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出版社:科學出版社 ISBN:9787030464101 商品編碼:10109250057 品牌:文軒 出版時間:2016-01-01 代碼:138 作者:李子纔,黃宏財,魏益民,程宏達
" 作 者:李子纔,黃宏財,魏益民,程宏達 著 定 價:138 出 版 社:科學出版社 出版日期:2016年01月01日 頁 數:351 裝 幀:精裝 ISBN:9787030464101 ●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 Chapter 14 Small Sample Statistical Condition Estimation for the Generalized Sylvester Equation Appendix A Definitions and Formulas Epilogue Bibliography Index 本書主要介紹偏微分方程數值解的有效條件數。首先介紹有效條件數的概念,與經典條件數概念的差異,接著將有效條件數運用於TREFFTZ方法;我們還討論了有限差分方法的有效條件數,最後研究了截斷奇異值分解和TIKHONOV正則化的有效條件數。第二版擬增加三章:Laplace方程混合邊界值問題基本解的穩定性分析;奇攝動微分方程迎風差分格式的穩定性分析;廣義Sylvester方程的有效條件數。
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