Boundary Element Methods (eBook)

Prof. Dr. rer. nat. Stefan Sauter Born in 1964, Heidelberg, Germany. Studies of mathematics and physics at the University of Heidelberg (1985-1990). Scientific assistant at the University of Kiel (PhD 1993). 1993/94 PostDoc at the University of College Park. Until 1998, senior assistant at the University of Kiel (Habilitation 1998). Chair in Mathematics at the University of Leipzig (1998/99). Since 1999 Ordinarius in Mathematics at the Universität Zürich. Prof. Christoph Schwab, PhD Born in 1962, Flörsheim, Germany. Studies of mathematics, mechanics, and aerospace engineering in Darmstadt and College Park, Maryland, USA (1982-1989). PhD in Applied Mathematics, University of Maryland, College Park 1989. Postdoctoral fellow (1990/91) University of Westminster, London, UK. Assistant professor (1991-1994) and associate professor (1995) of Mathematics, University of Maryland, Baltimore County, USA. Extraordinarius (1995-1998) and Ordinarius (1998-) for mathematics at the ETH Zürich. The authors were organizing various conferences and minisymposia on fast boundary element methods, e.g., at Oberwolfach, MAFELAP conferences at Brunel UK, Zurich Summer Schools, and were speakers on these topics at numerous international conferences.

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