Boundary element methods:
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| Main Authors: | , |
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| Format: | Book |
| Language: | English German |
| Published: |
Berlin
Springer
[2011]
|
| Series: | Springer series in computational mathematics
39 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | xvii, 561 Seiten Illustrationen 235 mm x 155 mm |
| ISBN: | 9783540680925 |
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Record in the Search Index
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IMAGE 1
CONTENTS
1 INTRODUCTION 1
1.1 THE CONCEPT OF THE BOUNDARY ELEMENT METHOD 1
1.1.1 BASIC TERMINOLOGY 1
1.1.2 A PHYSICAL EXAMPLE 3
1.1.3 FUNDAMENTAL SOLUTIONS 7
1.1.4 POTENTIALS AND BOUNDARY INTEGRAL OPERATORS 7
1.2 NUMERICAL ANALYSIS O F BOUNDARY INTEGRAL EQUATIONS 10
1.2.1 GALERKIN METHOD 10
1.2.2 EFFICIENT METHODS FOR THE SOLUTION OF THE GALERKIN EQUATIONS 12
1.2.2.1 QUADRATURE METHODS 12
1.2.2.2 SOLVING THE LINEAR SYSTEM OF EQUATIONS 13
1.2.2.3 CLUSTER METHOD 14
1.2.2.4 SURFACE APPROXIMATION 15
1.2.2.5 A POSTERIORI ERROR ESTIMATION 17
2 ELLIPTIC DIFFERENTIAL EQUATIONS 21
2.1 ELEMENTARY FUNCTIONAL ANALYSIS 21
2.1.1 BANACH AND HILBERT SPACES 21
2.1.1.1 NORMED SPACES 21
2.1.1.2 LINEAR OPERATORS 22
2.1.1.3 BANACH SPACES 23
2.1.1.4 EMBEDDINGS 24
2.1.1.5 HILBERT SPACES 24
2.1.2 DUAL SPACES 25
2.1.2.1 DUAL SPACE OF A NORMED, LINEAR SPACE 25
2.1.2.2 DUAL OPERATOR 2 6
2.1.2.3 ADJOINT OPERATOR 27
2.1.2.4 GELFAND TRIPLE 29
2.1.2.5 WEAK CONVERGENCE 30
2.1.3 COMPACT OPERATORS 3 0
2.1.4 FREDHOLM-RIESZ-SCHAUDER THEORY 31
2.1.5 BILINEAR AND SESQUILINEAR FORMS 32
XI
HTTP://D-NB.INFO/1006507566
IMAGE 2
XII C O N T E N T S
2.1.6 EXISTENCE THEOREMS 35
2.1.7 INTERPOLATION SPACES 4 6
2.2 GEOMETRIC TOOLS 47
2.2.1 FUNCTION SPACES 47
2.2.2 SMOOTHNESS OF DOMAINS 5 0
2.2.3 NORMAL VECTOR 52
2.2.4 BOUNDARY INTEGRALS 53
2.3 SOBOLEV SPACES ON DOMAINS 2 5 4
2.4 SOBOLEV SPACES ON SURFACES T 57
2.4.1 DEFINITION O F SOBOLEV SPACES ON T 57
2.4.2 SOBOLEV SPACES ON TO C T 59
2.5 EMBEDDING THEOREMS 60
2.6 TRACE OPERATORS 63
2.7 GREEN'S FORMULAS AND NORMAL DERIVATIVES 6 6
2.8 SOLUTION OPERATOR 7 2
2.9 ELLIPTIC BOUNDARY VALUE PROBLEMS 7 6
2.9.1 CLASSICAL FORMULATION O F ELLIPTIC BOUNDARY VALUE PROBLEMS 7 6
2.9.1.1 INTERIOR DIRICHLET PROBLEM (IDP) 7 6
2.9.1.2 INTERIOR NEUMANN PROBLEM (INP) 7 6
2.9.1.3 INTERIOR MIXED BOUNDARY VALUE PROBLEM (IMP) 77
2.9.1.4 EXTERIOR DIRICHLET PROBLEM (EDP) 77
2.9.1.5 EXTERIOR NEUMANN PROBLEM (ENP) 7 8
