Schwab Boundary Element Methods

1. Auflage 2010
ISBN: 978-3-540-68093-2
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 561 Seiten

This work presents a thorough treatment of boundary element methods (BEM) for solving strongly elliptic boundary integral equations obtained from boundary reduction of elliptic boundary value problems  in $\mathbb{R}^3$. The book is self-contained, the prerequisites on elliptic partial differential and integral equations being presented in Chapters 2 and 3. The main focus is on the development, analysis, and implementation of Galerkin boundary element methods, which is one of the most flexible and robust numerical discretization methods for integral equations. For the efficient realization of the Galerkin BEM, it is essential to replace time-consuming steps in the numerical solution process with fast algorithms. In Chapters 5-9 these methods are developed, analyzed, and formulated in an algorithmic way.

Autoren/Hrsg.

Schwab, Christoph

Inhaltsverzeichnis

1;Preface
;8
2;Contents
;12
3;Chapter 1: Introduction
;19
3.1;1.1 The Concept of the Boundary Element Method;19
3.1.1;1.1.1 Basic Terminology;19
3.1.2;1.1.2 A Physical Example;21
3.1.3;1.1.3 Fundamental Solutions;25
3.1.4;1.1.4 Potentials and Boundary Integral Operators;25
3.2;1.2 Numerical Analysis of Boundary Integral Equations;28
3.2.1;1.2.1 Galerkin Method;28
3.2.2;1.2.2 Efficient Methods for the Solution of the Galerkin Equations;30
3.2.2.1;1.2.2.1 Quadrature Methods;30
3.2.2.2;1.2.2.2 Solving the Linear System of Equations;31
3.2.2.3;1.2.2.3 Cluster Method;32
3.2.2.4;1.2.2.4 Surface Approximation;33
3.2.2.5;1.2.2.5 A Posteriori Error Estimation;35
4;Chapter 2: Elliptic Differential Equations
;38
4.1;2.1 Elementary Functional Analysis;38
4.1.1;2.1.1 Banach and Hilbert Spaces;38
4.1.1.1;2.1.1.1 Normed Spaces;38
4.1.1.2;2.1.1.2 Linear Operators ;39
4.1.1.3;2.1.1.3 Banach Spaces;40
4.1.1.4;2.1.1.4 Embeddings;41
4.1.1.5;2.1.1.5 Hilbert Spaces;41
4.1.2;2.1.2 Dual Spaces;42
4.1.2.1;2.1.2.1 Dual Space of a Normed, Linear Space;42
4.1.2.2;2.1.2.2 Dual Operator;43
4.1.2.3;2.1.2.3 Adjoint Operator;44
4.1.2.4;2.1.2.4 Gelfand Triple;46
4.1.2.5;2.1.2.5 Weak Convergence;47
4.1.3;2.1.3 Compact Operators;47
4.1.4;2.1.4 Fredholm–Riesz–Schauder Theory;48
4.1.5;2.1.5 Bilinear and Sesquilinear Forms;49
4.1.6;2.1.6 Existence Theorems;52
4.1.7;2.1.7 Interpolation Spaces;63
4.2;2.2 Geometric Tools;64
4.2.1;2.2.1 Function Spaces;64
4.2.2;2.2.2 Smoothness of Domains;67
4.2.3;2.2.3 Normal Vector;69
4.2.4;2.2.4 Boundary Integrals;70
4.3;2.3 Sobolev Spaces on Domains ;71
4.4;2.4 Sobolev Spaces on Surfaces Gamma
;74
4.4.1;2.4.1 Definition of Sobolev Spaces on Gamma
;74
4.4.2;2.4.2 Sobolev Spaces on Gamma0 subset Gamma
;76
4.5;2.5 Embedding Theorems;77
4.6;2.6 Trace Operators;80
4.7;2.7 Green's Formulas and Normal Derivatives;83
4.8;2.8 Solution Operator;89
4.9;2.9 Elliptic Boundary Value Problems;93
4.9.1;2.9.1 Classical Formulation of Elliptic Boundary Value Problems;93
4.9.1.1;2.9.1.1 Interior Dirichlet Problem (IDP);93
4.9.1.2;2.9.1.2 Interior Neumann Problem (INP);93
4.9.1.3;2.9.1.3 Interior Mixed Boundary Value Problem (IMP);94
4.9.1.4;2.9.1.4 Exterior Dirichlet Problem (EDP);94
