Preface to the Second Edition
Preface
Acknowledgements
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 posteriori 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 The GCTM Using Piecewise Particular Solutions
2.7 Stability Analysis of the 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 The CTM using particular solutions for Motz's problem
2.10.2 The MFS and the 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 approximation
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 The Method of Fundamental Solutions for Mixed Boundary Value Problems of Laplace's Equation
6.1 Introduction
6.2 Method of Fundamental Solutions
6.3 Dirichlet Problems on Disk Domains
6.3.1 Eigenvalues of the MFS
6.3.2 New approaches
6.3.3 Eigenvalues in terms of power series
6.3.4 Asymptotes of Cond
6.4 Neumann Problems in Disk Domains
6.4.1 Description of algorithms
6.4.2 Condition numbers of the MFS
6.5 Mixed Boundary Problems in Bounded Simply-Connected Domains
6.5.1 Trefftz methods
6.5.2 The collocation Trefftz methods
6.5.3 Bounds of condition numbers and effective condition numbers
6.5.4 Developments and evaluations on the MFS
6.5.5 The inverse inequality (6.5.9)
6.6 Numerical Experiments
Chapter 7 Finite Difference Method
7.1 Introduction
7.2 Shortley-Weller Difference Approximation
7.2.1 A Lemma
7.2.2 Bounds for Cond_EE
7.2.3 Bounds for Cond_eff
Chapter 8 Boundary Penalty Techniques of FDM
8.1 Introduction
8.2 Finite Difference Method
8.2.1 Shortley-Weller difference approximation
8.2.2 Superconvergence of solution derivatives
8.2.3 Bounds for Cond_eff
8.3 Penalty-Integral Techniques
8.4 Penalty-Collocation Techniques
8.5 Relations Between Penalty-Integral and Penalty- Collocation Techniques
8.6 Concluding Remarks
Chapter 9 Boundary Singularly Problems by FDM
9.1 Introduction
9.2 Finite Difference Method
9.3 Local Refinements of Difference Grids
9.3.1 Basic results
9.3.2 Nonhomogeneous Dirichlet and Neumann boundary conditions ..
9.3.3 A remark
9.3.4 A view on assumptions A1-A4
9.3.5 Discussions and comparisons
9.4 Numerical Experiments
9.5 Concluding Remarks
Chapter 10 Singularly Perturbed Differential Equations by the Upwind Difference Scheme
10.1 Introduction
10.2 The Upwind Difference Scheme
10.3 Properties of the Operator of SPDE and its Discretization
10.4 Stability Analysis
10.4.1 The traditional condition number
10.4.2 Effective condition number
10.4.3 Via the maximum principle
10.5 Numerical Experiments and Concluding Remarks
Chapter 11 Finite Element Method Using Local Mesh Refinements
11.1 Introduction
11.2 Optimal Convergence Rates
11.3 Homogeneous Boundary Conditions
11.4 Nonhomogeneous Boundary Conditions
11.5 Intrinsic View of Assumption A2 and Improvements of Theorem 11.4.1
11.5.1 Intrinsic view of assumption A2
11.5.2 Improvements of Theorem 11.4.1
11.6 Numerical Experiments
Chapter 12 Hermite FEM for Biharmonic Equations
12.1 Introduction
12.2 Description of Numerical Methods
12.3 Stability Analysis
12.3.1 Bounds of Cond
12.3.2 Bounds of Cond_eff
12.4 Numerical Experiments
Chapter 13 Truncated SVD and Tikhonov Regularization
13.1 Introduction
13.2 Algorithms of Regularization
13.3 New Estimates of Cond and Cond_eff
13.4 Brief Error Analysis
Chapter 14 Small Sample Statistical Condition Estimation for the Generalized Sylvester Equation
14.1 Introduction
14.2 Effective Condition Numbers
14.3 Small Sample Statistical Condition Estimation
14.3.1 Normwise perturbation analysis
14.3.2 Mixed and componentwise perturbation analysis
14.4 Numerical Examples
14.5 Concluding Remarks
Appendix A Definitions and Formulas
A.1 Square Systems
A.I.1 Symmetric and positive 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
A.5 Regularization
A.5.1 The Truncated singular value decomposition
A.5.2 The Tikhonov regularization
A.6 p-Norms
A.7 Conclusions
Epilogue
Bibliography
Index
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