t|2Hpw + c2 h2s2 −1 | u|2Hpw , t− th 2H −1/2 (Γ ) + (Γ ) (Γ )

s1 s2 when assuming t ∈ Hpw (Γ ) for some s1 ∈ [−1/2, 1], and u ∈ Hpw (Γ ) for some s2 ∈ [1/2, 2]. Since, in general, those regularity estimates result from a regularity estimate for the solution u ∈ H s (Ω) of the mixed boundary value s−1/2 s−3/2 problem (1.34), we obtain γ0int u ∈ Hpw (Γ ) and γ1int u ∈ Hpw (Γ ) by applying the trace theorems, and, therefore, s1 = s − 3/2 and s2 = s − 1/2. Thus, if u ∈ H s (Ω) is the solution of the mixed boundary value problem (1.34) for some s ∈ [1, 5/2], we then obtain the error estimate

u−u h 2H 1/2 (Γ ) ≤ c h2(s−1) |u|2H s (Ω) . t− th 2H −1/2 (Γ ) + As for the Dirichlet and for the Neumann boundary value problem, applying the Aubin–Nitsche trick (for σ ∈ [−2, 1/2)) and an inverse inequality argument (for σ ∈ (−1/2, 0]), we obtain the error estimate u−u h 2H σ+1 (Γ ) ≤ c h2(s−σ)−3 |u|2H s (Ω) , t− th 2H σ (Γ ) +

(2.39)

when assuming u ∈ H s (Ω) for some s ∈ [1, 5/2] and σ ∈ [−2, 0]. Inserting the computed Galerkin solutions th ∈ Sh0 (ΓD ) and u h ∈ Sh1 (ΓN ) into the representation formula (1.6), this gives an approximate representation formula ∗ int u∗ (x, y) u h (y) + g(y) dsy u (x) = u (x, y) th (y) + f (y) dsy − γ1,y Γ

Γ

for x ∈ Ω. The above formula describes an approximate solution of the mixed boundary value problem (1.34). For an arbitrary x ∈ Ω, the error is given by u(x) − u (x) = int u∗ (x, y) u u∗ (x, y) t(y) − th (y) dsy − γ1,y (y) − u h (y) dsy . ΓN

ΓD

Using a duality argument, the error estimate |u(x) − u (x)| ≤

t− th H σ1 (Γ ) + γ1int u∗ (x, ·) u∗ (x, ·)H −σ1 (Γ )

H −σ2 (Γ )

u−u h H σ2 (Γ )

82

2 Boundary Element Methods

for some σ1 , σ2 ∈ R follows. Combining this with the error estimate (2.39) for the minimal values σ1 = −2 and σ2 = −1, we obtain the pointwise error estimate |u(x) − u (x)| ≤ c h2s+1 u∗ (x, ·)H 2 (Γ ) + γ1int u∗ (x, ·)H 1 (Γ ) |u|H s (Ω) , when assuming u ∈ H s (Ω) for some s ∈ [1, 5/2]. In particular, for s = 5/2, we obtain the optimal order of convergence, |u(x) − u (x)| ≤ c(x) h3 |u|H 5/2 (Ω) .

(2.40)

Note that the error estimate (2.40) is based on the exact use of the given boundary data g ∈ H 1/2 (ΓD ) and f ∈ H −1/2 (ΓN ), and their extensions g ∈ H 1/2 (Γ ) and f ∈ H −1/2 (Γ ). Starting from an approximation uh ∈ Sh1 (Γ ) of the complete Dirichlet datum γ0int u, uh =

M

u j ϕj =

j=1

MN

u j ϕj +

j=1

M

u j ϕj = u h + gh ,

j=MN +1

we ﬁrst have to ﬁnd the coeﬃcients uj for j = MN + 1, . . . , M of the approximate Dirichlet datum gh ∈ Sh1 (Γ ) ∩ H 1/2 (ΓN ). This can be done, e.g., by applying the L2 projection, M

uj

j=MN +1

ϕj (x)ϕi (x)dx =

ΓD

g(x)ϕi (x)dsx

for i = MN + 1, . . . , M.

ΓD

In a similar way, we obtain an approximation fh ∈ Sh0 (ΓN ) of the given Neumann datum f ∈ H −1/2 (ΓN ), N =ND +1

t

ψ (x)ψk (x)dx =

ΓN

f (x)ψk (x)dsx

for k = ND + 1, . . . , N.

ΓN

Hence, we have to ﬁnd the remaining Cauchy data th ∈ Sh0 (ΓD )

and u h ∈ Sh1 (ΓN )

from the variational problem h ; ψk , ϕi ) = F(ψk , ϕi ) a( th , u for k = 1, . . . , ND and i = 1, . . . , MN , where the perturbed linear form is now given by

2.3 Laplace Equation

83

(x − y, n(y)) gh (y)dsy dsx |x − y|3 τk τk Γ 1 1 1 fh (y)dsy dsx + − fh (x)ϕi (x)dsx 4π |x − y| 2 τk ΓN ΓN 1 (y − x, n(x)) − ϕi (x) fh (y)dsy dsx 4π |x − y|3 ΓN ΓN curlΓ ϕi (x) , curlΓ gh (y) 1 dsy dsx . − 4π |x − y|

1 F(ψk , ϕi ) = 2

gh (x)dsx +

1 4π

ΓN Γ

The above perturbed variational problem is now equivalent to a linear system of equations 1 ¯ ¯h f Vh −Kh + K M −V¯h t h 2 = 1 ¯ . (2.41) ¯ ¯ g Kh Dh −Dh u 2 Mh − K h Note that the right hand side of this system diﬀers from the one in (2.36). The blocks on the right have the following dimensions: V¯h ∈ RND ×(N −ND ) ,

¯ h ∈ RND ×(M −MN ) , M

¯ h ∈ RND ×(M −MN ) K

and the following entries 1 1 ¯ Vh [k, ] = dsy dsx , 4π |x − y| τk τ ¯ h [k, j] = ϕj (x)dsx , M τk

(x − y, n(y)) ϕj (y)dsy dsx , |x − y|3 τk Γ curlΓ ϕj (y) , curlΓ ϕi (x) 1 ¯ dsy dsx Dh [i, j] = 4π |x − y|

¯ h [k, j] = 1 K 4π

Γ Γ

for = ND + 1, . . . , N , k = 1, . . . , ND , j = MN + 1, . . . , M , i = 1, . . . , MN . ¯ h, K ¯ h , and D ¯ h are also submatrices of the Note that the matrices V¯h , M stiﬀness matrices already used in (2.16) and (2.26) to handle the Dirichlet and Neumann boundary value problem, respectively. The solution of the perturbed linear system (2.41) can be realised as for the linear system (2.36). The error estimates for the resulting approximations

84

2 Boundary Element Methods

can be obtained as in the previous cases, however, the approximations of the given boundary data have to be recognised accordingly. This can be done as for the Dirichlet boundary value problem and as for the Neumann boundary value problem. In particular, the error estimate (2.39) holds for σ ∈ [−1, 0], and instead of (2.40), we obtain only the pointwise error estimate |u(x) − u (x)| ≤ c(x) h2 |u|H 5/2 (Ω)

(2.42)

for x ∈ Ω, when assuming u ∈ H 5/2 (Ω). 2.3.4 Interface Problem We consider the interface problem (1.56)–(1.58), i.e., the system of partial diﬀerential equations (1.56), −αi Δui (x) = f (x) for x ∈ Ω,

−αe Δue (x) = 0 for x ∈ Ω e ,

the transmission conditions (1.57), γ0int ui (x) = γ0ext ue (x),

αi γ1int ui (x) = αe γ1ext ue (x)

for x ∈ Γ,

and the radiation condition (1.58) with u0 = 0, 1 as |x| → ∞ . |ue (x)| = O |x| Introducing u ¯ = γ0int ui = γ0ext ue ∈ H 1/2 (Γ ), we have to solve the resulting variational problem (1.59), u, v Γ = S int γ0int up − γ1int up , v Γ (αi S int + αe S ext )¯ for all v ∈ H 1/2 (Γ ), where up is a particular solution satisfying −Δup = f in Ω. Using a sequence of ﬁnite dimensional subspaces Sh1 (Γ ) ⊂ H 1/2 (Γ ) spanned by piecewise linear and continuous basis functions, an associated approximate solution M u ¯h = u ¯j ϕj ∈ Sh1 (Γ ) j=1

can be found as the unique solution of the Galerkin equations uh , ϕi Γ = S int γ0int up − γ1int up , ϕi Γ (αi S int + αe S ext )¯ for i = 1, . . . , M . This is equivalent to a system of linear equations, ¯ = f, Sh u

(2.43)

2.3 Laplace Equation

85

with Sh ∈ RM ×M and f ∈ RM with the entries Sh [i, j] = (αi S int + αe S ext )ϕj , ϕi Γ , fi = S int γ0int up − γ1int up , ϕi Γ for i, j = 1, . . . , M . Since the Steklov–Poincar´e operators 1 ¯(x) (S int u ¯)(x) = V −1 I + K u 2 1 1 I + K V −1 I + K u ¯(x), = D+ 2 2 1 (S ext u ¯(x) ¯)(x) = V −1 − I + K u 2 1 1 u ¯(x) = D + − I + K V −1 − I + K 2 2 do not allow a direct evaluation of both, the stiﬀness matrix and the right hand side, additional approximations are required. The application of the Steklov– Poincar´e operator S int related to the interior Dirichlet boundary value problem can be written as 1 1 I + K V −1 I + K u ¯(x) ¯)(x) = D + (S int u 2 2 1 = (D¯ u)(x) + I + K ti (x) , 2 where ti = V −1

1

I +K u ¯ ∈ H −1/2 (Γ )

2 is the unique solution of the variational problem V ti , w Γ =

1 2

I +K u ¯, w

for all w ∈ H −1/2 (Γ ).

Γ

Let ti,h ∈ Sh0 (Γ ) be the unique solution of the Galerkin variational problem V ti,h , wh

Γ

=

1 2

I +K u ¯ , wh

Γ

for all wh ∈ Sh0 (Γ ).

Then, (Sint u ¯)(x) = (D¯ u)(x) +

1 2

I + K ti,h (x)

deﬁnes an approximate Steklov–Poincar´e operator associated to the interior Dirichlet boundary value problem. In the same way, we deﬁne an approximate Steklov–Poincar´e operator

86

2 Boundary Element Methods

1 − I + K te,h (x) , 2 which is associated to the exterior Dirichlet boundary value problem, and where te,h ∈ Sh0 (Γ ) is the unique solution of the Galerkin equations 1 V te,h , wh Γ = ¯ , wh − I +K u for all wh ∈ Sh0 (Γ ). 2 Γ (Sext u ¯)(x) = (D¯ u)(x) +

Now, instead of the variational problem (2.43), we consider the perturbed problem (αi Sint + αe Sext ) uh , ϕi Γ = Sint up,h − tp,h , ϕi Γ

(2.44)

and up,h ∈ are suitable for i = 1, . . . , M . In (2.44), tp,h ∈ approximations (L2 projections) of the Cauchy data of the particular solution up , i.e., tp,h , ψk L2 (Γ ) = γ1int up , ψk L2 (Γ ) Sh0 (Γ )

Sh1 (Γ )

for k = 1, . . . , N and up,h , ϕi L2 (Γ ) = γ0int up , ϕi L2 (Γ ) for i = 1, . . . , M . From (2.44), we then obtain the linear system 1 −1 1 αi Dh + M + Kh V h Mh + Kh 2 h 2 1 1 −1 +αe Dh + − Mh + Kh Vh − Mh + Kh u = 2 2 1 1 Dh + Mh + Kh Vh−1 Mh + Kh up − Mh tp , 2 2

(2.45)

where Vh ∈ RN ×N ,

Mh ∈ RN ×M ,

Kh ∈ RN ×M ,

Dh ∈ RM ×M

are the Galerkin stiﬀness matrices, which have already been used for the Dirichlet and for the Neumann boundary value problems. The entries of these matrices are deﬁned as 1 1 dsy dsx , Vh [k, ] = 4π |x − y| τk τ Mh [k, j] = ϕj (x)dsx , τk

1 Kh [k, j] = 4π

τk Γ

Dh [i, j] =

1 4π

Γ Γ

(x − y, n(y)) ϕj (y)dsy dsx , |x − y|3 (curlΓ ϕj (y), curlΓ ϕi (x)) dsy dsx |x − y|

2.4 Lame Equations

87

for k, = 1, . . . , N and i, j = 1, . . . , M . Instead of the linear system (2.45) we may also solve the equivalent coupled system ⎞⎛ ⎞ ⎛ 0 −αi ( 12 Mh + Kh ) α i Vh ti ⎟⎜ ⎟ ⎜ 1 −αe (− 2 Mh + Kh ) ⎠ ⎝ te ⎠ 0 αe Vh ⎝ 1 1 u αi ( 2 Mh + Kh ) αe (− 2 Mh + Kh ) (αi + αe )Dh ⎞ −( 12 Mh + Kh )up ⎟ ⎜ 0 = ⎝ ⎠, Dh up − Mh tp ⎛

(2.46)

which is of the same structure as the linear system (2.36), i.e. block skew symmetric but positive deﬁnite. Note that (2.45) is the Schur complement system of (2.46). As for the Neumann boundary value problem, we conclude the error estimate ¯ u−u h H 1/2 (Γ ) ≤ c1 +c2

inf

0 (Γ ) wh ∈Sh

inf

1 (Γ ) vh ∈Sh

¯ u − vh H 1/2 (Γ )

S int u ¯ − wh H −1/2 (Γ ) + c3

inf

0 (Γ ) wh ∈Sh

S ext u ¯ − wh H −1/2 (Γ ) .

2 1 Hence, assuming u ¯ ∈ Hpw (Γ ) and S int/ext u ¯ ∈ Hpw , (Γ ), we obtain the error estimate int u ext u 2 (Γ ) + S 1 (Γ ) + S 1 (Γ ) uHpw , ¯ u−u h H 1/2 (Γ ) ≤ c h3/2 ¯ ¯Hpw ¯Hpw

and by applying the Aubin–Nitsche trick, we get ¯ u−u h L2 (Γ ) ≤ c(¯ u) h2 . When the Dirichlet datum u ¯h is known, one can compute the remaining Neumann datum by solving both, the interior and exterior Dirichlet boundary value problems. Since those boundary value problems are Dirichlet boundary value problems with approximated boundary data, the corresponding error estimates are still valid.

2.4 Lame Equations For a simply connected domain Ω ⊂ R3 , we consider the mixed boundary value problem (1.79)

88

2 Boundary Element Methods

−

(γ1int u)i (x) =

3 ∂ σij (u, x) = 0 ∂xj j=1

3

for x ∈ Ω ,

γ0int ui (x) = gi (x)

for x ∈ ΓD,i ,

σij (u, x)nj (x) = fi (x)

for x ∈ ΓN,i ,

j=1

for i = 1, 2, 3. Note that we assume Γ = Γ N,i ∪ Γ D,i ,

ΓN,i ∩ ΓD,i = ∅ ,

meas ΓD,i > 0

for i = 1, 2, 3. To ﬁnd the yet unknown Cauchy data (γ1int u)i on ΓD,i and γ0int ui on ΓN,i , we consider the variational problem (1.80), which is related to the symmetric formulation of boundary integral equations. Hence, we have to ﬁnd −1/2 (ΓD,i ) ti = (γ1int u)i − fi ∈ H and

1/2 (ΓN,i ) u i = γ0int ui − gi ∈ H

such that

a( t, u ; w, v) = F (w, v)

−1/2 (ΓD,i ) and vi ∈ H 1/2 (ΓN,i ) for i = 1, 2, 3. Note is satisﬁed for all wi ∈ H that the bilinear form is given by ; w, v) = a( t, u

3

(V Lame t )i , wi

ΓD,i

i=1

+

3 ti , (K Lame v )i i=1

ΓN,i

−

3

(K Lame u )i , wi

ΓD,i

i=1

+

3

(DLame u )i , vi

i=1

, ΓN,i

while the linear form is F (w, v) = 3 1 Lame Lame gi , wi + + (K g )i , wi − (V f )i , wi 2 ΓD,i ΓD,i ΓD,i i=1 3 1 fi , vi . − fi , (K Lame v )i − (DLame g )i , vi 2 ΓN,i ΓN,i ΓN,i i=1 As for the Laplace equation, we ﬁrst deﬁne suitable trial spaces, ND,i −1/2 (ΓD,i ) = span ψi Sh0 (ΓD,i ) = Sh0 (Γ ) ∩ H , =1 MN,i 1/2 (ΓN,i ) = span ϕi Sh1 (ΓN,i ) = Sh1 (Γ ) ∩ H j j=1

2.4 Lame Equations

89

for i = 1, 2, 3. The Galerkin formulation of the variational problem (1.80) is i,h ∈ Sh1 (ΓN,i ) such that to ﬁnd ti,h ∈ Sh0 (ΓD,i ) and u a( th , u h ; wh , v h ) = F (wh , v h ) is satisﬁed for all wi ∈ Sh0 (ΓD,i ) and vi ∈ Sh1 (ΓM,i ) for i = 1, 2, 3. This formulation is equivalent to a linear system of equations ⎞ ⎛ ¯ Lame V¯hLame −K h ⎠ t = g , ⎝ (2.47) ¯ Lame ¯ Lame f u K D h

h

having the blocks V¯hLame ∈ RND ×ND , where ND =

¯ hLame ∈ RND ×MN , K 3

ND,i ,

MN =

i=1

3

¯ hLame ∈ RMN ×MN , D

MN,i .

i=1

While the blocks in the linear system (2.47) recover only the unknown coeﬃi,j , an implementation based on the complete stiﬀness matrices cients ti, and u may be advantageous. Let N M , Sh1 (Γ ) = span ϕj Sh0 (Γ ) = span ψ =1

j=1

be the boundary element spaces spanned by piecewise constant and piecewise linear continuous basis functions, respectively. Note that both Sh0 (Γ ) and Sh1 (Γ ) are deﬁned with respect to a boundary element mesh of the complete surface Γ . By Pi : RN → RND,i and Qi : RM → RMN,i , we denote some nodal projection operators describing the imbedding wi = Pi w ∈ RND,i for w ∈ RN with

ND,i

whi (x)

=

wi ψi (x) ∈ Sh0 (ΓD,i ),

wh (x) =

=1

N

w ψ (x) ∈ Sh0 (Γ )

=1

as well as the imbedding v i = Qi v ∈ RMN,i for v ∈ RM with

MN,i

vhi (x) =

vji ϕij (x) ∈ Sh1 (ΓN,i ),

vh (x) =

j=1

N

vj ϕj (x) ∈ Sh1 (Γ ).

j=1

From this we obtain the representations ¯ Lame = P K Lame Q , V¯hLame = P VhLame P , K h h

¯ Lame = QDLame Q , D h h

where the stiﬀness matrices VhLame , KhLame , and DhLame correspond to the Galerkin discretisation of the associated boundary integral operators V Lame ,

90

2 Boundary Element Methods

K Lame and DLame with respect to the boundary element spaces [Sh0 (Γ )]3 and [Sh1 (Γ )]3 . In particular, for the discrete single layer potential V¯h we have the representation VhLame =

⎛

⎛

⎞

⎛

⎞⎞

(2.48)

Vh 0 0 V11,h V21,h V13,h 1 1 1+ν ⎝ (3 − 4ν) ⎝ 0 Vh 0 ⎠ + ⎝ V21,h V22,h V23,h ⎠⎠ 2E 1−ν 0 0 Vh V31,h V32,h V33,h with the matrix Vh ∈ RN ×N having the entries 1 1 dsy dsx , Vh [k, ] = 4π |x − y|

(2.49)

τk τ

and six further matrices Vij,h ∈ RN ×N deﬁned by 1 (xi − yi )(xj − yj ) dsy dsx Vij,h [k, ] = 4π |x − y|3 τk τ 1 ∂ 1 = dsy dsx (xi − yi ) 4π ∂yj |x − y|

(2.50)

τk τ

for k, = 1, . . . , N and i, j = 1, 2, 3. Note that Vh is just the Galerkin stiﬀness matrix of the single layer potential for the Laplace operator, while the matrix entries Vij,h [, k] are similar to the Galerkin discretisation of the double layer potential for the Laplace operator. From Lemma 1.16, we ﬁnd the representation for the double layer potential K Lame E Lame V (K Lame v)(x) = (Kv)(x) − V M (∂, n)v (x) + M (∂, n)v (x) 1+ν for x ∈ Γ , and, therefore, the matrix representation KhLame = ⎛ ⎞ ⎛ ⎞ Kh 0 0 Vh 0 0 ⎝ 0 Kh 0 ⎠ − ⎝ 0 Vh 0 ⎠ T + E VhLame T , 1+ν 0 0 Kh 0 0 Vh

(2.51)

where Vh and Kh are the Galerkin matrices related to the single and double layer potential of the Laplace operator. Furthermore, T is a transformation matrix related to the matrix surface curl operator M (∂, n). Using the representation of the bilinear form of the hypersingular boundary integral operator DLame as given in Lemma 1.18, one can derive a similar representation for the Galerkin matrix DhLame , which is based on the transformation matrix T and on the Galerkin matrices related to the single layer potential of both, the Laplace operator and the system of linear elastostatics.

2.5 Helmholtz Equation

91

2.5 Helmholtz Equation 2.5.1 Interior Dirichlet Problem The solution of the interior Dirichlet boundary value problem (cf. (1.104)), γ0int u(x) = g(x)

−Δu(x) − κ2 u(x) = 0 for x ∈ Ω,

is given by the representation formula (cf. (1.95)) ∗ int u∗ (x, y)g(y)ds uκ (x, y)t(y)dsy − γ1,y u(x) = y κ Γ

for x ∈ Γ ,

for x ∈ Ω,

Γ

where the unknown Neumann datum t = γ1int u ∈ H −1/2 (Γ ) is the unique solution of the boundary integral equation (cf. (1.105)) (Vκ t)(x) =

1 g(x) + (Kκ g)(x) 2

for x ∈ Γ.

Note that for the unique solvability, we have to assume that κ2 is not an eigenvalue of the Dirichlet eigenvalue problem (1.108). Then, t ∈ H −1/2 (Γ ) is the unique solution of the variational problem (cf. (1.106))

Vκ t, w

= Γ

1 2

I + Kκ g, w

for all w ∈ H −1/2 (Γ ).

Γ

Using a sequence of ﬁnite dimensional subspaces Sh0 (Γ ) spanned by piecewise constant basis functions, associated approximate solutions th =

N

t ψ ∈ Sh0 (Γ )

=1

are obtained from the Galerkin equations

Vκ th , ψk

= Γ

1 2

I + Kκ g, ψk

for k = 1, . . . , N.

(2.52)

Γ

Hence, we ﬁnd the coeﬃcient vector t ∈ CN as the unique solution of the linear system Vκ,h t = f with 1 Vκ,h [k, ] = 4π

τk τ

for k, = 1, . . . , N , and

eıκ|x−y| dsy dsx , |x − y|

(2.53)

92

fk =

2 Boundary Element Methods

1 2

g(x)dsx + τk

1 4π

τk Γ

(x − y, n(y)) 1 − ı κ |x − y| eı κ|x−y| g(y)dsy dsx |x − y|3

for k = 1, . . . , N . Since the single layer potential Vκ : H −1/2 (Γ ) → H 1/2 (Γ ) is coercive, i.e. Vκ satisﬁes (1.97), and since Vκ is injective when κ2 is not an eigenvalue of the Dirichlet eigenvalue problem (1.108), we conclude the unique solvability of the Galerkin variational problem (2.52), as well as the quasi optimal error estimate, i.e. Cea’s lemma, t − th H −1/2 (Γ ) ≤ c

inf

0 (Γ ) wh ∈Sh

t − wh H −1/2 (Γ ) .

Combining this with the approximation property (2.5) for σ = −1/2, we get 1

s (Γ ) , t − th H −1/2 (Γ ) ≤ c hs+ 2 |t|Hpw

s when assuming t ∈ Hpw (Γ ) and s ∈ [0, 1]. Applying the Aubin–Nitsche trick (for σ < −1/2) and the inverse inequality argument (for σ ∈ (−1/2, 0]), we also obtain the error estimate s (Γ ) , t − th H σ (Γ ) ≤ c hs−σ |t|Hpw

(2.54)

s (Γ ) for some s ∈ [0, 1] and σ ∈ [−2, 0]. when assuming t ∈ Hpw Inserting the computed Galerkin solution th ∈ Sh0 (Γ ) into the representation formula (1.95), this gives an approximate representation formula γ0int u∗κ (x, y)th (y)dsy − γ1int u∗κ (x, y)g(y)dsy , (2.55) u (x) = Γ

Γ

for x ∈ Ω, describing an approximate solution of the Dirichlet boundary value problem (1.104). Note that u satisﬁes the Helmholtz equation, but the Dirichlet boundary conditions are satisﬁed only approximately. For an arbitrary x ∈ Ω, the error is given by u(x) − u (x) = u∗κ (x, y) t(y) − th (y) dsy . Γ

Using a duality argument, the error estimate |u(x) − u (x)| ≤ u∗κ (x, ·)H −σ (Γ ) t − th H σ (Γ ) for some σ ∈ R follows. Combining this with the error estimate (2.54) for the minimal value σ = −2, we obtain the pointwise error estimate s (Γ ) . |u(x) − u (x)| ≤ c hs+2 u∗κ (x, ·)H 2 (Γ ) |t|Hpw

2.5 Helmholtz Equation

93

1 Hence, if t ∈ Hpw (Γ ) is suﬃciently smooth, we obtain the optimal order of convergence for s = 1, 1 (Γ ) . |u(x) − u (x)| ≤ c h3 u∗κ (x, ·)H 2 (Γ ) |t|Hpw

(2.56)

Again, the error estimate (2.56) involves the position of the observation point x ∈ Ω, and, therefore, it is not valid in the limiting case x ∈ Γ . As for the Dirichlet problem for the Laplace equation, the computation of fk requires the evaluation of the integrals 1 1 (x − y, n(y)) 1 − ı κ |x − y| eı κ|x−y| fk = g(x)dsx + g(y)dsy dsx . 2 4π |x − y|3 τk

τk Γ

When using a piecewise linear approximation gh ∈ Sh1 (Γ ) of the given Dirichlet t ∈ CN from the datum g ∈ H 1/2 (Γ ), we ﬁnd a perturbed solution vector linear system 1 Mh + Kκ,h g t = (2.57) Vκ,h 2 with additional matrices deﬁned by the entries Mh [k, j] = ϕj (x)dsx , τk

and 1 Kκ,h [k, j] = 4π

τk Γ

(x − y, n(y)) 1 − ı κ |x − y| eı κ|x−y| ϕj (y)dsy dsx (2.58) |x − y|3

for k = 1, . . . , N and j = 1, . . . , M . Then, the exact Galerkin solution th has to be replaced by the perturbed solution th to obtain an approximate solution of the Dirichlet problem (1.104) for x ∈ Ω, u (x) = u∗κ (x, y) th (y)dsy − γ1int u∗κ (x, y)gh (y)dsy . Γ

Γ

Thus, we obtain the optimal error estimate |u(x) − u (x)| ≤ c(x, t, g) h3 ,

(2.59)

when using a L2 projection to approximate the boundary conditions, and 1 2 (Γ ) and g ∈ Hpw (Γ ). when assuming t ∈ Hpw 2.5.2 Interior Neumann Problem Next we consider the interior Neumann boundary value problem (1.109),

94

2 Boundary Element Methods

−Δu(x) − κ2 u(x) = 0 for x ∈ Ω,

γ1int u(x) = g(x)

for x ∈ Γ.

The solution is given by the representation formula for x ∈ Ω (cf. (1.95)) ∗ int u∗ (x, y)γ int u(y)ds . uκ (x, y)g(y)dsy − γ1,y u(x) = y 0 κ Γ

Γ

We assume that κ2 is not an eigenvalue of the Neumann eigenvalue problem (1.113). In this case, the unknown Dirichlet datum u ˆ = γ0int u ∈ H 1/2 (Γ ) is the unique solution of the boundary integral equation (1.111), ˆ)(x) = (Dκ u

1 g(x) − (Kκ g)(x) 2

for x ∈ Γ ,

or of the equivalent variational problem (1.112), 1 I − Kκ g, v Dκ u ˆ, v = for all v ∈ H 1/2 (Γ ) . 2 Γ Γ Using a sequence of ﬁnite dimensional subspaces Sh1 (Γ ) spanned by piecewise linear continuous basis functions, associated approximate solutions u ˆh =

M

u ˆj ϕj ∈ Sh1 (Γ )

j=1

are obtained from the Galerkin equations 1 I − Kκ g, ϕi Dκ u ˆh , ϕi = 2 Γ Γ

for i = 1, . . . , M.

(2.60)

Hence, we ﬁnd the coeﬃcient vector u ˆ ∈ CM as the unique solution of the linear system ˆ = f (2.61) Dκ,h u with Dκ,h [i, j] = Dκ ϕj , ϕi Γ ıκ|x−y| 1 e (curlΓ ϕj (y), curlΓ ϕi (x))dsy dsx = 4π |x − y|

(2.62)

Γ Γ

κ2 − 4π

Γ Γ

eıκ|x−y| ϕj (y)ϕi (x)(n(x), n(y))dsy dsx , |x − y|

for i, j = 1, . . . , M , and 1 fi = g(x)ϕi (x)dsx 2 Γ 1 (x − y, n(y)) 1 − ı κ |x − y| eı κ|x−y| − ϕi (x) g(y)dsy dsx 4π |x − y|3 Γ

Γ

2.5 Helmholtz Equation

95

for i = 1, . . . , M . Note that for the computation of the matrix entries Dκ,h [i, j], we can reuse the discrete single layer potential Vκ,h for picewise constant basis functions, but we also need to have the Galerkin discretisation with piecewise linear continuous basis functions of the operator ıκ|x−y| e (n(x), n(y)) u(y)dsy , (2.63) (Cκ u)(x) = |x − y| Γ

which is similar to the single layer potential operator. Since the hypersingular integral operator Dκ : H 1/2 (Γ ) → H −1/2 (Γ ) is coercive, i.e. Dκ satisﬁes (1.98), and since Dκ is injective when κ2 is not an eigenvalue of the Neumann eigenvalue problem (1.113), we conclude the unique solvability of the Galerkin variational problem (2.60), as well as the quasi optimal error estimate, i.e. Cea’s lemma, ¯ u−u ¯h H 1/2 (Γ ) ≤ c

inf

1 (Γ ) vh ∈Sh

¯ u − vh H 1/2 (Γ ) .

Combining this with the approximation property (2.10) for σ = 1/2, we get 1

s (Γ ) , ¯ u−u ¯h H 1/2 (Γ ) ≤ c hs− 2 ¯ uHpw

s when assuming u ¯ ∈ Hpw (Γ ) and s ∈ [1, 2]. Applying the Aubin–Nitsche trick we also obtain the error estimate s (Γ ) , ¯ u−u ¯h H σ (Γ ) ≤ c hs−σ ¯ uHpw

(2.64)

s (Γ ) for some s ∈ [1, 2] and σ ∈ [−1, 1/2]. when assuming u ¯ ∈ Hpw Inserting the computed Galerkin solution u ˆh ∈ Sh1 (Γ ) into the representation formula (1.95), this gives an approximate representation formula for x ∈ Ω, int u∗ (x, y)ˆ u (x) = u∗κ (x, y)g(y)dsy − γ1,y uh (y)dsy , (2.65) κ Γ

Γ

describing an approximate solution of the Neumann boundary value problem (1.109). Note that u satisﬁes the Helmholtz equation, but the Neumann boundary conditions are satisﬁed only approximately. For an arbitrary x ∈ Ω, the error is given by int u∗ (x, y) u γ1,y ˆh (y) − u ˆ(y) dsy . u(x) − u (x) = κ Γ

Using a duality argument, the error estimate

96

2 Boundary Element Methods

|u(x) − u (x)| ≤ u∗κ (x, ·)H −σ (Γ ) ¯ u−u ¯h H σ (Γ ) for some σ ∈ R follows. Combining this with the error estimate (2.64) for the minimal value σ = −1, we obtain the pointwise error estimate s (Γ ) . u|Hpw |u(x) − u (x)| ≤ c hs+1 u∗κ (x, ·)H 1 (Γ ) |ˆ

2 Hence, if u ˆ ∈ Hpw (Γ ) is suﬃciently smooth, we get the optimal order of convergence for s = 2, 2 (Γ ) . u|Hpw |u(x) − u (x)| ≤ c h3 u∗κ (x, ·)H 1 (Γ ) |¯

(2.66)

Again, the error estimate (2.66) involves the position of the observation point x ∈ Ω, and, therefore, is not valid in the limiting case x ∈ Γ . When using a piecewise constant approximation gh ∈ Sh0 (Γ ) of the given Neumann datum g ∈ H −1/2 (Γ ), we can compute a perturbed piecewise linear approximation u h ∈ Sh1 (Γ ) from the Galerkin equations

ˆh , ϕi Dκ u

=

1 2

Γ

I − Kκ gh , ϕi

for i = 1, . . . , M

Γ

or from the equivalent linear system Dκ,h u = with Mh [i, ]

2

Mh − Kκ,h g

=

ϕi (x)dsx = Mh [, i], τ

Kκ,h [i, ]

1

1 = 4π

ϕi (x)

Γ

τ

(x − y, n(y)) 1 − ı κ |x − y| eı κ|x−y| dsy dsx . |x − y|3

An approximate solution of the interior Neumann boundary value problem is then given for x ∈ Ω, ∗ int u∗ (x, y) u (x) = uκ (x, y)gh (y)dsy − γ1,y uh (y)dsy . κ Γ

Γ

As for the perturbed linear system (2.31) for the Neumann boundary value problem of the Laplace equation, we obtain the error estimate |u(x) − u (x)| ≤ c(x, t, g) h2 ,

(2.67)

when using a L2 projection to approximate the boundary conditions, when 1 2 (Γ ) and u ¯ ∈ Hpw (Γ ). assuming g ∈ Hpw

2.5 Helmholtz Equation

97

2.5.3 Exterior Dirichlet Problem The solution of the exterior Dirichlet boundary value problem (cf. (1.114)) −Δu(x) − κ2 u(x) = 0 for x ∈ Ω e ,

γ0ext u(x) = g(x)

for x ∈ Γ,

where, in addition, we have to require the Sommerfeld radiation condition (1.101), is given by the representation formula for x ∈ Ω e (cf. 1.103) ext u∗ (x, y)g(y)ds . u(x) = − u∗κ (x, y)t(y)dsy + γ1,y y κ Γ

Γ

Again we assume that κ2 is not an eigenvalue of the Dirichlet eigenvalue problem (1.108). The unknown Neumann datum t = γ1ext ∈ H −1/2 (Γ ) is then the unique solution of the boundary integral equation (cf. (1.115)) 1 (Vκ t)(x) = − g(x) + (Kκ g)(x) 2

for x ∈ Γ.

To compute an approximate solution of this boundary integral equation, and, therefore, of the exterior Dirichlet problem, we can proceed as in the case of the interior Dirichlet problem. In particular, when using a piecewise linear approximation gh ∈ Sh1 (Γ ), we ﬁnd a perturbed piecewise constant approximation th ∈ Sh0 (Γ ) from the Galerkin equations Vκ th , ψk

Γ

=

1 − I + Kκ gh , ψk 2 Γ

for k = 1, . . . , N .

Hence, we obtain the coeﬃcient vector t ∈ CN as the unique solution of the linear system 1 t = − Mh + Kκ,h g, Vκ,h 2 and an approximate solution of the exterior Dirichlet problem for x ∈ Ω, ext u∗ (x, y)g (y)ds . u (x) = − u∗κ (x, y) th (y)dsy + γ1,y (2.68) h y κ Γ

Γ

Moreover, as for the interior Dirichlet problem, there holds the optimal error estimate (2.69) |u(x) − u (x)| ≤ c(x, t, g) h3 , when using a L2 projection to approximate the boundary conditions, and 1 2 when assuming t ∈ Hpw (Γ ) and g ∈ Hpw (Γ ).

98

2 Boundary Element Methods

2.5.4 Exterior Neumann Problem The solution of the exterior Neumann boundary value problem (cf. (1.120)) −Δu(x) − κ2 u(x) = 0

for x ∈ Ω e ,

γ1ext u(x) = g(x)

for x ∈ Γ,

where, in addition, we have to require the Sommerfeld radiation condition (1.101), is given by the representation formula for x ∈ Ω c (cf. (1.95)) ext u∗ (x, y)γ ext u(y)ds . u(x) = − u∗κ (x, y)g(y)dsy + γ1,y y κ 0 Γ

Γ

Again, we assume that κ2 is not eigenvalue of the Neumann eigenvalue problem (1.113). The unknown Dirichlet datum u ¯ = γ0ext u ∈ H 1/2 (Γ ) is then the unique solution of the boundary integral equation (cf. (1.121)) 1 ¯)(x) = − g(x) − (Kκ g)(x) (Dκ u 2

for x ∈ Γ.

To compute an approximate solution of this boundary integral equation, and, therefore, of the exterior Neumann problem, we can proceed as in the case of the interior Neumann problem. In particular, when using a piecewise constant approximation gh ∈ Sh0 (Γ ) of the given Neumann datum g, we ﬁnd a perturbed piecewise linear approximation u h ∈ Sh1 (Γ ) from the Galerkin equations

Dκ u h , ϕi

= Γ

1 − I − Kκ gh , ϕi 2 Γ

for i = 1, . . . , M.

Hence, we obtain the coeﬃcient vector u ∈ CM as the unique solution of the linear system 1 g, = − Mh − Kκ,h Dκ,h u 2 and an approximate solution of the exterior Neumann problem for x ∈ Ω, ext u∗ (x, y) uh (y)dsy . u (x) = − u∗κ (x, y)gh (y)dsy + γ1,y κ Γ

Γ

Moreover, we obtain the error estimate |u(x) − u (x)| ≤ c(x, t, g) h2 ,

(2.70)

when using the L2 projection to approximate the boundary conditions, and 1 2 (Γ ) and u ˆ ∈ Hpw (Γ ). when assuming g ∈ Hpw

2.6 Bibliographic Remarks

99

2.6 Bibliographic Remarks The numerical analysis of boundary element methods was introduced independently by J.–C. N´ed´elec and J. Planchard [79] and by G. C. Hsiao and W. L. Wendland [57]. While the stability and error analysis of the Galerkin boundary element methods follow as in the case of the ﬁnite element methods, the stability of the collocation boundary element methods for general Lipschitz boundaries is still open, see [4, 5, 100, 101] for some special cases. The Aubin–Nitsche trick to obtain higher order error estimates for boundary element methods was ﬁrst given in [58]. Since the implementation of boundary element methods often requires numerical integration techniques, an appropriate numerical analysis is mandatory. Galerkin collocation schemes were ﬁrst discussed in [54, 68]. Further investigations on the use of numerical integration schemes were made in [45, 97, 98, 102]. In [76], the inﬂuence on an additional boundary approximation was considered. Further references on boundary element methods are, for example, [12, 15, 21, 40, 50, 104, 117] and [99, 105].

