Contents Chapter 1 E.ective Condition Number1 1.1 Introduction1 1.2 Preliminary3 1.3 Symmetric Matrices5 1.3.1 De-nitions of e.ective condition numbers6 1.3.2 A posteriori computation8 1.4 Overdetermined Systems10 1.4.1 Basic algorithms10 1.4.2 Re-nements of (1.4.10)14 1.4.3 Criteria16 1.4.4 Advanced re-nements17 1.4.5 E.ective condition number in p-norms19 1.5 Linear Algebraic Equations by GE or QR21 1.6 Application to Numerical PDE23 1.7 Application to Boundary Integral Equations33 1.8 Weighted Linear Least Squares Problems40 1.8.1 E.ective condition number41 1.8.2 Perturbation bounds44 1.8.3 Applications and comparisons45 Chapter 2 Collocation Tre.tz Methods 47 2.1 Introduction47 2.2 CTM for Motz's Problem48 2.3 Bounds of E.ective Condition Number51 2.4 Stability for CTM of Rp = 1 56 2.5 Numerical Experiments57 2.5.1 Choice of Rp 57 2.5.2 Extreme accuracy of D0 60 2.6 The GCTM Using Piecewise Particular Solutions60 2.7 Stability Analysis of the GCTM64 2.7.1 Tre.tz methods64 2.7.2 Collocation Tre.tz methods66 2.8 Method of Fundamental Solutions67 2.9 Collocation Methods Using RBF71 2.10 Comparisons Between Cond e. and Cond73 2.10.1 The CTM using particular solutions for Motz's problem73 2.10.2 The MFS and the CM-RBF74 2.11 A Few Remarks74 Chapter 3 Simpli-ed Hybrid Tre.tz Methods6 3.1 The Simpli-ed Hybrid TM76 3.1.1 Algorithms76 3.1.2 Error analysis80 3.1.3 Integration approximation80 3.2 Stability Analysis for Simpli-ed Hybrid TM81 Chapter 4 Penalty Tre.tz Method Coupled with FEM88 4.1 Introduction88 4.2 Combinations of TM and Adini's Elements90 4.2.1 Algorithms90 4.2.2 Basic theorem93 4.2.3 Global superconvergence95 4.3 Bounds of Cond e. for Motz's Problem99 4.4 E.ective Condition Number of One and In-nity Norms105 4.5 Concluding Remarks107 Chapter 5 Tre.tz Methods for Biharmonic Equations with Crack Singularities109 5.1 Introduction109 5.2 Collocation Tre.tz Methods110 5.2.1 Three crack models110 5.2.2 Description of the method112 5.2.3 Error bounds113 5.3 Stability Analysis114 5.3.1 Upper bound for .max(F)114 5.3.2 Lower bound for .min(F)115 5.3.3 Upper bound for Cond e. and Cond118 5.4 Proofs of Important Results Used in Section 5.3 119 5.4.1 Basic theorem119 5.4.2 Proof of Lemma 5.4.3123 5.4.3 Proof of Lemma 5.4.4 125 5.5 Numerical Experiments130 5.6 Concluding Remarks133 Chapter 6 The Method of Fundamental Solutions for Mixed Boundary Value Problems of Laplace's Equation135 6.1 Introduction135 6.2 Method of Fundamental Solutions137 6.3 Dirichlet Problems on Disk Domains140 6.3.1 Eigenvalues of the MFS140 6.3.2 New approaches142 6.3.3 Eigenvalues in terms of power series144 6.3.4 Asymptotes of Cond147 6.4 Neumann Problems in Disk Domains148 6.4.1 Description of algorithms148 6.4.2 Condition numbers of the MFS151 6.5 Mixed Boundary Problems in Bounded Simply-Connected Domains155 6.5.1 Tre.tz methods155 6.5.2 The collocation Tre.tz methods159 6.5.3 Bounds of condition numbers and e.ective condition numbers161 6.5.4 Developments and evaluations on the MFS162 6.5.5 The inverse inequality (6.5.9) 163 6.6 Numerical Experiments166 Chapter 7 Finite Di.erence Method171 7.1 Introduction171 7.2 Shortley-Weller Di.erence Approximation171 7.2.1 A Lemma173 7.2.2 Bounds for Cond EE175 7.2.3 Bounds for Cond e179 Chapter 8 Boundary Penalty Techniques of FDM184 8.1 Introduction184 8.2 Finite Di.erence Method185 8.2.1 Shortley-Weller di.erence approximation186 8.2.2 Superconvergence of solution derivatives186 8.2.3 Bounds for Cond e188 8.3 Penalty-Integral Techniques188 8.4 Penalty-Collocation Techniques194 8.5 Relations Between Penalty-Integral and Penalty- Collocation Techniques200 8.6 Concluding Remarks200 Chapter 9 Boundary Singularly Problems by FDM202 9.1 Introduction202 9.2 Finite Di.erence Method203 9.3 Local Re-nements of Di.erence Grids204 9.3.1 Basic results205 9.3.2 Nonhomogeneous Dirichlet and Neumann boundary conditions211 9.3.3 A remark 214 9.3.4 A view on assumptions A1.A4 216 9.3.5 Discussions and comparisons217 9.4 Numerical Experiments 217 9.5 Concluding Remarks224 Chapter 10 Singularly Perturbed Di.erential Equations by the Upwind Di.erence Scheme226 10.1 Introduction226 10.2 The Upwind Di.erence Scheme227 10.3 Properties of the Operator of SPDE and its Discretization229 10.4 Stability Analysis231 10.4.1 The traditional condition number 232 10.4.2 E.ective condition number234 10.4.3 Via the maximum principle236 10.5 Numerical Experiments and Concluding Remarks238 Chapter 11 Finite Element Method Using Local Mesh Re-nements243 11.1 Introduction 243 11.2 Optimal Convergence Rates244 11.3 Homogeneous Boundary Conditions250 11.4 Nonhomogeneous Boundary Conditions255 11.5 Intrinsic View of Assumption A2 and Improvements of Theorem 11.4.1 259 11.5.1 Intrinsic view of assumption A2 259 11.5.2 Improvements of Theorem 11.4.1 260 11.6 Numerical Experiments262 Chapter 12 Hermite FEM for Biharmonic Equations267 12.1 Introduction267 12.2 Description of Numerical Methods268 12.3 Stability Analysis270 12.3.1 Bounds of Cond 270 12.3.2 Bounds of Cond e271 12.4 Numerical Experiments274 Chapter 13 Truncated SVD and Tikhonov Regularization280 13.1 Introduction280 13.2 Algorithms of Regularization283 13.3 New Estimates of Cond and Cond e 284 13.4 Brief Error Analysis 290 Chapter 14 Small Sample Statistical Condition Estimation for the Generalized Sylvester Equation295 14.1 Introduction295 14.2 E.ective Condition Numbers299 14.3 Small Sample Statistical Condition Estimation303 14.3.1 Normwise perturbation analysis305 14.3.2 Mixed and componentwise perturbation analysis307 14.4 Numerical Examples308 14.5 Concluding Remarks315 Appendix A De-nitions and Formulas316 A.1 Square Systems316 A.1.1 Symmetric and positive de-nite matrices317 A.1.2 Symmetric and nonsingular matrices319 A.1.3 Nonsingular matrices319 A.2 Overdetermined Systems320 A.3 Underdetermined Systems321 A.4 Method of Fundamental Solutions322 A.5 Regularization323 A.5.1 The Truncated singular value decomposition324 A.5.2 The Tikhonov regularization324 A.6 p-Norms325 A.7 Conclusions326 Epilogue 327 Bibliography329 Index345
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