2.9.1.6 EXTERIOR MIXED BOUNDARY VALUE PROBLEM (EMP) 78
2.9.1.7 TRANSMISSION PROBLEM (TP) 78
2.9.2 VARIATIONAL FORMULATION O F ELLIPTIC BOUNDARY VALUE PROBLEMS 7 9
2.9.2.1 INTERIOR DIRICHLET PROBLEM (IDP) 7 9
2.9.2.2 INTERIOR NEUMANN PROBLEM (INP) 80
2.9.2.3 INTERIOR MIXED BOUNDARY VALUE PROBLEM (IMP) 80
2.9.2.4 FUNCTION SPACES FOR EXTERIOR PROBLEMS 81
2.9.2.5 EXTERIOR DIRICHLET PROBLEM (EDP) 83
2.9.2.6 EXTERIOR NEUMANN PROBLEM (ENP) 84
2.9.2.7 EXTERIOR MIXED BOUNDARY VALUE PROBLEM (EMP) 85
2.9.2.8 TRANSMISSION PROBLEM (TP) 85
2.9.3 EQUIVALENCE O F STRONG AND WEAK FORMULATION 86
2.9.3.1 INTERIOR PROBLEMS 86
2.9.3.2 EXTERIOR PROBLEMS 87
2.10 EXISTENCE AND UNIQUENESS 89
2.10.1 INTERIOR PROBLEMS 91
2.10.1.1 INTERIOR DIRICHLET PROBLEM 91
IMAGE 3
CONTENTS X I I I
2.10.1.2 INTERIOR NEUMANN PROBLEM 92
2.10.1.3 INTERIOR MIXED BOUNDARY VALUE PROBLEM 93
2.10.2 EXTERIOR PROBLEMS 93
2.10.2.1 GENERAL ELLIPTIC OPERATOR WITH A M J N C ||B|| 2 9 3
2.10.2.2 LAPLACE OPERATOR 94
2.10.2.3 HELMHOLTZ EQUATION 99
3 ELLIPTIC BOUNDARY INTEGRAL EQUATIONS 101
3.1 BOUNDARY INTEGRAL OPERATORS 101
3.1.1 NEWTON POTENTIAL 103
3.1.2 MAPPING PROPERTIES O F THE BOUNDARY INTEGRAL OPERATORS 112 3.2
REGULARITY O F THE SOLUTIONS O F THE BOUNDARY INTEGRAL EQUATIONS 114 3.3
JUMP RELATIONS O F THE POTENTIALS AND EXPLICIT REPRESENTATION FORMULAS
115
3.3.1 JUMP PROPERTIES OF THE POTENTIALS 115
3.3.2 EXPLICIT REPRESENTATION O F THE BOUNDARY INTEGRAL OPERATOR V 117
3.3.3 EXPLICIT REPRESENTATION O F THE BOUNDARY INTEGRAL OPERATORS K AND
K ' 122
3.3.4 EXPLICIT REPRESENTATION OF THE BOUNDARY INTEGRAL OPERATOR W 132
3.4 INTEGRAL EQUATIONS FOR ELLIPTIC BOUNDARY VALUE PROBLEMS 139
3.4.1 THE INDIRECT METHOD 140
3.4.1.1 INTERIOR PROBLEMS 140
3.4.1.2 EXTERIOR PROBLEMS 144
3.4.1.3 TRANSMISSION PROBLEM 144
3.4.2 THE DIRECT METHOD 145
3.4.2.1 INTERIOR PROBLEMS 145
3.4.2.2 EXTERIOR PROBLEMS 147
3.4.3 COMPARISON BETWEEN DIRECT AND INDIRECT METHOD 148
3.5 UNIQUE SOLVABILITY OF THE BOUNDARY INTEGRAL EQUATIONS 149
3.5.1 EXISTENCE AND UNIQUENESS FOR CLOSED SURFACES AND DIRICHLET OR
NEUMANN BOUNDARY CONDITIONS 149
3.5.2 EXISTENCE AND UNIQUENESS FOR THE MIXED BOUNDARY VALUE PROBLEM 153
3.5.3 SCREEN PROBLEMS 156
3.6 CALDERON PROJECTOR 157
3.7 POINCARE-STEKLOV OPERATOR 160
3.8 INVERTIBILITY O F BOUNDARY INTEGRAL OPERATORS OF THE SECOND KIND 162
3.9 BOUNDARY INTEGRAL EQUATIONS FOR THE HELMHOLTZ EQUATION 168
3.9.1 HELMHOLTZ EQUATION 168
3.9.2 INTEGRAL EQUATIONS AND RESONANCES 169