3 Approximation of Boundary Element Matrices

When using boundary element methods for the numerical solution of boundary value problems for three-dimensional second order partial diﬀerential equations, one has to deal with two main diﬃculties. First of all, almost all matrices involved are dense, i.e. all their entries do not vanish in general, leading to an asymptotically quadratic memory requirement for the whole procedure. Thus, classical boundary element realisations are applicable only for a rather moderate number N of boundary elements. Fortunately, all boundary element matrices can be decomposed into a hierarchical system of blocks which can be approximated by the use of low rank matrices. This approximation will be the main content of this chapter. The second diﬃculty is the complicated form of the matrix entries to be generated. The Galerkin method, for example, requires the evaluation of double surface integrals for each matrix entry. This can not be done analytically in general. Thus, combined semi-analytical computations will be used to generate the single entries of the matrices. A more detailed description of the corresponding procedures is presented in Appendix C.

3.1 Hierarchical Matrices The formal deﬁnition and description of hierarchical matrices as well as operations involving those matrices can be found in [41, 42]. In this section we give a more intuitive introduction to this topic. 3.1.1 Motivation Let K : [0, 1] × [0, 1] → R be a given function of two scalar variables and let A ∈ RN ×M be a given matrix having the entries ak = K(xk , y ) , k = 1, . . . , N , = 1, . . . , M ,

(3.1)

102

3 Approximation of Boundary Element Matrices

with (xk , y ) ∈ [0, 1] × [0, 1]. It is obvious, that the asymptotic memory requirement for the dense matrix A is Mem(A) = O(N M ), and the asymptotic number of arithmetical operations required for a matrix-vector multiplication is Op(A s) = O(N M ) as N, M → ∞. This quadratic amount is too high for modern computers, already for moderate values of N and M . However, if we agree to store just an approximation A˜ of the matrix A and to deal with the product A˜ s instead of the exact value A s, the situation may change. But then it is necessary to control the error, i.e. to guarantee the error bound ˜ F ≤ εAF , A − A

(3.2)

for some prescribed accuracy ε, where AF denotes the Frobenius norm of the matrix A, AF =

N M

1/2 a2k

.

(3.3)

k=1 =1

Singular value decomposition The best possible approximation of the matrix A ∈ RN ×M is given by its partial singular value decomposition ˜ A ≈ A˜ = A(r) =

r

σi ui vi ,

(3.4)

i=1

where σi ∈ R+ , ui ∈ RN , vi ∈ RM , i = 1, . . . , r are the biggest singular values and the corresponding singular vectors of the matrix A. The rank r = r(ε) is chosen corresponding to the condition

σi2 ≤ ε2

i=r+1

min(N,M )

min(N,M )

˜ 2 ≤ A − A F

σi2 = ε2 A2F .

(3.5)

i=1

Unfortunately, the complete singular value decomposition of the matrix A requires O(N 3 ) arithmetical operations when assuming N ∼ M , and, therefore, it is too expensive for practical computations. However, the singular value decomposition can be perfectly used for the illustration of the main ideas. As an example, let us consider the following function on [0, 1] × [0, 1], K(x, y) =

1 , α+x+y

(3.6)

where α > 0 is some real parameter. For small values of the parameter α the function K gets an artiﬁcial “singularity” at the corner (0, 0) of the square [0, 1] × [0, 1].

3.1 Hierarchical Matrices

The domain [0, 1] × [0, 1] is uniformly discretised using the nodes 1 1 (xk , y ) = (k − 1)hx , ( − 1)hy , hx = , hy = N −1 M −1

103

(3.7)

for k = 1, . . . , N and = 1, . . . , M . In Fig. 3.1, the logarithmic plots of the singular values of the matrix (3.1) (i.e. the quantities log10 σi , i = 1, . . . , N ) for N = M = 32 (left plot) and N = M = 1024 (right plot) are presented for α = 10−4 . It can easily be seen that only very few singular values are needed to represent the matrix A in its singular value decomposition (3.4). For N = 1024, almost all singular values are close to the computer zero, and, ˜ therefore, this matrix can be well approximated by a low rank matrix A.

0

0

-5

-5

-10

-10

-15

-15 5

10

15

20

25

30

0

200

400

600

800

1000

Fig. 3.1. Singular values for N = 32 and N = 1024

The number of signiﬁcant singular values slowly increases with the dimension, as it can be seen in Fig. 3.2. Here the ﬁrst 32 singular values (logarithmic plot) for N = M = 32, 128, 256 (left plot, from below) and for N = M = 256, 512, 1024 (right plot, from below) are shown. The accuracy of 5

5

0

0

-5

-5

-10

-10

-15

-15 0

5

10

15

20

25

30

0

5

10

15

20

25

30

Fig. 3.2. First 32 singular values for N = 32, 64, 128, 256, 512, and N = 1024

˜ the low rank approximation A(r) of the matrix A is illustrated in Fig. 3.3, where the logarithmic plot of the function ε(r) =

˜ A − A(r) F AF

104

3 Approximation of Boundary Element Matrices

0

0

-5

-5

-10

-10

-15

-15

-20

-20 0

5

10

15

20

25

30

0

5

10

15

20

25

30

Fig. 3.3. Accuracy of the low lank approximation

for r = 1, . . . , 32 is depicted. The left plot in Fig. 3.3 corresponds again to the dimensions N = 32, 64, and N = 128 (from below), while the right plot shows the results for N = 256, 512, and N = 1024 (from below). Thus, the behaviour of the singular values determines the quality of the low rank approximation (3.4). The results shown do not really depend on the parameter α. If α becomes smaller, the results are even better. The situation changes if the “singularity” of the function K is more serious. As a further example, let us consider the following function on [0, 1] × [0, 1], K(x, y) =

1 , α + (x − y)2

(3.8)

where α > 0 is again some real parameter. For small values of the parameter α the function K gets an artiﬁcial “singularity” along the diagonal {(x, x)} of the square [0, 1] × [0, 1]. In Fig. 3.4 (left plot), the rank r(ε) for ε = 10−6 and N = M = 256 is shown as a function of the parameter α. The horizontal axis corresponds to the values − log2 (α), while α changes from 20 till 2−8 . Thus, the rank of the matrix strongly depends on the parameter α. However, if we “separate” the 70

70

60

60

50

50

40

40

30

30

20

20

10

10 0

2

4

6

8

0

2

4

6

8

Fig. 3.4. Rang of the matrix A˜

variables x and y, i.e. consider only the quarter [0, 0.5] × [0.5, 1] of the square [0, 1]×[0, 1], then the situation is much better. The right plot in Fig. 3.4 shows the same curve for separated x and y, which is more or less constant now.

3.1 Hierarchical Matrices

105

The logarithmic plots of the singular values of the matrix A for α = 10−1 (lower curve) and for α = 10−8 (upper curve) are shown in Fig. 3.5. The left plot in this ﬁgure corresponds to the whole square [0, 1]×[0, 1], while the right plot shows the results for the separated variables x and y, i.e. if we consider only the quarter [0, 0.5] × [0.5, 1] of the square [0, 1] × [0, 1]. Now the main

0

0

-5

-5

-10

-10

-15

-15 0

50

100

150

200

250

0

50

100

150

200

250

Fig. 3.5. Singular values for N = 256

idea of hierarchical methods is quite clear. If we decompose the whole matrix A into four blocks corresponding to the domains [0, 0.5] × [0, 0.5], [0, 0.5] × [0.5, 1], [0.5, 1] × [0, 0.5], [0.5, 1] × [0.5, 1] , we will be able to approximate two of these four blocks eﬃciently. The two remaining, main diagonal blocks have the same structure as the original matrix, but only half of the size and their rank will be smaller. In Fig. 3.6, the left diagram corresponds to the whole matrix and its rank r(ε) = 73 is obtained for α = 2−9 , ε = 10−6 and N = M = 256. The 2 × 2 block matrix together with the ranks of the blocks is shown in the second diagram of Fig. 3.6. The

38

9

73 9

38

20

8

9

8 20

9

20

8

8 20

12 7 7 12

8

8

12 7 7 12

9

9 12 7 7 12

8

8

12 7 7 12

Fig. 3.6. Original matrix and its hierarchical decomposition in blocks

approximation of the separated blocks is now acceptable, and we continue to decompose only the blocks on the main diagonal. The results can be seen in the third and in the fourth diagram of Fig. 3.6. The memory requirements for these four matrices are quite diﬀerent: the ﬁrst matrix needs 146N words of memory, the second 94N , the third 74N , and, ﬁnally, we will need 72N words of memory for the last block matrix in Fig. 3.6. Thus, a hierarchical

106

3 Approximation of Boundary Element Matrices

decomposition into blocks and their separate approximation using a singular value decomposition leads to a drastic reduction of memory requirements (the latter decomposition requires less than 50%) even for this rather small matrix having a “diagonal singularity”. Note that the rank of the blocks on the main diagonal increases almost linear with the dimension: 12 − 20 − 38 − 73, while the rank of the separated blocks has at most a logarithmic growth: 7 − 8 − 9. Thus, a hierarchical approximation of large dense matrices arising from some generating function having diagonal singularity consists of three steps: • Construction of clusters for variables x and y, • Finding of possible admissible blocks (i.e. blocks with separated x and y), • Low rank approximation of admissible blocks. In the above example, the clusters were simply the sets of points xk which belong to smaller and smaller intervals. Fortunately, the decomposition problem is only slightly more complicated for general, three-dimensional irregular point sets. Also, the admissible blocks in the above example are very natural. They are just blocks outside of the main diagonal. In the general case, we will need some permutations of rows and columns of the matrix to construct admissible blocks. Finally, the singular value decomposition approximation we have used, is not applicable for more realistic examples. We will need more eﬃcient algorithms, namely the Adaptive Cross Approximation (ACA), to approximate admissible blocks. Degenerated approximation In the above example, the approximation of the blocks for separated variables x and y is based on the smoothness of the function K for x = y. However, if the function K is degenerated, i.e. it is a ﬁnite sum of products of functions depending only on x and y, K(x, y) =

r

pi (x)qi (y) ,

(3.9)

i=1

then the rank of the matrix A deﬁned in (3.1) is equal to r independent of its dimension. Thus for N, M r, the matrix A is a low rank matrix. This property is independent of the smoothness of the functions pi , qi in (3.9). The low rank representation of the matrix A is then A=

r

ui vi ,

i=1

with (ui )k = pi (xk ) , (vi ) = qi (y )

3.1 Hierarchical Matrices

107

for k = 1, . . . , N and = 1, . . . , M . Note that this representation is not the singular value decomposition (3.4). Another possibility to obtain a low rank approximation of a matrix of the form (3.1) is based on the smoothness. If the function K is smooth enough, then we can use its Taylor series (cf. [44]) with respect to the variable x in some ﬁxed point x∗ , K(x, y) =

r 1 ∂ i K(x∗ , y) (x − x∗ )i + Rr (x, y) , i i ! ∂ x i=0

to obtain a degenerated approximation A ≈ A˜ =

r

ui vi ,

(3.10)

i=0

with 1 ∂ i K(x∗ , y ) i! ∂ xi for k = 1, . . . , N and = 1, . . . , M . Again, (3.10) is not the partial singular value decomposition (3.4) of the matrix A. If the remainder term Rr is uniformly bounded by the original function K satisfying Rr (x, y) ≤ εK(x, y) (ui )k = (xk − x∗ )i , (vi ) =

for all x and y with some r = r(ε), then we can guarantee the accuracy of the low rank matrix approximation ˜ F ≤ εAF A − A

(3.11)

for all dimensions N and M . The rank r+1 of the matrix A˜ is also independent ˜ = of its dimension. Thus, for N ≈ M , the matrix A˜ requires only Mem(A) O(N ) words of computer memory. However, an eﬃcient construction of the Taylor series for a given function in the three-dimensional case is practically impossible. Thus, it is rather an illustration for the fact that there exist low rank approximations of dense matrices, which are not based on the singular value decomposition. A further example of a low rank approximation of a given function is the decomposition of the fundamental solution of the Laplace equation u∗ (x, y) =

1 1 4 π |x − y|

for x, y ∈ R3

(cf. (1.7)) into spherical harmonics in some point x∗ with |x − x∗ | < |y − x∗ | n ∞ |x − x∗ | 1 1 Pn (ex , ey ) , u∗ (x, y) = ∗ ∗ 4 π |y − x | n=0 |y − x | ex =

x − x∗ , |x − x∗ |

ey =

y − x∗ , |y − x∗ |

108

3 Approximation of Boundary Element Matrices

where the Legendre polynomials are deﬁned for |u| ≤ 1 as follows, P0 (u) = 1 ,

Pn (u) =

1 dn 2 (u − 1)n , n! dun

2n

for n ≥ 1 .

(3.12)

Note that the Legendre polynomials allow the following separation of variables n Ynm (ex )Yn−m (ey ) , Pn (ex , ey ) = m=−n

where Ynm are the spherical harmonics. See [39, 93] for more details. 3.1.2 Hierarchical clustering Large dense matrices arising from integral equations have no explicit structure in general. As a rule, because of the singularity of the kernel function on the diagonal, i.e. for x = y, these matrices are also not of low rank. However, it is possible to ﬁnd a permutation, so that the matrix with permuted rows and columns contains rather large blocks close to some low-rank matrices with respect to the Frobenius norm (cf. (3.2)). Cluster tree To ﬁnd a suitable permutation, a cluster tree is constructed by a recursive partitioning of some weighted, pairwise disjunct, characteristic points

(xk , gk ) , k = 1, . . . , N ⊂ R3 × R+ (3.13) and

(y , q ) , = 1, . . . , M

⊂ R3 × R+

(3.14)

in order to separate the variables x and y. A large distance between two characteristic points results in a large diﬀerence of the respective column or row numbers. When dealing with boundary element matrices, the characteristic points can be the mid points x∗k of the triangle elements τk with weights gk = Δk = |τk |, when using piecewise constant basis functions ψk (cf. (2.3)), • the nodes xk of the grid with weights gk = |supp ϕk |, when using piecewise linear continuous basis functions ϕk (cf. (2.9)).

•

A group of weighted points is called cluster if the points are “close” to each other with respect to the usual distance. A given cluster

Cl = (xk , gk ) , k = 1, . . . , n with n > 1 can be separated in two sons using the following algorithm.

3.1 Hierarchical Matrices

109

Algorithm 3.1 1. Mass of the cluster G=

n

gk ∈ R+ ,

k=1

2. Centre of the cluster 1 gk xk ∈ R3 , G n

X=

k=1

3. Covariance matrix of the cluster C=

n

gk (xk − X) (xk − X) ∈ R3×3 ,

k=1

4. Eigenvalues and eigenvectors C vi = λi vi , i = 1, 2, 3 , λ1 ≥ λ2 ≥ λ3 ≥ 0 , 5. Separation 5.1 initialisation Cl1 := ∅ , Cl2 := ∅ , 5.2 for k = 1, . . . , n if (xk − X, v1 ) ≥ 0 then Cl1 := Cl1 ∪ (xk , gk ) else Cl2 := Cl2 ∪ (xk , gk ) . The eigenvector v1 of the matrix C corresponds to the largest eigenvalue of this matrix and shows(in the direction of the longest ) expanse of the cluster. The separation plane x ∈ R3 : (x − X, v1 ) = 0 goes through the centre X of the cluster and is orthogonal to the eigenvector v1 . Thus, Algorithm 3.1 divides a given arbitrary cluster of weighted points into two more or less equal sons. In Fig. 3.7, the ﬁrst two separation levels of a given, rather complicated, surface are shown. The separation of a given cluster in two sons deﬁnes a permutation of the points in the cluster. The points in the ﬁrst son will be numbered ﬁrst and then the ones in the second son. Algorithm 3.1 will be applied recursively to the sons, until they contain less or equal than some prescribed (small and independent of N ) number nmin of points. Cluster pairs Next, cluster pairs which are geometrically well separated are identiﬁed. They will be regarded as admissible cluster pairs, as e.g. the clusters in Fig. 3.8. An

110

3 Approximation of Boundary Element Matrices

0.05 0 -0.05 0.1

-0.1 0 0 0.1 0.2 0 2

0.05 0 -0.05 0.1

-0.1 0 0 0.1 0.2 0 2 Fig. 3.7. Clusters of the ﬁrst two levels

3.1 Hierarchical Matrices

111

0.05 0 -0.05 0.1

-0.1 0 0 0.1 0.2 0 2 Fig. 3.8. An admissible cluster pair

appropriate admissibility criterion is the following simple geometrical condition. A pair of clusters (Clx , Cly ) with nx > nmin and my > nmin elements is admissible if min diam(Clx ), diam(Cly ) ≤ η dist(Clx , Cly ) , (3.15) where 0 < η < 1 is a given parameter. Although the criterion (3.15) is quite simple, a rather large computational eﬀort (quadratic with respect to the number of elements in the clusters Clx and Cly ) is required for calculating the exact values diam(Clx ) = max |xk1 − xk2 | , k1 ,k2

diam(Cly ) = max |y1 − y2 | , 1 ,2

dist(Clx , Cly ) = min |xk − y | . k,

In practice, one can use rougher, more restrictive, but easily computable bounds diam(Clx ) ≤ 2 max |X − xk | , k

diam(Cly ) ≤ 2 max |Y − y | ,

112

3 Approximation of Boundary Element Matrices

dist(Clx , Cly ) ≥ |X − Y | −

1 diam(Clx ) + diam(Cly ) , 2

where X and Y are the already computed centres (cf. Algorithm 3.1) of the clusters Clx and Cly , for the admissibility condition. If a cluster pair is not admissible, but nx > nmin and my > nmin are satisﬁed, then there exist sons of both clusters Clx = Clx,1 ∪ Clx,2 , Cly = Cly,1 ∪ Cly,2 . For simplicity, let us assume that the cluster Clx is bigger than Cly , i.e. diam(Clx ) ≥ diam(Cly ). In this case, we have to check the following two new pairs Clx,1 , Cly , Clx,2 , Cly for admissibility, and so on. This recursive procedure stops if nx ≤ nmin or my ≤ nmin is satisﬁed. The corresponding block of the matrix is small, and it will be computed exactly. The cluster trees for the variables x and y together with the set of the admissible cluster pairs, as well as the set of the small cluster pairs allow to split the matrix into a collection of blocks of various sizes. The hierarchical block structure of the Galerkin matrix for the single layer potential on the surface from Figs. 3.7–3.8 is shown in Fig. 3.9. The colours of the blocks indicate the “quality” of the approximation. The light grey colour corresponds to well approximated blocks, while the dark grey colour indicates a less good approximation. The small blocks are computed exactly and they are depicted in black. Thus, the remaining main problem is how to approximate the blocks which correspond to the admissible cluster pairs, without using the singular value decomposition. The corresponding procedures will be described in the following section.

3.2 Block Approximation Methods 3.2.1 Analytic Form of Adaptive Cross Approximation Let X, Y ⊂ R3 be two non–empty domains, and let K : X × Y → R be a given function. The following abstract Adaptive Cross Approximation algorithm constructs a degenerated approximation of the function K using nodal interpolation in some points ) ( ) ( x1 , x2 , . . . ⊂ X , y1 , y2 , . . . ⊂ Y , which will be determined during the realisation of the algorithm on an adaptive way.

3.2 Block Approximation Methods

Fig. 3.9. Matrix decomposition

Algorithm 3.2 1. Initialisation R0 (x, y) = K(x, y) , S0 = 0 . 2. For i = 0, 1, 2, . . . compute 2.1. pivot element (xi+1 , yi+1 ) = ArgMax|Ri (x, y)| , 2.2. normalising constant γi+1 = (Ri (xi+1 , yi+1 ))

−1

,

2.3. new functions ui+1 (x) = γi+1 Ri (x, yi+1 ) , vi+1 (y) = Ri (xi+1 , y) , 2.4. new residual Ri+1 (x, y) = Ri (x, y) − ui+1 (x)vi+1 (y) , 2.5. new approximation Si+1 (x, y) = Si (x, y) + ui+1 (x)vi+1 (y) .

113

114

3 Approximation of Boundary Element Matrices

The stopping criterion for the above algorithm can be realised in Step 2.1. corresponding to the condition |Rr (x, y)| ≤ ε|K(x, y)| for (x, y) ∈ X × Y .

(3.16)

Algorithm 3.2 produces a sequence of approximations {Si } and an associated sequence of residuals {Ri } possessing the approximation property K(x, y) = Ri (x, y) + Si (x, y) , (x, y) ∈ X × Y , i = 0, 1, . . . , r

(3.17)

and the interpolation property Rm (xi , y) = 0 , y ∈ Y , Rm (x, yi ) = 0 , x ∈ X for i = 1, 2, . . . , m and for m = 1, 2, . . . , r. Furthermore, if the function K(x, y) is harmonic for x = y then its approximations Si (x, y) are also harmonic for all i. The residuals {Ri } accumulate zeros, and, therefore, the sequence of the functions {Si } interpolates the given function K(x, y) in more and more points corresponding to (3.17). If we are interested in computing an approximation A˜ for a matrix A ∈ RN ×M having the entries xk , y˜ ) , k = 1, . . . , N , = 1, . . . , M ak = K(˜

(3.18)

for some points (˜ xk , y˜ ) ∈ X × Y , then the approximation Sr (x, y) of the function K(x, y) can be used to obtain xk , y˜ ) ≈ ak . a ˜k = Sr (˜ Due to the stopping criterion (3.16), and due to the approximation property (3.17), the following estimate obviously holds ˜ F ≤ εAF , A − A where · F denotes the Frobenius norm of a matrix (cf. (3.3)). Remark 3.3. Step 2.1. of Algorithm 3.2 should be discussed in more details. It can be very diﬃcult, if not impossible, to solve the maximum problem formulated there. There are two possibilities to proceed. First, we can look for the maximum only in a ﬁnite set of given points (˜ xk , y˜ ). In this case only the original entries of the matrix A will be used for its approximation. The algorithm will coincide with the algebraic fully pivoted ACA algorithm as described in Subsection 3.2.2. The other possibility is to choose some artiﬁcial points and to look for the maximum there. These points can be the zeros of the three-dimensional Chebyshev polynomials in corresponding bounding boxes for the sets X, Y . In this case the ACA approximation will be similar to the best possible polynomial interpolation.

3.2 Block Approximation Methods

115

Remark 3.4. The stopping criterion (3.16) can be applied only if the function K(x, y) is smooth on (X, Y ). If this is not the case, but if the function K is asymptotically smooth (cf. (3.19)), then we have to decompose the domains X and Y into two systems of clusters and to approximate the function on each admissible cluster pair (Clx , Cly ) separately using Algorithm 3.2. This decomposition implies the corresponding decomposition of the matrix in a hierarchical system of blocks. Using the theory of polynomial multidimensional interpolation, the following result was proven in [7]. Theorem 3.5. Let the function K(x, y) be asymptotically smooth with respect to y, i.e. K(x, ·) ∈ C ∞ (R3 \{x}) for all x ∈ R3 , satisfying |∂yα K(x, y)| ≤ cp |x − y|g−p , p = |α|

(3.19)

for all multiindices α ∈ N30 with a constant g < 0. Moreover, the matrix A ∈ RN ×M with entries (3.18) is decomposed into blocks corresponding to the admissibility condition diam(Cly ) ≤ η dist(Clx , Cly ) , η < 1 . Then the matrix A with M ∼ N can be approximated up to an arbitrary given accuracy ε > 0 using a system of given points (˜ xk , y˜ ), ˜ F ≤ εAF , A − A and ˜ = Op(A˜ s) = Mem(A) ˜ = O(N 1+δ ε−δ ) Op(A)

for all δ > 0 .

˜ denotes the number of arithmetical operations reIn Theorem 3.5, Op(A) ˜ Op(A˜ s) is the asymptotic number quired for the generation of the matrix A, of arithmetical operations required for the matrix-vector multiplication with ˜ and Mem(A) ˜ = r(M + N ) is the asymptotic memory requirethe matrix A, ment for the matrix A˜ as N → ∞. Thus, Theorem 3.5 states that an almost linear complexity is achieved for these important quantities. Let us now consider matrices arising from collocation boundary element methods (cf. Chapter 2). Let Γ be a Lipschitz boundary, let

ϕ : Γ → R, = 1, . . . , M be a given system of basis functions, and let

x∗k ∈ Γ , k = 1, . . . , N be a set of collocation points. If, for example, Γ is the union of plane triangles (cf. (2.1)),

116

3 Approximation of Boundary Element Matrices

Γ =

N '

τ¯ ,

=1

then the most simple collocation method with piecewise constant basis functions . 1 , y ∈ τ , ψ (y) = 0,y∈ / τ can be used. In this case, the collocation points x∗k are the midpoints of the triangles τk . The corresponding collocation matrix A ∈ RN ×M with ak = K(x∗k , y) ψ (y) dsy , k = 1, . . . , N , = 1, . . . , M (3.20) Γ

for some kernel function K can be approximated using the following fully pivoted ACA algorithm. Algorithm 3.6 1. Initialisation R0 (x, y) = K(x, y) , S0 = 0 . 2. For i = 0, 1, 2, . . . compute 2.1. pivot element ∗ (ki+1 , i+1 ) = ArgMax Ri (xk , y) ψ (y) dsy , Γ

2.2. normalising constant ⎛ ⎞−1 γi+1 = ⎝ Ri (x∗ki+1 , y) ψi+1 (y) dsy ⎠ , Γ

2.3. new functions ui+1 (x) = γi+1 Ri (x, y) ψi+1 (y) dsy , vi+1 (y) = Ri (x∗ki+1 , y) , Γ

2.4. new residual Ri+1 (x, y) = Ri (x, y) − ui+1 (x)vi+1 (y) , 2.5. new approximation Si+1 (x, y) = Si (x, y) + ui+1 (x)vi+1 (y) .

3.2 Block Approximation Methods

117

Note that the approximation property (3.17) remains valid for Algorithm 3.6, while the interpolation property (3.18) remains valid only with respect to the variable x, Rm (x∗ki , y) = 0, y ∈ Γ , i = 1, 2, . . . , m, m = 1, 2, . . . , r .

(3.21)

The interpolation property with respect to y changes to the orthogonality Rm (x, y) ψi (y)dsy = 0, x ∈ Γ, i = 1, . . . , m, m = 1, 2, . . . r. (3.22) Γ

For the analysis of Algorithm 3.6, it is useful to introduce the following functions U (x) = K(x, y) ϕ (y)dsy , = 1, . . . , M , Γ

having the property ak = U (x∗k ) , (cf. (3.20)). Using the properties (3.21) and (3.22), we can conclude that the functions ˜ (x) = Sr (x, y) ψ (y)dsy U (3.23) Γ

coincide with U for ∈ {1 , . . . , r }. Moreover, all other functions U , i.e. for ∈ / {1 , . . . , r }, are interpolated by the functions (3.23) at points x∗k , k ∈ {k1 , . . . , kr }. The approximation A˜ of the collocation matrix A is then given by the entries ak ≈ a ˜k = Sr (x∗k , y) ψ (y)dsy . Γ

In [9], the interpolation theory of multidimensional Chebyshev polynomials was used in order to prove Theorem 3.5 for collocation matrices. A straightforward modiﬁcation of Algorithm 3.6 leads to an algorithm for the Galerkin matrix A ∈ RN ×M with elements ak = K(x, y) ϕ (y) ψk (x) dsy dsx (3.24) Γ Γ

for k = 1, . . . , N and = 1, . . . , M . In (3.24), a system of basis functions

ϕ : Γ → R, = 1, . . . , M , may diﬀer from the system of test functions

ψk : Γ → R, k = 1, . . . , N . The Galerkin matrix A can be approximated using the following ACA algorithm.

118

3 Approximation of Boundary Element Matrices

Algorithm 3.7 1. Initialisation R0 (x, y) = K(x, y) , S0 = 0 . 2. For i = 0, 1, 2, . . . compute 2.1. pivot element Ri (x, y) ϕ (y) ψk (x) dsy dsx , (ki+1 , i+1 ) = ArgMax Γ Γ

2.2. normalising constant ⎛ γi+1 = ⎝

⎞−1 Ri (x, y) ϕi+1 (y) ψki+1 (x) dsy dsx ⎠

,

Γ Γ

2.3. new functions

ui+1 (x) = γi+1 vi+1 (y) =

Ri (x, y) ϕi+1 (y) dsy , Γ

Ri (x, y) ψki+1 (x) dsx , Γ

2.4. new residual Ri+1 (x, y) = Ri (x, y) − ui+1 (x)vi+1 (y) , 2.5. new approximation Si+1 (x, y) = Si (x, y) + ui+1 (x)vi+1 (y) . The approximation property (3.17) remains valid for Algorithm 3.7, which, instead of the interpolation property (3.18), possesses the following orthogonalities for i = 1, . . . , m , m = 1, 2, . . . r: Rm (x, y) ϕi (y) dsy = 0 , x ∈ Γ , Γ

Rm (x, y) ψki (x) dsx = 0 , y ∈ Γ . Γ

It is practically impossible to compute the elements of the Galerkin matrices corresponding to (3.24) analytically in a general setting. Even in the simplest

3.2 Block Approximation Methods

119

situation, e.g. for plane triangles τk and by using piecewise constant basis functions, some numerical integration is involved (cf. Chapter 4). If both integrals in (3.24) are computed numerically, i.e. ωk,kx ω,ky K(xk,kx , y,ky ) ϕ (y,ky )ψk (xk,kx ), (3.25) ak ≈ ak = kx

ky

where ωk,kx , ω,ky are the weights of the quadrature rule (including Jacobians) and xk,kx , y,ky are the corresponding integration points, then not the exact Galerkin matrix A, but its quadrature approximation A¯ will be further approximated by ACA. The matrix A¯ is, corresponding to (3.25), a ﬁnite sum of matrices as deﬁned in (3.18), multiplied by degenerated diagonal matrices. Therefore, Theorem 3.5 remains valid for Galerkin matrices, if the relative accuracy of the numerical integration (3.25) is higher than the approximation of the matrix A¯ by ACA. 3.2.2 Algebraic Form of Adaptive Cross Approximation On the matrix level, all three algorithms formulated in Section 3.2.1 can be written in the fully pivoted ACA form. Fully pivoted ACA algorithm Let A ∈ RN ×M be a given matrix. Algorithm 3.8 1. Initialisation R0 = A , S0 = 0 . 2. For i = 0, 1, 2, . . . compute 2.1. pivot element (ki+1 , i+1 ) = ArgMax |(Ri )k | , 2.2. normalising constant −1 γi+1 = (Ri )ki+1 i+1 , 2.3. new vectors ui+1 = γi+1 Ri ei+1 , vi+1 = Ri eki+1 , 2.4. new residual Ri+1 = Ri − ui+1 vi+1 ,

120

3 Approximation of Boundary Element Matrices

2.5. new approximation . Si+1 = Si + ui+1 vi+1

In Algorithm 3.8, ej denotes the jth column of the identity matrix I. The whole residual matrix Ri is inspected in Step 2.1 of Algorithm 3.8 for its maximal entry. Thus, its Frobenius norm can easily be computed in this step, and the appropriate stopping criterion for a given ε > 0 at step r would be Rr F ≤ εAF . Note that the crosses built from the column-row pairs with the indices ki , i for i = 1, . . . , r will be computed exactly Sm = aki ,l , l = 1, ..., M , k ,l i = ak,li , k = 1, ..., N Sm k,li

for i = 1, . . . , m , m = 1, . . . , r, while all other elements are approximated. The number of operations required to generate the approximation A˜ = Sr is O(r2 N M ). The memory requirement for Algorithm 3.8 is O(N M ), since the whole matrix A is assumed to be given at the beginning. Thus, Algorithm 3.8 is much faster than a singular value decomposition, but still rather expensive for large matrices. The eﬃciency of Algorithm 3.8 will now be illustrated using the following examples. First, we consider the matrix A, generated as in (3.1), for the function 1 , α = 10−2 K(x, y) = α+x+y (cf. (3.6)) on the uniform grid (3.7) for N = M = 32. In Table 3.1 the results of the application of Algorithm 3.8 are presented. The plot of the initial residual R0 , i.e. of the function K on the grid, is shown in Fig. 3.10, while the next three Figs. 3.11–3.13 show the residual Rk for k = 3, 6, and k = 9. The three-dimensional plots of the residuals Rk (x, y) are presented on the left, while the corresponding matrices are depicted on the right. Note that the exactly computed crosses are shown in black, while the remaining grey scales are adapted to the actual values of the residuals, and, therefore, are diﬀerent for all pictures. This example illustrates the behaviour of the fully pivoted ACA algorithm very clearly. The generation function (3.6) has only a weak “singularity” at the corner of the computational domain. This singularity is not important for the low rank approximation and is completely removed after the ﬁrst two iterations. Further iterations quickly reduce the relative error of the approximation. However, if the “singularity” is on the diagonal as for the function K(x, y) =

1 , α + (x − y)2

α = 10−2

(3.26)

3.2 Block Approximation Methods

121

Table 3.1. Fully Pivoted ACA algorithm for the function (3.6) Step

Pivot row

1 2 3 4 5 6 7 8 9 10

Pivot column

Pivot value

Relative error

1 2 6 28 3 13 4 20 9 32

1.00 · 10 7.91 · 10+0 1.10 · 10+0 2.25 · 10−1 6.10 · 10−2 9.87 · 10−3 3.91 · 10−4 1.02 · 10−4 6.32 · 10−6 1.97 · 10−6

3.43 · 10−1 1.62 · 10−1 3.66 · 10−2 2.26 · 10−3 8.40 · 10−4 2.28 · 10−5 8.85 · 10−6 2.69 · 10−7 3.30 · 10−8 1.13 · 10−9

1 2 6 28 3 13 4 20 9 32

+2

0

100

8

16

24

32

0

0

8

8

16

16

24

24

75 30

50 25 0

20 10 10 20

32 30

32 0

8

16

24

32

24

32

Fig. 3.10. Initial residual for the function (3.6) 0

0.2

8

16

0

0

8

8

16

16

24

24

30

0.1 0 20 10 10 20

32 30

32 0

8

Fig. 3.11. Residual R3 for the function (3.6)

16

24

32

122

3 Approximation of Boundary Element Matrices 0

0.0004 0.0002

8

16

24

32

0

0

8

8

16

16

24

24

30

0 -0.0002

20 10

10 20 30

32

32 0

8

16

24

32

16

24

32

Fig. 3.12. Residual R6 for the function (3.6) 0

8

0

0

8

8

16

16

24

24

32

32 0

8

16

24

32

Fig. 3.13. Residual R9 for the function (3.6)

(cf. (3.8)), then, as we have already seen, the situation changes. The results of the computations are presented in Table 3.2 and in the four Figs. 3.14–3.17. The convergence of the ACA algorithm is now slow, the crosses chosen can not approximate the main diagonal, because they are too small there. This illustrates once again the necessity of the hierarchical clustering. The next function we consider, K(x, y) = sin6 (π(2x + y)) ,

(3.27)

is degenerated corresponding to the deﬁnition (3.9), having the exact low rank r = 7. This function is obviously inﬁnitely smooth. But it is oscillating and the convergence of Algorithm 3.8 is slow again. The convergence is, of course, better than for the singular function (3.26), but not really suﬃcient. However, after exactly 7 iterations, the error is equal to computer zero. It means that Algorithm 3.8 has correctly detected the low rank of the function (3.27). The numerical results can be seen in Table 3.3 and in the four Figs. 3.18–3.21,

3.2 Block Approximation Methods

123

Table 3.2. Fully Pivoted ACA algorithm for the function (3.8) Step

Pivot row

1 2 3 4 5 6 7 8 9 10

Pivot column

Pivot value

Relative error

32 1 17 9 25 5 21 13 29 3

1.00 · 10 9.99 · 10+1 9.69 · 10+1 9.63 · 10+1 9.53 · 10+1 7.32 · 10+1 7.32 · 10+1 7.31 · 10+1 6.24 · 10+1 2.53 · 10+1

9.35 · 10−1 8.68 · 10−1 7.02 · 10−1 5.65 · 10−1 4.00 · 10−1 3.45 · 10−1 2.78 · 10−1 1.92 · 10−1 1.12 · 10−1 1.02 · 10−1

32 1 17 9 25 5 21 13 29 3

+2

0

100

8

16

24

32

0

0

8

8

16

16

24

24

75 30

50 25 0

20 10 10 20

32 30

32 0

8

16

24

32

24

32

Fig. 3.14. Initial residual for the function (3.8) 0

75 50 25 0

8

16

0

0

8

8

16

16

24

24

30

20 10 10 20

32 30

32 0

8

Fig. 3.15. Residual R3 for the function (3.8)

16

24

32

124

3 Approximation of Boundary Element Matrices 0

60 40 20 0

8

16

24

32

0

0

8

8

16

16

24

24

30

20 10 10 20

32 30

32 0

8

16

24

32

16

24

32

Fig. 3.16. Residual R6 for the function (3.8) 0

8

0

0

8

8

16

16

24

24

20 30

10 0 20 10 10 20

32 30

32 0

8

16

24

32

Fig. 3.17. Residual R9 for the function (3.8)

where the initial residual and three sequential residuals (for k = 2, 4, and k = 6) are shown. Table 3.3. Fully Pivoted ACA algorithm for the function (3.27) Step

Pivot row

Pivot column

Pivot value

Relative error

1 2 3 4 5 6 7

32 24 28 20 30 22 26

19 3 27 11 23 7 31

1.00 · 10+0 1.00 · 10+0 9.69 · 10−1 9.68 · 10−1 3.05 · 10−1 2.88 · 10−1 1.17 · 10−1

8.44 · 10−1 6.51 · 10−1 4.81 · 10−1 2.16 · 10−1 1.30 · 10−1 6.19 · 10−2 2.91 · 10−15

3.2 Block Approximation Methods 0

1 0.75 0.5 0.25 0

8

16

24

125 32

0

0

8

8

16

16

24

24

30 20 10

10 20 32

30

32 0

8

16

24

32

24

32

Fig. 3.18. Initial residual for the function (3.27) 0

0.5

8

16

0

0

8

8

16

16

24

24

30

0 20 10 10 20 32 30

32 0

8

16

24

32

24

32

Fig. 3.19. Residual R2 for the function (3.27) 0

8

16

0

0

8

8

16

16

24

24

0.2 30 0 20

-0.2 10 10 20

32 30

32 0

8

16

Fig. 3.20. Residual R4 for the function (3.27)

24

32

126

3 Approximation of Boundary Element Matrices 0

0.1 0.05 0 -0.05 -0.1

8

16

24

32

0

0

8

8

16

16

24

24

30 20 10

10 20 32

30

32 0

8

16

24

32

Fig. 3.21. Residual R6 for the function (3.27)

Partially pivoted ACA algorithm If the matrix A has not yet been generated, but if there is a possibility of generating its entries ak individually, then the following partially pivoted ACA algorithm can be used for the approximation: Algorithm 3.9 1. Initialisation S0 := 0 , I := ∅ , J := ∅ , c := 0 ∈ RN , r := 0 ∈ RM . 2. Restart with the next not yet generated row If #I = N or #J = M then STOP else ki+1 := min {k : k ∈ / I} , CrossT ype := Row , 3. Generate cross 3.1 Type of the cross If CrossT ype == Row then 3.1.1 Generate row, Update control vector a := A eki+1 , I := I ∪ {ki+1 } , r := r + |a| , 3.1.2 Test If |a| = 0 then GOTO 2.

