基本解方法的理論與應用 (英文版) (精) 9787030749161 李子才 黃宏財 魏益民 張理評著

目錄

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 <
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精彩書摘
Chapter 1 Introduction
Everything is simpler than you think
and al the 8ame time more complex than you imagine
Johann Wolfgang von Goethe， 1749-1832
This introductory chapter is organized as follows First， the history of the method of fundamental solutions (MFS) is