3.9.3 EXISTENCE O F SOLUTIONS O F THE EXTERIOR PROBLEM 172
3.9.4 MODIFIED INTEGRAL EQUATIONS 175
3.10 BIBLIOGRAPHICAL REMARKS ON VARIATIONAL BIES 177
IMAGE 4
X I V C O N T E N T S
4 BOUNDARY ELEMENT METHODS 1 83
4.1 BOUNDARY ELEMENTS FOR THE POTENTIAL EQUATION IN R 3 184
4.1.1 MODEL PROBLEM 1: DIRICHLET PROBLEM 184
4.1.2 SURFACE MESHES 186
4.1.3 DISCONTINUOUS BOUNDARY ELEMENTS 191
4.1.4 GALERKIN BOUNDARY ELEMENT METHOD 193
4.1.5 CONVERGENCE RATE O F DISCONTINUOUS BOUNDARY ELEMENTS. 197 4.1.6
MODEL PROBLEM 2: NEUMANN PROBLEM 201
4.1.7 CONTINUOUS BOUNDARY ELEMENTS 202
4.1.8 GALERKIN BEM WITH CONTINUOUS BOUNDARY ELEMENTS 211 4.1.9
CONVERGENCE RATES WITH CONTINUOUS BOUNDARY ELEMENTS . .212 4.1.10 MODEL
PROBLEM 3: MIXED BOUNDARY VALUE PROBLEM 218 4.1.11 MODEL PROBLEM 4:
SCREEN PROBLEMS 220
4.2 CONVERGENCE O F ABSTRACT GALERKIN METHODS 222
4.2.1 ABSTRACT VARIATIONAL PROBLEM 222
4.2.2 GALERKIN APPROXIMATION 223
4.2.3 COMPACT PERTURBATIONS 226
4.2.4 CONSISTENT PERTURBATIONS: STRANG'S LEMMA 231
4.2.5 AUBIN-NITSCHE DUALITY TECHNIQUE 236
4.2.5.1 ERRORS IN FUNCTIONALS O F THE SOLUTION 237
4.2.5.2 PERTURBATIONS 241
4.3 PROOF O F THE APPROXIMATION PROPERTY 246
4.3.1 APPROXIMATION PROPERTIES ON PLANE PANELS 247
4.3.2 APPROXIMATION ON CURVED PANELS 254
4.3.3 CONTINUITY O F FUNCTIONS I N / / P W ( R ) F O R I 1 258
4.3.4 APPROXIMATION PROPERTIES O F SG 1 2 5 9
4.3.5 APPROXIMATION PROPERTIES OF SG' 261
4.4 INVERSE ESTIMATES 278
4.5 CONDITION OF THE SYSTEM MATRICES 284
4.6 BIBLIOGRAPHICAL REMARKS AND FURTHER RESULTS 285
5 GENERATING THE MATRIX COEFFICIENTS 289
5.1 KERNEL FUNCTIONS AND STRONGLY SINGULAR INTEGRALS 290
5.1.1 GEOMETRIC CONDITIONS 290
5.1.2 CAUCHY-SINGULAR INTEGRALS 294
5.1.3 EXPLICIT CONDITIONS ON CAUCHY-SINGULAR KERNEL FUNCTIONS 297
5.1.4 KERNEL FUNCTIONS IN LOCAL COORDINATES 299
5.2 RELATIVE COORDINATES 304
5.2.1 IDENTICAL PANELS 305
5.2.2 COMMON EDGE 312
5.2.3 COMMON VERTEX 315
5.2.4 OVERVIEW: REGULARIZING COORDINATE TRANSFORMATIONS 316 5.2.5
EVALUATING THE RIGHT-HAND SIDE AND THE INTEGRAL-FREE TERM 320
IMAGE 5
CONTENTS X V
5.3 NUMERICAL INTEGRATION 321
5.3.1 NUMERICAL QUADRATURE METHODS 321
5.3.1.1 SIMPLE QUADRATURE METHODS 322
5.3.1.2 TENSOR-GAUSS QUADRATURE 323
5.3.2 LOCAL QUADRATURE ERROR ESTIMATES 324
5.3.2.1 LOCAL ERROR ESTIMATES FOR SIMPLE QUADRATURE METHODS 324
5.3.2.2 DERIVATIVE FREE QUADRATURE ERROR ESTIMATES FOR ANALYTIC
INTEGRANDS 329
5.3.2.3 ESTIMATES OF THE ANALYTICITY ELLIPSES OF THE REGULARIZED
INTEGRANDS 331