(zero row)

3.2 Block Approximation Methods

127

3.1.3 Row of the residual and the pivot column rv := a −

i

(um )ki+1 vm ,

m=1

i+1 := ArgMax |(rv ) | ,

3.1.4 Test If |rv | = 0 then GOTO 4. 3.1.5 Normalising constant

(linear depending row)

γi+1 := (rv )−1 i+1 , 3.1.6 Generate column, Update control vector b := A ei+1 , J := J ∪ {i+1 } , c := c + |b| , 3.1.7 Column of the residual and the pivot row ru := b −

i

(vm )i+1 um ,

m=1

ki+2 := ArgMax |(ru )k | , 3.1.8 New vectors ui+1 := ru , vi+1 := γi+1 rv , else 3.2.1 Generate column, Update control vector b := Aei+1 , J := J ∪ {i+1 } , c := c + |b| , 3.2.2 Test If |b| = 0 then GOTO 2. (zero column) 3.2.3 Column of the residual and the pivot row ru := b −

i

(vm )i+1 um ,

m=1

ki+1 := ArgMax |(ru )k | , 3.2.4 Test If |ru | = 0 then GOTO 4. 3.2.5 Normalising constant

(linear depending column)

γi+1 := (ru )−1 ki+1 ,

128

3 Approximation of Boundary Element Matrices

3.2.6 Generate row, Update control vector a := A eki+1 , I := I ∪ {ki+1 } , r := r + |a| , 3.2.7 Row of the residual and the pivot column rv := a −

i

(um )ki+1 vm ,

m=1

i+2 := ArgMax |(rv ) | , 3.2.8 New vectors ui+1 := γi+1 ru , vi+1 := rv , 3.3 New approximation , Si+1 := Si + ui+1 vi+1

3.4 Frobenius norm of the approximation Si+1 2F

=

Si 2F

+2

i

2 2 u i+1 um vm vi+1 + ui+1 F vi+1 F .

m=1

3.5 Test If ui+1 F vi+1 F < εSi+1 F then GOTO 4 else i := i + 1, GOTO 3 4. Check control vectors / I and ci∗ = 0 then If ∃i∗ ∈ i := i + 1 , ki+1 = i∗ , CrossT ype = Row , GOTO 3 or / J and rj ∗ = 0 then If ∃j ∗ ∈ i := i + 1 , i+1 = j ∗ , CrossT ype = Column , GOTO 3 else STOP Algorithm 3.9 starts to compute an approximation for the matrix A by generating its ﬁrst row. Then, the ﬁrst column will be chosen automatically. If a cross is successfully computed, the next row index is prescribed, and the procedure repeats. If a zero row is generated, then it is not possible to ﬁnd the column, and the algorithm restarts in Step 2. Since the matrix A will not be generated completely, we can use the norm of its approximant Si to deﬁne a stopping criterion. This norm can be computed recursively as it is described in Step 3.4. However, since the whole matrix A will not be generated while

3.3 Bibliographic Remarks

129

using the partially pivoted ACA algorithm, it is necessary to check the control vectors c and r before stopping the algorithm. Note that these vectors contain the sums of absolute values of all elements generated. If, for example, there / I with ci∗ = 0 then the row i∗ has not yet contributed to is some index i∗ ∈ the matrix. It can happen that this row contains relevant information, and, therefore, we have to restart the algorithm in Step 3. The same argumentation is valid for the columns. The only diﬀerence is, that the crosses after this restart will be generated on the diﬀerent way: ﬁrst prescribed column and then automatically chosen row. Thus, Algorithm 3.9 can be used not only for dense matrices but also for reducible, and even for sparse matrices containing only few non-zero entries. Algorithm 3.9 requires only O(r2 (N + M )) arithmetical operations and its memory requirement is O(r(N + M )). Thus, this algorithm is perfect for large matrices. All approximations of boundary element matrices of the next chapter will be generated with the help of Algorithm 3.9.

3.3 Bibliographic Remarks The history of asymptotically optimal approximations of dense matrices is now about 20 years old. It starts with the paper [93] by V. Rokhlin. The boundary value problem for a partial diﬀerential equation was transformed to a Fredholm boundary integral equation of the second kind. The Nystr¨ om method was used for the discretisation, leading to a dense large system of linear equations. This system was solved iteratively using the generalised conjugate residual algorithm. This algorithm requires matrix-vector multiplications, which were realised in a “fast” manner, leading to optimal costs of order O(N ), or O(N log(N )) for the whole procedure. Then, the method, which was called Fast Multipole Method, was developed in the papers [20, 37, 38] for large-scale particle simulations in problems of plasma physics, ﬂuid dynamics, molecular dynamics, and celestial mechanics. The method was signiﬁcantly improved in [39]. Later, the Fast Multipole Method was successfully applied to a variety of problems. In [39, 84], for example, we can ﬁnd its application to the Laplace equation. In [83], the authors apply the Fast Multipole Method to the system of linear elastostatics discretised by the use of a Galerkin Boundary Element Method. Many papers on the Fast Multipole Method are devoted to the Helmholtz equation, see, for example, [2, 3], where the problem of acoustic scattering was considered, and [94]. The next method, introduced in [44], is called Panel clustering. This method was also applied to the potential problems in [43, 44, 46] and for the system of linear elastostatics in [51]. A further possibility to solve boundary integral equations on an asymptotically optimal way is based on the use of Wavelets. This research starts with the papers [1, 11], where the dense matrices arising from discretisation

130

3 Approximation of Boundary Element Matrices

of integral operators were transformed into a sparse form using orthogonal or bi-orthogonal systems of compactly supported wavelets. The cost of the matrix-vector multiplication was reduced from the straightforward O(N 2 ) number of operations to O(N log(N )) or even O(N ). In two papers [27, 28], the authors study the stability and the convergence of the wavelet method for pseudodiﬀerential operator equations as well as their fast solution based on the matrix approximation. A diﬀerent wavelet technique was applied in [115, 116], leading in [67] to an optimal algorithm with O(N ) complexity. For recent results, see also [48, 49]. In [36], the authors consider an algebraic approach for the approximation of dense matrices based on the use of some their original entries. The Adaptive Cross Approximation method was introduced in [7] for Nystr¨ om type matrices and in [9] for collocation matrices arising from boundary integral equations. This method uses a hierarchical decomposition (cf. [41, 42]) of the matrix in a system of blocks. There are several applications of this method to diﬀerent problems. The ACA was applied to potential problems in [9, 87]. In [8, 10], the ACA was used for the approximation of matrices arising from the radiation heat transfer equation. The applications of the ACA to electromagnetic problems can be found in [16, 63, 64, 65]. In [16] and in [31], a comparison of the ACA method with the Fast Multipole Method was given. In [110, 111, 113, 119], the boundary integral equations arising from the Helmholtz equation were solved using the ACA method. Finally, we refer to [69], where an algebraic multigrid preconditioners were constructed for the boundary integral formulations of potential problems approximated by using the Adaptive Cross Approximation algorithm.

4 Implementation and Numerical Examples

4.1 Geometry Description In this section we describe some surfaces which will be used for numerical examples in the following sections. We show the geometry of these surfaces and give the number of elements and nodes. Furthermore, the corresponding cluster structures will be shown. 4.1.1 Unit Sphere The most simple smooth surface Γ = ∂Ω for Ω ⊂ R3 is the surface of the unit sphere,

(4.1) Γ = x ∈ R3 : |x| = 1 . As an appropriate discretisation of Γ , we consider the icosahedron that is uniformly triangulated before being projected onto the circumscribed unit sphere. On this way we obtain a sequence {ΓN } of almost uniform meshes on the unit sphere, which are shown in Figs. 4.1–4.2 for diﬀerent numbers of boundary elements N . This sequence allows to study the convergence of boundary element methods for diﬀerent examples. In Fig. 4.3 the clusters of the levels 1 and 2 obtained with Alg. 3.1 for N = 1280 are presented. In Fig. 4.4 a typical admissible cluster pair is shown. 4.1.2 TEAM Problem 10 Now we consider the TEAM problem 10 (cf. [75]). TEAM is an acronym for Testing Electromagnetic Analysis Methods, which is a community that creates benchmark problems to test ﬁnite element analysis software. An exciting coil is set between two steel channels and a thin steel plate is inserted between the channels. Thus, the domian consists of four disconnected parts. The coarsest

132

4 Implementation and Numerical Examples 1

1 0.5

0.5 0

0 -0.5

-0.5 -1 1 1

-1 1

0.5

0.5

0

0

-0.5 -0.5 -1 -1

-1

-0.5 -0.5

0 0

0.5 0.5

1

Fig. 4.1. Discretisation of the unit sphere for N = 80 and N = 320 1

1

0.5

0.5

0

0

-0.5

-0.5

-1 1 1

-1 1 1

0.5

0.5

0

0

-0.5

-0.5

-1 -1

-1 -1 -0.5

-0.5 0

0 0.5

0.5 1

1

Fig. 4.2. Discretisation of the unit sphere for N = 1280 and N = 5120

mesh of this model contains N = 4928 elements. We perform two uniform mesh reﬁnements in order to get meshes with N = 19712 and N = 78848 elements, respectively. The initial mesh for N = 4928 is shown in Fig. 4.5. The speciality of this model is an extremely thin chink (less then 0.2% of the model size) between the steel plate and the channels. Another speciality of it is the very ﬁne discretisation close to the edges of the channels. In Fig. 4.6 the clusters of the levels 1 and 2 for N = 4928 are shown.

4.1 Geometry Description -1 -0.5 0 0.5 1 1

0.5

0 -0.5 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 1

0.5

0 -0.5 -1 -1 -0.5 0 0.5 1

Fig. 4.3. Clusters of the level 1 and 2 for N = 1280

133

134

4 Implementation and Numerical Examples

1 0.5 0 -0.5 -1 1

0.5

0 -0.5 -1 1 0.5 0 -0.5 -1 Fig. 4.4. An admissible cluster pair for N = 1280

4.1.3 TEAM Problem 24 The test rig consists of a rotor and a stator (cf. [92]). The stator poles are ﬁtted with coils as shown in Fig. 4.7. The rotor is locked at 22◦ with respect to the stator, providing only a small overlap between the poles. The coarsest mesh of this model is shown in Fig. 4.7 and contains N = 4640 elements. We perform two uniform mesh reﬁnements in order to get meshes with N = 18560 and N = 74240 elements, respectively. This model consists of four independent parts. In Fig. 4.8 the clusters of the levels 1 and 2 for N = 4928 are shown. 4.1.4 Relay The simply connected domain shown in Fig. 4.9 will be considered as model for a relay. The speciality of this domain is the small air gap between the kernel and the armature. Its surface contains N = 4944 elements. We perform two uniform mesh reﬁnements in order to get meshes with N = 19776 and N = 79104 elements, respectively. In Fig. 4.10 the corresponding clusters can be seen.

4.2 Laplace Equation

135

50 100

0

-50

0

-100 100 -50 0 -100

50 100

Fig. 4.5. TEAM problem 10 for N = 4928

4.1.5 Exhaust manifold A simpliﬁed model of an exhaust manifold is shown in Fig. 4.11. Its surface contains N = 2264 elements. We perform two uniform mesh reﬁnements in order to get meshes with N = 9056 and N = 36224 elements, respectively. The clusters of the level 1 and 2 are presented in Fig. 4.12.

4.2 Laplace Equation In this section we consider some numerical examples for the Laplace equation −Δu(x) = 0 where u is an analytically given harmonic function. 4.2.1 Analytical solutions Particular solutions of the Laplace equation (4.2) are, for example,

(4.2)

136

4 Implementation and Numerical Examples

50 100

0

-50

0

-100 100 -50 0 -100

50 100

50 100

0

-50

0

-100 100 -50 0 -100

50 100

Fig. 4.6. Clusters of the level 1 and 2, TEAM problem 10 for N = 4928

4.2 Laplace Equation

40 20 0 -20 -40 -100 100

137

100 50 0 -50 -50

0 50 100

-100

Fig. 4.7. TEAM problem 24 for N = 4640

Φk1 ,k2 ,k3 (x) = exp k1 x1 + k2 x2 + k3 x3 , k12 + k22 + k32 = 0 , Φ0,k2 ,k3 (x) = (a + b x1 ) exp k2 x2 + k3 x3 , k22 + k32 = 0 ,

(4.3)

Φ0,0,0 (x) = (a1 + b1 x1 )(a2 + b2 x2 )(a3 + b3 x3 ) . Here, k1 , k2 , and k3 are arbitrary complex numbers satisfying the corresponding conditions. Thus, diﬀerent products of real valued linear, exponential, trigonometric and hyperbolic functions can be chosen for numerical tests, if we consider interior boundary value problems in a three-dimensional, open, and bounded domain Ω ⊂ R3 . Furthermore, the fundamental solution of the Laplace equation (cf. (1.7)) u∗ (x, y˜) =

1 1 , 4π |x − y˜|

(4.4)

can be considered as a particular solution of the Laplace equation for both, interior (x ∈ Ω , y˜ ∈ Ω e = R3 \ Ω), and exterior (x ∈ Ω e , y˜ ∈ Ω) boundary value problems. 4.2.2 Discretisation, Approximation and Iterative Solution We solve the interior Dirichlet, Neumann and mixed boundary value problems, as well as an interface problem using a Galerkin boundary element method

138

4 Implementation and Numerical Examples

40 20 0 -20 -40 -100 100

100 50 0 -50 -50

0 50 100

-100

40 20 0 -20 -40 -100 100

100 50 0 -50 -50

0 50 100

-100

Fig. 4.8. Clusters of the level 1 and 2, TEAM problem 24 for N = 4640

(cf. Section 2). Piecewise linear basis functions ϕ will be used for the approximation of the Dirichlet datum γ0int u and piecewise constant basis functions ψk for the approximation of the Neumann datum γ1int u. We will use the L2 projection for the approximation of the given part of the Cauchy data. The boundary element matrices Vh , Kh and Dh are generated in approximative form using the partially pivoted ACA algorithm with a variable relative accuracy ε1 . The resulting systems of linear equations are solved using some variants of the Conjugate Gradient method (CGM) with or without preconditioning up to a relative accuracy ε2 = 10−8 .

4.2 Laplace Equation

139

5 0 -5 10 7.5 5 2.5 0 0 5 10 15

Fig. 4.9. Relay for N = 4944

4.2.3 Generation of Matrices The most important matrices to be generated while using the Galerkin boundary element method are the single layer potential matrix Vh and the double layer potential matrix Kh , having the entries 1 1 dsy dsx for k, = 1, . . . , N , Vh [k, ] = (4.5) 4π |x − y| τk τ

and 1 Kh [k, j] = 4π

τk Γ

(x − y, n(y)) ϕj (y) dsy dsx |x − y|3

(4.6)

for k = 1, . . . , N and j = 1, . . . , M (cf. Section 2.3). The analytical evaluation of these integrals seems to be impossible in general. Thus, some numerical quadrature rules have to be involved. These quadrature formulae produce some additional numerical errors in the whole procedure. However, it is possible to compute the inner integrals of the entries (4.5)–(4.6), namely the integrals 1 1 dsy (4.7) S(τ, x) = 4π |x − y| τ

and

140

4 Implementation and Numerical Examples 5 0 -5 10 7.5 5 2.5 0 0 5 10 15 5 0 -5 10 7.5 5 2.5 0 0 5 10 15

Fig. 4.10. Clusters of the level 1 and 2, Relay for N = 4944

4.2 Laplace Equation

141

0.05 0 -0.05 0.1

-0.1 0 0 0.1 0.2 0 2

Fig. 4.11. Exhaust manifold for N = 2264

1 Di (τ, x) = 4π

τ

(x − y, n(y)) ψτ,i (y) dsy |x − y|3

for i = 1, 2, 3 .

(4.8)

Here, τ ⊂ R3 is a plane triangle having the nodes x1 , x2 , x3 , and ψτ,i is the piecewise linear function (2.6) which corresponds to the node xi , i.e. ψτ,i (xj ) = δij , j = 1, 2, 3 . The explicit form of these functions can be seen in Appendix C.2. By the use of the functions (4.7)–(4.8), the matrix entries (4.5)–(4.6) can be rewritten as follows: 1 Vh [k, ] = (4.9) S(τ , x) dsx + S(τk , x) dsx 2 τk

τ

for k, = 1, . . . , N and Kh [k, j] =

Di : xi (τ )=xj (τ, x) dsx

(4.10)

τ ∈I(j) τk

for k = 1, . . . , N , j = 1, . . . , M . Note that we have used the symmetrisation for the entries of the single layer potential in (4.9). In (4.10), the summation takes place over all triangles τ containing the node xj . The index i ∈ {1, 2, 3} of the function Di to be integrated over τk , is chosen in such a way, that the node i of the triangle τ is xj . We are not going to compute the latter integrals in (4.9)–(4.10) in a closed form. Thus, some numerical integration

142

4 Implementation and Numerical Examples

0.05 0 -0.05 0.1

-0.1 0 0 0.1 0.2 0 2

0.05 0 -0.05 0.1

-0.1 0 0 0.1 0.2 0 2

Fig. 4.12. Clusters of the level 1 and 2, Exhaust manifold for N = 2264

4.2 Laplace Equation

143

has to be involved. If we denote by NG the number of integration points, by ωm the weights of the quadrature, and by xτ,m the integration points within the triangle τ , then the exact entries (4.9)–(4.10) can be approximated by ⎛ ⎞ Ng 1 ωm S(τ , xτk ,m ) + S(τk , xτ ,m ) ⎠ Vh [k, ] ≈ ⎝ (4.11) 2 m=1 for k, = 1, . . . , N and Kh [k, j] ≈

Ng

ωm Di : xi (τ )=xj (τ, xτk ,m )

(4.12)

τ ∈I(j) m=1

for k = 1, . . . , N and j = 1, . . . , M . For our numerical tests, we have used a 7-point quadrature rule, see Appendix C.1 for more details. 4.2.4 Interior Dirichlet Problem Here we solve the Laplace equation (4.2) together with the boundary condition γ0int u(x) = g(x) for x ∈ Γ , where Γ is a given surface. The variational problem (1.16) 1 I + K g, w = for all w ∈ H −1/2 (Γ ) V t, w 2 Γ Γ is discretised, which leads to a system of linear equations (2.16) 1 Vh Mh + Kh g . t = 2 Since the matrix Vh is symmetric and positive deﬁnite, the classical Conjugate Gradient method (CGM) is used as solver. Unit sphere The analytical solution is taken in the form (4.3). For x = (x1 , x2 , x3 ) ∈ Ω we consider the harmonic function u(x) = Re Φ0,2π,ı2π = (1 + x1 ) exp(2π x2 ) cos(2π x3 )

(4.13)

as a test solution of the Laplace equation (4.2). The results of the computations are shown in Tables 4.1 and 4.2. The number of boundary elements is listed in the ﬁrst column of these tables. The second column contains the number of nodes, while in the third column of Table 4.1, the prescribed accuracy for the ACA algorithm for the approximation of both matrices Kh ∈ RN ×M and Vh ∈ RN ×N is given. The fourth column of this table shows the memory requirements in MByte for the approximate double layer potential matrix Kh . The quality of this approximation in percentage of the original matrix is listed

144

4 Implementation and Numerical Examples Table 4.1. ACA approximation of the Galerkin matrices Kh and Vh N

80 320 1280 5120 20480 81920

M 42 162 642 2562 10242 40962

ε1 −2

1.0 · 10 1.0 · 10−3 1.0 · 10−4 1.0 · 10−5 1.0 · 10−6 1.0 · 10−7

MByte(Kh )

%

MByte(Vh )

%

0.03 0.26 2.45 20.05 149.19 1085.0

97.8 65.6 39.1 20.0 9.3 4.2

0.02 0.21 1.94 15.72 115.83 837.50

48.7 27.2 15.5 7.9 3.6 1.6

Fig. 4.13. Partitioning of the BEM matrices for N = 5120 and M = 2562

in the next column. The corresponding values for the single layer potential matrix Vh can be seen in the columns six and seven. The partitioning of the matrix for N = 5120 as well as the quality of the approximation of single blocks is shown in Fig. 4.13. The left diagram in Fig. 4.13 shows the symmetric single layer potential matrix Vh , while the rectangular double layer potential matrix Kh is depicted in the right diagram. The legend indicates the percentage of memory needed for the ACA approximation of the blocks compared to the full memory. Further numerical results are shown in Table 4.2. The third column in Table 4.2 shows the number of Conjugate Gradient iterations needed to reach the prescribed accuracy ε2 . The relative L2 error for the Neumann datum Error1 =

γ1int u − t˜h L2 (Γ ) γ1int uL2 (Γ )

(4.14)

is given in the fourth column. The next column represents the rate of convergence for the Neumann datum, i.e. the quotient between the errors in two

4.2 Laplace Equation

145

Table 4.2. Accuracy of the Galerkin method, Dirichlet problem N

M

80 320 1280 5120 20480 81920

42 162 642 2562 10242 40962

Iter 22 32 45 56 72 94

Error1 −1

9.34 · 10 5.06 · 10−1 2.23 · 10−1 1.04 · 10−1 5.11 · 10−2 2.53 · 10−2

CF1 – 1.85 2.27 2.14 2.03 2.02

Error2 −0

7.29 · 10 3.29 · 10−1 3.53 · 10−2 3.54 · 10−3 4.11 · 10−4 4.30 · 10−5

CF2 – 22.16 9.32 9.97 8.61 9.56

consecutive lines of column four. Finally, the last two columns show the absolute error (cf. (2.19)) in a prescribed inner point x∗ ∈ Ω, ˜(x∗ )| , x∗ = (0.250685, 0.417808, 0.584932) , (4.15) Error2 = |u(x∗ ) − u for the value u ˜(x∗ ) obtained using an approximate representation formula (2.18). Table 4.2 obviously shows a linear convergence O(N −1/2 ) = O(h) of the Galerkin boundary element method for the Neumann datum in the L2 norm. It should be noted that this theoretically guaranteed convergence order can already be observed when approximating the matrices Kh and Vh with much less accuracy as it was used to obtain the results in Table 4.1. However, this high accuracy is necessary in order to be able to observe the third order (or even better) pointwise convergence rate within the domain Ω presented in the last two columns of Table 4.2. Especially for N = 81920, a very high accuracy of ε1 = 1.0 · 10−7 of the ACA approximation is necessary. In Figs. 4.14-4.15, the given Dirichlet datum and computed Neumann datum for N = 5120 boundary elements and M = 2562 nodes are presented. The numerical curve obtained when using an approximate representation formula in comparison with the curve of the exact values (4.13) along the line ⎞ ⎞ ⎛ ⎛ 0.6 −0.3 (4.16) x(t) = ⎝ −0.5 ⎠ + t ⎝ 1.0 ⎠ , 0 ≤ t ≤ 1 1.4 −0.7 inside of the domain Ω is shown in Fig. 4.16 for N = 80 (left plot) and for N = 320 (right plot). The values of the numerical solution u ˜ and of the analytical solution u have been computed in 512 points uniformly placed on the line (4.16). The thick dashed line represents in these ﬁgures the course of the analytical solution (4.13), while the thin solid line shows the course of the numerical solution u ˜. The values of the variable x1 along the line (4.16) are used for the axis of abscissas. The next Fig. 4.17 shows these curves for N = 1280 (left plot) and on the zoomed interval [0.2, 0.3] (right plot) with respect to the variable x1 in order to see the diﬀerence between them. It is almost impossible to see any optical diﬀerence between the numerical and analytical curves for higher values of N . Note that the point x∗ in (4.15) is

146

4 Implementation and Numerical Examples 5.8149E02

1

0.5

0 -1

1.4856E02

-0.5

-0.5 -1 -1

0 -0.5 0.5

0 0.5 1 1

2.8438E02

Fig. 4.14. Given Dirichlet datum for the unit sphere, N = 5120 -1

3.6811E03 -0.5 0 0.5 1 1

0.5 7.7897E02 0 -0.5 -1 -1 -0.5 0 0.5 1

2.1232E03

Fig. 4.15. Computed Neumann datum for the unit sphere, N = 5120

4.2 Laplace Equation 0

147

0

-5

-2.5

-10

-5

-15

-7.5 -10

-20

-12.5 -25 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

-15 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

Fig. 4.16. Numerical and analytical curves for N = 80 and N = 320, Dirichlet problem 0

-10

-2.5

-11

-5 -12 -7.5 -13

-10

-14

-12.5 -15 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

-15 0.2

0.22

0.24

0.26

0.28

0.3

Fig. 4.17. Numerical and analytical curves for N = 1280, Dirichlet problem

chosen close to the minimum of the function u along the line, where the error seems to reach its maximum. TEAM Problem 10 The analytical solution is now taken in the form (4.4) with y˜ = (0, 60, 50) . The results of the computations are shown in Tables 4.3 and 4.4. The third Table 4.3. ACA approximation of the Galerkin matrices Kh and Vh , Dirichlet problem N 4928 19712 78848

M 2470 9862 39430

ε1 −2

1.0 · 10 1.0 · 10−3 1.0 · 10−4

MByte(Kh )

%

MByte(Vh )

%

14.03 131.85 1190.00

15.11 8.9 5.0

7.46 65.22 604.56

4.0 2.2 1.3

column shows the number of iterations required by the Conjugate Gradient method with diagonal preconditioning

(4.17) D = diag |τ | , = 1, . . . , N .

148

4 Implementation and Numerical Examples Table 4.4. Accuracy of the Galerkin method, Dirichlet problem N

4928 19712 78848

M 2470 9862 39430

Iter 91 183 248

Error1

CF1

−1

6.02 · 10 1.99 · 10−1 1.13 · 10−1

– 3.02 1.76

Error2 −5

2.55 · 10 4.69 · 10−6 5.22 · 10−7

CF2 – 5.44 8.98

The last two columns of Table 4.4 show the absolute error (cf. (2.19)) in a prescribed inner point x∗ ∈ Ω, ˜(x∗ )| , x∗ = (0.0, 90.0, 49.7943) , Error2 = |u(x∗ ) − u

(4.18)

for the value u ˜(x∗ ) obtained using the approximate representation formula (2.18). All other entries in these tables have the same meaning as those displayed in Tables 4.1-4.2. In Figs. 4.18–4.19 the given Dirichlet datum and the computed Neumann datum for N = 4928 boundary elements and M = 2470 nodes are presented. The numerical curve obtained when using the approxi8.4120E03 -100 -50 0 50 100

4.3880E03 50 0 0

-100 -10

-50 0

100 3.6407E04

Fig. 4.18. Given Dirichlet datum for the TEAM problem 10

mate representation formula in comparison with the curve of the exact values (4.4) along the line

4.2 Laplace Equation

149

4.6686E04 -100 -50 0 50 100

3.7515E05 50 0 0

-100 -10

-50 0

100 3.9183E04

Fig. 4.19. Computed Neumann datum for the TEAM problem 10

⎛

⎞ ⎛ ⎞ 0.0 0.0 x(t) = ⎝ 90.0 ⎠ + t ⎝ 0.0 ⎠ , 0 ≤ t ≤ 1 −49.99 99.98

(4.19)

inside of the domain Ω is shown in Fig. 4.20 for N = 4928. The values of the numerical solution u ˜ and of the analytical solution u have been computed in 512 points uniformly placed on the line (4.19). The thick dashed line represents the course of the analytical solution (4.4), while the thin solid line shows the course of the numerical solution u ˜. The values of the variable x3 along the line (4.19) are used for the axis of abscissas. The right plot in this ﬁgure shows a zoomed picture on the interval [40, 49.99] with respect to the variable x3 . Note that the end of the line (4.19) is very close to the boundary of the domain, which lies at x3 = 50. Thus, the loss of accuracy of the numerical representation formula close to the boundary can be clearly seen. The courses of the numerical solutions obtained for N = 19712 (left plot) and N = 78848 (right plot) are shown in Fig. 4.21. They do not distinguish optically from the course of the analytical solution on the whole interval. Thus, we show only the zoomed pictures. 4.2.5 Interior Neumann Problem We consider the interior Neumann boundary value problem with the boundary condition γ1int u(x) = g(x) for x ∈ Γ . The variational problem (1.29)

150

4 Implementation and Numerical Examples

0.0025

0.002675

0.002

0.002625

0.0015

0.002575

0.00265

0.0026

0.00255 0.001

0.002525 -40

-20

0

20

40

40

42

44

46

48

50

Fig. 4.20. Numerical and analytical curves for N = 4928, Dirichlet problem 0.00266 0.00264

0.00264

0.00262

0.00262 0.0026

0.0026

0.00258

0.00258

0.00256

0.00256

0.00254

0.00254 0.00252

0.00252 40

42

44

46

48

40

50

42

44

46

48

50

Fig. 4.21. Numerical and analytical curves for N = 19712 and N = 78848

D¯ u, v Γ

v, 1 + u ¯, 1 Γ

Γ

=

1 2

I − K g, v + α v, 1 Γ

Γ

1/2

for all v ∈ H∗∗ (Γ ), is discretised and leads to a system of linear equations (cf. (2.31)) 1 Mh − Kh g + α a , = Dh + a a u 2 where the vector a ∈ RM contains the integrals of the piecewise linear basis functions ϕ over the surface Γ , a =

1 |supp ϕ | , = 1, . . . , M . 3

The symmetric and positive deﬁnite system is then solved using a Conjugate Gradient method up to the relative accuracy ε2 = 10−8 . Unit Sphere We consider again the harmonic function (4.13) as the exact solution. The results for the ACA approximation of the matrix Dh ∈ RM ×M are presented in Table 4.5. The corresponding results for the matrix Kh are identical to those already presented in Table 4.1. Note that in this example, the Galerkin matrix with piecewise linear basis functions for the hypersingular operator is generated according to (1.9). The partitioning of the matrices Kh and Dh for

4.2 Laplace Equation

151

Table 4.5. ACA approximation of the Galerkin matrix Dh , Neumann problem N 80 320 1280 5120 20480 81920

M

ε1

MByte(Dh )

%

0.01 0.10 1.02 8.67 64.75 446.13

51.2 48.1 32.3 17.3 8.09 3.49

−2

1.0 · 10 1.0 · 10−3 1.0 · 10−4 1.0 · 10−5 1.0 · 10−6 1.0 · 10−7

42 162 642 2562 10242 40962

N = 5120 and M = 2562 as well as the quality of the approximation of the single blocks are shown in Fig. 4.22. The left diagram in Fig. 4.22 shows the

Fig. 4.22. Partitioning of the BEM matrices for N = 5120 and M = 2562.

rectangular double layer potential matrix KhT ∈ RM ×N , while the symmetric hypersingular matrix Dh is depicted in the right diagram. The legend indicates the percentage of memory needed for the ACA approximation of the blocks compared to the full memory. The accuracy obtained for the whole numerical Table 4.6. Accuracy of the Galerkin method, Neumann problem N 80 320 1280 5120 20480 81920

M 42 162 642 2562 10242 40962

Iter 10 14 17 25 35 51

Error1 −1

7.63 · 10 2.72 · 10−1 6.02 · 10−2 1.37 · 10−2 3.28 · 10−3 8.02 · 10−4

CF1 – 2.81 4.52 4.39 4.18 4.09

Error2 −1

9.37 · 10 1.95 · 10−0 4.27 · 10−1 1.01 · 10−1 2.49 · 10−2 6.13 · 10−3

CF2 – – 4.56 4.22 4.08 4.06

procedure is presented in Table 4.6. The numbers in this table have the same meaning as in Table 4.2. The third column shows the number of iterations required by the Conjugate Gradient method without preconditioning. Note

152

4 Implementation and Numerical Examples

that the convergence of the Galerkin method for the unknown Dirichlet datum in the L2 norm Error1 =

h L2 (Γ ) γ0int u − u

(4.20)

γ0int uL2 (Γ )

is now quadratic corresponding to the error estimate (2.32). Also in the inner point x∗ (cf. (4.15)) we now observe the quadratic convergence (7th column) as it was predicted in (2.34) instead of the cubic order obtained for the Dirichlet problem (cf. Table 4.2). This fact is clearly illustrated in Figs. 4.23–4.24, where the convergence of the boundary element method can be seen optically. The results obtained for N = 80 are plotted in Fig. 4.23 (left plot). The numerical

0

0

-5 -5 -10 -10 -15 -15 -20 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Fig. 4.23. Numerical and analytical curves for N = 80 and N = 320, Neumann problem

curve in Fig. 4.23 (right plot) is notedly better than the previous one. However, its quality is not as high as the one of the corresponding curve obtained solving the Dirichlet problem (cf. Fig. 4.16). The next Fig. 4.24 shows the same curves for N = 1280 and for N = 5120. Here, we do not need to zoom the pictures in order to see the diﬀerence between the numerical and the analytical curves.

0

0

-2.5

-2.5

-5

-5

-7.5

-7.5

-10

-10

-12.5

-12.5

-15 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

-15 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

Fig. 4.24. Numerical and analytical curves for N = 1280 and N = 5120, Neumann problem

4.2 Laplace Equation

153

Exhaust Manifold The analytical solution is taken in the form (4.4) with y˜ = (0, 0, 0.06) . The results of the computations are reported in Tables 4.7 and 4.8. The third Table 4.7. ACA approximation of the Galerkin matrices Kh and Vh , Neumann problem N

M

ε1

MByte(Kh )

%

MByte(Vh )

%

2264 9056 36224

1134 4530 18114

1.0 · 10−3 1.0 · 10−4 1.0 · 10−5

6.99 62.27 500.66

35.7 19.9 10.0

3.94 34.99 282.44

10.1 5.6 2.8

Table 4.8. Accuracy of the Galerkin method, Neumann problem N 2264 9056 36224

M 1134 4530 18114

Iter 72 110 163

Error1 −2

2.12 · 10 4.95 · 10−3 1.13 · 10−3

CF1 – 4.3 4.4

Error2 −3

2.69 · 10 5.36 · 10−4 1.07 · 10−4

CF2 – 5.0 5.0

column shows the number of iterations required by the Conjugate Gradient method with diagonal preconditioning,

D = diag |supp ϕ | , = 1, . . . , M . (4.21) The last two columns of Table 4.8 show the absolute error (cf. (2.34)) in a prescribed inner point x∗ ∈ Ω, Error2 = |u(x∗ ) − u ˜(x∗ )| , x∗ = (−0.0112524, 0.1, −0.05) ,

(4.22)

for the value u ˜(x∗ ) obtained using an approximate representation formula (2.33). All other entries in these tables have the usual meaning. The quadratic convergence of the Dirichlet datum in the L2 norm as well as the quadratic convergence (or even slightly better) in the inner point x∗ can be observed again. In Figs. 4.25–4.26 the given Dirichlet datum and computed Neumann datum for N = 2264 boundary elements and M = 1134 nodes are presented. The numerical curve obtained when using an approximate representation formula in comparison with the curve of the exact values (4.4) along the line ⎛ ⎞ ⎛ ⎞ −0.05 0.2 x(t) = ⎝ 0.1 ⎠ + t ⎝ 0.0 ⎠ , 0 ≤ t ≤ 1 (4.23) −0.05 0.0

154

4 Implementation and Numerical Examples 4.4393E00

0.1 0

0.05 2.3679E00 0 -0.05 -0.1 0 0.1 0.2 0 2

2.9645E01

Fig. 4.25. Computed Dirichlet datum, exhaust manifold 2.4101E02

0.05 1.0975E02 0 -0.05 -0.1

0.1 0 0

0.1 0.2 0 2

Fig. 4.26. Given Neumann datum, exhaust manifold

2.1509E01

4.2 Laplace Equation

155

inside of the domain Ω is shown in Fig. 4.27 for N = 2254 and N = 9056, while Fig. 4.28 shows the results obtained for N = 36224 (left plot). The right plot in this ﬁgure presents the same curve on a zoomed interval [−0.05, 0.05]. The values of the numerical solution u ˜ and of the analytical solution u have been computed in 512 points uniformly placed on the line (4.23). The thick dashed line represents the course of the analytical solution (4.4), while the thin solid line shows the course of the numerical solution u ˜. The values of the variable x1 along the the line (4.23) are used for the axis of abscissas. Again, a quite high accuracy of the Galerkin BEM can be observed.

