偏微分方程数值解的有效条件数

章节摘录

Chapter 1E?ective Condition NumberIn this beginning chapter,new computational formulas are provided for the exective condition number Cond e?,and new error bounds involved in both Cond and Cond e? are derived. A theoretical analysis is provided to support some conclusions in Banoczi et al. [14]. For the linear algebraic equations solved by Gaussian elimination (GE) or the QR factorization (QR),the direction of the right-hand vector is insigniˉcant for the solution errors,but such a conclusion is invalid for the ˉnite di- ?erence method for Poisson0s equation. The e?ective condition number is important to the numerical partial di?erential equations,because the discretization errors are dominant. The materials of this chapter are adapted from [95,127,134,138,218].1.1 IntroductionThe deˉnition of the traditional condition number was given in Wilkinson [227], and then used in many books and papers. To solve the overdetermined system of the linear algebraic equations Fx = b,the traditional condition number in the 2-norm is deˉned by Cond = ?1=?n,where ?1 and ?n are the maximal and the minimal singular values of matrix F,respectively. The condition number is used to provide the bounds of the solution errors from the perturbation of both matrix F and vector b. The new computational formulas are derived in this chapter for the e?ective condition numbers,denoted by Cond e?,Cond E and Cond EE. The Cond e? denotes the enlarged factor of the solution errors over the residual errors. For further application,in this chapter we also derive the new error bound involved in both Cond and Cond e?. In this new bound,the Cond and the Cond e? denote basically the enlarged factors of the errors over the perturbation errors of matrix F and vector b,respectively. Hence,when the matrix and the vector errors are dominant,Cond and Cond e? will play an important role,respectively.First,we apply Cond e? to the linear algebraic equations,and a theoretical jus- ti- cation is given to support some conclusions in [14]. For the solution errors of the algebraic equations by Gaussian elimination (GE) or the QR factorization (QR),the direction of the right-hand vector is insigniˉcant. Then we apply Cond e? to the numerical partial di?erential equations (PDE) and the Tre?tz method (TM) for Poisson0s equation. Since the discretization errors are dominant,the e?ective condi- tion number is important. Moreover,since the TM solution is highly accurate,the small e?ective condition number well explains the high accuracy of the solutions, and strengthens the collocation Tre?tz method (CTM) in [155],where only error bounds and numerical experiments are provided without the stability analysis. It is explored in this chapter and the entire book that the huge Cond is often mislead-ing,but the e?ective condition number is the appropriate criterion for the stabilityanalysis.Here let us illustrate the references of the condition number and the e?ective con-dition number. The traditional condition number Cond was ˉrst given in Wilkinson [227],and then used in many textbooks,such as Strang [197],Atkinson [3],Schwarz [191],Datta [44],Gill et al. [65],Golub and van Loan [66],Stewart [195],Horn and Johnson [90],Sun [199],Wang et al. [212],and Higham [88]. The condition num-ber for eigenvalues is introduced in Parlett [180] and Frayss?e and Toumazou [61], and there are more discussions in Gulliksson and Wedin [71],and Elsner et al. [55]. The Cond is applied to the numerical partial di?erential equations in Strang and Fix [198],Quarteroni and Valli [184] and Li [115]. On the other hand,the e?ective condition number was deˉned and studied in Chan and Foulser [27]. However,its algorithm was ˉrst proposed in Rice [186] in 1981,but the natural condition number was called. Only a few reports,such as Christiansen and Hansen [37],Christiansen and Saranen [38],Banoczi et al. [14],and Axelsson and Kaporin [7,8],follow the line of the e?ective condition number. The error estimates for the solution of the linearal gebraic system in Brezinski [20] are also related to the e?ective condition number.Recently,the new computational formulas of e?ective condition number have been studied and applied to symmetric and positive deˉnite matrices,which are obtained from Poisson0s equation by the ˉnite di?erence method (FDM) in [127].