5.3.2.4 QUADRATURE ORDERS FOR REGULARIZED KERNEL FUNCTIONS 341
5.3.3 THE INFLUENCE OF QUADRATURE ON THE DISCRETIZATION ERROR . .342
5.3.4 OVERVIEW O F THE QUADRATURE ORDERS FOR THE GALERKIN METHOD WITH
QUADRATURE 351
5.3.4.1 INTEGRAL EQUATIONS O F NEGATIVE ORDER 351
5.3.4.2 EQUATIONS O F ORDER ZERO 351
5.3.4.3 EQUATIONS O F POSITIVE ORDER 352
5.4 ADDITIONAL RESULTS AND QUADRATURE TECHNIQUES 352
6 SOLUTION O F LINEAR SYSTEMS O F EQUATIONS 353
6.1 EG METHOD 354
6.1.1 EG BASIC ALGORITHM 354
6.1.2 PRECONDITIONING METHODS 356
6.1.3 ORTHOGONALITY RELATIONS 357
6.1.4 CONVERGENCE RATE O F THE EG METHOD 358
6.1.5 GENERALIZATIONS 360
6.2 DESCENT METHODS FOR NON-SYMMETRIC SYSTEMS 361
6.2.1 DESCENT METHODS 361
6.2.2 CONVERGENCE RATE OF M R AND ORTHOMIN(/:) 362
6.3 ITERATIVE SOLVERS FOR EQUATIONS OF NEGATIVE ORDER 364
6.4 ITERATIVE SOLVERS FOR EQUATIONS OF POSITIVE ORDER 367
6.4.1 INTEGRAL EQUATIONS O F POSITIVE ORDER 367
6.4.2 ITERATIVE METHODS 370
6.4.3 MULTI-GRID METHODS 374
6.4.3.1 MOTIVATION 375
6.4.3.2 MULTI-GRID METHOD FOR INTEGRAL EQUATIONS O F POSITIVE ORDER 378
6.4.3.3 NESTED ITERATIONS 381
6.4.3.4 CONVERGENCE ANALYSIS FOR MULTI-GRID METHODS. .382 6.5
MULTI-GRID METHODS FOR EQUATIONS O F NEGATIVE ORDER 399
6.6 FURTHER REMARKS AND RESULTS ON ITERATIVE SOLVERS OF BIES 402
IMAGE 6
X V ; C O N T E N T S
7 CLUSTER METHODS 403
7.1 THE CLUSTER ALGORITHM 4 0 4
7.1.1 CONDITIONS ON THE INTEGRAL OPERATOR 404
7.1.2 CLUSTER TREE AND ADMISSIBLE COVERING 405
7.1.3 APPROXIMATION OF THE KERNEL FUNCTION 409
7.1.3.1 CEBYSEV INTERPOLATION 410
7.1.3.2 MULTIPOLE EXPANSION 414
7.1.3.3 ABSTRACT CLUSTER APPROXIMATION 415
7.1.4 THE MATRIX-VECTOR MULTIPLICATION IN THE CLUSTER FORMAT 416 7.1.4.1
COMPUTATION THE FAR-FIELD COEFFICIENTS 4 2 0
7.1.4.2 CLUSTER-CLUSTER INTERACTION 421
7.1.4.3 EVALUATING THE CLUSTER APPROXIMATION O F A MATRIX-VECTOR
MULTIPLICATION 421
7.1.4.4 ALGORITHMIC DESCRIPTION O F THE CLUSTER METHOD 4 2 3
7.2 REALIZATION O F THE SUBALGORITHMS 425
7.2.1 ALGORITHMIC REALIZATION O F THE CEBYSEV APPROXIMATION . .425
7.2.2 EXPANSION WITH VARIABLE ORDER 431
7.3 ERROR ANALYSIS FOR THE CLUSTER METHOD 433
7.3.1 LOCAL ERROR ESTIMATES 4 3 3
7.3.1.1 LOCAL ERROR ESTIMATES FOR THE CEBYSEV INTERPOLATION 433
7.3.2 GLOBAL ERROR ESTIMATES 448
7.3.2.1 L 2 -ESTIMATE FOR THE CLUSTERING ERROR WITHOUT INTEGRATION BY
PARTS 449
7.3.2.2 L 2 -ESTIMATES FOR THE CLUSTER METHOD WITH INTEGRATION BY PARTS
452
7.3.2.3 STABILITY AND CONSISTENCY OF THE CLUSTER METHOD 453
7.4 THE COMPLEXITY O F THE CLUSTER METHOD 454