0.525

0.525

0.5

0.5

0.475

0.475

0.45

0.45

0.425

0.425

0.4 0.375 -0.05

0.4 0

0.05

0.1

0.375 0.15 -0.05

0

0.05

0.1

0.15

Fig. 4.27. Numerical and analytical curves for N = 4944 and N = 19776

0.535 0.525 0.53

0.5 0.475

0.525

0.45

0.52

0.425

0.515

0.4 0.375 -0.05

0.51 0

0.05

0.1

0.15

-0.04

-0.02

0

0.02

0.04

Fig. 4.28. Numerical and analytical curves for N = 36224

4.2.6 Interior Mixed Problem Here, we consider the interior mixed boundary value problem (cf. (1.34)) −Δu(x) = 0

for x ∈ Ω ,

γ0int u(x) = g(x)

for x ∈ ΓD ,

γ1int u(x) = f (x)

for x ∈ ΓN ,

(4.24)

where the function g is the interior trace of the exact solution on the boundary Γ , while the function f denotes its interior conormal derivative on Γ . After

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4 Implementation and Numerical Examples

discretisation, the variational problem (1.35) leads to the skew symmetric system of linear equations (cf. (2.36)) 1 ¯ ¯h f Vh −Kh + K M −V¯h t h 2 = 1 ¯ ¯ ¯ g Kh Dh −Dh u 2 Mh − K h The matrix of the single layer potential Vh and the matrix of the double layer potential Kh are generated in an approximative form using the partially pivoted ACA algorithm 3.9 with increasing accuracy. The system of linear equations (2.41) is then solved using the Conjugate Gradient method for the Schur complement system as in (2.37) up to the relative accuracy ε2 = 10−8 . Note that this realisation requires an additional solution of a linear system with the single layer potential matrix in each iteration step. This system is solved again using a Conjugate Gradient method up to the relative accuracy ε2 = 10−8 . The matrix of the hypersingular operator is not generated explicitly. Its multiplication with a vector is realised using the matrix of the single layer potential as it is described in (2.27). Unit Sphere In the ﬁrst example, we prescribe the Dirichlet datum on the upper part of the unit sphere ΓD = {x ∈ Γ : x3 ≥ 0} and the Neumann datum on the lower part ΓN = {x ∈ Γ : x3 < 0}. The analytical solution is taken in the form (4.13) and the numerical results are presented in Tables 4.9–4.10. Of course, Table 4.9. Accuracy of the Galerkin method on the boundary, mixed problem N 80 320 1280 5120 20480

M 42 162 642 2562 10242

Iter1 8 11 16 24 36

Iter2 18-19 25-27 36-38 49-53 64-75

D − Error1 −1

5.97 · 10 2.33 · 10−1 5.00 · 10−2 1.14 · 10−2 2.73 · 10−3

CFD – 2.56 4.66 4.39 4.18

N − Error1 −1

8.78 · 10 4.54 · 10−1 2.11 · 10−1 1.02 · 10−1 5.07 · 10−2

CFN – 1.93 2.15 2.07 2.01

the results for the ACA approximation of the matrices Vh and Kh are the same as for the Dirichlet problem (cf. Table 4.1). The accuracy obtained for the mixed boundary value problem is presented in Table 4.9. The numbers in this table have the following meaning: The third column in Table 4.9 shows the number of Conjugate Gradient iterations without preconditioning needed to reach the prescribed accuracy ε2 for the Schur complement, while the fourth column indicates the numbers of Conjugate Gradient iterations without preconditioning needed in each iteration step for the system with the single layer potential matrix. Thus, the total number of iterations when solving a mixed

4.2 Laplace Equation

157

boundary value problem is much higher than for solving a pure Dirichlet or Neumann boundary value problem. The error for the Dirichlet datum and the convergence factor are shown in columns 5 and 6, while the corresponding error for the Neumann datum can be seen in columns 7 and 8. Note that the convergence of the Galerkin method for the unknown Dirichlet datum in the L2 norm (4.20) is quadratic (6th column), while the convergence of the Neumann datum (4.14) is linear (8th column). Corresponding to Table 4.1, the matrices Kh and Vh together with some additional memory will require more than 2 Gbyte of memory for N = 81902. Thus, we are not able to store both these matrices on a regular workstation. The error in the inner point x∗ Table 4.10. Accuracy of the Galerkin method in the inner point x∗ , mixed problem N

M

80 320 1280 5120 20480

42 162 642 2562 10242

Error2

CF2

−0

5.82 · 10 4.94 · 10−1 1.43 · 10−1 3.68 · 10−2 9.31 · 10−3

– 11.77 3.46 3.88 3.95

(cf. (4.15)) can be observed in Table 4.10. Here, quadratic convergence can be observed, at least asymptotically. TEAM Problem 10 Here, we consider the mixed boundary value problem for the Laplace equation in the domain presented in Fig. 4.5. The Dirichlet part of the boundary Γ is deﬁned by ΓD = {x ∈ Γ : x3 = 50}. Thus, Dirichlet boundary conditions are given only on the “top” of the coil. The analytical solution is taken in the form (4.4) with y˜ = (0.0, 60.0, 50.0) ∈ / Ω and the numerical results are presented in Table 4.11. The meaning of the values presented in this table is Table 4.11. Accuracy of the Galerkin method on the boundary, TEAM problem 10 N 4928 19712

M 2470 9862

Iter1 1061 1732

Iter2 90-95 118-123

D − Error1 −2

3.04 · 10 7.60 · 10−3

CFD – 4.00

N − Error1 −1

6.04 · 10 2.03 · 10−1

CFN – 2.98

the same as in Table 4.9. The third column in Table 4.11 shows the number of Conjugate Gradient iterations with diagonal preconditioning (4.21) needed to reach the prescribed accuracy ε2 for the Schur complement, while the fourth

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4 Implementation and Numerical Examples

column indicates the number of Conjugate Gradient iterations with diagonal preconditioning (4.17) needed in each iteration step for the system with the single layer potential matrix. Note that the number of iterations is rather high for this geometrically very complicated example. Thus, a more eﬀective preconditioning is required. The courses of the numerical solution inside of the domain Ω along the line (4.19) is very similar to those presented in Figs. 4.20–4.21. The Cauchy data can be seen in Figs. 4.18–4.19. TEAM Problem 24 Here, we consider the mixed boundary value problem for the Laplace equation in the domain presented in Fig. 4.7. The Dirichlet part of the boundary Γ is deﬁned by ΓD = {x ∈ Γ : x3 ≥ 0}. Thus, the Dirichlet boundary condition is given on the upper part of the symmetric surface Γ . The analytical solution / Ω and the numerical is taken in the form (4.4) with y˜ = (0.0, −80.0, 20.0) ∈ results are presented in Table 4.12 The meaning of the values presented in this Table 4.12. Accuracy of the Galerkin method on the boundary, TEAM problem 24 N 4640 18560

M 2320 9280

Iter1 48 71

Iter2 72-78 93-102

D − Error1 −2

2.56 · 10 4.67 · 10−3

CFD – 5.48

N − Error1 −1

3.08 · 10 1.48 · 10−1

CFN – 2.08

table is the same as in Table 4.11. We have used the same preconditioning as in the previous example. The number of iterations reported in the third and in the fourth columns of the table is now much less. In Figs. 4.29–4.30 the Dirichlet datum and the Neumann datum for N = 4944 boundary elements and M = 2474 nodes are presented. The numerical curve obtained when using an approximate representation formula in comparison with the curve of the exact values (4.4) along the line ⎞ ⎞ ⎛ ⎛ 0.0 0.0 (4.25) x(t) = ⎝ −100.0 ⎠ + t ⎝ 40.0 ⎠ , 0 ≤ t ≤ 1 0 0 inside of the domain Ω is shown in Fig. 4.31 for N = 4640. The values of the numerical solution u ˜ and of the analytical solution u have been computed in 512 points uniformly placed on the line (4.25). The thick dashed line represents the course of the analytical solution (4.4), while the thin solid line shows the course of the numerical solution u ˜. The values of the variable x2 along the the line (4.25) are used for the axis of abscissas. The course of the numerical approximation does not optically distinguish from the exact solution on the left plot of Fig. 4.31. Thus, we show the zoom of these curves on the interval [−84, −76] (right plot).

4.2 Laplace Equation

159

1.1279E02

5.8440E03

40 20 0 -20 -40 -100 100

100 50 0 -50 -50

0 50 100

-100 4.0923E04

Fig. 4.29. Computed Dirichlet datum for the TEAM problem 24 1.3945E03

5.5593E04

40 20 0 -20 -40 -100 100

100 50 0 -50 -50

0 50 100

-100 2.8372E04

Fig. 4.30. Computed Neumann datum for the TEAM problem 24

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4 Implementation and Numerical Examples

0.004

0.00398

0.0038 0.00396 0.0036 0.0034

0.00394

0.0032 0.00392 0.003 0.0028 -100

-90

-80

-70

-84

-60

-82

-80

-78

-76

Fig. 4.31. Numerical and analytical curves for N = 4640, Mixed problem

4.2.7 Inhomogeneous Interface Problem The purpose of this subsection is twofold. The ﬁrst goal is to illustrate the numerical solution of the Poisson equation by the use of a particular solution (cf. Subsection 1.1.7). The second goal is to illustrate the numerical solution for an interface problem by Boundary Element Methods (cf. Subsection 1.1.8). We consider the following interface problem −αi Δui (x) = fi (x)

for x ∈ Ω,

−αe Δue (x) = 0 for x ∈ Ω e ,

(4.26)

with transmission conditions describing the continuity of the potential and of the ﬂux, respectively, γ0int ui (x) = γ0ext ue (x) , αi γ1int ui (x) = αe γ1ext ue (x) and with the radiation condition ue (x) = O

1 |x|

for x ∈ Γ ,

(4.27)

as |x| → ∞.

If a particular solution upi of the interior Poisson equation is known, −αi Δupi (x) = fi (x)

for x ∈ Ω ,

then the above interface problem can be reformulated as follows (cf. Subsection 1.1.8). Introduce a new unknown function u∗i by ui = u∗i + upi and rewrite the interface problem in terms of the functions u∗i and ue −αi Δu∗i (x) = 0 for x ∈ Ω,

−αe Δue (x) = 0 for x ∈ Ω e ,

with new transmission conditions γ0int u∗i (x) = γ0ext ue (x) − γ0int upi (x) , αi γ1int u∗i (x) = αe γ1ext ue (x) − αi γ1int upi (x)

for x ∈ Γ .

4.2 Laplace Equation

161

Then, by the use of the interior and exterior Steklov-Poincar´e operators S int and S ext (cf. (1.13), (1.46)) γ1int u∗i = S int γ0int u∗i ,

γ1ext ue = −S ext γ0ext ue ,

we rewrite the interface problem as (cf. (1.59)) αi S int + αe S ext γ0int u∗i = −αi γ1int upi − αe S ext γ0int upi .

(4.28)

Once the Dirichlet datum γ0int u∗i is found, we solve the interior Dirichlet boundary value problem for the Neumann datum γ1int u∗i . The Cauchy data for the unknown functions ui and ue are then obtained via αe ext γ ue = γ1int u∗i + γ1int upi . γ0int ui = γ0ext ue = γ0int u∗i + γ0int upi , γ1int ui = αi 1 Unit Sphere Let Γ be the surface of the unit sphere (4.1). The constants αi , αe and the right hand side fi in (4.26) are αi = αe = 1 ,

fi (x) = 1 ,

for x ∈ Ω .

The exact solution of this simple model problem is ui (x) =

3 − |x|2 , x∈Ω, 6

ue (x) =

1 , x ∈ Ωe . 3 |x|

(4.29)

Consider the function 1 upi (x) = − x21 , for x ∈ Ω 2 as a particular solution of the Poisson equation. The Galerkin method with piecewise linear basis functions ϕ for the Dirichlet data γ0int ui = γ0ext ue and γ0int u∗i and with piecewise constant basis functions ψk for the Neumann data γ1int ui = γ1ext ue and γ1int u∗i will be used. The matrix of the single layer potential Vh and the matrix of the double layer potential Kh are generated in an approximative form using the partially pivoted ACA algorithm 3.9 with increasing accuracy ε1 . The resulting system of linear equations (cf. (2.45)) is then solved using the Conjugate Gradient method without preconditioning up to the relative accuracy ε2 = 10−8 . The accuracy obtained for the analytical solution (4.29) is presented in Table 4.13. The numbers in this table have the following meaning. The third column in Table 4.13 shows the number of Conjugate Gradient iterations needed to reach the prescribed accuracy ε2 for the linear system (2.45). The error for the Dirichlet datum and the convergence factor are shown in columns 4 and 5, while the corresponding error for the Neumann datum can be seen in columns 6 and 7. Note that the convergence of the Galerkin method for the unknown Dirichlet datum in the L2 norm (4.20) is quadratic (5th column), while the convergence of the Neumann datum (4.14) is linear (7th column).

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4 Implementation and Numerical Examples

Table 4.13. Accuracy of the Galerkin method on the boundary, interface problem N 80 320 1280 5120 20480

M 42 162 642 2562 10242

Iter 8 12 17 22 33

D − Error1 −2

8.47 · 10 2.22 · 10−2 5.59 · 10−3 1.40 · 10−3 3.50 · 10−4

CFD – 3.82 3.97 3.99 4.00

N − Error1 −1

1.65 · 10 7.99 · 10−2 3.88 · 10−2 1.92 · 10−2 9.55 · 10−3

CFN – 2.07 2.06 2.02 2.01

4.3 Linear Elastostatics In this section we consider two numerical examples for the mixed boundary value problem of linear elastostatics (cf. (1.79)) −

3 ∂ σij (u, x) = 0 ∂x j j=1

for x ∈ Ω, i = 1, 2, 3,

γ0int u(x) = g(x)

for x ∈ ΓD ,

γ1int u(x) = f (x)

for x ∈ ΓN ,

(4.30)

where u is the displacement ﬁeld of an elastic body initially occupying some bounded open domain Ω ∈ R3 with boundary Γ = Γ D ∪ Γ N . 4.3.1 Generation of Matrices The most important matrices to be generated while using the Galerkin boundary element method for the mixed boundary value problem (4.30) are the single layer potential matrix VhLame and the double layer potential matrix KhLame (cf. 2.4), having the representation (2.48) and (2.51), respectively. Thus, in addition to the single and double layer potential matrices (Vh and Kh ) for the Laplace operator, six additional dense matrices Vij,h ∈ RN ×N for 1 ≤ i ≤ j ≤ 3 have to be generated corresponding to (cf. (2.50)) 1 (xi − yi )(xj − yj ) Vij,h [k, ] = dsy dsx . 4π |x − y|3 τk τ

Using the abbreviation (cf. Appendix C.2.3) 1 (xi − yi )(xj − yj ) dsy , Sij (τ, x) = 4π |x − y|3 τ

the above entries can be written in a symmetrised form

4.3 Linear Elastostatics

Vij,h [k, ] =

1 2

Sij (τ , x) dsx +

τk

163

Sij (τk , x) dsx .

τ

The explicit form of the functions Sij can be seen in Appendix C.2.3. The remaining integrals in the above symmetric form of the matrix entries Vij,h can be computed numerically using a 7-point quadrature rule, see Appendix C.1. 4.3.2 Relay The geometry of the domain is shown in Fig. 4.32. The bottom of the relay is chosen to be the Dirichlet part ΓD of the boundary Γ and the boundary condition is homogeneous, i.e. ( ) γ0int u(x) = 0 , for x ∈ ΓD = x ∈ Γ : x3 = 0 . The remaining part of the boundary is then considered as the Neumann boundary, where only on the top of the domain inhomogeneous boundary conditions are formulated, 0 , x ∈ Γ : x3 < 10 , γ1int u(x) = 1 , x ∈ Γ : x3 = 10 . We choose the Young modulus E = 114 000 and the Poisson ratio ν = 0.24 that correspond to the values of steel. The original domain is shown in Fig. 4.32 for N = 4944 The matrix of the single layer potential Vh for the Laplace operator (cf. (2.49)), six matrices of the single layer potential Vh for the Lame operator (cf. (2.50)), and the matrix of the double layer potential Kh (cf. (2.51)) for the Laplace operator are generated in an approximative form using the partially pivoted ACA algorithm 3.9 with increasing accuracy. The system of linear equations is then solved using the Conjugate Gradient method for the Schur complement of the system (cf. (2.47)) up to the relative accuracy ε2 = 10−8 . Note that this realisation requests an additional solution of a linear system with the single layer potential matrix in each iteration step. This system is solved again using Conjugate Gradient method up to the relative accuracy ε2 = 10−8 . The matrix of the hypersingular operator is not generated explicitly. Its multiplication with a vector is realised using the matrix of the single layer potential as it is described in Section 2.4. The results of the approximation are presented in Tables 4.14–4.16. The number of boundary elements is listed in the ﬁrst column of these tables. The second column contains the number of nodes, while the prescribed accuracy for the ACA algorithm for the approximation of all matrices Kh ∈ RN ×M and Vh , Vk,h ∈ RN ×N , k, = 1, 2, 3 is given in the third column. The pairs of further columns of these tables show the memory requirements in MByte and the percentage of memory compared to the original matrix. The deformed

164

4 Implementation and Numerical Examples

5 0

5 10

0

15 -5 15

10

5

0

Fig. 4.32. Relay for N = 4944 Table 4.14. ACA approximation of the Galerkin matrices Vh and Kh N

M

ε1

Vh

%

Kh

%

4944 19776

2474 9890

1.0 · 10−4 1.0 · 10−5

37.24 258.65

20.0 8.7

52.96 326.45

56.8 10.9

Table 4.15. ACA approximation of the Galerkin matrices V11,h , V12,h and V13,h N 4944 19776

M 2474 9890

ε1 −4

1.0 · 10 1.0 · 10−5

V11,h

%

V12,h

%

V13,h

%

46.03 435.61

24.7 14.6

46.75 433.91

25.1 14.5

45.74 402.00

24.5 13.5

4.3 Linear Elastostatics

165

Table 4.16. ACA approximation of the Galerkin matrices V22,h , V23,h and V33,h N

M

4944 19776

2474 9890

ε1 −4

1.0 · 10 1.0 · 10−5

5

V22,h

%

V23,h

%

V33,h

%

47.21 463.43

25.3 15.5

46.78 415.98

25.1 13.9

45.75 421.66

24.5 14.1

0

5.3390E01 5 10

0

15 -5 15

10

2.5840E01

5

0

1.7101E02 Fig. 4.33. Deformation of the relay for N = 4944 Table 4.17. Number of iterations, Relay problem N

M

Iter1

Iter2

4944 19776

2474 9890

286 368

26-28 25-29

166

4 Implementation and Numerical Examples

domain can be seen from the same point of view in Fig. 4.33. In this ﬁgure, the real deformation is ampliﬁed by a factor 10. In Table 4.17, the number of iterations required by the Conjugate Gradient method is shown. The third column of this table shows the number of iterations for the Schur complement equation (2.47), while the fourth column shows the number of iterations required for the iterative solution of the linear system for the single layer potential in each iteration step. The required accuracy was ε2 = 10−8 for both systems. 4.3.3 Foam The geometry of the domain, which is a model for a metal foam, is shown in Fig. 4.34. The speciality of this domain is its multiple connectivity and rather small volume compared to its surface. There is only one discretisation of the domain with N = 28952 surface elements. The bottom and the top of the foam are chosen to be the Dirichlet part ΓD of the boundary Γ , and the boundary condition is homogeneous on the bottom, i.e. γ0int u(x) = 0 ,

for x ∈ Γ : x3 = 0 ,

while a prescribed constant displacement is posed on the top, i.e. γ0int u(x) = (0, 0, 0.1) ,

for x ∈ Γ : x3 = 15 .

The remaining part of the boundary is then considered as the Neumann boundary, where homogeneous boundary conditions are formulated: γ1int u(x) = 0 ,

for x ∈ Γ : 0 < x3 < 15 .

We choose the Young modulus E = 114 000 and the Poisson ratio ν = 0.24 that correspond to the values of steel. The original domain is shown in Fig. 4.34 for N = 28952. The matrix of the single layer potential Vh for the Laplace operator (cf. (2.49)), six matrices of the single layer potential Vh for the Lame operator (cf. (2.50)), and the matrix of the double layer potential Kh (cf. (2.51)) for the Laplace operator are generated in an approximative form using the partially pivoted ACA algorithm 3.9. The system of linear equations is then solved using a Conjugate Gradient method for the Schur complement system (cf. (2.47)) up to the relative accuracy ε2 = 10−8 . Note that this realisation requires an additional solution of a linear system with the single layer potential matrix in each iteration step. This system is solved again using a Conjugate Gradient method up to the relative accuracy ε2 = 10−8 . The matrix of the hypersingular operator is not generated explicitly. Its multiplication with a vector is realised using the matrix of the single layer potential as it is described in Section 2.4. The results of the approximation are presented in Tables 4.18–4.20. The number of boundary elements is listed in the ﬁrst column of these tables.

4.3 Linear Elastostatics

30 0

167

10 20

20

30 10 0 40

30

20

10 0

Fig. 4.34. Foam for N = 28952

The second column contains the number of nodes while in the third column the prescribed accuracy for the ACA algorithm for the approximation of all matrices Kh ∈ RN ×M and Vh , Vk,h ∈ RN ×N , k, = 1, 2, 3 is given. The pairs of further columns of these tables show the memory requirements in MByte and the percentage of memory compared to the original matrix. The Table 4.18. ACA approximation of the Galerkin matrices Vh and Kh N 28952

M 14152

ε1 −4

1.0 · 10

Vh

%

Kh

%

260.66

4.1

496.58

15.9

deformed domain can be seen from the same point of view in Fig. 4.35. In this ﬁgure, the real deformation is ampliﬁed by a factor 100. The number of iterations required by the Conjugate Gradient method is shown in Table 4.21. In this table, the third column shows the number of iterations for the Schur complement equation (2.47), while the number of iterations required for the

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4 Implementation and Numerical Examples

Table 4.19. ACA approximation of the Galerkin matrices V11,h , V12,h and V13,h N

M

28952

ε1 1.0 · 10

14152

−4

V11,h

%

V12,h

%

V13,h

%

398.36

6.2

417.94

6.5

418.55

6.5

Table 4.20. ACA approximation of the Galerkin matrices V22,h , V23,h and V33,h N

M

28952

ε1 1.0 · 10

14152

30 0

−4

V22,h

%

V23,h

%

V33,h

%

402.36

6.3

415.22

6.5

398.93

6.2

1.0000E01 10

20

20 30

10 0 40

30 4.4986E02 20

10 0

1.0028E02 Fig. 4.35. Deformation of the foam for N = 28952

iterative solution of the linear system for the single layer potential in each iteration step can be seen in the fourth column. The required accuracy was ε2 = 10−8 for both systems.

4.4 Helmholtz Equation In this section we consider some numerical examples for the Helmholtz equation

4.4 Helmholtz Equation

169

Table 4.21. The number of iterations, Foam problem N

M

Iter1

Iter2

28952

14152

253

19-21

−Δu(x) − κ2 u(x) = 0 ,

(4.31)

where u is an analytically given function. 4.4.1 Analytical Solutions Particular solutions of the Helmholtz equation (4.31) are, for example, Φk1 ,k2 ,k3 (x) = exp ı (k1 x1 + k2 x2 + k3 x3 ) , k12 + k22 + k32 = κ2 , (4.32) Φ0,k2 ,k3 (x) = (a + b x1 ) exp ı (k2 x2 + k3 x3 ) , k22 + k32 = κ2 , Φ0,0,0 (x) = (a1 + b1 x1 )(a2 + b2 x2 ) exp ı κ x3 . Here k1 , k2 and k3 are arbitrary complex numbers satisfying the corresponding conditions. Thus, diﬀerent products of linear, exponential, trigonometric, and hyperbolic functions can be chosen for numerical tests, if we consider interior boundary value problems in a three-dimensional open bounded domain Ω ⊂ R3 . Furthermore, the fundamental solution u∗ (x, y˜) =

1 eı κ |x−˜y| 4π |x − y˜|

(4.33)

can be considered as a particular solution of the Helmholtz equation (4.31) for both, interior (x ∈ Ω , y˜ ∈ Ω e = R3 \ Ω), and exterior (x ∈ Ω e , y˜ ∈ Ω) boundary value problems. 4.4.2 Discretisation, Approximation and Iterative Solution We solve the interior and exterior Dirichlet and Neumann boundary value problems using a Galerkin boundary element method (cf. Section 2). Piecewise linear basis functions ϕ will be used for the approximation of the Dirichlet datum γ0int u and piecewise constant basis functions ψk for the approximation of the Neumann datum γ1int u. We will use the L2 projection for the approximation of the given part of the Cauchy data. The boundary element matrices Vh , Kh , and Ch are generated in an approximative form using the complex valued version of the partially pivoted ACA algorithm 3.9 with a variable relative accuracy ε1 . The resulting systems of linear equations are solved using the GMRES method with or without preconditioning up to a relative accuracy ε2 = 10−8 .

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4 Implementation and Numerical Examples

4.4.3 Generation of Matrices The most important matrices to be generated while using the Galerkin boundary element method for boundary value problems for the Helmholtz equation (4.31) are the single layer potential matrix Vκ,h (cf. (2.53)), 1 Vκ,h [k, ] = 4π

τk τ

eıκ|x−y| dsy dsx , |x − y|

and the double layer potential matrix Kκ,h (cf. 2.58), 1 (x − y, n(y)) 1 − ı κ |x − y| eı κ|x−y| ϕj (y)dsy dsx . Kκ,h [k, j] = 4π |x − y|3 τk Γ

Furthermore, when solving the Neumann boundary value problem for the Helmholtz equation, the matrix of the hypersingular operator (cf. (2.62)) Dκ,h [i, j] = ıκ|x−y| 1 e (curlΓ ϕj (y), curlΓ ϕi (x))dsy dsx − 4π |x − y| Γ Γ

2

κ 4π

Γ Γ

eıκ|x−y| ϕj (y)ϕi (x)(n(x), n(y))dsy dsx |x − y|

has to be involved. The ﬁrst part of this formula corresponds for κ = 0 to the hypersingular operator for the Laplace equation, and, therefore, can be handled in the same way, i.e. these entries are some linear combinations of the entries of the matrix of the single layer potential Vκ,h . It remains to generate an additional matrix Cκ,h , having the entries Cκ,h [i, j] = Γ Γ

eıκ|x−y| ϕj (y)ϕi (x)(n(x), n(y))dsy dsx . |x − y|

(4.34)

To generate the entries of the single layer potential matrix Vκ,h numerically, we rewrite (2.53) as follows: Vκ,h [k, ] = V0,h [k, ] +

1 4π

τk τ

eıκ|x−y| − 1 dsy dsx . |x − y|

In the above, the entries V0,h [k, ] are the entries of the single layer potential matrix of the Laplace operator, and, therefore, can be computed as it was discussed in Section 4.2.3. The remaining double integral has no singularity for x → y, and can be computed numerically, using the 7-point quadrature rule (cf. Appendix C.1) for each triangle.

4.4 Helmholtz Equation

171

For the double layer potential matrix, the same idea leads to the following decomposition: Kκ,h [k, j] = 1 K0,h [k, j] + 4π

(1 − ı κ|x − y|)eı κ|x−y| − 1

τk Γ

(x − y, n(y)) |x − y|3

ϕj (y)dsy dsx .

Again, the ﬁrst part of this decomposition belongs to the double layer potential matrix of the Laplace operator, while the second part has no singularity for x → y. Thus, the 7-point quadrature rule can be applied again. However, the numerical integration with respect to the variable y has to be done over all triangles in the support of the basis function ϕj for each integration point with respect to the variable x. Therefore, the generation of the matrix entries for the double layer potential matrix for the Helmholtz equation is by far more complicated than for the Laplace equation. However, the most complicated numerical procedure is required when generating the entries of the matrix Cκ,h corresponding to (4.34). Using the same decomposition as for the previous matrices, we get ıκ|x−y| e −1 ϕj (y)ϕi (x)(n(x), n(y))dsy dsx . Cκ,h [i, j] = C0,h [i, j] + |x − y| Γ Γ

In the above, the second summand has no singularity for x → y, and the 7point quadrature rule can be applied again. Note that in this case, the quadrature rule has to be applied to every triangle in the support of the function ϕi , and, for each of its integration points, to every triangle in the support of the function ϕj . Furthermore, a symmetrisation is necessary in order to keep the symmetry of the matrix Cκ,h . Fortunately, the ﬁrst summand C0,h [i, j] does not require some additional analytical work. Since the normal vectors n(x) and n(y) are constant within the single triangles in the supports of the functions ϕi and ϕj , these integrals can be computed using the symmetrised combination of the analytical integration and of the 7-point quadrature rule. 4.4.4 Interior Dirichlet Problem Here we solve the Helmholtz equation (4.31) together with the Dirichlet boundary condition γ0int u(x) = g(x) for x ∈ Γ , where Γ is a given surface. The variational problem (1.106) 1 I + Kκ g, w for all w ∈ H −1/2 (Γ ) Vκ t, w Γ = 2 Γ is discretised and leads to a system of linear equations 1 Mh + Kκ,h g . t = Vκ,h 2

(4.35)

172

4 Implementation and Numerical Examples

The matrix Vκ,h of this system is symmetric. This property can be used in order to save computer memory while generating the matrix. However, the matrix is not selfadjoint, and, therefore, the Conjugate Gradient method can not be used. Thus, for an iterative solution of the system (4.35), the complex GMRES method will be used instead. Unit Sphere The analytical solution is taken in the form (4.32). For x = (x1 , x2 , x3 ) ∈ Ω we consider the function √ (4.36) u(x) = Φ0,−ı 2√3,4 = 4 x1 exp(2 3 x2 ) exp(ı 4 x3 ) , which satisﬁes the Helmholtz equation (4.31) for κ = 2. The results of the computations for this rather moderate wave number are shown in Tables 4.22 and 4.23. The number of boundary elements is listed in the ﬁrst column of Table 4.22. ACA approximation of the Galerkin matrices Kκ,h and Vκ,h N 80 320 1280 5120 20480

M 42 162 642 2562 10242

ε1 −2

1.0 · 10 1.0 · 10−3 1.0 · 10−4 1.0 · 10−5 1.0 · 10−6

MByte(Kh )

%

MByte(Vh )

%

0.05 0.66 6.08 49.05 357.90

99.9 84.0 48.5 24.5 11.2

0.05 0.55 5.02 39.46 280.60

50.4 35.0 20.1 9.86 4.47

these tables. The second column contains the number of nodes, while in the third column of Table 4.22 the prescribed accuracy for the ACA algorithm for the approximation of both matrices Kκ,h ∈ CN ×M and Vκ,h ∈ CN ×N is given. In this table, the fourth column shows the memory requirements in MByte for the approximate double layer potential matrix Kκ,h . The quality of this approximation in percentage of the original matrix is listed in the next column, whereas the corresponding values for the single layer potential matrix Vκ,h can be seen in the columns six and seven. The third column in Table 4.23 shows the number of GMRES iterations needed to reach the prescribed accuracy ε2 , while the relative L2 error for the Neumann datum, Error1 =

γ1int u − t˜h L2 (Γ ) γ1int uL2 (Γ )

,

is given in the fourth column. The next column represents the rate of convergence for the Neumann datum, i.e. the quotient of the errors in two consecutive lines of column four. Finally, the last two columns show the absolute error (cf. (2.56)) in a prescribed inner point x∗ ∈ Ω,

4.4 Helmholtz Equation

173

Table 4.23. Accuracy of the Galerkin method, Dirichlet problem N 80 320 1280 5120 20480

M 42 162 642 2562 10242

Iter 21 29 37 46 57

Error1 −1

6.98 · 10 3.13 · 10−1 1.45 · 10−1 7.03 · 10−2 3.48 · 10−2

CF1 – 2.23 2.16 2.06 2.02

Error2 −0

3.12 · 10 8.68 · 10−2 7.42 · 10−3 7.37 · 10−4 7.74 · 10−5

CF2 – 36.00 11.70 10.06 9.52

Error2 = |u(x∗ ) − u ˜(x∗ )| , x∗ = (0.28591, 0.476517, 0.667123) , (4.37) for the value u ˜(x∗ ) obtained using the approximate representation formula (2.55). Table 4.23 obviously shows a linear convergence O(N −1/2 ) = O(h) of the Galerkin boundary element method for the Neumann datum in the L2 norm. It should be noted that this theoretically guaranteed convergence order can already be observed when approximating the matrices Kκ,h and Vκ,h with much less accuracy as it was used to obtain the results in Table 4.22. However, this high accuracy is necessary in order to be able to observe the third order (or even better) pointwise convergence rate within the domain Ω presented in the last two columns of Table 4.23. In Figs. 4.36–4.37, the given Dirichlet datum (real and imaginary parts) for N = 1280 boundary elements is presented. The computed Neumann datum is presented in Figs. 4.38 (real part) and 4.39 (imaginary part). The numerical curves obtained when using an approximate representation formula in comparison with the curve of the exact values (4.36) along the line ⎛ ⎞ ⎛ ⎞ −0.3 0.6 x(t) = ⎝ −0.5 ⎠ + t ⎝ 1.0 ⎠ , 0 ≤ t ≤ 1 (4.38) −0.7 1.4 inside of the domain Ω are shown in Fig. 4.40 for N = 80 and in Fig. 4.41 for N = 320. The values of the numerical solution u ˜ and of the analytical solution u have been computed in 512 points uniformly placed on the line (4.38). In these ﬁgures, the thick dashed line represents the course of the analytical solution (4.36), while the thin solid line shows the course of the numerical solution u ˜. The values of the variable x2 along the the line (4.38) are used for the axis of abscissas. The left plots in these ﬁgures correspond to the real parts of the solutions while the imaginary parts are shown on the right. The next Fig. 4.42 shows these curves for N = 1280, but on the zoomed interval [0.3, 0.5] with respect to the variable x2 in order to see the very small diﬀerence between them. It is almost impossible to see any optical diﬀerence between the numerical and analytical curves for higher values of N . Note that the point x∗ in (4.37) is chosen close to the maximum of the function Im u along the line where the error seems to reach its maximum. Thus, for the moderate value of the wave number κ = 2, the quality of numerical results on the unit sphere is almost the same as for the Laplace

174

4 Implementation and Numerical Examples -1

4.0924E01 -0.5 0 0.5 1 1

0.5 0.0000E00 0 -0.5 -1 -1 -0.5 0 0.5 4.0924E01

1

Fig. 4.36. Given Dirichlet datum (real part) for the unit sphere, N = 1280 -1

3.0617E01 -0.5 0 0.5 1 1

0.5 4.4670E01 0 -0.5 -1 -1 -0.5 0 0.5 1

3.1510E01

Fig. 4.37. Given Dirichlet datum (imaginary part) for the unit sphere, N = 1280

4.4 Helmholtz Equation -1

175

1.6062E02 -0.5 0 0.5 1 1

0.5 0.0000E00 0 -0.5 -1 -1 -0.5 0 0.5 1.6062E02

1

Fig. 4.38. Computed Neumann datum (real part) for the unit sphere, N = 1280 -1

1.3744E02 -0.5 0 0.5 1 1

0.5 2.1111E00 0 -0.5 -1 -1 -0.5 0 0.5 1

1.3322E02

Fig. 4.39. Computed Neumann datum (imaginary part) for the unit sphere, N = 1280

176

4 Implementation and Numerical Examples

0

6

-1

5

-2

4

-3

3

-4

2

-5

1

-6 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

Fig. 4.40. Numerical and analytical curves for N = 80, Dirichlet problem 0

3

-1

2.5

-2

2

-3

1.5

-4

1

-5

0.5

-6 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

Fig. 4.41. Numerical and analytical curves for N = 320, Dirichlet problem -1

3

-2 2.8 -3 -4

2.6

-5 2.4 -6 0.2

0.22

0.24

0.26

0.28

0.3

0.2

0.22

0.24

0.26

0.28

0.3

Fig. 4.42. Numerical and analytical curves for N = 1280, Dirichlet problem

equation. The ACA approximation is good, the number of GMRES iterations is low without any preconditioning, and it grows corresponding to the theory and, ﬁnally, the theoretical linear convergence order of the Neumann datum on the surface Γ as well as the cubic convergence order in the inner points of the domain Ω are perfectly illustrated. Unit Sphere. Multifrequency Analysis Since the Helmholtz equation provides an additional parameter, the wave number κ, it is especially interesting and important to study the behaviour of our numerical methods with respect to this parameter. The quality of the matrix approximation, the number of iterations needed to solve the correspond-

4.4 Helmholtz Equation

177

ing linear systems, and, of course, the accuracy of the whole procedure are of special interest. We will now solve the Helmholtz equation for a ﬁxed discretisation of the surface Γ but for a sequence of wave numbers κ ∈ [κmin , κmax ]. If Im κ = 0, then the inner Dirichlet boundary value problem is uniquely solvable. The situation is diﬀerent for Im κ = 0. In this case the uniqueness holds only if κ2 is not an eigenvalue of the Laplace operator subjected to homogeneous Dirichlet boundary conditions, −Δu(x) = λ u(x)

for x ∈ Ω,

γ0int u(x) = 0

for x ∈ Γ ,

(cf. Section 1.4). For general Γ , the eigenvalues of the Laplace operator are not known and it can happen that one or even several of them belong to the interval [κmin , κmax ]. In this case some diﬃculties will occur when solving the discrete problem. Now we are going to illustrate the situation. The exact eigenfunctions and eigenvalues on the unit ball Ω are analytically known and can be represented in spherical coordinates ⎛ ⎞ cos ϕ sin θ x = ⎝ sin ϕ sin θ ⎠ , 0 ≤ < 1 , 0 ≤ ϕ < 2π , 0 ≤ θ ≤ π cos θ as follows uk,n,m (, ϕ, θ) =

Jn+1/2 (μn,m ) Pn,|k| (cos θ) eı kϕ , √

(4.39)

with m ∈ N,

n ∈ N0 ,

|k| ≤ n .

In (4.39), μn,m are the zeros of the Bessel functions Jn+1/2 . Pn,k are the associated Legendre polynomials k/2 dk Pn (u) Pn,k (u) = (−1)k 1 − u2 dxk deﬁned for |u| ≤ 1 ,

k = 0, . . . , n ,

n ∈ N0 .

The Legendre polynomials Pn are given in (3.12). The corresponding eigenvalues are λn,m = μ2n,m . For n = k = 0 and m ∈ N, the eigenvalues and the eigenfunctions are of an especially simple form. In this case we use / 2 sin z √ J1/2 (z) = π z and obtain

178

4 Implementation and Numerical Examples

u0,0,m (, ϕ, θ) = cm

sin(μ0,m ) ,

m ∈ N.

(4.40)

Thus, the corresponding critical values of κ are κ = μ0,m = π m ,

m ∈ N.

In particular, κ = π is a critical value. We solve the boundary value problem for the Dirichlet boundary value problem for the Helmholtz equation (4.31) having the analytical solution / Ω. We will use 17 uniformly distributed val(4.33) with y˜ = (1.1, 0.0, 0.0) ∈ ues of κ on the interval [3.1, 3.2]. The discretisation of the boundary will be a polyhedron having N = 320 boundary elements (cf. Fig. 4.1). The following ﬁgures illustrate the results: In Fig. 4.43, the L2 error of the Neumann datum is shown as a function of κ in the left plot. The right plot shows the number of GMRES iterations needed to reach the relative accuracy ε2 = 10−8 of the numerical solution of the linear system (2.57). Thus, a signiﬁcant jump of the 42 0.74 41 0.735 40 0.73 39 0.725 3.1

3.12

3.14

3.16

3.18

3.2

38 3.1

3.12

3.14

3.16

3.18

3.2

Fig. 4.43. Multifrequency computation for N = 320, Dirichlet problem

accuracy is displayed for the value κ = 3.1750. Also, a signiﬁcant increase of the number of iterations can be seen close to the critical value κ = 3.1750. The quality of the ACA approximation of the matrices Kκ,h and Vκ,h is more or less the same for all values of the parameter κ on this rather small interval. Thus we can deduce that the value of the parameter κ2 = 3.17502 is close to the eigenvalue of the Dirichlet boundary value problem for the Laplace equation in the polyhedron Ωh with N = 320 elements. This value is remarkably close to the ﬁrst eigenvalue π 2 of the continuous problem. However, the discrete values of the parameter κ will never meet the correct eigenvalue exactly and, probably, the closeness to the eigenvalue will not be detected. In this situation quite wrong results can be obtained. We illustrate this fact in the next three ﬁgures where the real parts (left plots) and the imaginary parts (right plots) of the analytical solution (thick dashed lines) and of the numerical solution (thin solid lines) are presented for three values of the parameter κ = 3.15, 3.175, 3.2. We can see that the numerical solution diﬀers signiﬁcantly from the analytical one for κ = 3.175, while the numerical results for κ = 3.15 and κ = 3.2 are quite good for this rather rough discretisation.

