基本解方法最早由V.D. Kupradze 在文章Potential methods in elasticity J.N.Sneddon 和 R.Hill (Eds), Progress in Solid Mechanics, Vol.III, Amsterdam, pp.1-259, 1963 中提出。自 1963 年开始，出现大量基本解方法的计算，但鲜有对基本解方法的分析。本书中，给出基本解方法的数值算法、特点，主要着力于建立其误差和稳定性的理论分析。 本书中的严格分析（以及源节点的选择）为MFS提供了坚实的理论基础，使其成为偏微分方程（PDE）的有效且称职的数值方法。内容源于作者已经发表的论文，本书介绍了MFS的基本和重要要素。
样章试读
目录
- Contents
Preface
Acknowledgements
CHAPTER1 Introduction 1
1.1 Historic Review 1
1.2 Basic Algorithms 3
1.3 Numerical Experiments 5
1.4 Characteristics of the MFS 11
Part I Laplace’s Equation 15
CHAPTER2 Dirichlet Problems 19
2.1 Basic Algorithms of MFS 19
2.2 Preliminary Lemmas 21
2.3 Main Theorems 27
2.4 Stability Analysis for DiskDomains 32
2.5 Proof Methodology 39
CHAPTER3 Neumann Problems 41
3.1 Introduction 41
3.2 Method of Fundamental Solutions 42
3.2.1 Description of Algorithms 42
3.2.2 Main Results of Analysisand Their Applications 44
3.3 Stability Analysis of DiskDomains 45
3.4 Stability Analysis for BoundedSimply-Connected Domains 49
3.4.1 Trefftz Methods 50
3.4.2 Collocation Trefftz Methods 52
3.5 Error Estimates 54
3.6 Concluding Remarks 58
CHAPTER4 Other Boundary Problems 61
4.1 Mixed Boundary Condition Problems 61
4.2 Interior Boundary Conditions 66
4.3 Annular Domains 70
CHAPTER5 Combined Methods 77
5.1 Combined Methods 77
5.2 Variant Combinations of FS and PS 79
5.2.1 Simplified Hybrid Combination 79
5.2.2 Hybrid Plus Penalty Combination 81
5.2.3 Indirect Combination 84
5.3 Combinations of MFS with Other Domain Methods 86
5.3.1 Combined with FEM 86
5.3.2 Combined with FDM 87
5.3.3 Combined with Radial Basis Functions 90
5.4 Singularity Problems by Combination of MFS and MPS 91
CHAPTER 6 Source Nodes on Elliptic Pseudo-Boundaries 99
6.1 Introduction 99
6.2 Algorithms of MFS 101
6.3 Error Analysis 103
6.3.1 Preliminary Lemmas 103
6.3.2 Error Bounds 107
6.4 Stability Analysis 113
6.5 Selection of Pseudo-Boundaries 119
6.6 Numerical Experiments 121
6.7 Concluding Remarks 124
Part II. Helmholtz’s Equations and Other Equations 125
CHAPTER7 Helmholtz Equationsin Simply-Connected Domains 127
7.1 Introduction 127
7.2 Algorithms 128
7.3 Error Analysis for Bessel Functions 131
7.3.1 Preliminary Lemmas 131
7.3.2 Error Bounds with Small k 134
7.3.3 Exploration of Bounded k 140
7.4 Stability Analysis for Disk Domains 146
7.5 Application to BKM 149
CHAPTER8 Exterior Problems of Helmholtz Equation 155
8.1 Introduction 155
8.2 Standard MFS 157
8.2.1 Basic Algorithms 157
8.2.2 Brief Error Analysis 159
8.3 Numerical Characteristics of Spurious Eigenvalues by MFS 161
8.4 Modified MFS 165
8.5 Error Analysis for Modified MFS 166
8.5.1 Preliminary Lemmas 167
8.5.2 Error Bounds 175
8.6 Stability Analysis for Modified MFS 179
8.7 Numerical Experiments 181
8.7.1 Circular Pseudo-Boundaries by Two MFS 181
8.7.2 Non-Circular Pseudo-Boundaries by Modified MFS 186
8.8 Concluding Remarks 188
CHAPTER9 Helmholtz Equations in Bounded Multiply-Connected Domains 191
9.1 Introduction 191
9.2 Bounded Simply-Connected Domains 192
9.2.1 Algorithms 192
9.2.2 Brief Error Analysis 193
9.3 Bounded Multiply-Connected Domains 197
9.3.1 Algorithms 197
9.3.2 ErrorAnalysis 198
9.4 Stability Analysis for Ring Domains 201
9.5 Numerical Experiments 210
9.6 Concluding Remarks 214
CHAPTER10 Biharmonic Equations 215
10.1 Introduction 215
10.2 Preliminary Lemmas 217
10.3 Error Bounds 224
10.4 Stability Analysis for Circular Domains 228
10.4.1 Approaches for Seeking Eigenvalues 228
10.4.2 Eigenvalues λk(Φ) and λk(DΦ) 231
10.4.3 Bounds of Condition Number 236
10.5 Numerical Experiments 242
CHAPTER11 Elastic Problems 247
11.1 Introduction 247
11.2 Linear Elastostatics Problemsin2D 247