7.4.1 NUMBER O F CLUSTERS AND BLOCKS 455
7.4.2 THE ALGORITHMIC COMPLEXITY O F THE CLUSTER METHOD 460
7.5 CLUSTER METHOD FOR COLLOCATION METHODS 463
7.6 REMARKS AND ADDITIONAL RESULTS 464
8 P -PARAMETRIC SURFACE APPROXIMATION 467
8.1 DISCRETIZATION O F BOUNDARY INTEGRAL EQUATIONS WITH SURFACE
APPROXIMATIONS 467
8.1.1 P-PARAMETRIC SURFACE MESHES FOR GLOBALLY SMOOTH SURFACES 467
8.1.2 (K, /?)-BOUNDARY ELEMENT SPACES WITH P-PARAMETRIC SURFACE
APPROXIMATION 471
IMAGE 7
C O N T E N T S X V I I
8.1.3 DISCRETIZATION O F BOUNDARY INTEGRAL EQUATIONS
WITH /7-PARAMETRIC SURFACE APPROXIMATION 472
8.2 CONVERGENCE ANALYSIS 475
8.3 OVERVIEW O F THE ORDERS O F THE /^-PARAMETRIC SURFACE APPROXIMATIONS
495
8.4 ELEMENTARY DIFFERENTIAL GEOMETRY 498
9 A POSTERIORI ERROR ESTIMATION 517
9.1 PRELIMINARIES 518
9.2 LOCAL ERROR INDICATORS AND A POSTERIORI ERROR ESTIMATORS 521
9.2.1 OPERATORS O F NEGATIVE ORDER 521
9.2.2 OPERATORS O F NON-NEGATIVE ORDER 523
9.3 PROOF OF EFFICIENCY AND RELIABILITY 523
9.3.1 ANALYSIS OF OPERATORS O F NEGATIVE ORDER 524
9.3.2 ANALYSIS OF OPERATORS O F NON-NEGATIVE ORDER 534
9.3.3 BIBLIOGRAPHICAL REMARKS, FURTHER RESULTS AND OPEN PROBLEMS 543
REFERENCES 545
INDEX O F SYMBOLS 555
INDEX 559 |
| any_adam_object | 1 |
| author | Sauter, Stefan 1964- Schwab, Christoph 1962- |
| author_GND | (DE-588)129230006 (DE-588)129230049 |
| author_facet | Sauter, Stefan 1964- Schwab, Christoph 1962- |
| author_role | aut aut |
| author_sort | Sauter, Stefan 1964- |
| author_variant | s s ss c s cs |
| building | Verbundindex |
| bvnumber | BV036866013 |
| classification_rvk | SK 560 SK 640 SK 920 |
| ctrlnum | (OCoLC)729934218 (DE-599)DNB1006507566 |
| dewey-full | 518.64 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis |
| dewey-raw | 518.64 |
| dewey-search | 518.64 |
| dewey-sort | 3518.64 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| genre | (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV036866013 |
| illustrated | Illustrated |
| indexdate | 2025-01-30T09:01:10Z |
| institution | BVB |
| isbn | 9783540680925 |
| language | English German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020781682 |
| oclc_num | 729934218 |
| open_access_boolean | |
| owner | DE-83 DE-29T DE-703 DE-11 |
| owner_facet | DE-83 DE-29T DE-703 DE-11 |
| physical | xvii, 561 Seiten Illustrationen 235 mm x 155 mm |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Springer |
| record_format | marc |
| series | Springer series in computational mathematics |