4.9.1.5;2.9.1.5 Exterior Neumann Problem (ENP);95
4.9.1.6;2.9.1.6 Exterior Mixed Boundary Value Problem (EMP);95
4.9.1.7;2.9.1.7 Transmission Problem (TP);95
4.9.2;2.9.2 Variational Formulation of Elliptic Boundary Value Problems;96
4.9.2.1;2.9.2.1 Interior Dirichlet Problem (IDP);96
4.9.2.2;2.9.2.2 Interior Neumann Problem (INP);97
4.9.2.3;2.9.2.3 Interior Mixed Boundary Value Problem (IMP);97
4.9.2.4;2.9.2.4 Function Spaces for Exterior Problems;98
4.9.2.5;2.9.2.5 Exterior Dirichlet Problem (EDP);100
4.9.2.6;2.9.2.6 Exterior Neumann Problem (ENP);101
4.9.2.7;2.9.2.7 Exterior Mixed Boundary Value Problem (EMP);102
4.9.2.8;2.9.2.8 Transmission Problem (TP);102
4.9.3;2.9.3 Equivalence of Strong and Weak Formulation;103
4.9.3.1;2.9.3.1 Interior Problems;103
4.9.3.2;2.9.3.2 Exterior Problems;104
4.10;2.10 Existence and Uniqueness;106
4.10.1;2.10.1 Interior Problems;108
4.10.1.1;2.10.1.1 Interior Dirichlet Problem;108
4.10.1.2;2.10.1.2 Interior Neumann Problem;109
4.10.1.3;2.10.1.3 Interior Mixed Boundary Value Problem;110
4.10.2;2.10.2 Exterior Problems;110
4.10.2.1;2.10.2.1 General Elliptic Operator with amin c >||
b||2;110
4.10.2.2;2.10.2.2 Laplace Operator;111
4.10.2.3;2.10.2.3 Helmholtz Equation;116
5;Chaptre 3: Elliptic Boundary Integral Equations
;118
5.1;3.1 Boundary Integral Operators;118
5.1.1;3.1.1 Newton Potential;120
5.1.2;3.1.2 Mapping Properties of the Boundary Integral Operators;129
5.2;3.2 Regularity of the Solutions of the Boundary Integral Equations;131
5.3;3.3 Jump Relations of the Potentials and Explicit Representation Formulas;132
5.3.1;3.3.1 Jump Properties of the Potentials;132
5.3.2;3.3.2 Explicit Representation of the Boundary Integral Operator V;134
5.3.3;3.3.3 Explicit Representation of the Boundary Integral Operators K and K';139
5.3.4;3.3.4 Explicit Representation of the Boundary Integral Operator W;149
5.4;3.4 Integral Equations for Elliptic Boundary Value Problems;156
5.4.1;3.4.1 The Indirect Method;157
5.4.1.1;3.4.1.1 Interior Problems;157
5.4.1.2;3.4.1.2 Exterior Problems;161
5.4.1.3;3.4.1.3 Transmission Problem;161
5.4.2;3.4.2 The Direct Method;162
5.4.2.1;3.4.2.1 Interior Problems;162
5.4.2.2;3.4.2.2 Exterior Problems;164
5.4.3;3.4.3 Comparison Between Direct and Indirect Method;165
5.5;3.5 Unique Solvability of the Boundary Integral Equations;166
5.5.1;3.5.1 Existence and Uniqueness for Closed Surfaces and Dirichlet or Neumann Boundary Conditions;166
5.5.2;3.5.2 Existence and Uniqueness for the Mixed Boundary Value Problem;170
5.5.3;3.5.3 Screen Problems;173
5.6;3.6 Calderón Projector;174
5.7;3.7 Poincaré–Steklov Operator;177
5.8;3.8 Invertibility of Boundary Integral Operators of the Second Kind;179
5.9;3.9 Boundary Integral Equations for the Helmholtz Equation;185
5.9.1;3.9.1 Helmholtz Equation;185
5.9.2;3.9.2 Integral Equations and Resonances;186
5.9.3;3.9.3 Existence of Solutions of the Exterior Problem;189
5.9.4;3.9.4 Modified Integral Equations;192
5.10;3.10 Bibliographical Remarks on Variational BIEs;194
6;Chapter 4: Boundary Element Methods
;199
6.1;4.1 Boundary Elements for the Potential Equation in R3;200
6.1.1;4.1.1 Model Problem 1: Dirichlet Problem;200
6.1.2;4.1.2 Surface Meshes;202
6.1.3;4.1.3 Discontinuous Boundary Elements;207