11.2.1 Basic Theory 247
11.2.2 Traction Boundary Conditions 249
11.2.3 Fundamental Solutions 250
11.2.4 Particular Solutions 251
11.3 HTM,MFS and MPS 252
11.3.1 Algorithms of HTM 252
11.3.2 Algorithms of MFS and MPS 252
11.4 Errors Between FS and PS 254
11.4.1 Preliminary Lemmas 254
11.4.2 Polynomials Pn Approximated by *and * 257
11.4.3 Other Proof for Theorem11.4.1 258
11.4.4 The Polynomials LPn Approximated by Principal FS 261
11.5 Error Bounds for MFS and HTM 264
11.5.1 The MFS 264
11.5.2 The HTM Using FS 266
11.6 Numerical Experiments 268
11.7 Appendix:Addition Theorems of FS in Linear Elastostatics 271
11.7.1 Preliminary Lemmas 271
11.7.2 Addition Theorems 277
CHAPTER12 Cauchy Problems 281
12.1 Introduction 281
12.2 Algorithms of Collocation Trefftz Methods 281
12.3 Characteristics 284
12.3.1 Existence and Uniqueness 284
12.3.2 Ill-Posedness of Inverse Problems 287
12.4 Error and Stability Analysis 290
12.4.1 Error Analysis 290
12.4.2 Stability Analysis 291
12.5 Applications to Cauchy Data 295
12.5.1 Errors on Cauchy Boundary 295
12.5.2 Sensitivity of Solutionson Cauchy Data 296
12.6 Numerical Experiments and ConcludingRemarks 297
CHAPTER13 3D Problems 301
13.1 Introduction 301
13.2 Method of Particular Solutions 302
13.3 Method of Fundamental Solutions 309
13.3.1 Algorithms 309
13.3.2 LinktoMPS 310
13.4 Error Analysisfor MFS 313
13.4.1 Preliminary Lemmas 314
13.4.2 ErrorBounds 321
13.5 Numerical Experiments 324
13.5.1 Collocation Equations on Γ 324
13.5.2 By MFS 325
13.5.3 By MPS 330
13.6 Concluding Remarks 331
13.7 Appendix: 3D Problems of Helmholtz Equations 332
13.7.1 Interior Dirichlet Problems 332
13.7.2 Exterior Dirichlet Problems 333
Part III. Selection of Source Nodes and Related Topics 335
CHAPTER 14 Comparisons of MFSandMPS 339
14.1 Introduction 339
14.2 TwoBasis Boundary Methods 340
14.2.1 Method of Particular Solutions 340
14.2.2 Method of Fundamental Solutions 342
14.3 The MFS-QR 346
14.3.1 Algorithms in Elliptic Coordinates 346
14.3.2 Characteristics of MFS-QR 349
14.4 Numerical Experiments and Comparisons 354
14.4.1 Highly Smooth Boundary Data 355
14.4.2 Boundary Data with Strong Singularity 356
14.4.3 Better Pseudo-Boundaries 358
14.5 Concluding Remarks 360
CHAPTER 15 Stability Analysis for Smooth Closed Pseudo-Boundaries 361
15.1 Introduction 361
15.2 Relations Between FS and PS 362
15.3 Bounds of Cond for Non-Elliptic Pseudo-Boundaries 365
CHAPTER 16 Singularity Problems from Source Functions; Removal Techniques 375
16.1 Introduction 376
16.2 Analytical Framework for CTM in[169] 378
16.3 Error Bounds for Singular Solutions from (16.1.3) 380
16.4 Singularity for Polygonal Domains and Arbitrary Domains 383
16.5 Removal Techniques for Laplace’s Equation 384
16.5.1 For the Case of Q OutsideΓ 384
16.5.2 For the Case of Q InsideΓ under the Image Node Existing 386
16.6 Numerical Experiments 388
16.7 Applications to Amoeba-Like Domains 390
16.7.1 Numerical Results 390
16.7.2 Removal Techniques Linked to Source Identification Problems 394
16.8 Concluding Remarks 399
CHAPTER 17 Source Nodeson Pseudo Radial-Lines 401
17.1 Introduction 401
17.2 Pseudo Radial-Lines 404
17.2.1 One Pseudo Radial-Line 404
17.2.2 Two Pseudo Radial-Lines 408
17.3 Stability Analysis 409
17.3.1 Lower Bound Estimates of Cond for Basic Case 409
17.3.2 Upper Bound Estimates of Cond for Variant Case by CaseII 412
17.4 Numerical Experiments 415
17.4.1 Disk Domains 415
17.4.2 Non-Disk Domains 420
17.5 Concluding Remarks 424
Epilogue 427
References 431
Glossary of Symbols 443
Index 449
Book list of the Series in Information and Computational Science 455
京公网安备 11010102004214号