| series2 | Springer series in computational mathematics |
| spelling | Sauter, Stefan 1964- Verfasser (DE-588)129230006 aut Randelementmethoden (@ 2004) Boundary element methods Stefan A. Sauter, Christoph Schwab Berlin Springer [2011] © 2011 xvii, 561 Seiten Illustrationen 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Springer series in computational mathematics 39 Galerkin-Methode (DE-588)4155831-5 gnd rswk-swf Randelemente-Methode (DE-588)4076508-8 gnd rswk-swf Panel Clustering (DE-588)4788033-8 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Randelemente-Methode (DE-588)4076508-8 s Galerkin-Methode (DE-588)4155831-5 s Panel Clustering (DE-588)4788033-8 s DE-604 Schwab, Christoph 1962- Verfasser (DE-588)129230049 aut Erscheint auch als Online-Ausgabe 978-3-540-68093-2 Springer series in computational mathematics 39 (DE-604)BV000012004 39 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3531998&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020781682&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Sauter, Stefan 1964- Schwab, Christoph 1962- Boundary element methods Springer series in computational mathematics Galerkin-Methode (DE-588)4155831-5 gnd Randelemente-Methode (DE-588)4076508-8 gnd Panel Clustering (DE-588)4788033-8 gnd |
| subject_GND | (DE-588)4155831-5 (DE-588)4076508-8 (DE-588)4788033-8 (DE-588)4123623-3 |
| title | Boundary element methods |
| title_alt | Randelementmethoden (@ 2004) |
| title_auth | Boundary element methods |
| title_exact_search | Boundary element methods |
| title_full | Boundary element methods Stefan A. Sauter, Christoph Schwab |
| title_fullStr | Boundary element methods Stefan A. Sauter, Christoph Schwab |
| title_full_unstemmed | Boundary element methods Stefan A. Sauter, Christoph Schwab |
| title_short | Boundary element methods |
| title_sort | boundary element methods |
| topic | Galerkin-Methode (DE-588)4155831-5 gnd Randelemente-Methode (DE-588)4076508-8 gnd Panel Clustering (DE-588)4788033-8 gnd |
| topic_facet | Galerkin-Methode Randelemente-Methode Panel Clustering Lehrbuch |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3531998&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020781682&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000012004 |
| work_keys_str_mv | AT sauterstefan randelementmethoden2004 AT schwabchristoph randelementmethoden2004 AT sauterstefan boundaryelementmethods AT schwabchristoph boundaryelementmethods |