4.4 Helmholtz Equation 0.02

-0.01

0

-0.02

-0.02

-0.03

-0.04

179

-0.04

-0.06 -0.05 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Fig. 4.44. Numerical and analytical curves for κ = 3.15 and N = 320, Dirichlet problem -0.01 0.02 -0.02

0 -0.02

-0.03

-0.04

-0.04

-0.06 -0.3

-0.05 -0.2

-0.1

0

0.1

0.2

0.3

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Fig. 4.45. Numerical and analytical curves for κ = 3.175 and N = 320, Dirichlet problem 0.02 -0.02 0 -0.03

-0.02 -0.04

-0.04

-0.06 -0.05 -0.08 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Fig. 4.46. Numerical and analytical curves for κ = 3.2 and N = 320, Dirichlet problem

In the next example, we solve the Dirichlet boundary value problem on the polyhedron Ωh with N = 1280 elements for 65 values of the wave number κ uniformly distributed on the interval [0, 16]. Thus, the ﬁrst value corresponds to the Laplace operator. In Figs. 4.47–4.48 we show how the ACA approximation quality of the matrices Kκ,h and Vκ,h depends on the wave number. The left plots in these ﬁgures present the memory requirements in MByte, while the right plots show the same result in percentage compared to the full memory for ε1 = 10−4 . The linear dependence of the memory requirement of the wave number is

180

4 Implementation and Numerical Examples

clearly indicated by these numerical tests. Also this example shows the loss 100 11 80 10 60 9 8

40

7

20

6 0

2.5

5

7.5

10

12.5

15

0

2.5

5

7.5

10

12.5

15

Fig. 4.47. Approximation of the double layer potential matrix Kκ,h for N = 1280

100 9 80 8 60 7 40 6 20 5 0

2.5

5

7.5

10

12.5

15

0

2.5

5

7.5

10

12.5

15

Fig. 4.48. Approximation of the single layer potential matrix Vκ,h for N = 1280

of the accuracy close to the critical values of the parameter κ. In Fig. 4.49, we present again the L2 norm of the error for the Neumann datum (left plot) and the number of GMRES iterations (right plot) as functions of the wave number κ. The left plot in Fig. 4.49 clearly shows a total loss of accuracy close to the 1.6 1.4

200

1.2 150

1 0.8

100

0.6 50

0.4 0

2.5

5

7.5

10

12.5

15

0

2.5

5

7.5

10

12.5

15

Fig. 4.49. Multifrequency computation for N = 1280, Dirichlet problem

wave number κ = 13.0. If we plot the analytical and the numerical values

4.4 Helmholtz Equation

181

of the solution of the boundary value problem for three subsequent points 12.75, 13.0 and 13.25 for the parameter κ we can see this loss of accuracy optically. The results are presented in Figs. 4.50–4.52. It is remarkable that 0.08

0.06

0.06

0.04

0.04

0.02

0.02

0

0 -0.02 -0.02 -0.04 -0.04 -0.06 -0.4

-0.2

0

0.2

0.4

-0.06 -0.4

-0.2

0

0.2

0.4

Fig. 4.50. Numerical and analytical curves for κ = 12.75 and N = 1280, Dirichlet problem

0.04

0.1

0.02

0.05

0 0 -0.02 -0.05

-0.04 -0.06

-0.1 -0.4

-0.2

0

0.2

0.4

-0.4

-0.2

0

0.2

0.4

Fig. 4.51. Numerical and analytical curves for κ = 13.0 and N = 1280, Dirichlet problem

0.06

0.04

0.04

0.02

0.02

0

0 -0.02 -0.02 -0.04 -0.04 -0.06

-0.06 -0.4

-0.2

0

0.2

0.4

-0.4

-0.2

0

0.2

0.4

Fig. 4.52. Numerical and analytical curves for κ = 13.25 and N = 1280, Dirichlet problem

only one critical value (close to 4π) of the wave number κ was detected on

182

4 Implementation and Numerical Examples

the interval [0, 16]. This fact is due to the rather big step of 0.25 with respect to κ, which was used in the above example. Exhaust Manifold The analytical solution is taken in the form (4.33) with y˜ = (0, 0, 0.06) and κ = 80, which is moderate compared to the rather small dimension of the domain (cf. Fig. 4.11). The results of the computations are shown in Tables 4.24 and 4.25. The third column shows the number of iterations required Table 4.24. ACA approximation of the Galerkin matrices Kκ,h and Vκ,h N 2264 9056 36224

M 1134 4530 18114

ε1 −3

1.0 · 10 1.0 · 10−4 1.0 · 10−5

MByte(Kκ,h )

%

MByte(Vκ,h )

%

16.62 137.86 1046.80

42.4 22.0 10.5

11.82 97.08 696.65

15.1 7.8 3.5

Table 4.25. Accuracy of the Galerkin method, Dirichlet problem N 2264 9056 36224

M 1134 4530 18114

Iter 177 208 244

Error1 −1

3.10 · 10 1.40 · 10−1 5.83 · 10−2

CF1 2.2 2.4

Error2 −3

8.88 · 10 1.08 · 10−3 9.27 · 10−5

CF2 8.2 11.7

by the GMRES method without preconditioning. The fourth column displays the L2 error of the Neumann datum, while the next column shows its linear convergence. The last pair of columns of Table 4.25 shows the absolute error (cf. (2.56)) in a prescribed inner point x∗ ∈ Ω, ˜(x∗ )| , x∗ = (0.145303, 0.1, −0.05) Error2 = |u(x∗ ) − u

(4.41)

for the value u ˜(x∗ ) obtained using an approximate representation formula (2.55). Finally, the last column of this table indicates the cubic (or even better) convergence of this quantity. In Figs. 4.53– 4.54, the real part and the imaginary part of the given Dirichlet datum are presented. The computed Neumann datum is presented in Figs. 4.55–4.56, where again the left plot corresponds to the real part of the Neumann datum, while the imaginary part is shown on the right. The numerical curve, obtained when using an approximate representation formula in comparison with the curve of the exact values (4.33) along the line

4.4 Helmholtz Equation

183

1.2070E00

0.05 5.9464E01

0 -0.05 0.1

-0.1 0 0 0.1 0.2 0 2

2.3963E00

Fig. 4.53. Given Dirichlet datum (real part) for the exhaust manifold 4.7306E00

0.05 1.5706E00 0 -0.05 0.1

-0.1 0 0 0.1 0.2 0 2

1.5894E00

Fig. 4.54. Given Dirichlet datum (imaginary part) for the exhaust manifold

184

4 Implementation and Numerical Examples 3.6341E02

0.05 1.3407E02 0 -0.05 0.1

-0.1 0 0 0.1 0.2 0 2

9.5273E01

Fig. 4.55. Computed Neumann datum (real part) for the exhaust manifold 1.9587E02

0.05 6.7482E01 0 -0.05 0.1

-0.1 0 0 0.1 0.2 0 2

6.0909E01

Fig. 4.56. Computed Neumann datum (imaginary part) for the exhaust manifold

4.4 Helmholtz Equation

185

⎛

⎞ ⎛ ⎞ −0.05 0.2 x(t) = ⎝ 0.1 ⎠ + t ⎝ 0.0 ⎠ , 0 ≤ t ≤ 1 −0.05 0.0

(4.42)

inside of the domain Ω is shown in Figs. 4.57-4.58 for N = 2264 and correspondingly for N = 9056. The values of the numerical solution u ˜ and of the analytical solution u have been computed in 512 points uniformly placed on the line (4.42). The thick dashed line represents the course of the analytical solution (4.33) while the thin solid line shows the course of the numerical solution u ˜. The values of the variable x1 along the line (4.42) are used for the axis of abscissas. Note that the numerical solution for N = 9056 perfectly

0.4

0.4

0.2

0.2

0

0

-0.2 -0.4 -0.05

-0.2

0

0.05

0.1

0.15

-0.05

0

0.05

0.1

0.15

Fig. 4.57. Numerical and analytical curves for the exhaust manifold for N = 2264

0.4

0.4 0.2

0.2

0

0

-0.2 -0.4 -0.05

-0.2

0

0.05

0.1

0.15

-0.05

0

0.05

0.1

0.15

Fig. 4.58. Numerical and analytical curves for the exhaust manifold for N = 9056

coincides with the analytical curves. 4.4.5 Interior Neumann Problem We consider the interior Neumann boundary value problem for the Helmholtz equation with the boundary condition γ1int u(x) = g(x) for x ∈ Γ . The variational problem (1.109) 1 Dκ u I − Kκ g, v ¯, v = for all v ∈ H 1/2 (Γ ) 2 Γ Γ

186

4 Implementation and Numerical Examples

is discretised and leads to a system of linear equations (cf. (2.61)) Dκ,h u =

1 2

Mh − Kκ,h g.

The symmetric but complex valued system is then solved using the GMRES method up to the relative accuracy ε2 = 10−8 . Unit Sphere We consider again the harmonic function (4.36) as the exact solution. The results for the ACA approximation of the matrix Cκ,h ∈ CM ×M are presented for the compuin Table 4.26. The corresponding results for the matrices Kκ,h tation of the right hand side of the above system and Vκ,h , which will be used for the multiplication with the matrix Dκ,h are identical to those already presented in Table 4.22. Note that in this example the complex valued Galerkin Table 4.26. ACA approximation of the Galerkin matrix Cκ,h , Neumann problem N

M

ε1

MByte(Cκ,h )

%

80 320 1280 5120 20480

42 162 642 2562 10242

1.0 · 10−2 1.0 · 10−3 1.0 · 10−4 1.0 · 10−5 1.0 · 10−6

0.01 0.20 2.15 17.53 126.29

51.2 49.1 34.2 17.5 7.89

matrix Cκ,h with piecewise linear basis functions is generated. This is a rather time consuming procedure. Thus, a quite good approximation of this matrix is especially important when using the ACA algorithm. The accuracy obtained Table 4.27. Accuracy of the Galerkin method, Neumann problem N 80 320 1280 5120 20480

M 42 162 642 2562 10242

Iter 10 17 25 37 46

Error1 −1

4.94 · 10 1.25 · 10−1 2.75 · 10−2 6.41 · 10−3 1.55 · 10−3

CF1 – 3.95 4.55 4.29 4.14

Error2 −0

2.15 · 10 4.72 · 10−1 1.09 · 10−1 2.60 · 10−2 6.41 · 10−3

CF2 – 4.67 4.32 4.19 4.05

for the whole numerical procedure is presented in Table 4.27. The numbers in this table have the usual meaning. The third column shows the number of iterations required by the GMRES method without preconditioning. Note that

4.4 Helmholtz Equation

187

the convergence of the Galerkin method for the unknown Dirichlet datum in L2 norm, h L2 (Γ ) γ int u − u , Error1 = 0 int γ0 uL2 (Γ ) is now quadratic corresponding to the estimate (2.67). In the inner point x∗ , we now observe quadratic convergence (7th column), as it was predicted in (2.67), instead of the cubic order obtained for the Dirichlet problem (cf. Table 4.23). This fact is clearly illustrated in Figs. 4.59–4.61, where the convergence of the boundary element method can be seen optically. The results obtained for N = 80 are plotted in Fig. 4.59, where the left plot shows the course of the real part of the solution, while the imaginary part is presented on the right. The numerical curves in Fig. 4.60 are notedly better than the previous ones. However, their quality is not as high as of the corresponding curves obtained while solving the Dirichlet problem (cf. Fig. 4.41).

0 4 -1 3

-2 -3

2 -4 1

-5 -6 -0.05

0

0.05

0.1

0 0.15 -0.05

0

0.05

0.1

0.15

Fig. 4.59. Numerical and analytical curves for N = 80, Neumann problem

0

3

-1

2.5

-2

2

-3

1.5

-4

1

-5

0.5

-6 -0.05

0

0.05

0.1

0.15

0 -0.05

0

0.05

0.1

0.15

Fig. 4.60. Numerical and analytical curves for N = 320, Neumann problem

Exhaust Manifold The analytical solution is taken in the form (4.33) with y˜ = (0, 0, 0.06) and κ = 10, which is rather small compared with the dimension of the domain (cf.

188

4 Implementation and Numerical Examples

0

3

-1

2.5

-2

2

-3

1.5

-4

1

-5

0.5

-6 -0.05

0

0.05

0.1

0.15

0 -0.05

0

0.05

0.1

0.15

Fig. 4.61. Numerical and analytical curves for N = 1280, Neumann problem

Fig. 4.11). We will consider bigger values of the wave number, when studying the multifrequency behaviour of the problem. This small value of κ is well situated to demonstrate convergence properties of the Galerkin BEM for a regular value of the wave number. The results of the computations are shown in Tables 4.28 and 4.29. The quality of the ACA approximation of the maTable 4.28. ACA approximation of the Galerkin matrices Kκ,h and Vκ,h N 2264 9056 36224

M 1134 4530 18114

ε1 −3

1.0 · 10 1.0 · 10−4 1.0 · 10−5

MB(Kκ,h )

%

MB(Vκ,h )

%

MB(Cκ,h )

%

13.81 124.86 998.19

35.3 20.0 9.9

8.29 72.14 580.32

10.6 5.8 2.9

4.54 40.33 354.61

23.1 12.9 7.1

trix Cκ,h can be seen in columns eight and nine of the Table 4.28. The third Table 4.29. Accuracy of the Galerkin method, Neumann problem N 2264 9056 36224

M 1134 4530 18114

Iter 177 201 244

Error1 −2

2.26 · 10 5.20 · 10−3 1.21 · 10−3

CF1 4.3 4.3

Error2 −3

1.40 · 10 2.92 · 10−4 4.90 · 10−5

CF2 4.8 5.9

column shows the number of iterations required by the GMRES method with diagonal preconditioning (4.21). The fourth column displays the L2 error of the computed Dirichlet datum, and the next column shows its quadratic convergence. Column six of Table 4.29 displays the absolute error (cf. (2.67)) in a prescribed inner point x∗ ∈ Ω, ˜(x∗ )| , x∗ = (0.0740705, 0.1, −0.05) , Error2 = |u(x∗ ) − u

(4.43)

4.4 Helmholtz Equation

189

for the value u ˜(x∗ ) obtained using an approximate representation formula (2.65). Finally, the last column of this table indicates the quadratic (or even better) convergence of this quantity. The Cauchy data obtained when solving the Neumann boundary value problem is optically the same as for the Dirichlet boundary value problem and can be seen in Figs. 4.53– 4.54. The numerical curve obtained when using an approximate representation formula in comparison with the curve of the exact values (4.36) along the line (4.42) inside of the domain Ω is shown in Figs. 4.62-4.63 for N = 2264, and, correspondingly, for N = 9056. The values of the numerical solution u ˜ and of the analytical solution u have been computed in 512 points uniformly placed on the line (4.42). The thick dashed line represents the course of the analytical solution (4.33) while the thin solid line shows the course of the numerical solution u ˜. The values of the variable x1 along the the line (4.16) are used for the axis of abscissas. Note that 0.05 0.5

0 -0.05

0.45

-0.1

0.4

-0.15 0.35 -0.2 -0.05

0

0.05

0.1

0.15

-0.05

0

0.05

0.1

0.15

Fig. 4.62. Numerical and analytical curves for the Neumann Problem, N = 2264

0.05 0.5

0 -0.05

0.45

-0.1

0.4

-0.15 -0.2 -0.05

0.35 0

0.05

0.1

0.15

-0.05

0

0.05

0.1

0.15

Fig. 4.63. Numerical and analytical curves for the Neumann Problem, N = 9056

the numerical solution for N = 9056 perfectly coinsides with the analytical curves.

190

4 Implementation and Numerical Examples

Exhaust Manifold. Multifrequency Analysis Here we solve the Neumann boundary value problem for the Helmholtz equation on the surface, depicted in Fig. 4.11, for 17 uniformly distributed values of the wave number κ on the interval [16, 24]. In Fig. 4.64, we present the L2 norm of the error for the computed Dirichlet datum (left plot) and the number of GMRES iterations (right plot) as functions of the wave number κ. The left plot in Fig. 4.64 clearly shows a total loss of accuracy close to the 92.5 0.3 90 0.25 87.5 0.2

85

0.15

82.5

0.1

80

0.05

77.5 16

18

20

22

24

16

18

20

22

24

Fig. 4.64. Multifrequency computation for N = 2264, Neumann problem

wave numbers κ = 17.5 and κ = 20.5. If we plot the analytical and the numerical values of the solution of the boundary value problem for three subsequent points 17.0, 17.5, and 18.0 for the parameter κ, we can see this loss of accuracy optically. The results are presented in Figs. 4.65–4.67. Thus, the numerical -0.34

0.3

-0.36 0.2 -0.38 -0.4

0.1

-0.42 0

-0.44 -0.46 -0.48 -0.05

-0.1 0

0.05

0.1

0.15

-0.05

0

0.05

0.1

0.15

Fig. 4.65. Numerical and analytical curves for κ = 17.0 and N = 2264, Neumann problem

solution is quite wrong for κ = 17.5, while it is acceptable for κ = 17.0 and κ = 18.0. The picture is similar, if we consider the numerical and the analytical solutions for κ = 20.0, 20.5, and κ = 21.0. Note that the quality of the approximation of the boundary element matrices Kκ,h , Vκ,h , and Cκ,h is more or less constant for all values of the wave number κ on the whole interval [16, 24].

4.4 Helmholtz Equation

191

0.3 -0.25

0.2

-0.3 0.1 -0.35 0 -0.4 -0.1 -0.45 -0.05

0

0.05

0.1

0.15

-0.2 -0.05

0

0.05

0.1

0.15

Fig. 4.66. Numerical and analytical curves for κ = 17.5 and N = 2264, Neumann problem 0.2

-0.3

0.1 -0.35 0 -0.4 -0.1 -0.45 -0.2 -0.05

0

0.05

0.1

0.15

-0.05

0

0.05

0.1

0.15

Fig. 4.67. Numerical and analytical curves for κ = 18.0 and N = 2264, Neumann problem

4.4.6 Exterior Dirichlet Problem Here, we solve the Helmholtz equation (4.31) in Ω e = R3 \ Ω together with the boundary condition γ0ext u(x) = g(x) for x ∈ Γ , where Γ = ∂Ω is the boundary of the surface Ω. The variational problem (1.116) 1 = − I + Kκ g, w for all w ∈ H −1/2 (Γ ) Vκ t, w 2 Γ Γ is discretised and leads to a system of linear equations (cf. (2.57)) 1 t = − Mh + Kκ,h g . Vκ,h 2 The matrix Vκ,h of this system is identical with the corresponding matrix of the interior boundary value problem. Thus, in this subsection, we will choose diﬀerent values of the wave number κ compared with those used in Subsection 4.4.6. Unit Sphere The analytical solution is taken in the form (4.33) for y˜ = (0.9, 0, 0) ∈ Ω, i.e. close to the boundary of the domain Ω. The results of the computations

192

4 Implementation and Numerical Examples Table 4.30. ACA approximation of the Galerkin matrices Kκ,h and Vκ,h N

80 320 1280 5120 20480

M

ε1

MByte(Kκ,h )

%

MByte(Vκ,h )

%

0.05 0.75 6.88 53.85 379.09

100.0 94.5 54.9 26.9 11.8

0.05 0.64 5.66 42.91 302.35

50.6 40.8 22.7 10.7 4.72

−2

1.0 · 10 1.0 · 10−3 1.0 · 10−4 1.0 · 10−5 1.0 · 10−6

42 162 642 2562 10242

for the wave number κ = 4, which is still moderate, are shown in Tables 4.30 and 4.31. The number of boundary elements is listed in the ﬁrst column of these tables. The second column contains the number of nodes, while in the third column of Table 4.30, the prescribed accuracy for the ACA algorithm for the approximation of both matrices Kκ,h ∈ CN ×M and Vκ,h ∈ CN ×N is given. The diﬀerence in the ACA approximation of these matrices for κ = 4 (Table 4.30) and for κ = 2 (Table 4.22) can be clearly seen. In Table 4.31, the Table 4.31. Accuracy of the Galerkin method, Dirichlet problem N

M

Iter

Error1

CF1

Error2

CF2

80 320 1280 5120 20480

42 162 642 2562 10242

28 41 52 63 75

9.43 · 10−1 6.95 · 10−1 3.68 · 10−1 1.64 · 10−1 7.79 · 10−2

– 1.36 1.89 2.24 2.11

1.88 · 10−1 4.39 · 10−2 7.48 · 10−3 8.04 · 10−4 6.89 · 10−5

– 4.28 5.87 9.30 11.68

third column shows the number of GMRES iterations without preconditioning needed to reach the prescribed accuracy ε2 = 10−8 . The relative L2 error for the Neumann datum, Error1 =

γ1int u − t˜h L2 (Γ ) γ1int uL2 (Γ )

,

is given in the fourth column. The next column represents the rate of convergence for the Neumann datum, i.e. the quotient of the errors in two consecutive lines of column four. We can see that linear convergence can be observed asymptotically. Finally, the last two columns show the absolute error (cf. (2.69)) in the point x∗ ∈ Ω e , ˜(x∗ )| , x∗ = (1.1, 0, 0) , Error2 = |u(x∗ ) − u

(4.44)

for the value u ˜(x∗ ) obtained using the approximate representation formula (2.68). Again, a rather high accuracy of the ACA approximation is necessary in

4.4 Helmholtz Equation

193

order to be able to observe a third order (asymptotically even better) pointwise convergence rate within the domain Ω e , shown in the last two columns of Table 4.31. In Figs. 4.68–4.69, the given Dirichlet datum (real and imaginary parts) for N = 320 boundary elements is presented. The computed Neumann datum 7.6908E01

1

0.5

3.2795E01

0

-1 -

-0.5 5

-0.5 -0 0 -1

0.5 -1

-0.5

0

0.5

1

1 1.1317E01

Fig. 4.68. Given Dirichlet datum (real part) for the unit sphere, N = 320

is shown in Figs. 4.70 (real part) and 4.71 (imaginary part). The numerical curves obtained when using an approximate representation formula in comparison with the curve of the exact values (4.33) along the line ⎛ ⎞ ⎛ ⎞ 1.1 0.0 x(t) = ⎝ 0.0 ⎠ + t ⎝ 0.0 ⎠ , 0 ≤ t ≤ 1 (4.45) −4.0 8.0 inside of the domain Ω e is shown in Fig. 4.72 for N = 80 and in Fig. 4.73 for N = 320. The values of the numerical solution u ˜ and of the analytical solution u have been computed in 512 points uniformly placed on the line (4.45). In these ﬁgures, the thick dashed line represents the course of the analytical solution (4.36), while the thin solid line shows the course of the numerical solution u ˜. The values of the variable x3 along the line (4.38) are used for the axis of abscissas. The left plots in these ﬁgures correspond to the real parts of the solutions, while the imaginary parts are shown on the right. The next Fig. 4.74 shows these curves for N = 1280, but on the zoomed interval

194

4 Implementation and Numerical Examples 3.3070E01

1

0.5

1.2859E01

0

-1 -

-0.5 5

-0.5 -0 0 -1

0.5 -1

-0.5

0

0.5

1

1 7.3514E02

Fig. 4.69. Given Dirichlet datum (imaginary part) for the unit sphere, N = 320 2.6568E01

1

0.5

1.2298E00

0

-1 -

-0.5 5

-0.5 -0 0 -1

0.5 -1

-0.5

0

0.5

1

1 2.7254E00

Fig. 4.70. Computed Neumann datum (real part) for the unit sphere, N = 320

4.4 Helmholtz Equation

195

1.8124E01

1

0.5

3.6691E02

0

-1 -

-0.5 5

-0.5 -0 0 -1

0.5 -1

-0.5

0

0.5

1

1 2.5462E01

Fig. 4.71. Computed Neumann datum (imaginary part) for the unit sphere, N = 320 0.25 0.2

0.2 0.15

0.1 0.1 0.05 0 0 -0.05

-0.1 -4

-2

0

2

-4

4

-2

0

2

4

Fig. 4.72. Numerical and analytical curves for N = 80, Dirichlet problem

0.25 0.2

0.2 0.15

0.1 0.1 0.05 0 0 -0.05

-0.1 -4

-2

0

2

4

-4

-2

0

2

Fig. 4.73. Numerical and analytical curves for N = 320, Dirichlet problem

4

196

4 Implementation and Numerical Examples

[−1, 1] with respect to the variable x3 , in order to see the very small diﬀerence between them. It is almost impossible to see any optical diﬀerence between

0.25 0.2 0.2 0.15 0.1 0.1 0.05

0

0 -0.05

-0.1 -1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

Fig. 4.74. Numerical and analytical curves for N = 1280, Dirichlet problem

the numerical and the analytical curves for higher values of N . Note that the point x∗ in (4.44) is chosen close to the maximum of the function Re u along the line where the error seems to reach its maximum. Thus, for the moderate value of the wave number κ = 4, the quality of the numerical results on the unit sphere is perfect. The ACA approximation is good, the number of GMRES iterations is low without any preconditioning and it grows corresponding to the theory, and, ﬁnally, the theoretical linear convergence order of the Neumann datum on the surface Γ as well as the cubic convergence order in the inner points of the domain Ω e is perfectly illustrated. 4.4.7 Exterior Neumann Problem We consider the exterior Neumann boundary value problem for the Helmholtz equation in Ω e = R3 \ Ω with the boundary condition γ1ext u(x) = g(x) for x ∈ Γ . The variational problem (1.122)

Dκ u ¯, v

Γ

1 I + Kκ g, v = − 2 Γ

for all v ∈ H 1/2 (Γ )

is discretised and leads to a system of linear equations (cf. (2.61)) 1 Dκ,h u g. = − Mh − Kκ,h 2 This symmetric, but complex valued system is then solved using the GMRES method up to the relative accuracy ε2 = 10−8 . Unit Sphere We consider again the analytical solution in the form (4.33) for an interior point y˜ = (0.9, 0, 0) ∈ Ω and κ = 4 as the exact solution. The results for the

4.4 Helmholtz Equation

197

Table 4.32. ACA approximation of the Galerkin matrix Cκ,h , Neumann problem N 80 320 1280 5120 20480

M

ε1

MByte(Cκ,h )

%

0.01 0.20 2.41 19.17 135.47

51.2 50.3 38.3 19.1 8.46

−2

1.0 · 10 1.0 · 10−3 1.0 · 10−4 1.0 · 10−5 1.0 · 10−6

42 162 642 2562 10242

Table 4.33. Accuracy of the Galerkin method, Neumann problem N

M

Iter

Error1

CF1

Error2

CF2

80 320 1280 5120 20480

42 162 642 2562 10242

16 23 31 44 62

6.78 · 10−1 1.91 · 10−1 5.81 · 10−2 1.42 · 10−2 3.27 · 10−3

– 3.55 3.29 4.09 4.34

3.24 · 10−2 4.99 · 10−3 1.24 · 10−3 2.90 · 10−4 7.16 · 10−5

– 6.49 4.02 4.28 4.04

ACA approximation of the matrices Kκ,h and Vκ,h are the same as for the exterior Dirichlet boundary value problems and can be seen in Table 4.30. The approximation results for the matrix Cκ,h are shown in Table 4.32. In Table 4.33, the accuracy obtained for the whole numerical procedure is presented and the numbers in this table have the usual meaning. The third column shows the number of iterations required by the GMRES method without preconditioning. Note that the convergence of the Galerkin method for the unknown Dirichlet datum in the L2 norm, Error1 =

h L2 (Γ ) γ0int u − u γ0int uL2 (Γ )

,

is close to quadratic. In the inner point x∗ = (1.1, 0.0, 1.0978) ∈ Ω e (close to a local minimum of the exact solution), we now observe quadratic convergence (7th column) as it was predicted in (2.70) instead of the cubic order obtained for the Dirichlet problem (cf. Table 4.31). This fact is clearly illustrated in Figs. 4.75–4.77, where the convergence of the boundary element method can be seen optically. The results obtained for N = 80 are plotted in Fig. 4.75, where the left plot shows the course of the real part of the solution, while the imaginary part is presented on the right. Thus, these curves are rather rough approximations of the exact solution. The numerical curves in Fig. 4.76 are notedly better than the previous ones. However, their quality is not as high as of the corresponding curves obtained while solving the Dirichlet problem (cf. Fig. 4.73). The numerical curves for N = 1280 are acceptable.

198

4 Implementation and Numerical Examples 0.4

0.2 0.3 0.1

0.2 0.1

0

0 -0.1 -4

-2

0

2

4

-0.1

-4

-2

0

2

4

Fig. 4.75. Numerical and analytical curves for N = 80, exterior Neumann problem

0.25 0.2

0.2 0.15

0.1 0.1 0.05 0 0 -0.05

-0.1 -4

-2

0

2

-4

4

-2

0

2

4

Fig. 4.76. Numerical and analytical curves for N = 320, exterior Neumann problem

0.25 0.2

0.2 0.15

0.1 0.1 0.05 0 0 -0.05

-0.1 -4

-2

0

2

4

-4

-2

0

2

4

Fig. 4.77. Numerical and analytical curves for N = 1280, exterior Neumann problem

A Mathematical Foundations

A.1 Function Spaces Let α = (α1 , . . . , αd ) ∈ Nd0 be a multiindex with |α| = α1 + . . . + αd and αd 1 α! = α1 ! . . . αd !, d = 2, 3. Moreover, for x ∈ Rd we deﬁne xα = xα 1 . . . xd as well as the partial derivatives Dα u(x) =

1 ∂xα 1

∂ |α| u(x1 , . . . , xd ). d . . . ∂xα d

For an open bounded domain Ω ⊂ Rd and k ∈ N0 , we denote by C k (Ω) the space of k times continuously diﬀerentiable functions equipped with the norm sup |Dα u(x)| ; uC k (Ω) = |α|≤k

x∈Ω

C ∞ (Ω) is deﬁned accordingly. The support of a given function is the closed set ( ) supp u = x ∈ Ω : u(x) = 0 . Then, C0∞ (Ω) is the space of inﬁnite times continuously diﬀerentiable functions with compact support, ( ) C0∞ (Ω) = u ∈ C ∞ (Ω) : supp u ⊂ Ω . For k ∈ N0 and κ ∈ (0, 1], we deﬁne the space C k,κ (Ω) of H¨ older continuous functions equipped with the norm uC k,κ (Ω) = uC k (Ω) +

|α|=k

sup x,y∈Ω

|Dα u(x) − Dα u(y)| . |x − y|κ

In particular, for k = 0 and κ = 1, we obtain the space C 0,1 (Ω) of Lipschitz continuous functions.

200

A Mathematical Foundations

The boundary Γ of an open and bounded set Ω ⊂ R3 is given as Γ = ∂Ω = Ω ∩ (R3 \Ω). We assume that the boundary Γ can be represented by a certain decomposition p ' Γi , (A.1) Γ = i=1

where each boundary segment Γi is described via a local parametrisation, ) ( (A.2) Γi = x ∈ R3 : x = χi (ξ) for ξ ∈ Ti ⊂ R2 , with respect to some open parameter domain Ti . A domain Ω is said to be a Lipschitz domain, when all functions χi in (A.2) are Lipschitz continuous for any arbitrary decomposition (A.1). For 1 ≤ p < ∞, we deﬁne Lp (Ω) as the space of all measurable functions u on Ω with uLp (Ω) < ∞, where · Lp (Ω) denotes the norm: ⎛ uLp (Ω) = ⎝

⎞1/p

|u(x)|p dx ⎠

.

Ω

In fact, two functions u, v ∈ Lp (Ω) are identiﬁed, if they diﬀer only on a set K with Lebesgue zero measure μ(K) = 0. L∞ (Ω) is the space of all measurable functions which are bounded almost everywhere, uL∞ (Ω) = ess sup |u(x)| = x∈Ω

inf

sup |u(x)|.

K⊂Ω,μ(K)=0 x∈Ω\K

Note that for u ∈ Lp (Ω) and v ∈ Lq (Ω) with adjoint parameters p, q satisfying 1 1 + = 1, p q the H¨older inequality holds: |u(x)v(x)| dx ≤ uLp (Ω) vLq (Ω) . Ω

Deﬁning the duality pairing u, v Ω =

u(x)v(x)dx , Ω

we then obtain vLq (Ω) =

sup 0=u∈Lp (Ω)

| u, v Ω | uLp (Ω)

for 1 ≤ p < ∞,

1 1 + = 1. p q

A.1 Function Spaces

201

In particular, for p = 2, the Hilbert space L2 (Ω) is the function space of square integrable functions. Finally, Lloc 1 (Ω) is the space of locally integrable functions. A function u ∈ Lloc 1 (Ω) is said to have a generalised partial derivative v=

∂ u ∈ Lloc 1 (Ω) , ∂xi

if it satisﬁes the equality ∂ v(x)ϕ(x)dx = − u(x) ϕ(x)dx ∂xi Ω

for all ϕ ∈ C0∞ (Ω).

(A.3)

Ω

In the same way, we may deﬁne the generalised derivative Dα u ∈ Lloc 1 (Ω) by Dα u(x)ϕ(x)dx = (−1)|α| u(x)Dα ϕ(x)dx for all ϕ ∈ C0∞ (Ω). Ω

Ω

Then, for k ∈ N0 and 1 ≤ p < ∞, ⎛ vWpk (Ω) = ⎝

⎞1/p Dα vpLp (Ω) ⎠

|α|≤k

deﬁnes a norm, while for p = ∞ we set α vW∞ k (Ω) = max D vL (Ω) . ∞ |α|≤k

Now we are able to deﬁne the Sobolev spaces Wpk (Ω) as the closure of C ∞ (Ω) with respect to the Sobolev norms as introduced above, ·W k (Ω)

Wpk (Ω) = C ∞ (Ω)

p

.

In particular, for any v ∈ Wpk (Ω), there exists a sequence {ϕj }j∈N ⊂ C ∞ (Ω) such that lim v − ϕj Wpk (Ω) = 0. j→∞

In the same way, we may also deﬁne the Sobolev spaces ◦

·W k (Ω)

k W p (Ω) = C0∞ (Ω)

p

.

Up to now, the above Sobolev spaces are deﬁned only for k ∈ N0 . However, the deﬁnition of the Sobolev norms · Wpk (Ω) , and, therefore, of the Sobolev spaces, can be generalised for arbitrary s ∈ R. For s > 0 with s = k + κ, k ∈ N0 , κ ∈ (0, 1), we deﬁne the Sobolev–Slobodeckii norm

202

A Mathematical Foundations

vWps (Ω) = with the semi-norm |v|pW s (Ω) p

1/p vpW k (Ω) + |v|pW s (Ω) p

p

|Dα v(x) − Dα v(y)|p = dxdy. |x − y|d+pκ |α|=k Ω Ω

For s < 0 and 1 < p < ∞, the Sobolev space Wps (Ω) is the dual space of ◦

W q−s (Ω), where 1/p + 1/q = 1. The corresponding norm is given by f Wps (Ω) =

sup

◦

v∈W q−s (Ω)

| f, v Ω | . vWq−s (Ω)

◦

Accordingly, W ps (Ω) is the dual space of Wq−s (Ω). Next we will collect some properties of the Sobolev spaces needed later. Theorem A.1 (Sobolev’s imbedding theorem). Let Ω ⊂ R3 be a bounded domain with Lipschitz boundary Γ = ∂Ω, and let s ≥ 3 for p = 1, and s > 3/p for p > 1. Then every function v ∈ Wps (Ω) is continuous, v ∈ C(Ω), satisfying vL∞ (Ω) ≤ c vWps (Ω) for all v ∈ Wps (Ω). In particular, we are interested in the Sobolev spaces W2s (Ω), i.e. for p = 2. For s = 1, the norm in W21 (Ω) is given by 1/2 . vW21 (Ω) = v2L2 (Ω) + ∇v2L2 (Ω) Now, we will derive equivalent norms in W21 (Ω). A norm · W21 (Ω),f is called equivalent to the norm · W21 (Ω) , if there are some positive constants c1 and c2 such that c1 vW21 (Ω) ≤ vW21 (Ω),f ≤ c2 vW21 (Ω)

for all v ∈ W21 (Ω).

Let f : W21 (Ω) → R be a bounded linear functional satisfying |f (v)| ≤ cf vW21 (Ω)

for all v ∈ W21 (Ω).

Let c ∈ R be an arbitrary constant. If we can always conclude c = 0 from f (c) = 0, then 1/2 vW21 (Ω),f = |f (v)|2 + ∇v2L2 (Ω) deﬁnes an equivalent norm in W21 (Ω). Examples are the norms ⎛ ⎞2 v2 1 = ⎝ v(x)dx⎠ + ∇v2L (Ω) W2 (Ω),Ω

2

Ω

A.1 Function Spaces

and v2W 1 (Ω),Γ 2

203

⎞2 ⎛ = ⎝ v(x)dsx ⎠ + ∇v2L2 (Ω) . Γ

As a ﬁrst consequence, the Sobolev norms ∇ · L2 (Ω) and · W21 (Ω) are ◦

equivalent norms in W pk (Ω). Secondly, there holds the Poincar´e inequality ⎞ ⎛⎛ ⎞2 2 ⎟ ⎜ |v(x)|2 dx ≤ cP ⎝⎝ v(x)dx⎠ + ∇v(x) dx⎠ for all v ∈ W21 (Ω). Ω

Ω

Ω

Theorem A.2 (Bramble–Hilbert–Lemma). Let f : W2p+1 (Ω) → R for p = 0, 1 be a bounded linear functional satisfying |f (v)| ≤ cf vW p+1 (Ω) 2

for all v ∈ W2p+1 (Ω).

Let P0 (Ω) = {q(x) = q0 : x ∈ Ω ⊂ Rd } be the space of constant polynomials and let P1 (Ω) = {q(x) = q0 + q1 x1 + . . . + qd xd : x ∈ Ω ⊂ Rd } be the space of linear polynomials deﬁned on Ω. If f (q) = 0 is satisﬁed for all q ∈ Pp (Ω), then it follows that |f (v)| ≤ cp cf |v|W p+1 (Ω) 2

for all v ∈ W2p+1 (Ω).

Recall that the deﬁnition of the Sobolev spaces W2s (Ω) is based on the generalised derivatives (cf. (A.3)) in the sense of Lloc 1 (Ω). In what follows, we will give a second deﬁnition of Sobolev spaces, which is based on derivatives of distributions. A distribution T ∈ D (Ω) is a complex continuous linear functional with respect to D(Ω) = C0∞ (Ω). For v ∈ Lloc 1 (Ω), the equality Tv (ϕ) = v(x)ϕ(x)dx for ϕ ∈ D(Ω) Ω

deﬁnes a regular distribution Tv ∈ D (Ω). The most famous distribution, which is not regular, is the Dirac δ−distribution satisfying δ0 (ϕ) = ϕ(0) ,

for all ϕ ∈ D(Ω) .

The equality (Dα Tv )(ϕ) = (−1)|α| Tv (Dα ϕ)

for all ϕ ∈ D(Ω)

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deﬁnes the derivative Dα Tv ∈ D (Ω) of a distribution Tv ∈ D (Ω). Next, we consider the Schwartz space S(Rd ) of smooth fast decreasing functions and its dual space S (Rd ) of tempered distributions. For ϕ ∈ S(Rd ), we deﬁne the Fourier transform F : S(Rd ) → S(Rd ) as ϕ(ξ) 0 = (F ϕ)(ξ) = (2π)−d/2 e−ı (x,ξ) ϕ(x)dx for ξ ∈ Rd , Rd

as well as the inverse Fourier transform ϕ(x) = (F

−1

−d/2

ϕ)(x) 0 = (2π)

eı (x,ξ) ϕ(ξ)dξ 0

for x ∈ Rd .

Rd

Note that for ϕ ∈ S(Rd ), we have Dα (Fϕ)(ξ) = (−i)|α| F(xα ϕ)(ξ),

ξ α (Fϕ)(ξ) = (−i)|α| F(Dα ϕ)(ξ).

For a distribution T ∈ S (Rd ), the Fourier transformation T0 ∈ S (Rd ) is deﬁned via T0(ϕ) = T (ϕ) 0 for all ϕ ∈ S(Rd ). For s ∈ R and v ∈ S(Rd ) we deﬁne the Bessel potential operator J s : S(Rd ) → S(Rd ) as s (J v)(x) = (1 + |ξ|2 )s/2 v0(ξ)eı (x,ξ) dξ for x ∈ Rd . Rd

The application of the Fourier transform yields (FJ s v)(ξ) = (1 + |ξ|2 )s/2 (Fv)(ξ) . Thus, J s acts similar to a diﬀerential operator of order s. For a distribution T ∈ S (Rd ), we can deﬁne J s T ∈ S (Rd ) via (J s T )(ϕ) = T (J s ϕ)

for all ϕ ∈ S(Rd ).

