Preface Acknowledgments Chapter 1 Effective Condition Number 1.1 Introduction 1.2 Preliminary 1.3 Symmetric Matrices 1.3.1 Deffnitions of Effective condition numbers 1.3.2 A posteriori computation 1.4 Overdetermined Systems 1.4.1 Basic algorithms 1.4.2 Reffnements of (1.4.10) 1.4.3 Criteria 1.4.4 Advanced reffnements 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 R_p=1 2.5 Numerical Experiments 2.5.1 Choice of R_p 2.5.2 Extreme accuracy of D_0 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 CTM using particular solutions for Motz’s problem 2.10.2 MFS and CM-RBF 2.11 A Few Remarks Chapter 3 Simpliffed Hybrid Trefftz Methods 3.1 The Simpliffed Hybrid TM 3.1.1 Algorithms 3.1.2 Error analysis 3.1.3 Integration approximation 3.2 Stability Analysis for Simpliffed Hybrid TM Chapter 4 Penalty Trefftz Method Coupled with FEM 4.1 Introduction 4.2 Combinations of TM and Adini0s 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 Inffnity 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 σ_max(F) 5.3.2 Lower bound for σ_min(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 Approximation 6.2.1 A Lemma 6.2.2 Bounds for Cond EE 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 approximation 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 Reffnements 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 9.4.1 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 Deffnitions and Formulas A.1 Square Systems A.1.1 Symmetric and positive deffnite 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 Truncated singular value decomposition A.5.2 Tikhonov regularization A.6 p-Norms A.7 Conclusions Epilogue Bibliography Index
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