6.1.4;4.1.4 Galerkin Boundary Element Method;209
6.1.5;4.1.5 Convergence Rate of Discontinuous Boundary Elements;213
6.1.6;4.1.6 Model Problem 2: Neumann Problem;217
6.1.7;4.1.7 Continuous Boundary Elements;218
6.1.8;4.1.8 Galerkin BEM with Continuous Boundary Elements;227
6.1.9;4.1.9 Convergence Rates with Continuous Boundary Elements;228
6.1.10;4.1.10 Model Problem 3: Mixed Boundary Value Problem;234
6.1.11;4.1.11 Model Problem 4: Screen Problems;236
6.2;4.2 Convergence of Abstract Galerkin Methods;238
6.2.1;4.2.1 Abstract Variational Problem;238
6.2.2;4.2.2 Galerkin Approximation;239
6.2.3;4.2.3 Compact Perturbations;242
6.2.4;4.2.4 Consistent Perturbations: Strang's Lemma;247
6.2.5;4.2.5 Aubin–Nitsche Duality Technique;252
6.2.5.1;4.2.5.1 Errors in Functionals of the Solution;253
6.2.5.2;4.2.5.2 Perturbations;257
6.3;4.3 Proof of the Approximation Property;262
6.3.1;4.3.1 Approximation Properties on Plane Panels;263
6.3.2;4.3.2 Approximation on Curved Panels;270
6.3.3;4.3.3 Continuity of Functions in Hpws(Gamma) for s>1;274
6.3.4;4.3.4 Approximation Properties of SGp,-1 ;275
6.3.5;4.3.5 Approximation Properties of SGp,0 ;277
6.4;4.4 Inverse Estimates;294
6.5;4.5 Condition of the System Matrices;300
6.6;4.6 Bibliographical Remarks and Further Results;301
7;Chapter 5: Generating the Matrix Coefficients
;304
7.1;5.1 Kernel Functions and Strongly Singular Integrals;305
7.1.1;5.1.1 Geometric Conditions;305
7.1.2;5.1.2 Cauchy-Singular Integrals ;309
7.1.3;5.1.3 Explicit Conditions on Cauchy-Singular KernelFunctions;312
7.1.4;5.1.4 Kernel Functions in Local Coordinates;314
7.2;5.2 Relative Coordinates ;319
7.2.1;5.2.1 Identical Panels;320
7.2.2;5.2.2 Common Edge;327
7.2.3;5.2.3 Common Vertex;330
7.2.4;5.2.4 Overview: Regularizing Coordinate Transformations;331
7.2.5;5.2.5 Evaluating the Right-Hand Side and the Integral-Free Term;335
7.3;5.3 Numerical Integration;336
7.3.1;5.3.1 Numerical Quadrature Methods ;336
7.3.1.1;5.3.1.1 Simple Quadrature Methods;337
7.3.1.2;5.3.1.2 Tensor-Gauss Quadrature ;338
7.3.2;5.3.2 Local Quadrature Error Estimates;339
7.3.2.1;5.3.2.1 Local Error Estimates for Simple Quadrature Methods;339
7.3.2.2;5.3.2.2 Derivative Free Quadrature Error Estimates for Analytic Integrands;344
7.3.2.3;5.3.2.3 Estimates of the Analyticity Ellipses of the Regularized Integrands;346
7.3.2.4;5.3.2.4 Quadrature Orders for Regularized Kernel Functions;356
7.3.3;5.3.3 The Influence of Quadrature on the Discretization Error;357
7.3.4;5.3.4 Overview of the Quadrature Orders for the Galerkin Method with Quadrature;366
7.3.4.1;5.3.4.1 Integral Equations of Negative Order;366
7.3.4.2;5.3.4.2 Equations of Order Zero;366
7.3.4.3;5.3.4.3 Equations of Positive Order;367
7.4;5.4 Additional Results and Quadrature Techniques;367
8;Chapter 6: Solution of Linear Systems of Equations
;368
8.1;6.1 cg Method ;369
8.1.1;6.1.1 cg Basic Algorithm;369
8.1.2;6.1.2 Preconditioning Methods;371
8.1.3;6.1.3 Orthogonality Relations;372
8.1.4;6.1.4 Convergence Rate of the cg Method;373
8.1.5;6.1.5 Generalizations;375
8.2;6.2 Descent Methods for Non-symmetric Systems;376
8.2.1;6.2.1 Descent Methods;376
8.2.2;6.2.2 Convergence Rate of MR and Orthomin(k);377