Now we are in a position to deﬁne the Sobolev space H s (Rd ) as a space of all distributions v ∈ S (Rd ) with J s v ∈ L2 (Rd ), and with the inner product u, v H s (Rd ) = J s u, J s v L2 (Rd ) , and the induced norm

v2H s (Rd ) = J s v2L2 (Rd ) =

(1 + |ξ|2 )s |0 v (ξ)|2 dξ. Rd

It turns out that H s (Rd ) = W2s (Rd ) for all s ∈ R. For a bounded domain Ω ⊂ Rd , the Sobolev space H s (Ω) is deﬁned by restriction

A.1 Function Spaces

(

H s (Ω) =

205

) v = v|Ω : v ∈ H s (Rd )

with the norm vH s (Ω) =

inf

v ∈H s (Rd ) : v |Ω =v

v H s (Rd ) .

Moreover, we introduce the Sobolev spaces s (Ω) = C ∞ (Ω)·H s (Rd ) , H 0

·H s (Ω)

H0s (Ω) = C0∞ (Ω)

and state the following result: s (Ω) ⊂ Theorem A.3. Let Ω ⊂ R3 be a Lipschitz domain and s ≥ 0. Then H s H0 (Ω). In particular, there holds s (Ω) = H0s (Ω) H

for s =

1 3 5 , , ,... . 2 2 2

Moreover, s (Ω) = H −s (Ω) , H

−s (Ω) H s (Ω) = H

for all s ∈ R.

Finally, we comment on the equivalence of the Sobolev spaces W2s (Ω) and H s (Ω), where we have to impose additional restrictions on the bounded domain Ω. In particular, let us assume that there is given a bounded linear extension operator EΩ : W2s (Ω) → W2s (Rd ). Note that this condition is satisﬁed, if an uniform cone condition holds for Ω. In fact, if Ω is a Lipschitz domain, we conclude H s (Ω) = W2s (Ω) for all s > 0. To deﬁne Sobolev spaces on the boundary Γ = ∂Ω of a bounded domain Ω ⊂ R3 we start with an arbitrary overlapping parametrisation Γ =

J '

Γi ,

Γi =

(

) x ∈ R3 : x = χi (ξ), ξ ∈ Ti ⊂ R2 .

i=1

We consider a partition of unity subordinated to the above decomposition, i.e. non–negative functions ϕi ∈ C0∞ (R3 ) satisfying J

ϕi (x) = 1

for x ∈ Γ,

ϕi (x) = 0 for x ∈ Γ \Γi .

i=1

Any function v on Γ can then be written as v(x) =

J i=1

with

ϕi (x)v(x) =

J i=1

vi (x)

for x ∈ Γ ,

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A Mathematical Foundations

vi (x) = ϕi (x)v(x)

for x ∈ Γi .

Inserting the local parametrisation, this gives vi (x) = ϕi (x)v(x) = ϕi (χi (ξ))v(χi (ξ)) = vi (ξ)

for ξ ∈ Ti ⊂ R2 .

For the above deﬁned functions vi (ξ), for ξ ∈ Ti ⊂ R2 , and s ≥ 0, we now convi H s (Ti ) . Hence, sider the Sobolev space H s (Ti ) associated with the norms we can deﬁne the Sobolev spaces H s (Γ ) equipped with the norm vHχs (Γ ) =

J

1/2 vi 2H s (Ti )

.

i=1

Note that derivatives Dξα vi (ξ) of the order |α| ≤ k require the existence of derivatives Dξα χi (ξ) due to the chain rule. Hence, we assume χi ∈ C k−1,1 (Ti ). Therefore, if Ω is a Lipschitz domain, we can deﬁne the Sobolev spaces H s (Γ ) on the boundary only for |s| ≤ 1. Note that the deﬁnition of the above Sobolev norm · Hχs (Γ ) depends on the parametrisation chosen, but all of the norms are equivalent. In particular, for s = 0 ⎞1/2 ⎛ vL2 (Γ ) = ⎝ |v(x)|2 dsx ⎠ Γ

is equivalent to vHχ0 (Γ ) . For s ∈ (0, 1), an equivalent norm is the Sobolev– Slobodeckii norm ⎛ ⎞1/2 2 |v(x) − v(y)| dsx dsy ⎠ . v2H s (Γ ) = ⎝v2L2 (Γ ) + |x − y|2+2s Γ Γ

Moreover,

vH 1/2 (Γ ),Γ

⎛⎛ ⎞1/2 ⎞2 2 |v(x) − v(y)| ⎜ ⎟ = ⎝⎝ v(x)dsx ⎠ + dsx dsy ⎠ |x − y|3 Γ

Γ Γ

deﬁnes an equivalent norm in H 1/2 (Γ ). For s < 0, Sobolev spaces H s (Γ ) = H −s (Γ ) are deﬁned by duality, wH s (Γ ) =

w, v Γ , 0=v∈H −s (Γ ) vH −s (Γ )

with respect to the duality pairing

sup

w, v Γ =

w(x)v(x)dsx . Γ

A.1 Function Spaces

207

Next, we consider open boundary parts Γ0 ⊂ Γ = ∂Ω. For s ≥ 0, we deﬁne ( ) H s (Γ0 ) = v = v|Γ0 : v ∈ H s (Γ ) , and vH s (Γ0 ) =

inf

v ∈H s (Γ ): v|Γ0 =v

v H s (Γ ) .

Correspondingly, s (Γ0 ) = H

(

) v = v|Γ0 : v ∈ H s (Γ ), supp v ⊂ Γ0 .

If s < 0, then by duality −s (Γ0 ) , H s (Γ0 ) = H

s (Γ0 ) = H −s (Γ0 ) . H

Finally, we consider a piecewise smooth boundary Γ =

J '

Γ i,

Γi ∩ Γj = ∅ for i = j ,

i=1

and deﬁne for s ≥ 0 the Sobolev spaces ( ) s (Γ ) = v ∈ L2 (Γ ) : v|Γi ∈ H s (Γi ), i = 1, . . . , J , Hpw equipped with the norm s (Γ ) = vHpw

J

1/2 v|Γi 2H s (Γi )

.

i=1

For s < 0, the corresponding Sobolev spaces are given as a product space s Hpw (Γ ) =

J

s (Γi ) , H

i=1

with the norm s (Γ ) = wHpw

J

w|Γi H s (Γ ) .

i=1 s Note that for all w ∈ Hpw (Γ ) and s < 0, we conclude w ∈ H s (Γ ), satisfying s (Γ ) . wH s (Γ ) ≤ wHpw

At the end of this subsection, we state some relations between the Sobolev spaces H s (Ω) in the domain Ω and H s (Γ ) on the boundary Γ = ∂Ω.

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Theorem A.4 (Trace Theorem). Let Ω ⊂ R3 be a bounded domain with boundary Γ ∈ C k−1,1 . The trace operator γ0int : H s (Ω) → H s−1/2 (Γ ) is continuous for 1/2 < s ≤ k, i.e. γ0int vH s−1/2 (Γ ) ≤ c T vH s (Ω)

for all v ∈ H s (Ω).

Theorem A.5 (Inverse Trace Theorem). Let Ω ⊂ R3 be a bounded domain with boundary Γ ∈ C k−1,1 . For 1/2 < s ≤ k, there exists a continuous extension operator E : H s−1/2 (Γ ) → H s (Ω) , i.e. EvH s (Ω) ≤ c IT vH s−1/2 (Γ )

for all v ∈ H s−1/2 (Γ ) ,

satisfying γ0int Ev = v.

A.2 Fundamental Solutions A.2.1 Laplace Equation The fundamental solution (1.7) of the Laplace equation can be found from the relation (1.5), which can be written as a partial diﬀerential equation in the distributional sense, −Δy u∗ (x, y) = δ0 (y − x)

for x, y ∈ R3 ,

where δ0 denotes the Dirac δ−distribution. Since the Laplace operator is invariant with respect to translations and rotations, we may use the transformation z = y − x to ﬁnd v satisfying −Δz v(z) = δ0 (z)

for z ∈ R3 .

The application of the Fourier transform together with the transformation rules for derivatives gives |ξ|2 v0(ξ) =

1 , (2π)3/2

and, therefore, v0(ξ) =

1 1 ∈ S (R3 ) . (2π)3/2 |ξ|2

The Fourier transform of the tempered distributions of the above form is well studied and can be found, for example, in [35, Chapter 2]. Thus, in d−dimensional space, it holds for λ = −d, −d − 2, . . .

A.2 Fundamental Solutions

209

λ+d −1 λ λ+d/2 Γ 2 |z|−λ−d . F |ξ| (z) = 2 Γ − λ2 For d = 3 and λ = −2, the above formula leads to / π −1 −1 −2 F |ξ| |z| , (z) = 2 where the the values Γ (1/2) = π/2 ,

Γ (1) = 1

of the Gamma function Γ have been used. Therefore, with z = x − y, the fundamental solution of the Laplace operator is 1 1 . 4π |x − y|

u∗ (x, y) =

A.2.2 Lame System The fundamental solution of linear elastostatics, given by the Kelvin tensor (1.66), can be found from (1.64), which can be written as a system (for = 1, 2, 3) of partial diﬀerential equations in the distributional sense: −μΔU ∗ (x, y) − (λ + μ)grad div U ∗ (x, y) = δ0 (y − x)e

for x, y ∈ R3 .

Deﬁning U ∗ (x, y) = Δw (x, y) −

λ+μ grad div w (x, y) λ + 2μ

for x, y ∈ R3 ,

the solution of the inhomogeneous system of linear elastostatics is equivalent to a system of scalar Bi–Laplace equations −μΔ2 w (x, y) = δ0 (y − x)e

for x, y ∈ R3 .

In particular, for = 1, we set w,2 (x, y) = w,3 (x, y) = 0, and it remains to solve −μΔ2 w,1 (x, y) = δ0 (y − x) for x, y ∈ R3 . Using the transformation z = y − x, we have to ﬁnd v satisfying −μΔ2 v(z) = δ0 (z)

for z ∈ R3

or −μΔϕ(z) = δ0 (z),

Δv(z) = ϕ(z) .

From this, we ﬁrst get ϕ(z) =

1 1 1 . μ 4π |z|

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A Mathematical Foundations

To determine v, we rewrite the Laplace equation in spherical coordinates as 1 1 1 1 ∂ 2 ∂ r v(r) = for r > 0 . r2 ∂r ∂r μ 4π r Thus, we obtain the general solution 1 a 1 1 r+ +b v(r) = μ 4π 2 r

for r > 0 ,

a, b ∈ R .

Choosing a = b = 0, we, therefore, get v(z) =

1 1 |z|. μ 8π

From this, we ﬁnd ∗ U1,1 (z) = Δv(z) −

λ + μ ∂2 v(z) , λ + 2μ ∂z12

∗ (z) = − U2,1

λ + μ ∂2 v(z) , λ + 2μ ∂z1 ∂z2

∗ (z) = − U3,1

λ+μ ∂ v(z) , λ + 2μ ∂z1 ∂z3

and, hence, ∗ (z) = U1,1

1 1 λ + 3μ 1 λ+μ z12 + , 8π μ(λ + 2μ) |z| 8π μ(λ + 2μ) |z|3

∗ (z) = U2,1

1 λ + μ z1 z2 , 8π μ(λ + 2μ) |z|3

∗ U3,1 (z) =

1 λ + μ z1 z3 . 8π μ(λ + 2μ) |z|3

Doing the same computations for = 2, 3, and inserting the Lam´e constants, this gives the Kelvin tensor as the fundamental solution of linear elastostatics, δk (yk − xk )(y − x ) 1 1 1+ν ∗ (3 − 4ν) + Uk (x, y) = 8π E 1 − ν |x − y| |x − y|3 for k, = 1, 2, 3. A.2.3 Stokes System To ﬁnd the fundamental solution for the Stokes system, we have to solve the following system of partial diﬀerential equations

A.2 Fundamental Solutions

−ΔU ∗ (x, y) + ∇q∗ (x, y) = δ0 (y − x)e ,

211

div U ∗ (x, y) = 0 for x, y ∈ R3 ,

for = 1, 2, 3, and −ΔU ∗4 (x, y) + ∇q4∗ (x, y) = 0,

div U ∗4 (x, y) = δ0 (y − x)

for x, y ∈ R3 .

As in the case of linear elastostatics, we set U ∗ (x, y) = Δw (x, y) − grad div w (x, y),

q∗ (x, y) = Δ div w (x, y)

for x, y ∈ R3 and for = 1, 2, 3. This implies divU ∗ (x, y) = 0, and −Δ2 w (x, y) = δ0 (y − x)e . For = 1, we ﬁnd Δw1,1 (x, y) = and w1,1 (x, y) =

1 1 |x − y|, 8π

1 1 1 4π |x − y|

w1,2 (x, y) = w1,3 (x, y) = 0 .

Using div w1 (x, y) =

1 1 ∂ 1 1 y1 − x1 , |x − y| = 8π ∂y1 8π |x − y|

we obtain ∗ U11 (x, y) = Δw1,1 (x, y) −

∂ div w1 (x, y) = ∂y1

1 1 ∂ y1 − x1 1 1 1 − = 4π |x − y| 8π ∂y1 |x − y| 1 (x1 − y1 )2 1 1 + = , 8π |x − y| |x − y|3 =

∗ U12 (x, y) = −

=− ∗ (x, y) = − U13

=− and

∂ div w1 (x, y) = ∂y2 1 1 (x1 − y1 )(x2 − y2 ) 1 1 ∂ y1 − x1 = , 8π ∂y2 |x − y| 8π |x − y|3 ∂ div w1 (x, y) = ∂y3 1 1 (x1 − y1 )(x3 − y3 ) 1 1 ∂ y1 − x1 = , 8π ∂y3 |x − y| 8π |x − y|3

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q1∗ (x, y) = divΔw1 (x, y) =

1 x1 − y1 1 ∂ 1 = . 4π ∂y1 |x − y| 4π |x − y|3

Doing the same computations for = 2, 3, we obtain the fundamental solution of the Stokes system as δk (xk − yk )(x − y ) 1 1 ∗ + Uk , (x, y) = 8π |x − y| |x − y|3 1 x − y q∗ (x, y) = 4π |x − y|3 for k, = 1, 2, 3. To ﬁnd the fundamental solution U ∗4 (x, y) and q4∗ (x, y), we have to solve the system −ΔU ∗4 (x, y) + ∇q4∗ (x, y) = 0,

div U ∗4 (x, y) = δ0 (y − x)

for x, y ∈ R3

in the distributional sense. Setting U ∗4 (x, y) = −∇ϕ(x, y)

for x, y ∈ R3 ,

−Δϕ(x, y) = δ0 (y − x)

for x, y ∈ R3

we get and therefore ϕ(x, y) =

1 1 . 4π |x − y|

Hence, ∗ (x, y) = − U4k

∂ 1 xk − yk ϕ(x, y) = , ∂yk 4π |x − y|3

k = 1, 2, 3 .

In addition, taking the divergence of the ﬁrst equation, this gives Δq4∗ (x, y) = Δdiv U ∗4 (x, y) = Δδ0 (y − x) , implying

q4∗ (x, y) = δ0 (y − x) .

A.2.4 Helmholtz Equation The fundamental solution u∗κ (x, y) satisﬁes the relation (1.94), − Δu∗κ (x, y) − κ2 u∗κ (x, y) u(y)dy = u(x) for x ∈ Ω , Ω

which can be written as a partial diﬀerential equation in the distributional sense,

A.3 Mapping Properties

−Δu∗κ (x, y) − κ2 u∗κ (x, y) = δ0 (y − x)

213

for x, y ∈ R3 .

Setting z = y − x, we have to ﬁnd v as a solution of for z ∈ R.

−Δv(z) − κ2 v(z) = δ0 (z)

Using spherical coordinates as well as v(z) = v(r) with r = |z|, this is equivalent to 1 2 ∂ 1 ∂ r2 v(r) − κ2 v(r) = 0 for r > 0 . − 2 r ∂r ∂r With the substitutions v(r) =

w(r) , r

∂ ∂ w(r) 1 ∂ 1 v(r) = = w(r) − 2 w(r) , ∂r ∂r r r ∂r r

we get ∂2 w(r) + κ2 w(r) = 0 for r > 0 , ∂r2 with the general solution w(r) = Aeı κr + Be−ı κr

for r > 0 .

Choosing A = 1/4π and B = 0, this gives the fundamental solution for the Helmholtz equation, u∗κ (x, y) =

1 eı κ|x−y| 4π |x − y|

for x, y ∈ R3 .

A.3 Mapping Properties In this section, we will summarise the mapping properties of all boundary integral operators used. For simplicity, we will restrict ourselves to the case of the Laplace equation. For a more detailed presentation, we refer, for example, to [24, 71, 105]. The Newton potential f )(x) = u∗ (x, y)f (y)dy for x ∈ Ω u(x) = (N Ω

is a generalised solution of the Poisson equation −Δu(x) = f (x) for x ∈ Ω. −1 (Ω), we have N f ∈ H 1 (Ω) satisfying In particular, for a given f ∈ H f H 1 (Ω) ≤ c f −1 uH 1 (Ω) = N H (Ω)

−1 (Ω). for all f ∈ H

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Taking the interior trace, we obtain f )(x) = (N f )(x) = γ0int (N

u∗ (x, y)f (y)dy

for x ∈ Γ .

Ω

Combining the mapping properties of the Newton potential :H −1 (Ω) → H 1 (Ω) N with those of the interior trace operator γ0int : H 1 (Ω) → H 1/2 (Γ ) , this gives

: H −1 (Ω) → H 1/2 (Γ ) . N = γ0int N

With this we can derive corresponding mapping properties of the single layer potential (V w)(x) = u∗ (x, y)w(y)dsy for x ∈ R3 \Γ . Γ

−1

For f ∈ H

(Ω), we have, by exchanging the order of integration, V w, f Ω = f (x) u∗ (x, y)w(y)dsy dx Ω

Γ

w(y)

= Γ

u∗ (x, y)f (x)dx dsy = w, N f Γ .

Ω

−1 (Ω), we ﬁnd w ∈ H −1/2 (Γ ) and Hence, due to N f ∈ H 1/2 (Γ ) for f ∈ H 1 V w ∈ H (Ω). In fact, the single layer potential V : H −1/2 (Γ ) → H 1 (Ω) is a bounded operator, and, combining this with the mapping property of the interior trace operator γ0int : H 1 (Ω) → H 1/2 (Γ ) , this gives

V = γ0int V : H −1/2 (Γ ) → H 1/2 (Γ ).

Moreover, for x ∈ Ω we have ∗ − Δx u∗ (x, y) w(y)dsy = 0, −Δ(V w)(x) = −Δ u (x, y)w(y)dsy = Γ

Γ

A.3 Mapping Properties

215

i.e. V w is a generalised solution of the Laplace equation for any w ∈ H −1/2 (Γ ). The above considerations are essentially based on the symmetry of the fundamental solution, i.e. u∗ (x, y) = u∗ (y, x) . Therefore, these considerations are valid only for self–adjoint partial diﬀerential operators. For more general partial diﬀerential operators, one has to incorporate also all the volume and surface potentials which are deﬁned by the fundamental solution of the formally adjoint partial diﬀerential operator, see, for example, [71]. It remains to describe an explicite representation of the single layer potential operator (V w)(x) = γ0int (V w)(x) for x ∈ Γ . For w ∈ L∞ (Γ ), we obtain (V w)(x) = γ0int (V w)(x) =

u∗ (x, y)w(y)dsy

Γ

as a weakly singular boundary integral (cf. Lemma 1.1). For ε > 0, we consider x ∈ Ω and x ∈ Γ with | x − x| < ε. Then, ∗ ∗ x, y)w(y)dsy − u (x, y)w(y)dsy lim u ( x →x Γ y∈Γ :|y−x|>ε ∗ ∗ u ( x, y) − u (x, y) w(y)dsy ≤ lim x →x y∈Γ :|y−x|>ε ∗ u ( x, y)w(y)dsy + lim x →x y∈Γ :|y−x|<ε u∗ ( x, y)w(y)dsy = lim x →x y∈Γ :|y−x|<ε ∗ x, y)dsy ≤ sup w(y) lim u ( x →x y∈Γ :|y−x|<ε

y∈Γ :|y−x|<ε

1 wL∞ (Γ ) lim ≤ x →x 4π

1 dsy | x − y|

y∈Γ :|y−x|<ε

1 wL∞ (Γ ) = 4π

y∈Γ :|x−y|<ε

1 dsy ≤ c wL∞ (Γ ) ε, |x − y|

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where we used polar coordinates to obtain the last inequality. Taking the limit ε → 0, this gives the deﬁnition of the single layer potential as a weakly singular integral. In the same way we get for the exterior trace γ0ext (V w)(x) = (V w)(x)

for x ∈ Γ.

Since V w ∈ H 1 (Ω) is a generalised solution of the Laplace equation for any w ∈ H 1/2 (Γ ), we can compute the associated conormal derivative of V w as γ1int V w ∈ H −1/2 (Γ ). For v ∈ H 1 (Ω), and using Green’s ﬁrst formula as well as interpreting the surface integral as weakly singular integral, we obtain ∇(V w)(x), ∇v(x) dx γ1int (V w)(x)γ0int v(x)dsx = Γ

Ω

=

∇ u∗ (x, y)w(y)dsy , ∇v(x) dx

Ω

Γ ∇ lim =

u∗ (x, y)w(y)dsy , ∇v(x) dx

ε→0 y∈Γ :|y−x|≥ε

Ω

=

w(y) lim

∇x u∗ (x, y), ∇v(x) dx dsy .

ε→0 x∈Ω:|x−y|≥ε

Γ

Again, Green’s ﬁrst formula gives ∇x u∗ (x, y), ∇x v(x) dx = x∈Ω:|x−y|≥ε

int u∗ (x, y)γ int v(x)ds γ1,x x 0

x∈Γ :|x−y|≥ε

int u∗ (x, y)γ int v(x)ds , γ1,x x 0

+

x∈Ω:|x−y|=ε

and, therefore, γ1int (V w)(x)γ0int v(x)dsx = w(y) lim Γ

Γ

=

int

γ1,x u ε→0 x∈Γ :|x−y|≥ε

+

w(y) lim Γ

lim

int

= Γ

w(y) lim Γ

∗

(x, y)γ0int v(x)dsx dsy

int

γ1,x u ε→0 x∈Ω:|x−y|=ε

γ1,x u ε→0 Γ y∈Γ :|y−x|≥ε +

∗

∗

(x, y)γ0int v(x)dsx dsy

(x, y)w(y)dsy γ0int v(x)dsx int

γ1,x u ε→0 x∈Ω:|x−y|=ε

∗

(x, y)v(x)dsx dsy

(K w)(x)γ0int v(x)dsx +

w(y)σ(y)v(y)dsy . Γ

A.3 Mapping Properties

In the above, the adjoint double layer potential int u∗ (x, y)w(y)ds γ1,x (K w)(x) = lim y ε→0 y∈Γ :|y−x|≥ε

217

for x ∈ Γ

has been introduced. Furthermore, we have used int u∗ (x, y)v(x)ds = σ(y)v(y) , lim γ1,x x ε→0 x∈Ω:|x−y|=ε

with

int

γ1,x u ε→0 x∈Ω:|x−y|=ε

σ(y) = lim

1 = lim ε→0 4π

∗

(x, y)dsx

(y − x, n(x)) 1 1 dsx = lim ε→0 4π ε2 |x − y|3

x∈Ω:|x−y|=ε

dsx ,

x∈Ω:|x−y|=ε

which is related to the interior angle of Ω in y ∈ Γ . In particular, if Γ is smooth in y ∈ Γ , we ﬁnd σ(y) = 1/2 (cf. Lemma 1.3). Summarising the above, for w ∈ H −1/2 (Γ ), we have γ1int V w = σ(x)w(x) + (K w)(x)

for x ∈ Γ

in the sense of H −1/2 (Γ ). Correspondingly, the application of the exterior conormal derivative gives γ1ext V w = (σ(x) − 1)w(x) + (K w)(x)

for x ∈ Γ ,

and, therefore, we obtain the jump relation of the adjoint double layer potential, [γ1 V w] = γ1ext (V w)(x) − γ1int (V w)(x) = −w(x) Next, we consider the double layer potential int u∗ (x, y)v(y)ds (W v)(x) = γ1,y y

for x ∈ Γ.

for x ∈ R3 \Γ .

Γ

−1 (Ω), by exchanging the order of integration, we have, For f ∈ H int u∗ (x, y)v(y)ds dx W v, f Ω = f (x) γ1,y y Ω

=

Γ

v(y)γ int

1,y

Γ

=

Ω

f )(y)ds v(y)γ int (N 1,y

Γ

u∗ (x, y)f (x)dx dsy

y

f Γ . = v, γ1int N

218

A Mathematical Foundations

From this, we ﬁnd W v ∈ H 1 (Ω) for any v ∈ H 1/2 (Γ ). Moreover, W v ∈ H 1 (Ω) is a generalised solution of the Laplace equation for any v ∈ H 1/2 (Γ ). The application of the interior trace operator γ0int : H 1 (Ω) → H 1/2 (Γ ) gives γ0int W : H 1/2 (Γ ) → H 1/2 (Γ ). It turns out (cf. Lemma 1.2) that γ0int (W v)(x) = (−1 + σ(x))v(x) + (Kv)(x) with the double layer potential

(Kv)(x) = lim (Kε v)(x) = lim

int

γ1,y u ε→0 y∈Γ :|y−x|≥ε

ε→0

∗

for x ∈ Γ ,

(x, y)v(y)dsy

for x ∈ Γ.

This representation follows, when considering x ∈ Ω and x ∈ Γ satisfying | x − x| < ε, from int u∗ ( int u∗ (x, y)v(y)ds x, y)v(y)dsy − γ1,y (W v)( x) − (Kε v)(x) = γ1,y y Γ

= y∈Γ :|y−x|≥ε

y∈Γ :|y−x|≥ε

int u∗ ( int u∗ (x, y) v(y)ds γ1,y x, y) − γ1,y y int u∗ ( γ1,y x, y) v(y) − v(x) dsy

+ y∈Γ :|y−x|≤ε

int u∗ ( γ1,y x, y)dsy .

+v(x) y∈Γ :|y−x|≤ε

Note that for all ε > 0 we have int u∗ ( int u∗ (x, y) v(y)ds = 0. γ1,y x, y) − γ1,y lim y x →x y∈Γ :|y−x|≥ε

In addition,

lim

ε→0 y∈Γ :|y−x|≤ε

int u∗ ( γ1,y x, y) v(y) − v(x) dsy = 0.

Moreover, for x ∈ Bε (x) = {z ∈ Ω : |z − x| < ε}, we have int ∗ int ∗ int u∗ ( γ1,y u ( x, y)dsy = γ1,y u ( x, y)dsy − γ1,y x, y)dsy . y∈Γ :|y−x|≤ε

∂Bε (x)

y∈Ω:|y−x|=ε

A.3 Mapping Properties

Taking into account that int u∗ ( γ1,y x, y)dsy = −1

219

for x ∈ Bε (x)

∂Bε (x)

as well as the direction of the normal vector along y ∈ Ω : |y − x| = ε, we ﬁnally obtain the above representation for γ0int W v. Correspondingly, the application of the exterior trace operator gives γ0ext (W v)(x) = σ(x)v(x) + (Kv)(x)

for x ∈ Γ ,

and, therefore, the jump relation of the double layer potential reads [γ0 W v] = γ0ext (W v)(x) − γ0int (W v)(x) = v(x)

for x ∈ Γ .

Since the double layer potential W v ∈ H 1 (Ω) is a solution of the Laplace equation for any v ∈ H 1/2 (Γ ), the application of the conormal derivative γ1int deﬁnes the bounded operator D = −γ1int W : H 1/2 (Γ ) → H −1/2 (Γ ). Note that for x ∈ Γ , we have (Dv)(x) = −γ1int (W v)(x) = 1 lim = 4π ε→0

lim

Ω x→x∈Γ

∇x (W v)( x), n(x)

(n(x), n(y)) (y − x, n(y))(y − x, n(x)) −3 3 |x − y| |x − y|5

v(y)dsy ,

y∈Γ :|y−x|≥ε

which does not exist. In what follows we will sketch the computations to obtain the alternative representation (1.9) (cf. Lemma 1.4). For the derivatives of the double layer potential for x ∈ Ω we ﬁrst have ∂ ∂ (W v)( x) = γ1int u∗ ( x, y)v(y)dsy ∂ xi ∂ xi Γ 1 ∂ 1 v(y)dsy n(y), ∇y = 4π ∂ xi | x − y| Γ 1 ∂ 1 v(y)dsy n(y), ∇y =− 4π ∂yi | x − y| Γ 1 1 v(y)dsy n(y), curly ei × ∇y = 4π | x − y| Γ 1 1 dsy . = v(y) curlΓ,y ei × ∇y 4π | x − y| Γ

220

A Mathematical Foundations

Now, using integration by parts, i.e. ϕ(y), curlΓ ψ(y) dsy , curlΓ ϕ(y)ψ(y)dsy = − Γ

we obtain

Γ

1 1 ∂ dsy curlΓ v(y), ei × ∇y (W v)( x) = − ∂ xi 4π | x − y| Γ 1 1 dsy , ei , curlΓ v(y) × ∇y = 4π | x − y| Γ

and, therefore, x) = ∇x (W v)(

1 4π

curlΓ v(y) × ∇y Γ

1 =− 4π

1 | x − y|

curlΓ v(y) × ∇x Γ

1 | x − y|

dsy dsy .

Hence, 1 1 , n(x) dsy curlΓ v(y) × ∇x 4π | x − y| Γ 1 1 dsy . curlΓ v(y), n(x) × ∇x = 4π | x − y|

(∇x (W v)( x), n(x)) = −

Γ

Taking the limes x → x, we obtain 1 1 dsy curlΓ v(y), n(x) × ∇x (Dv)(x) = − 4π |x − y| Γ 1 1 dsy , curlΓ v(y), curlΓ,x =− 4π |x − y| Γ

and, therefore, 1 1 dsy dsx curlΓ v(y), curlΓ,x u(x) 4π |x − y| Γ Γ 1 1 dsx dsy . = curlΓ,x u(x)curlΓ,y v(y) 4π |x − y|

Dv, u Γ = −

Γ Γ

With

A.3 Mapping Properties

221

curlΓ,x u(x)curlΓ,y v(y) = n(x), ∇x × u(x)curlΓ,y v(y) = n(x), ∇x u(x) × curlΓ,y v(y) = n(x) × ∇x u(x), curlΓ,y v(y) = curlΓ,x u(x), curlΓ,y v(y) , we ﬁnally obtain the alternative representation (1.9), 1 (curlΓ u(x), curlΓ v(y)) dsx dsy . Dv, u Γ = 4π |x − y| Γ Γ

Since the ellipticity of boundary integral operators is essential to derive results on the unique solvability of boundary integral equations, we will sketch the proof of the ellipticity of the single layer potential V , cf. Lemma 1.1. The single layer potential u∗ (x, y)w(y)dsy for x ∈ R3 \Γ u(x) = (V w)(x) = Γ

is a solution of the interior Dirichlet boundary value problem −Δu(x) = 0 for x ∈ Ω,

γ0int u(x) = γ0int (V w)(x) = (V w)(x)

for x ∈ Γ.

By applying Green’s ﬁrst formula (cf. (1.2)), we obtain |∇u(x)|2 dx = γ1int u(x)γ0int u(x)dsx aΩ (u, u) = Ω

Γ

=

γ1int (V w)(x)γ0int (V w)(x)dsx

Γ

=

1 w(x) + (K w)(x) (V w)(x)dsx . 2

Γ

From Green’s ﬁrst formula (1.2), we also ﬁnd the estimate c1 γ1int u2H −1/2 (Γ ) ≤ aΩ (u, u). Since u = V w is also a solution of the exterior Dirichlet boundary value problem, −Δu(x) = 0 for x ∈ Ω e , γ0ext u(x) = γ0ext (V w)(x) = (V w)(x) for x ∈ Γ , satisfying the radiation condition

222

A Mathematical Foundations

|u(x)| = O we also ﬁnd the relations

aΩ e (u, u) = −

as |x| → ∞ ,

γ1ext u(x)γ0ext u(x)dsx

Γ

=−

1 |x|

1 − w(x) + (K w)(x) (V w)(x)dsx , 2

Γ

and

c2 γ0ext u2H −1/2 (Γ ) ≤ aΩ e (u, u).

Hence, we obtain V w, w Γ =

w(x)(V w)(x)dsx Γ

=

1 w(x) + (K w)(x) (V w)(x)dsx 2 Γ 1 − w(x) + (K w)(x) (V w)(x)dsx − 2 Γ

= aΩ (u, u) + aΩ e (u, u) ≥ min{c1 , c2 } γ1int u2H −1/2 (Γ ) + γ1ext u2H −1/2 (Γ ) . Using 1 1 2 w2H −1/2 (Γ ) = w + K w − − w + K w −1/2 2 2 H (Γ ) = γ1int u − γ1ext u2H −1/2 (Γ ) ≤ 2 γ1int u2H −1/2 (Γ ) + γ1ext u2H −1/2 (Γ ) , we ﬁnally obtain the ellipticity estimate V w, w Γ ≥ cV1 w2H −1/2 (Γ )

for all w ∈ H −1/2 (Γ ).

Note that the ellipticity estimate of the hypersingular boundary integral operator D (cf. Lemma 1.4) follows almost in the same manner when considering the double layer potential u = W v. If Γ = ∂Ω is the boundary of a Lipschitz domain Ω ⊂ R3 , we then can extend the mapping properties of all boundary integral operators, see [24]. In particular, the boundary integral operators

A.3 Mapping Properties

V : H −1/2+s (Γ ) → H 1/2+s (Γ ), K : H 1/2+s (Γ ) → H 1/2+s (Γ ), K : H −1/2+s (Γ ) → H −1/2+s (Γ ), D : H 1/2+s (Γ ) → H −1/2+s (Γ ) are bounded for all s ∈ [−1/2, 1/2].

223

B Numerical Analysis

B.1 Variational Methods Let X be a Hilbert space with an inner product ·, · X , which induces a norm * vX = v, v X for all v ∈ X . Let X be the dual space of X with respect to the duality pairing · , · : X × X → R . Then, f X ≤

sup 0=v∈X

f, v

vX

for all f ∈ X .

Let A : X → X be a bounded linear operator with AvX ≤ cA 2 vX

for all v ∈ X,

(B.1)

which is symmetric, i.e. Au, v = Av, u for all u, v ∈ X. For a given f ∈ X , we consider an operator equation to ﬁnd u ∈ X such that Au = f

(B.2)

is satisﬁed in X . Then, 0 = Au − f X =

sup 0=v∈X

Au − f, v

, vX

and, therefore, u ∈ X is a solution of the variational problem Au, v = f, v for all v ∈ X.

(B.3)

226

B Numerical Analysis

For a given operator A : X → X , we deﬁne the functional F (v) =

1 Av, v − f, v . 2

If the operator A is assumed to be positive semi–elliptic, i.e. Av, v ≥ 0 for all v ∈ X , then the solution of the variational problem (B.3) is equivalent to the minimisation problem (B.4) F (u) = min F (v) . v∈X

Essential for the analysis of variational problems is the Riesz representation theorem [118], i.e. any bounded linear functional f ∈ X is of the form f, v = u, v X

for all v ∈ X.

(B.5)

The element u ∈ X is hereby uniquely determined and satisﬁes uX = f X . The Riesz representation theorem (see (B.5)) deﬁnes a linear and bounded operator J : X → X with Jf, v X = f, v

for all v ∈ X,

Jf X = f X .

Now, instead of the operator equation (B.2), Au = f in X , we consider the equivalent operator equation JAu = Jf

in X.

(B.6)

Since A : X → X is a bounded operator satisfying (B.1), we obtain JAvX = AvX ≤ cA 2 vX

for all v ∈ X,

i.e. the operator JA : X → X is bounded with JAX→X ≤ cA 2 . An operator A : X → X is called X–elliptic, if 2 Av, v ≥ cA 1 vX

for all v ∈ X .

(B.7)

Then, if A is X–elliptic, the estimate 2 JAv, v X = Av, v ≥ cA 1 vX

for all v ∈ X

follows, i.e. the operator JA : X → X is also X–elliptic. Therefore, instead of (B.6), we consider the ﬁx point equation

B.1 Variational Methods

u = u − J(Au − f ) = T u + Jf ,

227

(B.8)

with T = I − JA : X → X for some positive ∈ R+ . For cA 0 < < 2 1 2 cA 2 the operator T : X → X deﬁnes a contraction in X with T X→X < 1. Thus, the unique solvability of the ﬁx point equation (B.8) follows from Banach’s ﬁx point theorem. Hence, if A : X → X is bounded and X–elliptic, we conclude the unique solvability of the operator equation (B.2), and, therefore, of the equivalent variational problem (B.3), and of the minimisation problem (B.4). This result is just the well known Lax–Milgram lemma. Moreover, for the solution u ∈ X of the operator equation Au = f , we obtain from the ellipticity estimate (B.7) 2 cA 1 uX ≤ Au, u = f, u ≤ f X uX ,

and, therefore, uX ≤

1 f X . cA 1

Now we consider a sequence {XM }M ∈N of conformal trial spaces M XM = span ϕ ⊂X, =1

which is assumed to be dense in X. In particular, we assume the approximation property inf v − vM X = 0 for all v ∈ X. (B.9) lim M →∞ vM ∈XM

To deﬁne an approximate solution of the operator equation (B.2), or of the equivalent minimisation problem (B.4), we consider the ﬁnite dimensional minimisation problem F (uM ) =

min F (vM ) ,

vM ∈XM

with 1 AvM , vM − f, vM

2 M M M 1 = v vj Aϕ , ϕj − v f, ϕ = F(v) . 2 j=1

F (vM ) =

=1

From the necessary condition

=1

228

B Numerical Analysis

d F (v) = 0 for k = 1, . . . , M , dvk we then obtain M

v Aϕ , ϕk − f, ϕk = 0 for k = 1, . . . , M .

=1

Thus, uM ∈ XM is the solution of the Galerkin variational problem M

u Aϕ , ϕk = f, ϕk for k = 1, . . . , M,

(B.10)

=1

or, in the equivalent formulation, AuM , vM = f, vM for all vM ∈ XM .

(B.11)

The Galerkin variational problem (B.10) is obviously equivalent to a linear system AM u = f , with AM [k, ] = Aϕ , ϕk ,

fk = f, ϕk

for k, = 1, . . . , M . Note that (AM u, v) =

M M

A[k, ]u vk =

k=1 =1

M M

Aϕ , ϕk u vk = AuM , vM

k=1 =1

for all u, v ∈ RM , i.e. uM , vM ∈ XM . Therefore, 2 (AM v, v) = AvM , vM ≥ cA 1 vM X

for all v ∈ RM , i.e. vM ∈ XM ⊂ X. The Galerkin stiﬀness matrix AM is hence positive deﬁnite, and, therefore, invertible. In particular, the Galerkin variational problem (B.11) is uniquely solvable. Moreover, from 2 cA 1 uM X ≤ AuM , uM = f, uM ≤ f X uM X ,

we conclude the stability estimate uM X ≤

1 f X . cA 1

From (B.3), we obtain Au, vM = f, vM for all vM ∈ XM ⊂ X.