书籍目录

PrefaceAcknowledgmentsChapter 1 Effective Condition Number1.1 Introduction1.2 Preliminary1.3 Symmetric Matrices1.3.1 Deffnitions of Effective condition numbers1.3.2 A posteriori computation1.4 Overdetermined Systems1.4.1 Basic algorithms1.4.2 Reffnements of (1.4.10)1.4.3 Criteria1.4.4 Advanced reffnements1.4.5 Effective condition number in p-norms1.5 Linear Algebraic Equations by GE or QR1.6 Application to Numerical PDE1.7 Application to Boundary Integral Equations1.8 Weighted Linear Least Squares Problems1.8.1 Effective condition number1.8.2 Perturbation bounds1.8.3 Applications and comparisonsChapter 2 Collocation Trefftz Methods2.1 Introduction2.2 CTM for Motz's Problem2.3 Bounds of Effective Condition Number2.4 Stability for CTM of Rp=12.5 Numerical Experiments2.5.1 Choice of Rp2.5.2 Extreme accuracy of D02.6 GCTM Using Piecewise Particular Solutions2.7 Stability Analysis of GCTM2.7.1 Trefftz methods2.7.2 Collocation Trefftz methods2.8 Method of Fundamental Solutions2.9 Collocation Methods Using RBF2.10 Comparisons Between Cond_eff and Cond2.10.1 CTM using particular solutions for Motz's problem2.10.2 MFS and CM-RBF2.11 A Few RemarksChapter 3 Simpliffed Hybrid Trefftz Methods3.1 The Simpliffed Hybrid TM3.1.1 Algorithms3.1.2 Error analysis3.1.3 Integration approximation3.2 Stability Analysis for Simpliffed Hybrid TMChapter 4 Penalty Trefftz Method Coupled with FEM4.1 Introduction4.2 Combinations of TM and Adini0s Elements4.2.1 Algorithms4.2.2 Basic theorem4.2.3 Global superconvergence4.3 Bounds of Cond_eff for Motz's Problem4.4 Effective Condition Number of One and Inffnity Norms4.5 Concluding RemarksChapter 5 Trefftz Methods for Biharmonic Equations with Crack Singularities5.1 Introduction5.2 Collocation Trefftz Methods5.2.1 Three crack models5.2.2 Description of the method5.2.3 Error bounds5.3 Stability Analysis5.3.1 Upper bound for σmax(F)5.3.2 Lower bound for σmin(F)5.3.3 Upper bound for Cond_eff and Cond5.4 Proofs of Important Results Used in Section 5.35.4.1 Basic theorem5.4.2 Proof of Lemma 5.4.35.4.3 Proof of Lemma 5.4.45.5 Numerical Experiments5.6 Concluding RemarksChapter 6 Finite Difference Method6.1 Introduction6.2 Shortley-Weller Difference Approximation6.2.1 A Lemma6.2.2 Bounds for Cond EE6.2.3 Bounds for Cond_effChapter 7 Boundary Penalty Techniques of FDM7.1 Introduction7.2 Finite Difference Method7.2.1 Shortley-Weller Difference approximation7.2.2 Superconvergence of solution derivatives7.2.3 Bounds for Cond_eff7.3 Penalty-Integral Techniques7.4 Penalty-Collocation Techniques7.5 Relations Between Penalty-Integral and Penalty-Collocation Techniques7.6 Concluding RemarksChapter 8 Boundary Singularly Problems by FDM8.1 Introduction8.2 Finite Difference Method8.3 Local Reffnements of Difference Grids8.3.1 Basic results8.3.2 Nonhomogeneous Dirichlet and Neumann boundary conditions8.3.3 A remark8.3.4 A view on assumptions A1-A48.3.5 Discussions and comparisons8.4 Numerical Experiments8.5 Concluding RemarksChapter 9 Finite Element Method Using Local Mesh Refinements9.1 Introduction9.2 Optimal Convergence Rates9.3 Homogeneous Boundary Conditions9.4 Nonhomogeneous Boundary Conditions9.5 Intrinsic View of Assumption A2 and Improvements of Theorem 9.4.19.5.1 Intrinsic view of assumption A29.5.2 Improvements of Theorem 9.4.19.6 Numerical ExperimentsChapter 10 Hermite FEM for Biharmonic Equations10.1 Introduction10.2 Description of Numerical Methods10.3 Stability Analysis10.3.1 Bounds of Cond10.3.2 Bounds of Cond_eff10.4 Numerical ExperimentsChapter 11 Truncated SVD and Tikhonov Regularization11.1 Introduction11.2 Algorithms of Regularization11.3 New Estimates of Cond and Cond_eff11.4 Brief Error AnalysisAppendix Deffnitions and FormulasA.1 Square SystemsA.1.1 Symmetric and positive deffnite matricesA.1.2 Symmetric and nonsingular matricesA.1.3 Nonsingular matricesA.2 Overdetermined SystemsA.3 Underdetermined SystemsA.4 Method of Fundamental SolutionsA.5 RegularizationA.5.1 Truncated singular value decompositionA.5.2 Tikhonov regularizationA.6 p-NormsA.7 ConclusionsEpilogueBibliographyIndex

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