●Preface
Acknowledgments
Chapter 1 Effective Condition Number
1.1 Introduction
1.2 Preliminary
1.3 Symmetric Matrices
1.3.1 Definitions of effective condition numbers.
1.3.2 A teriori computation
1.4 Overdetermined Systems
1.4.1 Basic algorithms
1.4.2 Refinements of （1.4.10）
1.4.3 Criteria
1.4.4 Advanced refinements
1.4.5 Effective condition number in p-norms
1.5 Linear Algebraic Equations by GE or QR.
1.6 Application to Numerical PDE
1.7 Application to Boundary Integral Equations
1.8 Weighted Linear Least Squares Problems
1.8.1 Effective condition number
1.8.2 Perturbation bounds
1.8.3 Applications and comparisons
Chapter 2 Collocation Trefftz Methods
2.1 Introduction
2.2 CTM for Motz's Problem
2.3 Bounds of Effective Condition Number
2.4 Stability for CTM of Rp = 1
2.5 Numerical Experiments
2.5.1 Choice of Rp
2.5.2 Extreme accuracy of Do
2.6 GCTM Using Piecewise Particular Solutions
2.7 Stability Analysis of GCTM
2.7.1 Trefftz methods
2.7.2 Collocation Trefftz methods
2.8 Method of Fundamental Solutions
2.9 Collocation Methods Using RBF
2.10 Comparisons Between Cond eff and Cond
2.10.1 CTsing particular solutions for Motz's problem
2.10.2 MFS and CM-RBF
2.11 A Few Remarks
Chapter 3 Simplified Hybrid Trefftz Methods
3.1 The Simplified Hybrid TM
3.1.1 Algorithms
3.1.2 Error analysis
3.1.3 Integration appromation
3.2 Stability Analysis for Simplified Hybrid TM
Chapter 4 Penalty Trefftz Method Coupled with FEM
4.1 Introduction
4.2 Combinations of TM and Adini~s Elements
4.2.1 Algorithms
4.2.2 Basic theorem
4.2.3 Global superconvergence
4.3 Bounds of Cond_eff for Motz~s Problem
4.4 Effective Condition Number of One and Infinity Norms
4.5 Concluding Remarks
Chapter 5 Trefftz Methods for Biharmonic Equations with Crack Singularities
5.1 Introduction
5.2 Collocation Trefftz Methods
5.2.1 Three crack models
5.2.2 Description of the method
5.2.3 Error bounds
5.3 Stability Analysis
5.3.1 Upper bound for σmax（F）
5.3.2 Lower bound for σmin（F）
5.3.3 Upper bound for Cond_eff and Cond
5.4 Proofs of Important Results Used in Section 5.3
5.4.1 Basic theorem
5.4.2 Proof of Lemma 5.4.3
5.4.3 Proof of Lemma 5.4.4
5.5 Numerical Experiments
5.6 Concluding Remarks
Chapter 6 Finite Difference Method
6.1 Introduction
6.2 Shortley-Weller Difference Appromation
6.2.1 A Lemma
6.2.2 Bounds for Cond~E
6.2.3 Bounds for Cond_eff
Chapter 7 Boundary Penalty Techniques of FDM
7.1 Introduction
7.2 Finite Difference Method
7.2.1 Shortley-Weller difference appromation
7.2.2 Superconvergence of solution derivatives
7.2.3 Bounds for Cond_eff
7.3 Penalty-Integral Techniques
7.4 Penalty-Collocation Techniques
7.5 Relations Between Penalty-Integral and Penalty-Collocation Techniques
7.6 Concluding Remarks
Chapter 8 Boundary Singularly Problems by FDM
8.1 Introduction
8.2 Finite Difference Method
8.3 Local Refinements of Difference Grids
8.3.1 Basic results
8.3.2 Nonhomogeneous Dirichlet and Neumann boundary conditions
8.3.3 A remark
8.3.4 A view on assumptions A1-A4
8.3.5 Discussions and comparisons
8.4 Numerical Experiments
8.5 Concluding Remarks
Chapter 9 Finite Element Method Using Local Mesh Refinements
9.1 Introduction
9.2 Optimal Convergence Rates
9.3 Homogeneous Boundary Conditions
9.4 Nonhomogeneous Boundary Conditions
9.5 Intrinsic View of Assumption A2 and Improvements of Theorem
9.5.1 Intrinsic view of assumption A2
9.5.2 Improvements of Theorem 9.4.1
9.6 Numerical Experiments
Chapter 10 Hermite FEM for Biharmonic Equations
10.1 Introduction
10.2 Description of Numerical Methods
10.3 Stability Analysis
10.3.1 Bounds of Cond
10.3.2 Bounds of Cond_eff
10.4 Numerical Experiments
Chapter 11 Truncated SVD and Tikhonov Regularization.
11.1 Introduction
11.2 Algorithms of Regularization
11.3 New Estimates of Cond and Cond_eff
11.4 Brief Error Analysis
Appendix Definitions and Forlas
A.1 Square Systems
A.I.1 Symmetric and itive definite matrices
A.1.2 Symmetric and nonsingular matrices
A.1.3 Nonsingular matrices
A.2 Overdetermined Systems
A.3 Underdetermined Systems
A.4 Method of Fundamental Solutions
Regularization
.1 Truncated singular value decomition
.2 Tikhonov regularization
A.6 p-Norms
A.7 Conclusions
Epilogue
Bibliography
Index