B.1 Variational Methods

229

Subtracting from this the Galerkin variational problem (B.11), this gives the Galerkin orthogonality A(u − uM ), vM = 0

for all vM ∈ XM .

(B.12)

Using the ellipticity estimate (B.7), the Galerkin orthogonality (B.12), and the boundedness of A, we then obtain 2 cA 1 u − uM X ≤ A(u − uM ), u − uM

= A(u − uM ), u − vM + A(u − uM ), vM − uM

= A(u − uM ), u − vM

≤ A(u − uM )X u − vM X ≤ cA 2 u − uM X u − vM X , and, therefore, u − uM X ≤

cA 2 u − vM X cA 1

for all vM ∈ XM .

This results in Cea’s lemma, i.e. u − uM X ≤

cA 2 cA 1

inf

vM ∈XM

u − vM X .

(B.13)

Hence, we obtain convergence uM → u for M → ∞ from the approximation property (B.9). In many applications, e.g. for direct boundary integral formulations, the right hand side in (B.2) is given as f = Bg, where B : Y → X is some bounded operator satisfying BgX ≤ cB 2 gY

for all g ∈ Y.

Then the Galerkin formulation (B.11) requires the evaluation of fk = Bg, ϕk for all k = 1, . . . , M , which may be complicated for general given g ∈ Y . Hence, introducing an approximation N gj ψj ∈ YN ⊂ Y , gN = j=1

we may compute the perturbed vector values fk = BgN , ϕk =

N j=1

leading to the linear system

gj Bψj , ϕk =

N j=1

BN [k, j]gj ,

230

B Numerical Analysis

AM u = BN g . M Note that the solution vector u ∈ RM corresponds to the unique solution u of the perturbed variational problem A uM , vM = BgN , vM for all vM ∈ XM . Now, instead of the Galerkin orthogonality (B.12), we obtain A(u− uM ), vM = A(uM − uM ), vM = B(g−gN ), vM for all vM ∈ XM . From the ellipticity estimate (B.7), we then conclude cA M 2X ≤ A(uM − u M ), uM − u M

1 uM − u M ≤ cB M X , = B(g − gN ), uM − u 2 g − gN Y uM − u and, therefore, uM − u M X ≤

cB 2 g − gN Y . cA 1

Hence, we ﬁnd from the triangle inequality the error estimate u − u M X ≤ u − uM X + uM − u M X B c ≤ u − uM X + 2A g − gN Y . c1 Besides an approximation of the given right hand side f , we also have to consider an approximation of the given operator A, e.g., when applying numerical integration schemes. Hence, instead of the Galerkin formulation (B.11), we have to solve a perturbed variational problem to ﬁnd the solution u M ∈ XM satisfying uM , vM = f, vM for all vM ∈ XM . (B.14) A : X → X is a bounded linear operator satisfying We assume that A

X ≤ cA Av 2 vX

for all v ∈ X.

To ensure the unique solvability of the Galerkin formulation (B.14), we have i.e. to assume the XM –ellipticity of A,

M , vM ≥ cA vM 2 Av X 1

for all vM ∈ XM .

(B.15)

M , deﬁned by The perturbed stiﬀness matrix A M [k, ] = Aϕ , ϕk

A for k, = 1, . . . , M , is then positive deﬁnite, and, therefore, invertible. Subtracting the perturbed variational formulation (B.14) from the Galerkin variational problem (B.11), this gives the orthogonality

B.2 Approximation Properties

231

uM , vM = 0 for all vM ∈ XM . AuM − A we ﬁnd From this, and by using the XM –ellipticity of A, M −u cA M 2X ≤ A(u M ), uM − u M

1 uM − u

− A)uM , uM − u = (A M

− A)uM X uM − u ≤ (A M X , and, therefore, M X ≤ uM − u

1

cA 1

M X . (A − A)u

Hence, applying the triangle inequality, we ﬁnally obtain the error estimate u − u M X ≤ u − uM X + uM − u M X 1 M X ≤ u − uM X + (A − A)u cA 1 1 X + (A − A)(u ≤ u − uM X + (A − A)u − uM )X cA 1 A 1 cA + c X . ≤ 1 + 2 2 u − uM X + (A − A)u A cA c 1 1

B.2 Approximation Properties In this section, we will prove the approximation properties of piecewise polynomial basis functions deﬁned in Chapter 2. Let Γ = ∂Ω be a piecewise smooth Lipschitz boundary which is represented by a non–overlapping decomposition Γ =

J '

Γi ,

Γi ∩ Γj = ∅ for i = j,

i=1

where each boundary segment Γi is described via a local parametrisation (A.2), ) ( Γi = x ∈ R3 : x = χi (ξ) for ξ ∈ Ti ⊂ R2 , with respect to some parameter domain Ti . For a sequence of boundary element discretisations (cf. Chapter 2), Γ =

N ' =1

τ ,

232

B Numerical Analysis

we further assume that for each boundary element τ there exists exactly one boundary segment Γi with τ ⊂ Γi . Hence, there also exists an element qi such that τ = χi (qi ). For the area Δ of the boundary element τ , we then obtain * Δ = dsx = EG − F 2 dξ , τ

qi

with E=

2 2 3 3 3 ∂ ∂ ∂ ∂ xi (ξ) , G = xi (ξ) , F = xi (ξ) xi (ξ). ∂ξ ∂ξ ∂ξ ∂ξ 1 2 1 2 i=1 i=1 i=1

We assume that the parametrisation is uniformly bounded, i.e. * cχ1 ≤ EG − F 2 ≤ cχ2 for all ξ ∈ Ti , i = 1, . . . , J. Hence, we obtain cχ1 area qi ≤ Δ ≤ cχ2 area qi . Then, τ

≤

|v(x)|2 dsx =

v2L2 (τ ) =

cχ2

|v(χi (ξ))|2

*

qi

cχ Δ |v(χi (ξ))| dξ ≤ 2χ c1 area qi

EG − F 2 dξ |v(χi (ξ))|2 dξ .

2

qi

qi

Now, using a parametrisation qi = χi (τ ) with respect to the parameter domain τ (cf. Chapter 2), this gives |v(χi (ξ))|2 dξ = |v(χi (χi (η)))|2 |det χi | dη , τ

qi

and with area qi

=

|det χi | dη =

dξ = qi

1 |det χi | , 2

τ

we ﬁnally obtain v 2L2 (τ ) , v2L2 (τ ) ≤ 2c Δ

v (η) = v(χi (χi (η))) .

Note that this result would follow directly when considering a parametrisation of the boundary element τ with respect to the reference element τ . However, the above approach is needed when considering higher order Sobolev spaces, for example

B.2 Approximation Properties

|v|2H m (τ ) =

|α|=m

|Dξα v(χi (ξ))|2 dξ,

233

m ∈ N.

qi

For the parametrisation qi = χi (τ ) and for |α| = m, we now obtain the norm equivalence inequalities, 1 (area qi )1−m |Dηα v (η)|2 dη ≤ |Dξα v(χi (ξ))|2 dξ , cm τ

qi

and

|Dξα v(χi (ξ))|2 dξ

≤

|Dηα v (η)|2 dη .

cm (area qi )1−m τ

qi

Hence, we have c1 Δ1−m | v |2H m (τ ) ≤ |v|2H m (τ ) ≤ c2 Δ1−m | v |2H m (τ )

for m ∈ N .

Let Qh w ∈ Sh0 (Γ ) be the L2 projection satisfying the variational problem N =1

w

ψ (x)ψk (x)dsx =

Γ

w(x)ψk (x)dsx

for k = 1, . . . , N.

Γ

Due to the deﬁnition of the piecewise constant basis functions ψk , we obtain the Galerkin orthogonality w(x) − Qh w(x) dsx = 0 for = 1, . . . , N τ

as well as w =

1 Δ

w(y)dsy

for = 1, . . . , N.

τ

For the local error, we ﬁrst ﬁnd w − Qh w2L2 (τ ) ≤ 2c Δ w − Qτ w 2L2 (τ ) , where Qτ w (ξ) = Qh w(χi (ξ)). For an arbitrary but ﬁxed μ ∈ L2 (τ ), we deﬁne the linear functional w (η) − Qτ w ) = (η) μ(η)dη fμ (w τ

satisfying )| ≤ w − Qτ w L2 (τ ) μL2 (τ ) |fμ (w ≤ 2 w L2 (τ ) μL2 (τ ) ≤ 2 w H 1 (τ ) μL2 (τ ) ,

234

B Numerical Analysis

since the L2 projection Qτ : L2 (τ ) → L2 (τ ) is bounded. If q is given as a constant function, we obviously have Qτ q = q, and, therefore, fμ (q) = 0 for all q ∈ P0 (τ ). The Bramble–Hilbert lemma then implies )| ≤ c |w |H 1 (τ ) μL2 (τ ) . |fμ (w ∈ L2 (τ ), we then obtain In particular for μ ¯=w − Qτ w 2 w (η) − Qτ w 2L2 (τ ) = (η) dη = |fμ¯ (w )| w − Qτ w τ

≤ c |w |H 1 (τ ) ¯ μL2 (τ ) = c |w |H 1 (τ ) w − Qτ w L2 (τ ) , and, therefore, L2 (τ ) ≤ c |w |H 1 (τ ) . w − Qτ w Altogether, we ﬁnally obtain w − Qh w2L2 (τ ) = 2Δ w − Qτ w 2L2 (τ ) ≤ c Δ |w |2H 1 (τ ) ≤ c¯ h2 |w|2H 1 (τ ) , as well as w − Qh w2L2 (Γ ) ≤ c¯

N

h2 |w|2H 1 (τ )

=1

and 1 (Γ ) , w − Qh wL2 (Γ ) ≤ c h |w|Hpw

1 (Γ ). This is the error estimate (2.4) for s = 1. when assuming w ∈ Hpw To obtain the error estimate (2.4) for s ∈ (0, 1), we consider x ∈ τ , where we ﬁnd 1 1 w(y)dsy = (w(x) − w(y))dsy w(x) − Qh w(x) = w(x) − Δ Δ τ

τ

for the error, and, therefore, by applying the Cauchy–Schwarz inequality, 2 1 2 (w(x) − w(y))dsy |w(x) − Qh w(x)| = Δ τ 2 1 w(x) − w(y) 1+s = 2 |x − y| dsy Δ |x − y|1+s τ

B.2 Approximation Properties

≤ ≤

1 Δ2

(w(x) − w(y))2 dsy |x − y|2+2s

τ

d2+2s Δ

τ

235

|x − y|2+2s dsy τ

(w(x) − w(y)) dsy , |x − y|2+2s 2

where d is the element diameter (cf. Chapter 2). Integrating over τ , and since the boundary element τ is shape regular, i.e. d ≤ cB h , this gives with Δ = h2 (w(x) − w(y))2 2s (w(x) − Qh w(x))2 dsx ≤ c2+2s h dsy dsx . B |x − y|2+2s τ

τ τ

Taking the sum over all boundary elements τ , we obtain (w(x) − Qh w(x))2 dsx ≤ c2+2s B

N =1

Γ

h2s τ τ

(w(x) − w(y))2 dsy dsx , |x − y|2+2s

which is the error estimate (2.4) for s ∈ (0, 1). Hence, we have s (Γ ) , w − Qh wL2 (Γ ) ≤ c hs |w|Hpw

s when assuming w ∈ Hpw (Γ ) for some s ∈ [0, 1]. For σ ∈ [−1, 0), we then ﬁnd by duality,

w − Qh wH σ (Γ ) = =

w − Qh w, v Γ vH −σ (Γ ) 0=v∈H −σ (Γ ) sup sup 0=v∈H −σ (Γ )

=

w − Qh w, v − Qh v Γ vH −σ (Γ )

w − Qh wL2 (Γ ) v − Qh vL2 (Γ ) vH −σ (Γ ) 0=v∈H −σ (Γ ) sup

s (Γ ) , ≤ c h−σ w − Qh wL2 (Γ ) ≤ c hs−σ |w|Hpw

and, therefore, the approximation property (2.5). It remains to derive the approximation property (2.8) for piecewise linear but globally discontinuous basis functions. For this, we deﬁne the L2 projection Qh w ∈ Sh1,−1 (Γ ) satisfying the variational problem N 3 =1 i=1

w,i

ψ,i (x)ψk,j (x)dsx =

Γ

w(x)ψk,j (x)dsx Γ

for k = 1, . . . , N, j = 1, 2, 3. As for piecewise constant basis functions, we then obtain

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B Numerical Analysis

w − Qh w2L2 (τ ) ≤ 2c Δ w − Qτ w L2 (τ ) for the local error. The Bramble–Hilbert lemma now implies L2 (τ ) ≤ c |w |H 2 (τ ) . w − Qτ w Hence, we have |2H 2 (τ ) ≤ c Δ2 |w|2H 2 (τ ) = c h4 |w|2H 2 (τ ) , w − Qh w2L2 (τ ) ≤ c Δ |w and, therefore, c w − Qh w2L2 (Γ ) ≤

N

h4 |w|2H 2 (τ )

=1

as well as 2 (Γ ) , w − Qh wL2 (Γ ) ≤ c h2 |w|Hpw

2 (Γ ). Since when assuming w ∈ Hpw

Qh : L2 (Γ ) → L2 (Γ ) is bounded, we obtain, by applying an interpolation argument, the error estimate s , w − Qh wL2 (Γ ) ≤ c hs |w|Hpw(Γ ) s when assuming w ∈ Hpw (Γ ) for some s ∈ [0, 2]. By duality, we then ﬁnd s (Γ ) w − Qh wH σ (Γ ) ≤ c hs−σ |w|Hpw

for σ ∈ [−2, 0] and s ∈ [0, 2], which is the approximation property (2.8). Finally, we consider the piecewise linear, globally continuous L2 projection Qh w ∈ Sh1 (Γ ) as the unique solution of the variational problem Qh w(x)ϕj (x)dsx = w(x)ϕj (x)dsx for j = 1, . . . , M. Γ

Γ

Due to Sh1 (Γ ) ⊂ Sh1,−1 (Γ ), we immediately ﬁnd the approximation property (2.10) for σ ∈ [−2, 0] and s ∈ [0, 2]. To derive (2.10) for σ ∈ (0, 1], we introduce the H 1 projection of a given function u ∈ H 1 (Γ ), Ph u =

M

uj ϕj ∈ Sh1 (Γ ),

j=1

which minimises the error u − Ph u in the H 1 (Γ )–norm: Ph u = arg

min

1 (Γ ) vh ∈Sh

u − vh H 1 (Γ ) .

B.2 Approximation Properties

237

Thus, Ph u ∈ Sh1 (Γ ) is the unique solution of the variational problem Ph u, vh H 1 (Γ ) = u, vh H 1 (Γ )

for all vh ∈ Sh1 (Γ ).

As for the L2 projection, we ﬁnd the error estimate 2 (Γ ) , u − Ph uH 1 (Γ ) ≤ c h |u|Hpw

2 when assuming u ∈ Hpw (Γ ). Since

Ph : H 1 (Γ ) → H 1 (Γ ) is bounded, we also conclude the trivial estimate u − Ph uH 1 (Γ ) ≤ uH 1 (Γ ) . Then, using an interpolation argument, we also obtain the error estimate u − Ph uH 1 (Γ ) ≤ c hs−1 |u|H s (Γ ) , s (Γ ) for some s ∈ [1, 2]. Then, for σ ∈ [0, 1), we get when assuming u ∈ Hpw by duality,

u − Ph uH σ (Γ ) =

=

sup 0=v∈H 2−σ (Γ )

sup 0=v∈H 2−σ (Γ )

≤

sup 0=v∈H 2−σ (Γ )

u − Ph u, v H 1 (Γ ) vH 2−σ (Γ ) u − Ph u, v − Ph v H 1 (Γ ) vH 2−σ (Γ ) u − Ph uH 1 (Γ ) v − Ph vH 1 (Γ ) vH 2−σ (Γ )

≤ c h1−σ u − Ph uH 1 (Γ ) ≤ c hs−σ |u|H s (Γ ) , when assuming u ∈ H s (Γ ) for some s ∈ [1, 2]. This is the approximation property (2.10) for σ ∈ (0, 1] and s ∈ [1, 2].

C Numerical Algorithms

C.1 Numerical Integration For the boundary Γ = ∂Ω of a Lipschitz domain Ω ⊂ R3 , we consider a sequence of boundary element meshes (2.1), ΓN =

N '

τ .

=1

We assume that all boundary elements τ are plane triangles. Using the reference element ) ( τ = ξ ∈ R2 : 0 < ξ1 < 1 , 0 < ξ2 < 1 − ξ1 , the boundary element τ = χ (τ ) with nodes xi for i = 1, 2, 3 can be described via the parametrisation x(ξ) = χ (ξ) = x1 + ξ1 (x2 − x1 ) + ξ2 (x3 − x1 ) = x1 + J ξ ∈ τ for ξ ∈ τ . Using

Δ =

dsx , τ

we obtain for the integral of a given function f on τ , f (x)dsx = 2Δ f (ξ) dξ, f (ξ) = f (χ (ξ)). I = τ

τ

Hence, it is suﬃcient to consider numerical integration schemes 1 ωi f (ξi ) ≈ 2 i=1 M

τ

f (ξ) dξ

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C Numerical Algorithms

with respect to the reference element τ , which are exact for polynomials of a certain order. From this, we may ﬁnd 3M parameters (ξi,1 , ξi,2 , ωi ) for i = 1, . . . , M . Let ) ( Pk = span ξ1α1 ξ2α2 α1 +α2 ≤k be the space of polynomials of an order less or equal k. The dimensions of Pk and the minimal number M of integration points are given in Table C.1. The last line of this table shows the number P = 3M of free parameters. These parameters (ξi,1 , ξi,2 , ωi ) for i = 1, . . . , M are solutions of the nonlinear Table C.1. Dimensions of Pk and required number of integration nodes k dimPk M P

0 1

1 3 1 3

2 6 3 9

3 10 4 12

4 15

5 21 7 21

for α1 , α2 : α1 + α2 ≤ k .

(C.1)

equations τ

1 α1 α2 ωi ξi,1 ξi,2 2 i=1 M

ξ1α1 ξ2α2 dξ =

For k = 1 and M = 1, the equations (C.1) read 1 1 1 dξ = = ω1 , 2 2 τ 1 1 ξ1 dξ = = ω1 ξ1,1 , 6 2 τ 1 1 ξ2 dξ = = ω1 ξ1,2 , 6 2 τ

and, therefore, we obtain ω1 = 1,

ξ1,1 =

1 , 3

ξ1,2 =

1 . 3

The resulting formula is the midpoint quadrature rule f (x) dsx ≈ Δ f (x∗ ) , τ

which integrates linear functions exactly. For k = 2 and M = 3, we have from (C.1)

C.1 Numerical Integration

1 dξ = τ ξ1 dξ = τ

ξ2 dξ = τ ξ12 dξ = τ

ξ22 dξ = τ ξ1 ξ2 dξ =

241

1 1 = ω1 + ω 2 + ω 3 , 2 2 1 1 = ω1 ξ1,1 + ω2 ξ2,1 + ω3 ξ3,1 , 6 2 1 1 = ω1 ξ1,2 + ω2 ξ2,2 + ω3 ξ3,2 , 6 2 1 1 2 2 2 = ω1 ξ1,1 , + ω2 ξ2,1 + ω3 ξ3,1 12 2 1 1 2 2 2 = ω1 ξ1,2 , + ω2 ξ2,2 + ω3 ξ3,2 12 2 1 1 = ω1 ξ1,1 ξ1,2 + ω2 ξ2,1 ξ2,2 + ω3 ξ3,1 ξ3,2 . 24 2

τ

To ﬁnd a solution of the above system of nonlinear equations, we add some additional constraints with respect to the geometrical setting. Due to symmetry, we ﬁrst set 1 ω1 = ω 2 = ω 3 = . 3 Introducing some real parameter s ∈ R, we may deﬁne the integration nodes on the straight lines connecting the corner nodes with the midpoint, s 1 s −2 s 1 1 0 , ξ2 = , ξ3 = . + + ξ1 = 1 0 1 3 1 3 3 −2 Note that with this choice, the second and third equations are satisﬁed. For the remaining three equations, we obtain 4s2 − 8s + 3 = 0 , with the solutions

In particular, for s1 weights are ξ1 =

1 1 , , 6 6

1 s1/2 = 1 ± . 2 = 1/2, the integration nodes and the corresponding

ξ2 =

2 1 , , 3 6

ξ3 =

1 2 , , 6 3

ω1 = ω 2 = ω 3 =

1 , 3

ω1 = ω 2 = ω 3 =

1 . 3

while for s2 = 3/2 the solution is ξ1 =

1 1 , , 2 2

ξ2 =

1 , 0, 2

ξ3 =

1 2

,0 ,

Note that both integration rules are exact for quadratic polynomials. For k = 3 and M = 4, the nonlinear equations (C.1) read

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C Numerical Algorithms

1 α1 α2 ωi ξi,1 ξi,2 2 i=1 4

ξ1α1 ξ2α2 dξ = τ

for α1 , α2 : α1 + α2 ≤ 3 .

Hence, we have to ﬁnd 12 parameters (ξi,1 , ξi,2 , ωi ) satisfying 10 nonlinear equations. As in the previous case, we may introduce additional constraints to construct particular solutions of the above nonlinear system. Due to symmetry, we deﬁne the integration nodes as 1 1 s 1 s −2 s 1 1 0 , ξ2 = , ξ3 = , ξ4 = , + + ξ1 = 1 0 1 3 1 3 1 3 3 −2 and the associated integration weights as ω1 = ω,

ω2 = ω3 = ω4 .

From the equation for α1 = α2 = 0, we then ﬁnd ω1 = ω,

ω2 = ω3 = ω4 =

1 (1 − ω). 3

Note that both equations, for α1 , α2 : α1 + α2 = 1, are then satisﬁed. It turns out that it is suﬃcient to consider only two additional equations, the remaining equations are then satisﬁed automatically. For α1 = 2 and α2 = 0, it follows 1 (1 − ω) s2 − 2s + 1 = , 4 while for α1 = 3 and α2 = 0, we have 17 . (1 − ω) 8 − 18s + 12s2 − 2s3 = 10 Thus, there are two equations for the two remaining unknowns ω and s. We obtain 17 1 1 1 = , 1−ω = 4 s2 − 2s + 1 10 8 − 18s + 12s2 − 2s3 and, therefore, (s − 1)2 (5s − 3) = 0 , with the solutions

3 . 5 For s = 3/5, we then get the integration weights s1/2 = 1,

ω1 = − and the integration points

9 , 16

s3 =

ω2 = ω 3 = ω 4 =

25 48

C.1 Numerical Integration

1 3

1 1 1 , ξ2 = , 1 5 1

1 ξ3 = 5

1 1 3 , ξ4 = . 1 5 3

ξ1 =

243

However, the ﬁrst weight ω1 is negative, and, therefore, the above quadrature can not be used for practical computations. For k = 5 and M = 7, the nonlinear equations (C.1) read

1 α1 α2 ωi ξi,1 ξi,2 2 i=1 7

ξ1α1 ξ2α2 dξ = τ

for all α1 , α2 : α1 + α2 ≤ 5.

Hence, we have to ﬁnd 21 parameters (ξi,1 , ξi,2 , ωi ) satisfying 21 nonlinear equations. As in the previous cases, we introduce additional constraints to construct a solution of the above nonlinear equations. Due to symmetry, we deﬁne the integration nodes as 1 1 s 1 s −2 s 1 1 0 , ξ2 = , ξ3 = , ξ4 = + + ξ1 = 1 0 1 3 1 3 1 3 3 −2 and ξ5 =

1 2

t −1 t −1 t 1 1 1 0 1 2 + , ξ6 = + , ξ7 = + , 0 2 6 2 1 6 −1 2 1 6 −1

as well as the associated integration weights as ω1 = ω,

ω2 = ω3 = ω4 ,

ω5 = ω 6 = ω 7 .

From the equation for α1 = α2 = 0, we easily ﬁnd ω1 = 1 − 3ω2 − 3ω5 . With this, the equations for α1 , α2 : α1 + α2 = 1 are satisﬁed automatically. Moreover, all the remaining equations can be reduced to the four equations for the four parameters ω2 , ω5 , s, and t: 12ω2 (s − 1)2 + 3ω5 (t − 1)2 = 1 , 120ω2 (s − 1) (4 − s) + 15ω5 (t − 1)2 (t + 5) = 34 , 2

240ω2 (s − 1)2 (3s2 − 10s + 13) + 15ω5 (t − 1)2 (3t2 + 2t + 19) = 176 , 3360ω2 (s − 1)2 (8 − 11s + 6s2 − s3 ) + 105ω5 (13 − t + 3t2 + t3 ) = 1184 . From the ﬁrst two equations, we ﬁnd ω2 =

1 − 3ω5 (t − 1)2 , 12(s − 1)2

ω5 =

10s − 6 . 15(t − 1)2 (2s + t − 3)

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C Numerical Algorithms

Inserting this into the third equation, this gives s(9 − 5t) + 3(t − 1) = 0 , and, therefore,

t−1 . 5t − 9 The fourth equation is ﬁnally equivalent to s = 3

7t2 − 18t + 3 = 0 . Thus, t1/2

√ 9 ± 2 15 . = 7

Hence, we have

√ √ 9 − 2 15 6 − 15 t = , s = . 7 7 For the integration weights, we ﬁnd √ √ 155 + 15 155 − 15 9 ω5 = , ω2 = , ω1 = , 1200 1200 40 while for the integration nodes, we ﬁnally obtain √ √ √ 1 6 − 15 1 9 + 2 15 1 6 − √15 √ √ , ξ3 = , ξ4 = ξ2 = 21 6 − 15 21 6 − 15 21 9 + 2 15 and ξ5 =

1 21

√ √ √ 1 1 9 − 2 15 6 + √15 6 + √15 √ , ξ6 = , ξ7 = . 6 + 15 21 9 − 2 15 21 6 + 15

C.2 Analytic Integration For the boundary Γ = ∂Ω of a Lipschitz domain Ω ⊂ R3 , we consider a sequence of boundary element meshes (2.1), ΓN =

N '

τ .

=1

We assume that all boundary elements τ are plane triangles. In order not to overload the subsequent formulae, we consider a plane triangle τ ⊂ R3 given via its three corner points x1 , x2 , and x3 (cf. Fig. C.2), having in mind that this triangle is one of the boundary elements τ , = 1 . . . , N . We ﬁrst deﬁne a suitable local coordinate system connected to the boundary element τ with the origin in x1 and having the basis vectors

C.2 Analytic Integration

r1 x1 n

245

x3

r2

x∗

x2

Fig. C.1. Boundary element τ

r1 , r2 , n .

The unit vector r2 directed from x2 to x3 is deﬁned as follows r2 =

1 (x3 − x2 ) , tτ

tτ = |x3 − x2 | .

Here, tτ > 0 denotes the length of the side x2 x3 of the triangle τ . This is one of the characteristic quantities, we will need for the analytical integration. To ﬁnd the unit vector r1 which is directed along the line through the origin in x1 and perpendicular to r2 , we ﬁrst determine the intersection point x∗ , x∗ = x1 + sτ r1 = x2 + t∗ r2 , where sτ > 0, the height of the triangle τ , will be the second characteristic quantity. The parameter t∗ can be easily computed as t∗ = (x1 − x2 , r2 ) . Then, 1 ∗ (x − x1 ) , sτ = |x∗ − x1 | . sτ Finally, the corresponding normal vector is deﬁned by r1 =

n = r1 × r2 . Two further characteristic quantities, we will need for the analytical integration, are the tangents of the angles between the height x1 x∗ and the sides x1 x2 and x1 x3 , correspondingly. These quantities can be computed as α1 = −

t∗ , sτ

α2 =

t τ − t∗ . sτ

Thus, an appropriate parametrisation of the boundary element τ would be

τ = y = y(s, t) = x1 + s r1 + t r2 : 0 < s < sτ , α1 s < t < α2 s .

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C Numerical Algorithms

Any point x ∈ R3 can now be represented in the new coordinate system as x = x1 + sx r1 + tx r2 + ux n with sx = (x − x1 , r1 ),

tx = (x − x1 , r2 ),

ux = (x − x1 , n) .

From this, we ﬁnd the distance between any arbitrary point x ∈ R3 and a point y ∈ τ as follows: |x − y|2 = (s − sx )2 + (t − tx )2 + u2x . C.2.1 Single Layer Potential for the Laplace operator For x ∈ R3 , we ﬁrst consider the following function: 1 1 S(τ, x) = dsy 4π |x − y| τ

=

1 4π

sτ α2 s 0 α1 s

=

1 4π

sτ 3

1 * dt ds 2 (s − sx ) + (t − tx )2 + u2x

4α2 s * log t − tx + (s − sx )2 + (t − tx )2 + u2x ds α1 s

0

=

1 F (sτ , α2 ) − F (0, α2 ) − F (sτ , α1 ) + F (0, α1 ) . 4π

Hence, it is suﬃcient to compute the integral * F (s, α) = log αs − tx + (s − sx )2 + (αs − tx )2 + u2x ds * = log αs − tx + (1 + α2 )(s − p)2 + q 2 ds , with parameters p =

αtx + sx , 1 + α2

q 2 = u2x +

(tx − αsx )2 . 1 + α2

Integration by parts gives * F (s, α) = (s − sx ) log αs − tx + (1 + α2 )(s − p)2 + q 2 − s −

* (tx − αsx ) (1 + α2 )(s − p)2 + q 2 + (1 + α2 )(s − p)(p − s ) − q 2 * ds . (1 + α)2 (s − p)2 + q 2 + (αs − tx ) (1 + α2 )(s − p)2 + q 2

C.2 Analytic Integration

247

With the substitution s = p+ √

q sinh u, 1 + α2

q ds = √ cosh u , du 1 + α2

we obtain for the remaining integral * (tx − αsx ) (1 + α2 )(s − p)2 + q 2 + (1 + α2 )(s − p)(p − sx ) − q 2 * ds I= (1 + α)2 (s − p)2 + q 2 + (αs − tx ) (1 + α2 )(s − p)2 + q 2 √ =q

√ 1 + α2 (tx − αsx ) cosh u + α(tx − αsx ) sinh u − 1 + α2 q √ du. q(1 + α2 ) cosh u + α 1 + α2 q sinh u + (αsx − tx )

The second substitution u v = tanh , 2

sinh u =

2v , 1 − v2

cosh u =

1 + v2 , 1 − v2

2 du = dv 1 − v2

gives for the last integral √ √ 1 + α2 (tx − αsx )(1 + v 2 ) + 2α(tx − αsx )v − 1 + α2 q(1 − v 2 ) 1 √ 2q dv, 1 − v2 q(1 + α2 )(1 + v 2 ) + 2α 1 + α2 qv + (αsx − tx )(1 − v 2 ) and, therefore, I = 2q

1 B2 v 2 + B1 v + B0 dv , 1 − v 2 A2 v 2 + A1 v + A0

with parameters B2 =

*

1 + α2 (tx − αsx + q),

B1 = 2α(tx − αsx ), * B0 = 1 + α2 (tx − αsx − q), A2 = (1 + α2 )q − (αsx − tx ), * A1 = 2α 1 + α2 q, A0 = (1 + α2 )q + (αsx − tx ). Due to A21 − 4A0 A2 = −4(1 + α2 )u2x ≤ 0 , we decompose C2 C3 v + C4 C1 1 B2 v 2 + B1 v + B0 + + = 2 2 2 1 − v A2 v + A1 v + A0 1 − v 1 + v A2 v + A1 v + A0 to obtain

248

C Numerical Algorithms

C1 =

1 B0 + B1 + B2 , 2 A0 + A1 + A2

C2 =

1 B0 − B1 + B2 , 2 A0 − A1 + A2

C3 = A2 (C1 − C2 ), C4 = B0 − (C1 + C2 )A0 , and, therefore, tx − αsx C1 = C 2 = √ , 2q 1 + α2

√ C3 = 0,

C4 = −

1 + α2 u2x . q

Hence, * 1 tx − αsx 1 1 2 2 √ I = + − 2 1 + α ux dv . A2 v 2 + A1 v + A0 1 + α2 1 − v 1 + v Integrating the ﬁrst part and resubstituting, this gives 1 + v 1 1 − dv = log |v + 1| − log |v − 1| = log 1+v v−1 1 − v √ 1 + α2 (s − p) = 2 arctanh v = u = arcsinh q 5 √ 1 + α2 (s − p) (1 + α2 )(s − p)2 + +1 = log q q2 = log

*

1 + α2 (s − p) +

* (1 + α2 )(s − p)2 + q 2 − log q .

For the remaining integral, we obtain * 2A2 v + A1 1 2 1 + α2 u2x dv = 2ux arctan √ . 2 A2 v + A1 v + A0 2 1 + α 2 ux From sinh u = 2v/(1 − v 2 ), we ﬁnd * * (1 + α2 )(s − p)2 + q 2 − q 1 + sinh2 u − 1 √ = v = , sinh u 1 + α2 (s − p) and, therefore,

* A2 (1 + α2 )(s − p)2 + q 2 + α(1 + α2 )(s − p) − A2 q 2A2 v + A1 √ = (1 + α2 )(s − p)τ 2 1 + α 2 ux * x −tx q − αs (1 + α2 )(s − p)2 + q 2 + αs − tx − q q 1+α2 = . (s − p)ux

C.2 Analytic Integration

249

Collecting all terms together and ignoring constant parts, we ﬁnally obtain * F (s, α) = (s − sx ) log αs − tx + (s − sx )2 + (αs − tx )2 + u2x − s * * αsx − tx +√ log 1 + α2 (s − p) + (1 + α2 )(s − p)2 + q 2 1 + α2 * x −tx q − αs (1 + α2 )(s − p)2 + q 2 + αs − tx − q q 1+α2 +2ux arctan . (s − p)ux C.2.2 Double Layer Potential for the Laplace operator For piecewise linear basis functions we have to compute the local contributions of the integrals 1 (x − y, n(y)) ψτ,i (y)dsy Di (τ, x) = 4π |x − y|3 τ

1 = 4π

sτ

sτ − s sτ

α2 s

ux

dtds 2 + (s − s )2 + u2 3/2 (t − t ) x x x α1 s 0 6 7 α2 s sτ ux t − tx sτ − s 1 * = ds 4π sτ (s − sx )2 + u2x (t − tx )2 + (s − sx )2 + u2x α1 s

0

=

1 F (sτ , α2 ) − F (0, α2 ) − F (sτ , α1 ) + F (0, α1 ) . 4πsτ

Hence, it is suﬃcient to compute the integral (sτ − s)(αs − tx ) ds F (s, α) = ux * (s − sx )2 + u2x (1 + α2 )(s − p)2 + q 2 α(sτ − sx ) − (αsx − tx ) (s − sx ) + (sτ − sx )(αsx − tx ) + αu2x = ux ds * (s − sx )2 + u2x (1 + α2 )(s − p)2 + q 2 1 * ds −αux 2 (1 + α )(s − p)2 + q 2 α(sτ − sx ) − (αsx − tx ) (s − sx ) + (sτ − sx )(αsx − tx ) + αu2x = ux ds * (s − sx )2 + u2x (1 + α2 )(s − p)2 + q 2 * * α log 1 + α2 (s − p) + (1 + α2 )(s − p)2 + q 2 , −ux √ 1 + α2 with parameters p and q already used for the single layer potential: p=

αtx + sx , 1 + α2

q 2 = u2x +

(tx − αsx )2 . 1 + α2

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C Numerical Algorithms

With the substitution q ds =√ cosh u , du 1 + α2

q sinh u, s = p+ √ 1 + α2

we obtain for the remaining integral α(sτ − sx ) − (αsx − tx ) (s − sx ) + (sτ − sx )(αsx − tx ) + αu2x ds = I = ux * (s − sx )2 + u2x (1 + α2 )(s − p)2 + q 2 q αs +t −2αs sinh u+ √ 1 α(1 + α2 )q 2 +(s − s )(t − αs ) τ x x x τ x x 1+α2 √ du . ux q 2 sinh2 u + 2 1 + α2 q(p − sx ) sinh u + (1 + α2 ) (p − sx )2 + u2x The second substitution v = tanh

u , 2

sinh u =

2v , 1 − v2

cosh u =

1 + v2 , 1 − v2

2 du = dv 1 − v2

gives for the last integral 2q αs +t −2αs v+ √ 1 α(1+α2 )q 2+(s −s )(t −αs )(1−v 2 ) τ x x x τ x x 1+α2 √ dv, 2ux 4q 2 v 2 +4 1+α2 q(p−sx )v(1−v 2 ) + (1+α2 ) (p− sx )2 +u2x (1−v 2 )2 and, thereforfe, I = 2ux

C 2 v 2 + C 1 v − C2 dv , a1 v 4 − a2 v 3 + a3 v 2 + a2 v + a1

with parameters 1 (sτ − sx )(tx − αsx ) − α(1 + α2 )q 2 , C2 = √ 2 1+α C1 = 2q αsτ + tx − 2αsx , a1 = (1 + α2 ) u2x + (p − sx )2 = u2x + α2 q 2 > 0 , * 4αq(tx − αsx ) √ a2 = 4q 1 + α2 (p − ss ) = , 1 + α2 a3 = 4q 2 − 2a1 , or I =

2ux u2x + α2 q 2

with a = √

v4

C2 v 2 + C1 v − C2 dv , − av 3 + bv 2 + av + 1

4αq(tx − αsx ) , 1 + α2 u2x + α2 q 2

For the decomposition

b =

u2x

4q 2 − 2. + α2 q 2

C.2 Analytic Integration

v4

251

D1 v + E 1 D2 v + E2 C 2 v 2 + C 1 v − C2 = 2 + 2 , − av 3 + bv 2 + av + 1 v + A1 v + B1 v + A2 v + B2

we ﬁnd the coeﬃcients A1/2 B1/2

√ 2α 1 + α2 q tx − αsx =− 2 ±q , ux + α 2 q 2 1 + α2 2 tx − αsx 1 + α2 = 2 ± q , ux + α 2 q 2 1 + α2

as well as 1 2 ux + α 2 q 2 , 2 1 2 D2 = ux + α2 q 2 , 2 1 sτ − sx + αq tx − αsx + (1 + α2 )q , E1 = √ 2 2 1+α 1 E2 = √ sτ − sx − αq tx − αsx − (1 + α2 )q . 2 1 + α2 D1 = −

Hence, we have 2ux I = 2 ux + α 2 q 2

D1 v + E1 dv + v 2 + A1 v + B1

D2 v + E2 dv v 2 + A2 v + B2

,

and it remains to integrate D1/2 v + E1/2 2ux I1/2 = 2 dv ux + α 2 q 2 v 2 + A1/2 v + B1/2 1 2ux D1/2 log v 2 + A1/2 v + B1/2 = 2 2 2 ux + α q 2 1 1 dv + E1/2 − A1/2 B1/2 2 v 2 + A1/2 v + B1/2 1 = ∓ ux log v 2 + A1/2 v + B1/2 2 1 2ux 1 E1/2 − A1/2 B1/2 dv . + 2 2 2 2 ux + α q 2 v + A1/2 v + B1/2 Due to 1 (1 + α2 )u2x G21/2 = B1/2 − A21/2 = 2 4 u2x + α2 q 2 √

we ﬁnd G1/2 =

1 + α2 |ux | u2x + α2 q 2

tx − αsx ±q 1 + α2

tx − αsx q± 1 + α2

> 0.

