Table of 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 <