Contents
Preface
Acknowledgements
CHAPTER 1 Introduction
1.1 Historic Review
1.2 Basic Algorithms
1.3 Numerical Experiments
1.4 Characteristics of the MFS
Part Ⅰ Laplace's Equation
CHAPTER 2 Dirichlet Problems
2.1 Basic Algorithms of MFS
2.2 Preliminary Lemmas
2.3 Main Theorems
2.4 Stability Analysis for DiskDomains
2.5 Proof Methodology
CHAPTER 3 Neumann Problems
3.1 Introduction
3.2 Method of Fundamental Solutions
3.2.1 Description of Algorithms
3.2.2 Main Results of Analysisand Their Applications
3.3 Stability Analysis of DiskDomains
3.4 Stability Analysis for BoundedSimply-Connected Domains
3.4.1 Trefftz Methods
3.4.2 Collocation Trefftz Methods
3.5 Error Estimates
3.6 Concluding Remarks
CHAPTER 4 Other Boundary Problems
4.1 Mixed Boundary Condition Problems
4.2 Interior Boundary Conditions
4.3 Annular Domains
CHAPTER 5 Combined Methods
5.1 Combined Methods
5.2 Variant Combinations of FS and PS
5.2.1 Simplified Hybrid Combination
5.2.2 Hybrid Plus Penalty Combination
5.2.3 Indirect Combination
5.3 Combinations of MFS with Other Domain Methods
5.3.1 Combined with FEM
5.3.2 Combined with FDM
5.3.3 Combined with Radial Basis Functions
5.4 Singularity Problems by Combination of MFS and MPS
CHAPTER 6 Source Nodes on Elliptic Pseudo-Boundaries
6.1 Introduction
6.2 Algorithms of MFS
6.3 Error Analysis
6.3.1 Preliminary Lemmas
6.3.2 Error Bounds
6.4 Stability Analysis
6.5 Selection of Pseudo-Boundaries
6.6 Numerical Experiments
6.7 Concluding Remarks
Part Ⅱ.Helmholtz's Equations and Other Equations
CHAPTER 7 Helmholtz Equationsin Simply-Connected Domains
7.1 Introduction
7.2 Algorithms
7.3 Error Analysis for Bessel Functions
7.3.1 Preliminary Lemmas
7.3.2 Error Bounds with Small k
7.3.3 Exploration of Bounded k
7.4 Stability Analysis for Disk Domains
7.5 Application to BKM
CHAPTER 8 Exterior Problems of Helmholtz Equation
8.1 Introduction
8.2 Standard MFS
8.2.1 Basic Algorithms
8.2.2 Brief Error Analysis
8.3 Numerical Characteristics of Spurious Eigenvalues by MFS
8.4 Modified MFS
8.5 Error Analysis for Modified MFS
8.5.1 Preliminary Lemmas
8.5.2 Error Bounds
8.6 Stability Analysis for Modified MFS
8.7 Numerical Experiments
8.7.1 Circular Pseudo-Boundaries by Two MFS
8.7.2 Non-Circular Pseudo-Boundaries by Modified MFS
8.8 Concluding Remarks
CHAPTER 9 Helmholtz Equations in Bounded Multiply-Connected Domains
9.1 Introduction
9.2 Bounded Simply-Connected Domains
9.2.1 Algorithms
9.2.2 Brief Error Analysis
9.3 Bounded Multiply-Connected Domains
9.3.1 Algorithms
9.3.2 ErrorAnalysis
9.4 Stability Analysis for Ring Domains
9.5 Numerical Experiments
9.6 Concluding Remarks
CHAPTER 10 Biharmonic Equations
10.1 Introduction
10.2 Preliminary Lemmas
10.3 Error Bounds
10.4 Stability Analysis for Circular Domains
10.4.1 Approaches for Seeking Eigenvalues
10.4.2 Eigenvalues λk(Φ) and λk(DΦ)
10.4.3 Bounds of Condition Number
10.5 Numerical Experiments
CHAPTER 11 Elastic Problems
11.1 Introduction
11.2 Linear Elastostatics Problemsin2D
11.2.1 Basic Theory
11.2.2 Traction Boundary Conditions
11.2.3 Fundamental Solutions
11.2.4 Particular Solutions
11.3 HTM,MFS and MPS
11.3.1 Algorithms of HTM
11.3.2 Algorithms of MFS and MPS
11.4 Errors Between FS and PS
11.4.1 Preliminary Lemmas
11.4.2 Polynomials Pn Approximated by *and *
11.4.3 Other Proof for Theorem 11.4.1
11.4.4