2 ,

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C Numerical Algorithms

Hence, we obtain 1 E1/2 − A1/2 D1/2 2

1 dv v 2 + A1/2 v + B1/2 1 1 = E1/2 − A1/2 D1/2 dv 1 2 (v + 2 A1/2 )2 + G21/2 =

2v + A1/2 1 1 E1/2 − A1/2 D1/2 arctan G1/2 2 2G1/2

= ±

2v + A1/2 u2x + α2 q 2 (sτ − sx ) arctan . 2|ux | 2G1/2

Therefore, 2v + A1/2 1 ux arctan . I1/2 = ∓ ux log v 2 + A1/2 v + B1/2 ± (sτ − sx ) 2 |ux | 2G1/2 Recall that * * (1 + α2 )(s − p)2 + q 2 − q 1 + sinh2 u − 1 √ = . v = sinh u 1 + α2 (s − p) C.2.3 Linear Elasticity Single Layer Potential By the use of Lemma 1.15, it is suﬃcient to describe the computation of 1 (xi − yi )(xj − yj ) dsy . Sij (τ, x) = 4π |x − y|3 τ

With the local parametrisations y = x1 + s r1 + t r2 ,

x = x1 + sx r1 + tx r2 + ux n ,

the kernel function can be written as aij (xi − yi )(xj − yj ) = 1/2 + |x − y|3 2 (s − sx ) + (t − tx )2 + u2x bij (s − sx ) − cij ux (t − tx ) dij (s − sx )2 − eij (s − sx )ux + fij u2x 3/2 + 3/2 (s − sx )2 + (t − tx )2 + u2x (s − sx )2 + (t − tx )2 + u2x with the parameters

C.2 Analytic Integration

aij bij cij dij eij

253

= r2,i r2,j , = r1,i r2,j + r2,i r1,j , = r2,i nj + r2,j ni , = r1,i r1,j − aij , = r1,i nj + r1,j ni ,

fij = ni nj − aij . Hence, we have to compute 1 2 3 Sij (τ, x) = Sij (τ, x) + Sij (τ, x) + Sij (τ, x)

with 1 Sij (τ, x)

1 = aij 4π

sτ α2 s 0 α1 s

2 (τ, x) Sij

1 = 4π

1 (s − sx

)2

+ (t − tx )2 + u2x

1/2 dt ds,

sτ α2 s bij (s − sx ) − cij ux (t − tx ) dt ds, 2 + (t − t )2 + u2 3/2 (s − s ) x x x 0 α s 1

3 (τ, x) Sij

1 = 4π

sτ α2 s 0 α1 s

dij (s − sx )2 − eij (s − sx )ux + fij u2x 3/2 dt ds. (s − sx )2 + (t − tx )2 + u2x

1 Note that Sij (τ, x) corresponds to the single layer potential function for the Laplace equation. For the second entry we obtain

2 Sij (τ, x) =

1 4π

sτ α2 s bij (s − sx ) − cij ux (t − tx ) 3/2 (s − sx )2 + (t − tx )2 + u2x 0 α s 1

1 =− 4π

sτ 6 0

bij (s − sx ) − cij ux (s − sx )2 + (t − tx )2 + u2x

7α2 s 1/2

ds α1 s

1 2 Fij (sτ , α2 ) − Fij2 (sτ , α1 ) − Fij2 (0, α2 ) + Fij2 (0, α1 ) , =− 4π with the integral bij (s − sx ) − cij ux Fij2 (s, α) = 1/2 ds (s − sx )2 + (αs − tx )2 + u2x b (s − sx ) − cij ux * ij ds = (1 + α2 )(s − p)2 + q 2

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C Numerical Algorithms

= bij

s−p * ds + 2 (1 + α )(s − p)2 + q 2

*

bij (p − sx ) − cij ux (1 + α2 )(s − p)2 + q 2

ds

1/2 bij (1 + α2 )(s − p)2 + q 2 1 + α2 * * bij (p − sx ) − cij ux √ + log 1 + α2 (s − p) + (1 + α2 )(s − p)2 + q 2 . 1 + α2

=

For the third integral, we get 3 Sij (τ, x) =

1 4π

sτ α2 s 0 α1 s

sτ 6 = 0

=

dij (s − sx )2 − eij (s − sx )ux + fij u2x 3/2 dt ds (s − sx )2 + (t − tx )2 + u2x

t − tx dij (s − sx )2 − eij (s − sx )ux + fij u2x * (s − sx )2 + u2x (s − sx )2 + (t − tx )2 + u2x

7 α2 s ds α1 s

1 3 Fij (sτ , α2 ) − Fij3 (sτ , α1 ) − Fij3 (0, α2 ) + Fij3 (0, α1 ) 4π

with

dij (s − sx )2 − eij (s − sx )ux + fij u2x αs − tx * ds 2 2 (s − sx ) + ux (s − sx )2 + (αs − tx )2 + u2x dij (s − sx )2 + u2x − eij (s − sx )ux + fij − dij u2x αs − tx * ds = (s − sx )2 + u2x (s − sx )2 + (αs − tx )2 + u2x αs − tx * ds = dij (1 + α2 )(s − p)2 + q 2 αs − tx (s − sx )ux * ds −eij 2 2 (s − sx ) + ux (1 + α2 )(s − p)2 + q 2 αs − tx u2x * + fij − dij ds (s − sx )2 + u2x (1 + α2 )(s − p)2 + q 2 = dij F 3,1 (s, α) − eij F 3,2 (s, α) + fij − dij F 3,3 (s, α)

Fij3 (s, α) =

and F

3,1

(s, α) =

*

=α

αs − t (1 + α2 )(s − p)2 + q 2

ds

s−p * ds + (1 + α2 )(s − p)2 + q 2

αp − t * ds (1 + α2 )(s − p)2 + q 2

C.2 Analytic Integration

255

* α (1 + α2 )(s − p)2 + q 2 1 + α2 * * αp − t +√ log 1 + α2 (s − p) + (1 + α2 )(s − p)2 + q 2 . 1 + α2

= √

The remaining integrals can be computed in the same way as for the double layer potential of the Laplace operator. In particular, up to the sign and the choice sτ = s , the integral αs − tx (s − sx )ux 3,2 * F (s, α) = ds (s − sx )2 + u2x (1 + α2 )(s − p)2 + q 2 coincides with the integral of the double layer potential function of the Laplace operator. Furthermore, using the transformations as for the Laplace operator, we have αs − tx u2x * ds F 3,3 (s, α) = (s − sx )2 + u2x (1 + α2 )(s − p)2 + q 2 √ αq sinh u + 1 + α2 (αp − tx ) √ du = u2x q 2 sinh2 u + 2 1 + α2 q(p − sx ) sinh u + (1 + α2 )[u2x + (p − sx )2 ] √ αq2v+ 1+α2 (αp−tx )(1−v 2 ) √ dv = 2u2x 4q 2 v 2+4 1+α2 q(p−sx )v(1−v 2 )+(1+α2 ) u2x +(p − sx )2 (1−v 2 )2 C2 v 2 + C1 v − C 2 dv , = 2u2x 4 a 1 v − a 2 v 3 + a 3 v 2 + a 2 v + a1 with the parameters * C2 = − 1 + α2 (αp − tx ), and a1 = u2x + α2 q 2 ,

a2 =

C1 = 2αq ,

4αq(tx − αsx ) √ , 1 + α2

a3 = 4q 2 − 2a1 .

Hence, we get F

3,3

2u2 (s, α) = 2 x 2 2 ux + α q

C2 v 2 + C1 v − C2 dv (v 2 + A1 v + B1 )(v 2 + A2 v + B2 )

with √ 2 tx − αsx 2α 1 + α2 q tx − αsx 1 + α2 A1/2 = − 2 ± q , B = ± q . 1/2 ux + α 2 q 2 1 + α2 u2x + α2 q 2 1 + α2 For the decomposition

256

C Numerical Algorithms

(v 2

D1 v + E1 D2 v + E2 C2 v 2 + C1 v − C2 = 2 + 2 , + A1 v + B1 )(v 2 + A2 v + B2 ) v + A1 v + B1 v + A2 v + B2

we ﬁnd the coeﬃcients D1 = D2 = 0, E1 =

(tx − αsx ) + (1 + α2 )q (tx − αsx ) − (1 + α2 )q √ √ , E2 = . 2 2 1+α 2 1 + α2

Therefore, F

3,3

2τ 2 (s, α) = 2 2 2 τ + α q

E1 dv + v 2 + A1 v + B1

E2 dv v 2 + A2 v + B2

,

and it remains to integrate I1/2 =

u2x (tx − αsx ) ± (1 + α2 )q √ 2 2 2 ux + α q 1 + α2

(tx − αsx ) ± (1 + α2 )q u2 √ = 2 x2 2 ux + α q 1 + α2 =

v2

1 dv + A1/2 v + B1/2

1 (v +

1 2 2 A1/2 )

+ G21/2

dv

2v + A1/2 1 u2x (tx − αsx ) ± (1 + α2 )q √ arctan , 2 2 2 2 G1/2 ux + α q 2G1/2 1+α √

with G1/2 =

1 + α2 |τ | u2x + α2 q 2

q±

tx − αsx 1 + α2

> 0.

Finally, I1/2 = ±|ux | arctan

2v + A1/2 . 2G1/2

C.3 Iterative Solution Methods In this section we consider the iterative solution of a linear system Ax = f ,

A ∈ RN ×N ,

x, f ∈ RN .

(C.2)

Note that the dimension N of the system is usually connected to the discretisation parameter h (cf. (2.2)), and, therefore, we have a sequence of linear systems to be solved for N → ∞. We ﬁrst consider symmetric and positive deﬁnite systems and later general systems with regular matrices. C.3.1 Conjugate Gradient Method (CG) For a symmetric and positive deﬁnite matrix A ∈ RN ×N , we may deﬁne an inner product

C.3 Iterative Solution Methods

257

(u, v)A = (Au, v) for all u, v ∈ RN . A system of vectors ( )N −1 P = pk k=0 ,

pk = 0 , k = 0, . . . , N − 1

(C.3)

is called conjugate, or A–orthogonal, if there holds (Ap , pk ) = 0 for k = . Note that (Apk , pk ) > 0 since A is assumed to be positive deﬁnite. The vector system P forms an A–orthogonal basis of the vector space RN . Hence, the solution vector x ∈ RN of the linear system (C.2) can be written as x = x0 −

N −1

α p

=0

with an arbitrary given vector x0 ∈ RN . To ﬁnd the yet unknown coeﬃcients α , we consider the linear system Ax = Ax0 −

N −1

α Ap = f .

=0

Taking the Euclidean inner product with pk , this gives N −1

α (Ap , pk ) = (Ax0 − f , pk ) ,

for k = 0, . . . , N − 1 ,

=0

and, therefore, by using the A–orthogonality of the system P αk =

(Ax0 − f , pk ) (Apk , pk )

for k = 0, . . . , N − 1 .

(C.4)

Thus, if a system P of A–orthogonal vectors pk is given, we can compute the coeﬃcients αk from (C.4), and, therefore, the solution x of the linear system (C.2). With it, this method can be seen as a direct solution algorithm. To deﬁne an iterative process, we introduce approximate solutions as xk+1 = x0 −

k

α p = xk − αk pk

for k = 0, . . . , N − 1 .

=0

For the computation of the coeﬃcient αk , we obtain from (C.4) k−1 (Ax0 − f , pk ) (Axk − f , pk ) 1 0 k = Ax , − f − α Ap , p = αk = (Apk , pk ) (Apk , pk ) (Apk , pk ) =0

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C Numerical Algorithms

when using the A–orthogonality of the system (C.3). Therefore, αk =

(rk , pk ) (Apk , pk )

for k = 0, . . . , N − 1 ,

(C.5)

where rk = Axk − f is the residual vector induced by the approximate solution xk . Note that the following recursion holds: rk+1 = Axk+1 − f = A(xk − αk pk ) − f = rk − αk Apk for k = 0, . . . , N − 2. Moreover, we have (rk+1 , pk ) = (rk − αk Apk , pk ) = (rk , pk ) −

(rk , pk ) (Apk , pk ) = 0 (Apk , pk )

for k = 0, . . . , N − 2. From (rk+1 , p ) = (rk , p ) − αk (Apk , p ) , it follows by induction that (rk+1 , p ) = 0 for = 0, . . . , k,

k = 0, . . . , N − 2 .

(C.6)

It remains to construct an A–orthogonal vector system P via the orthogonalisation method of Gram–Schmidt. Let W be any system of linear independent vectors wk . Setting p0 = w0 , we can compute A–orthogonal vectors pk for k = 0, . . . , N − 2 as follows: pk+1 = wk+1 −

k

βk p ,

=0

βk =

(Awk+1 , p ) , (Ap , p )

for = 0, . . . , k . (C.7)

From (C.7) and (C.6), we further conclude (rk+1 , w ) = (rk+1 , p ) +

l−1

βj (rk+1 , pj ) = 0

(C.8)

j=0

for = 0, . . . , k. In particular, if the residual vector rk+1 does not vanish, the vectors {w0 , w1 , . . . , wk , rk+1 } are linear independent and we can choose wk+1 = rk+1 . If rk+1 = 0, then the system (C.2) is solved. In general, this will not happen and we can choose wk = rk

for k = 0, . . . , N − 1 .

C.3 Iterative Solution Methods

259

From (C.8), we then conclude (rk+1 , r ) = 0

for all = 0, . . . , k .

For the enumerator of the coeﬃcient αk deﬁned in (C.5), we obtain (r , p ) = (r , r ) − k

k

k

k

k−1

βk−1, (rk , p ) = (rk , rk ) = k .

=0

by using the recursion (C.7) and the orthogonality relation (C.6). From > 0, it follows α > 0 for = 0, . . . , k, and, therefore, we can write Ap =

1 r − r+1 . α

For the enumerator of the coeﬃcients βkj in (C.7), we then obtain (Awk+1 , p ) = (rk+1 , Ap ) =

1 k+1 (r , r − r+1 ) = 0 α

for = 0, . . . , k − 1 and (Awk+1 , pk ) = −

1 k+1 k+1 k+1 (r ,r ) = − αk αk

for = k. Hence, we have βk = 0 for = 0, . . . , k − 1 and βkk = βk = −

k+1 . αk (Apk , pk )

Using rk+1 = rk − αk Apk , we ﬁnally obtain αk (Apk , pk ) = (rk − rk+1 , pk ) = (rk , pk ) = k , and, therefore, pk+1 = rk+1 + βk pk ,

βk =

k+1 . k

Hence, we end up with the conjugate gradient method as summarised in Algorithm C.1. Algorithm C.1 1. Compute for an arbitrary given initial solution x0 ∈ RN r0 = Ax0 − f ,

p0 = r0 ,

0 = (r0 , r0 ).

2. Iterate for k = 0, . . . , N − 2 sk = Apk ,

σk = (sk , pk ),

αk =

k , σk

xk+1 = xk − αk pk

260

C Numerical Algorithms

and compute the new residual rk+1 = rk − αk sk ,

k+1 = (rk+1 , rk+1 ).

Stop, if k+1 ≤ ε2 0 is satisfied with some prescribed accuracy ε. Otherwise compute βk =

k+1 , k

pk+1 = rk+1 + βk pk .

For the error of the computed approximate solution xk+1 , we obtain xk+1 − x =

N −1

α Ap ,

=k+1

and using the A–orthogonality of the vector system P , we further conclude N −1 2 k+1 x − xA = A(xk+1 − x), xk+1 − x = α2 (Ap , p ) . =k+1

For an arbitrary given vector uk+1 = x0 −

k

γ p ,

(C.9)

=0

we obtain in the same way k N −1 k+1 2 2 u α − γ (Ap , p ) + − xA = α2 (Ap , p ) =0

=k+1

Hence, we ﬁnd the approximate solution xk+1 as the solution of the minimisation problem k+1 x − xA = min uk+1 − xA , uk+1

where the minimum is taken over all vectors uk+1 of the form (C.9). From the recursions p0 = r 0 ,

pk+1 = rk+1 + βk pk ,

rk+1 = rk − αk Apk ,

we ﬁnd representations p = ψ (A)r0 with a matrix polynomials ψ (A) of degree . Hence, we conclude with e0 = x0 − x

C.3 Iterative Solution Methods

uk+1 − x = x0 − x −

k

γ p = e0 −

k

=0

γ ψ (A) r0 = e0 −

=0

k

261

γ ψ (A) Ae0 ,

=0

and, therefore, uk+1 − x = pk+1 (A)e0 with some matrix polynomial pk+1 (A) of degree k + 1 and having the property pk+1 (0) = 1. The polynomial pk+1 (A) obtained for the vector xk+1 is, therefore, the solution of the minimisation problem k+1 x − xA = min pk+1 (A)e0 A , (C.10) pk+1 (A)

where the minimum is taken over all polynomials pk+1 (A) having the property pk+1 (0) = 1. The space ) ( ) ( Sk (A, r0 ) = span p0 , p1 , . . . , pk = span r0 , Ar, . . . , Ak r0 is called a Krylov space of the matrix A induced by the residual vector r0 . From the minimisation problem (C.10), we further conclude the estimate k+1 e ≤ min max pk+1 (λ) e0 , A A pk+1 (A)

λ

where the minimum is again taken over all polynomials pk+1 (A) having the property pk+1 (0) = 1, and the maximum over the spectrum of the matrix A, i.e. λ ∈ [λmin (A), λmax (A)] . The above min − max–problem will be solved by the scaled Tschebyscheﬀ polynomials Tk+1 (λ), and we ﬁnd min max pk+1 (λ) = max Tk+1 (λ) =

pk+1 (A)

with

λ

λ

2q k+1 , 1 + q 2(k+1)

* * * λmax (A) + λmin (A) κ2 (A) + 1 * = * , q = * λmax (A) − λmin (A) κ2 (A) − 1

where κ2 (A) =

λmax (A) λmin (A)

is the spectral condition number of the symmetric and positive deﬁnite matrix A. Using the Raleigh quotient λmin (A) = min x=0

(Ax, x) (Ax, x) ≤ max = λmax (A) , x = 0 (x, x) (x, x)

we ﬁnd from the spectral equivalence inequalities A cA 1 (x, x) ≤ (Ax, x) ≤ c2 (x, x)

for x ∈ RN

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C Numerical Algorithms

an upper bound for the spectral condition number κ2 (A) ≤

cA 2 . cA 1

Since the spectral condition number of the boundary element stiﬀness matrices may depend on mesh parameters such as the mesh size h or the mesh ratio hmax /hmin , an appropriate preconditioning is mandatory in many cases. Hence, we assume that there is a symmetric and positive deﬁnite matrix 1/2 1/2 1/2 CA ∈ RN ×N which can be factorised as CA = CA CA , where CA is again symmetric and positive deﬁnite. Instead of the linear system (C.2), we now consider the transformed system x = C −1/2 A C −1/2 C 1/2 x = C −1/2 f = f, A A A A A = C −1/2 A C −1/2 is again symmetwhere the transformed system matrix A A A x = f, we ric and positive deﬁnite. For the solution of the linear system A can apply Algorithm C.1 to obtain, by substituting the transformations, the precondioned conjugate gradient method as described in Algorithm C.2. Algorithm C.2 1. Compute for an arbitrary given initial solution x0 ∈ RN r0 = Ax0 − f ,

−1 0 v 0 = CA r ,

p0 = v 0 ,

0 = (v 0 , r0 ).

2. Iterate for k = 0, . . . , N − 2 sk = Apk ,

σk = (sk , pk ),

αk =

k , σk

xk+1 = xk − αk pk

and compute the new residual rk+1 = rk − αk sk ,

−1 k+1 v k+1 = CA r ,

k+1 = (v k+1 , rk+1 ).

Stop, if k+1 ≤ ε2 0 is satisfied with some prescribed accuracy ε. Otherwise compute βk =

k+1 , k

pk+1 = v k+1 + βk pk .

For the preconditioned conjugate gradient scheme, we then obtain the error estimate k+1 2q k+1 e ≤ e0 , A A 2(k+1) 1+q where

C.3 Iterative Solution Methods

5 q =

+1 κ2 (A) , −1 κ2 (A)

= κ2 (A)

263

λmax (A) cA ≤ 2 , λmin (A) cA 1

A and cA 1 , c2 are the positive constants from the spectral equivalence inequalities

x, x x, x ) ≤ (A ) ≤ cA x, x ) for all x ∈ RN . cA 1 ( 2 ( 1/2

Inserting x = CA x, this is equivalent to the spectral equivalence inequalities

A cA 1 (CA x, x) ≤ (Ax, x) ≤ c2 (CA x, x)

for x ∈ RN .

(C.11)

Hence, the quite often challenging problem is to ﬁnd a preconditioning matrix CA satisfying the spectral equivalence inequalities (C.11) and allowing a sim−1 as needed in Algorithm ple and eﬃcient application of the inverse matrix CA C.2. C.3.2 Generalised Minimal Residual Method (GMRES) For a symmetric and positive deﬁnite matrix A, we have used the Krylov space ) ( Sk (A, r0 ) = span r0 , Ar0 , . . . , Ak r0 to construct an A–orthogonal vector system P (cf. (C.3)). Formally, such a vector system can be deﬁned for any arbitrary matrix A ∈ RN ×N . However, a nonsymmetric and possibly indeﬁnite matrix A does not induce an inner product. Instead of an A–orthogonal vector system, we therefore deﬁne an orthonormal vector system ( )N −1 V = v k k=0 satisfying

k

(v , v ) =

1 0

for k = , for k =

using the method of Arnoldi as described in Algorithm C.3. Algorithm C.3 1. Compute for an arbitrary given initial solution x0 ∈ RN r0 = Ax0 − f ,

v0 =

r0 . r0 2

2. Iterate for k = 0, . . . , N − 1 vˆk+1 = Av k −

k =0

βk v ,

βk = (Av k , v ) .

264

C Numerical Algorithms

Stop, if ˆ v k+1 2 = 0 is satisfied. Otherwise, compute vˆk+1 v k+1 = . ˆ v k+1 2 Note that the method of Arnoldi (Algorithm C.3) may fail if ˆ v k+1 2 = 0 is satisﬁed. We will comment this break down situation later. Using the orthonormal basis vectors from the system V , we may deﬁne an approximate solution of the linear system Ax = f as xk+1 = x0 −

k

α v ,

=0

where we have to ﬁnd the yet unknown coeﬃcients α . To this end, we may require to minimise the residual rk+1 = Axk+1 −f with respect to the Euclidean vector norm, k k+1 = Axk+1 − f = Ax0 − f − r α Av 2 −→ min , 2 2 =0

using the parameters α0 , . . . , αk . From the method of Arnoldi (Algorithm C.3), we obtain Av = vˆ+1 +

βj v j =

j=0

+1

βj v j ,

β+1 = vˆ+1 2 .

j=0

Hence, we have rk+1 = r0 −

k =0

α

+1

βj v j = r0 − Vk+1 Hk α

j=0

with the orthogonal matrix Vk+1 = v 0 , v 1 , . . . , v k+1 ∈ RN ×(k+2) , and with a upper Hessenberg matrix Hk ∈ R(k+2)×(k+1) deﬁned by for j ≤ + 1, βj Hk [j, ] = 0 for j > + 1. Moreover, we can write r0 = r0 2 v 0 = r0 2 Vk+1 e0 , where the notation e0 = (1, 0, . . . , 0) ∈ Rk+2 has been used. Since Vk+1 is orthogonal, we deduce

C.3 Iterative Solution Methods

265

k+1 r = Vk+1 r0 2 e0 − Hk α 2 2 0 0 0 = r 2 e − Hk α 2 = r 2 Qk e0 − Qk Hk α2 , where Qk ∈ R(k+2)×(k+2) is an orthogonal matrix such that Rk = Qk Hk ∈ R(k+2)×(k+1) is an upper triangular matrix. Then, we obtain k+1 2 0 = r 2 Qk e0 − Rk α2 r 2 2 =

k+1

r0 2 Qk e0 − Rk α

=0

=

2

k 2 2 r0 2 Qk e0 − Rk α + r0 2 Qk e0

=0

k+1

2 = r0 2 Qk e0

,

k+1

if the coeﬃcient vector α ∈ Rk+1 is found from the upper triangular linear system Rk α = r0 2 Qk e0 . It remains to ﬁnd an orthogonal matrix Qk ∈ R(k+2)×(k+2) transforming the upper Hessenberg matrix ⎞ ⎛ β0,0 β1,0 . . . βk,0 ⎜ .. ⎟ ⎜ β0,1 β1,1 . ⎟ ⎟ ⎜ ⎜ .. ⎟ ∈ R(k+2)×(k+1) Hk = ⎜ 0 β1,2 . . . . ⎟ ⎟ ⎜ ⎟ ⎜ . . ⎠ ⎝ . β 0 k,k

βk,k+1 into an upper triangular matrix ⎞ ⎛ r0,0 r0,1 . . . r0,k ⎜ .. ⎟ ⎜ 0 r1,1 . ⎟ ⎟ ⎜ ⎟ ⎜ . Rk = Qk Hk = ⎜ 0 0 . . ... ⎟ ∈ R(k+2)×(k+1) . ⎟ ⎜ ⎟ ⎜ . . ⎝ . rk,k ⎠ 0 This can be done by the use of the Givens rotations. Let us ﬁrst consider the column vector

hj = (βj,0 , . . . , βj,j−1 , βj,j , βj,j+1 , 0, . . . , 0) ∈ Rk+2 , where we have to ﬁnd an orthogonal matrix Gj such that Gj hj =

βj,0 , . . . , βj,j−1 , βj,j , 0, 0, . . . , 0

266

C Numerical Algorithms

j ∈ is satisﬁed. For this, it is suﬃcient to consider the orthogonal matrix G 2×2 such that R βj,j βj,j Gj = βj,j+1 0 j ∈ R2×2 allows the general representation is fulﬁlled. The orthogonal matrix G aj b j , a2j + b2j = 1 , Gj = −bj aj where the coeﬃcients aj and bj can be found from the condition −bj βj,j + aj βj,j+1 = 0 as aj = 8

βj,j 2 βj,j

+

2 βj,j+1

,

bj = 8

and, therefore, when assuming βj,j+1 > 0, βj,j = aj βj,j + bj βj,j+1 =

βj,j+1 2 βj,j

,

2 + βj,j+1

8 2 + β2 βj,j j,j+1 > 0.

(C.12)

For j = 0, . . . , k the resulting orthogonal matrices Gj are of the form ⎛ ⎞ 1 ⎜ .. ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ ⎟ 1 ⎜ ⎟ ⎜ ⎟ aj b j ⎟ ∈ R(k+2)×(k+2) Gj = ⎜ ⎜ ⎟ −b a j j ⎜ ⎟ ⎜ ⎟ 1 ⎜ ⎟ ⎜ .. ⎟ ⎝ . ⎠ 1 with Gj [j, j] = Gj [j + 1, j + 1] = aj . Their recursive application ⎛ β0,0 β1,0 . . . ⎜ β0,1 β1,1 . . . ⎜ ⎜ .. Gk Gk−1 . . . G2 G1 G0 Hk = Gk Gk−1 . . . G2 G1 G0 ⎜ ⎜ 0 β1,2 . ⎜ . ⎝ 0 .. ⎛

β0,0 β1,0 ⎜ 0 β¯1,1 ⎜ ⎜ = Gk Gk−1 . . . G2 G1 ⎜ β1,2 ⎜ ⎜ ⎝

... ... .. . ..

gives βk,0 βk,1 .. . βk,k βk,k+1 ⎞

βk,0 β¯k,1 .. .

. βk,k βk,k+1

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

C.3 Iterative Solution Methods

⎛ β0,0 ⎜ 0 ⎜ ⎜ = Gk Gk−1 . . . G2 ⎜ ⎜ ⎜ ⎝ ⎛ β0,0 ⎜ 0 ⎜ ⎜ =⎜ ⎜ ⎜ ⎝

β1,0 . . . βk,0 β1,1 . . . βk,1 .. . 0 .. . .. . β

k,k

⎞ β1,0 . . . βk,0 β1,1 . . . βk,1 ⎟ ⎟ . . .. ⎟ . . ⎟ 0 ⎟ = Rk . .. ⎟ . βk,k ⎠ 0

267

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

βk,k+1

Hence, we have constructed the orthogonal matrix Qk = Gk Gk−1 . . . G1 G0 ∈ R(k+2)×(k+2) , which fulﬁls

⎛ ⎞ ⎛ ⎞ 1 a0 ⎜0⎟ ⎜ −b0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜0⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ 0 Qk e = Gk . . . G0 ⎜ . ⎟ = Gk . . . G1 ⎜ . ⎟ ⎜ .. ⎟ ⎜ .. ⎟ ⎜ ⎟ ⎜ ⎟ ⎝0⎠ ⎝ 0 ⎠ 0 0 ⎞ ⎞ ⎛ ⎛ a0 a0 ⎟ ⎜ a1 (−b0 ) ⎟ ⎜ a1 (−b0 ) ⎟ ⎟ ⎜ ⎜ ⎜ (−b0 )(−b1 ) ⎟ ⎜ a2 (−b0 )(−b1 ) ⎟ ⎟ ⎟ ⎜ ⎜ = Gk . . . G2 ⎜ ⎟ = ⎜ ⎟ ∈ Rk+2 . .. .. ⎟ ⎟ ⎜ ⎜ . . ⎟ ⎟ ⎜ ⎜ ⎠ ⎝ ⎝ ak (−b0 ) · · · (−bk−1 ) ⎠ 0 0 (−b0 ) · · · (−bk )

From this, we ﬁnd k k+1 = e0 2 (Qk e0 )k+1 = e0 2 bj . j=0

With the deﬁnition of βj,j+1 ˆ v j 2 = 8 , bj = 8 2 2 βj,j+1 + βj,j ˆ v j 22 + (Av j , v j )2 we conclude bj < 1 when assuming (Av j , v j ) = 0. Hence, the error is monotonic decreasing. In the case of the break down situation in the method

268

C Numerical Algorithms

of Arnoldi (Algorithm C.3), i.e. ˆ v k 2 = 0, we ﬁnd bk = 0, and, therefore, k+1 2 = 0. In particular, xk+1 = x is the solution of the linear k+1 = r system Ax = f . Summarising the above, we obtain the Generalised Method of the Minimal Residual (GMRES) as described in Algorithm C.4, see [96]. Algorithm C.4 1. Compute for an arbitrary given initial solution x0 ∈ RN r0 = Ax0 − f ,

0 = r0 2 ,

v0 =

1 0 r , 0

p0 = 0 .

2. Iterate for k = 0, . . . , N − 2 wk = Av k , vˆk+1 = wk −

k

βk v , βk = (wk , v ), βkk+1 = ˆ v k+1 2 .

=0

Go to 3. if βkk+1 = 0 is satisfied. Otherwise compute 1 v k+1 = vˆk+1 . βkk+1 For = 0, . . . , k − 1 compute βk = a βk + b βk+1 ,

βk+1 = −b βk + a βk+1

and ak = 8

βkk 2 + β2 βkk kk+1

,

βkk+1 bk = 8 , 2 + β2 βkk kk+1

βkk =

8 2 + β2 βkk kk+1

as well as pk+1 = −bk pk ,

k+1 = |pk+1 |.

pk = ak pk ,

Stop, if k+1 < ε0 is satisfied with some prescribed accuracy ε. 3. Compute the approximate solution, i.e. for = k, k − 1, . . . , 0 ⎛ ⎞ k 1 ⎝ βj αj ⎠ p − α = β j=+1

and xk+1 = x0 −

k =0

α v .

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Index

H 1 projection, 236 L2 projection, 61, 63, 70, 75, 82, 169 Adaptive Cross Approximation fully pivoted, 119 partially pivoted, 126 Admissibility condition, 112 Admissible cluster pairs, 109 Analytical solution, 143, 147, 153 Approximate representation formula, 67, 69, 71, 74, 76, 81, 92, 145, 153, 158, 173, 182, 189 Approximation property, 62, 64, 114, 117, 229, 231 Area, 59 Arnoldi method, 263 Aubin–Nitsche trick, 66, 69, 74, 76, 81, 87, 92, 95 Banach’s ﬁx point theorem, 227 Bessel function, 177 Bilinear form, 2, 14, 38, 41, 77 Boundary element, 59 Boundary element mesh, 59, 64, 239 Boundary value problem exterior Dirichlet, 21, 52, 191 exterior Neumann, 22, 54, 196 homogeneous Neumann, 29 interior Dirichlet, 10, 35, 42, 49, 65, 143 interior Neumann, 13, 36, 50, 72, 149 mixed, 17, 37, 77, 87, 155, 162 nonlinear Robin, 20 Robin, 19

Calderon projector, 9, 34 Cauchy data, 17, 19, 26, 77, 88, 138, 169 Cauchy–Schwarz inequality, 234 Cea’s lemma, 66, 69, 74, 81, 95, 229 Characteristic points, 108 Cluster, 108 Cluster tree, 108 Collocation Method, 66 Conormal derivative exterior, 21 interior, 2, 155 Contraction, 227 Convergence cubic, 145, 176, 182 linear, 145, 176, 192 quadratic, 152, 153, 187, 189, 197 Dirac δ−distribution, 208 Dirichlet datum, 12, 13, 19, 20, 70, 72, 98, 138, 169 Displacement ﬁeld, 27 Distribution, 203 Duality, 206, 235 Duality argument, 62, 64 Duality pairing, 31, 200, 206, 225 Eigenfunction, 14, 43, 177 Eigenvalue, 43, 109, 177 Eigenvector, 109 Element diameter, 60 Equation Bi–Laplace, 209 ﬁx point, 226

278

Index

Helmholtz, 44, 168 Laplace, 2, 10, 13, 135 operator, 225 Poisson, 24, 160 wave, 44 Equilibrium equations, 27 Error estimate, 62, 67 Exhaust manifold, 135, 153, 182, 187

Interface problem, 26, 84, 160 Interpolation, 70, 112 Interpolation property, 114, 117 Inverse inequality argument, 66, 69, 81, 92 Iterative solution, 256

Foam, 166 Formula Betti’s ﬁrst, 28 Betti’s second, 29 Green’s ﬁrst, 2, 41, 44 Green’s second, 2, 41, 44 Fourier transform, 204, 208 Frobenius norm, 102, 120 Function k times continuously diﬀerentiable, 199 degenerated, 106 H¨ older continuous, 199 harmonic, 3, 4 inﬁnite times continuously diﬀerentiable, 199 Lipschitz continuous, 199 piecewise constant, 233 with compact support, 199 Fundamental solution Helmholtz equation, 45, 169, 212 Laplace equation, 3, 107, 137, 208 linear elastostatics, 29, 209, 210 Stokes system, 42, 210

Lam´e constants, 28, 210 Lax–Milgram lemma, 227 Lebesgue measure, 200 Legendre polynomials, 108, 177 Lipschitz boundary, 2, 13, 115, 231 Lipschitz domain, 59, 66, 200, 239 Local coordinate system, 244 Low rank approximation, 103

Galerkin method, 67, 137, 169 Galerkin orthogonality, 62, 229 Galerkin solution, 69 Galerkin variational problem, 228 Generalised derivative, 201 Givens rotations, 265 Global trial space, 61, 63 Gram–Schmidt orthogonalisation, 258 G˚ arding’s inequality, 46, 47

Krylov space, 261

Matrix approximation, 102 dense, 102 Hessenberg, 265 hierarchical, 101 low rank, 106 orthogonal, 265 sparse, 129 stiﬀness, 228 symmetric and positive deﬁnite, 256 transformation, 74 triangular, 265 Mesh ratio, 60 Mesh size global, 60 local, 60 Midpoint quadrature rule, 240 Minimisation problem, 68, 72, 226 Multiindex, 115, 199

H¨ older inequality, 200 Hierarchical block structure, 112 Hooke’s law, 28

Navier system, 28, 40 Neumann datum, 14, 19, 20, 24, 25, 65, 75, 91, 98, 138, 169 Neumann series, 12, 14, 16, 23 Norm equivalent, 202 semi-, 62, 202 Sobolev, 201, 206 Sobolev–Slobodeckii, 201, 206 Numerical integration, 239

Inner product, 204, 225, 256

Operator

Index X–elliptic, 226 adjoint double layer potential, 6, 33, 45 Bessel potential, 204 boundary stress, 28 bounded, 225 double layer potential, 5, 32, 47 hypersingular, 7, 15, 33, 47 matrix surface curl, 32, 90 semi–elliptic, 226 single layer potential, 4, 30, 45 Steklov–Poincar´e, 9, 16, 19, 22, 27, 35, 85, 161 surface curl, 7, 74 trace, 208 Optimal order of convergence, 67, 70, 75, 76 Orthogonality, 23, 117, 118, 259 Parametrisation, 239 Particular solution, 25, 26, 84, 135, 160, 169 Partition of unity, 205 Poincar´e inequality, 203 Poisson ratio, 28, 163, 166 Potential adjoint double layer, 6 double layer, 4, 32, 46 Newton, 24 single layer, 3, 30, 42, 45 Preconditioning, 147, 153, 262 Pressure, 40 Radiation condition, 21, 22, 26, 84, 160 Raleigh quotient, 261 Reference element, 59 Relay, 134, 163 Representation formula, 3, 17, 21, 22, 24, 26, 36, 41, 45, 49, 52, 54, 65, 72, 77, 91, 94, 98 Riesz representation theorem, 226 Rigid body motions, 29 Scaling condition, 14, 37 Schur complement, 79, 87, 156 Shape function linear, 63, 64 piecewise constant, 61 Singular value, 102 Singular value decomposition, 102

279

Singular vector, 102 Singularity, 102, 104 Solvability condition, 13, 36, 43 Somigliana identity, 29, 35 Sommerfeld radiation condition, 48, 52, 54 Space conformal trial, 227 dual, 202, 225 Hilbert, 201, 225 Schwartz, 204 Sobolev, 3, 17, 201, 204, 206 Spectral condition number, 69, 73, 262 Spherical harmonics, 107 Stability, 66 Stiﬀness matrix, 66, 69 Stopping criterion, 114, 120 Stress–strain relation, 28 Support, 199 System linear, 228, 256 linear elastostatics, 27 Stokes, 40 Taylor series, 107 TEAM problem 10, 131, 147, 157 problem 24, 134, 158 Tempered distribution, 204 Tensor Kelvin, 29, 209, 210 Strain, 28 Stress, 28 Trace exterior, 21 interior, 2, 155 Transmission conditions, 26, 84, 160 Tschebyscheﬀ polynomials, 261 Uniform cone condition, 205 Unit sphere, 131, 143, 150, 156 Variational problem, 11, 14–16, 20, 23, 27, 35, 36, 40, 43, 61, 75, 77, 82, 84, 225, 233, 237 Velocity ﬁeld, 40 Viscosity, 40 Young modulus, 28, 163, 166