8.3;6.3 Iterative Solvers for Equations of Negative Order;379
8.4;6.4 Iterative Solvers for Equations of Positive Order;382
8.4.1;6.4.1 Integral Equations of Positive Order;382
8.4.2;6.4.2 Iterative Methods ;385
8.4.3;6.4.3 Multi-grid Methods;389
8.4.3.1;6.4.3.1 Motivation;390
8.4.3.2;6.4.3.2 Multi-grid Method for Integral Equations of Positive Order;393
8.4.3.3;6.4.3.3 Nested Iterations ;396
8.4.3.4;6.4.3.4 Convergence Analysis for Multi-grid Methods;397
8.5;6.5 Multi-grid Methods for Equations of Negative Order;414
8.6;6.6 Further Remarks and Results on Iterative Solvers of BIEs;417
9;Chapter 7: Cluster Methods
;418
9.1;7.1 The Cluster Algorithm;419
9.1.1;7.1.1 Conditions on the Integral Operator;419
9.1.2;7.1.2 Cluster Tree and Admissible Covering;420
9.1.3;7.1.3 Approximation of the Kernel Function;424
9.1.3.1;7.1.3.1 Cebyšev Interpolation;425
9.1.3.2;7.1.3.2 Multipole Expansion;429
9.1.3.3;7.1.3.3 Abstract Cluster Approximation;430
9.1.4;7.1.4 The Matrix-Vector Multiplication in the Cluster Format;431
9.1.4.1;7.1.4.1 Computation the Far-Field Coefficients ;435
9.1.4.2;7.1.4.2 Cluster–Cluster Interaction;436
9.1.4.3;7.1.4.3 Evaluating the Cluster Approximation of a Matrix-VectorMultiplication;436
9.1.4.4;7.1.4.4 Algorithmic Description of the Cluster Method;438
9.2;7.2 Realization of the Subalgorithms;440
9.2.1;7.2.1 Algorithmic Realization of the Cebyšev Approximation;440
9.2.2;7.2.2 Expansion with Variable Order;446
9.3;7.3 Error Analysis for the Cluster Method;448
9.3.1;7.3.1 Local Error Estimates;448
9.3.1.1;7.3.1.1 Local Error Estimates for the Cebyšev Interpolation;448
9.3.2;7.3.2 Global Error Estimates;463
9.3.2.1;7.3.2.1 L2-Estimate for the Clustering Error Without Integration by Parts;464
9.3.2.2;7.3.2.2 L2-Estimates for the Cluster Method with Integration by Parts;467
9.3.2.3;7.3.2.3 Stability and Consistency of the Cluster Method;468
9.4;7.4 The Complexity of the Cluster Method;469
9.4.1;7.4.1 Number of Clusters and Blocks;470
9.4.2;7.4.2 The Algorithmic Complexity of the Cluster Method;475
9.5;7.5 Cluster Method for Collocation Methods;478
9.6;7.6 Remarks and Additional Results;479
10;Chapter 8: p-Parametric Surface Approximation
;481
10.1;8.1 Discretization of Boundary Integral Equations with Surface Approximations;481
10.1.1;8.1.1 p-Parametric Surface Meshes for Globally Smooth Surfaces;481
10.1.2;8.1.2 (k,p)-Boundary Element Spaces with p-Parametric Surface Approximation;485
10.1.3;8.1.3 Discretization of Boundary Integral Equations with p-Parametric Surface Approximation;486
10.2;8.2 Convergence Analysis;489
10.3;8.3 Overview of the Orders of the p-Parametric Surface Approximations;509
10.4;8.4 Elementary Differential Geometry;512
11;Chapter 9: A Posteriori Error Estimation
;531
11.1;9.1 Preliminaries;532
11.2;9.2 Local Error Indicators and A Posteriori Error Estimators;535
11.2.1;9.2.1 Operators of Negative Order;535
11.2.2;9.2.2 Operators of Non-negative Order;537
11.3;9.3 Proof of Efficiency and Reliability;537
11.3.1;9.3.1 Analysis of Operators of Negative Order;538
11.3.2;9.3.2 Analysis of Operators of Non-negative Order;548
11.3.3;9.3.3 Bibliographical Remarks, Further Results and Open Problems;557
12;References;559
13;Index of Symbols;569
14;Index;573