椭圆方程有限元方法的整体超收敛及其应用（英文版）
出版时间：2011年版
内容简介
　　This book covers the advanced study on the global superconvegence of elliptic equations in both theory and computation， where the main materials are adapted from ourjournal papers published. A deep and rather completed analysis of global supperconvergence is explored for bilinear， biquadratic， Adini's and bi-cubic Hermite elements， as well as for the finite difference method. Poisson's and the biharmonic equations are included， and eigenvalue and semi-linear problems are discussed. The singularity problems， blending problems， coupling techniques， a posteriori interpolant techniques， and some physical and engineering problems are studied. Numerical examples are provided for verification of the analysis， and other numerical experiments can be found from our publications. This book has also summarized some important results of Lin， his colleagues and others. This book is written for researchers and graduate students of mathematics and engineering to study and apply the global superconvergence for numerical PDE.
目录
PrefaceAcknowledgementsChapter I Basic Approaches 1.1 Introduction 1.2 Simplified Hybrid Combined Methods 1.3 Basic Theorem for Global Superconvergenee 1.4 Bilinear Elements 1.5 Numerical Experiments 1.6 Concluding RemarksChapter 2 Adini's Elements 2.1 Introduction 2.2 Adini's Elements 2.3 Global Superconvergence 2.3.1 New error estimates 2.3.2 A posteriori interpolant formulas 2.4 Proof of Theorem 2.3.1 2.4.1 Preliminary lemmas 2.4.2 Main proof of Theorem 2.3.1 2.5 Stability Analysis 2.6 New Stability Analysis via Effective Condition Number. 2.6.1 Computational formulas 2.6.2 Bounds of effective condition number 2.7 Numerical Experiments and Concluding RemarksChapter 3 Biquadratic Lagrange Elements 3.1 Introduction 3.2 Biquadratic Lagrange Elements 3.3 Global Superconvergence 3.3.1 New error estimates 3.3.2 Proof of Theorem 3.3.1 3.3.3 Proof of Theorem 3.3.2 3.3.4 Error bounds for Q8 elements 3.4 Numerical Experiments and Discussio 3.4.1 Global superconvergence 3.4.2 Special case of h = k and 3.4.3 Compariso 3.4.4 Relation between Uh and 3.5 Concluding RemarksChapter 4 Simplified Hybrid Method for Motz's Problems 4.1 Introduction 4.2 Simplified Hybrid Combined Methods 4.3 Lagrange Rectangular Elements 4.4 Adini's Elements 4.5 Concluding RemarksChapter 5 Finite Difference Methods for Singularity Problem 5.1 Introduction 5.2 The Shortley-Weller Difference Approximation 5.3 Analysis for uD with no Error of Divergence Integration 5.4 Analysis for Uh with Approximation of DivergenceIntegration.. 5.5 Numerical Verification on Reduced Convergence Rates 5.5.1 The model on stripe domai 5.5.2 The Richardson extrapolation and the least squaresmethod 5.6 Concluding RemarksChapter 6 Basic Error Estimates for Biharmonic Equatio ..Chapter 7 Stability Analysis and Superconvergence of Blending Problems 7.1 Introduction 7.2 Description of Numerical Methods 7.3 Stability Analysis 7.3.1 Optimal convergence rates and the uniform V-ellipticinequality. 7.3.2 Bounds of condition number 7.3.3 Proof for Theorem 7.3.4 7.4 Global Superconvergence 7.5 Numerical Experiments and Other Kinds of Superconvergence..- 7.5.1 Verification of the analysis in Section 7.3 and Section7.4 7.5.2 New superconvergence of average nodal solutio 7.5.3 Superconvergence of L-norm 7.5.4 Global superconvergence of the a posteriori interpolantsolutio 7.6 Concluding RemarksChapter 8 Blending Problems in 3D with Periodical Boundary Conditio 8.1 Introduction 8.2 Biharmouic Equatio 8.2.1 Description of numerical methods 8.2.2 Global superconvergence 8.3 The BPH-FEM for Blending Surfaces 8.4 Optimal Convergence and Numerical Stability 8.5 SuperconvergenceChapter 9 Lower Bounds of Leading Eigenvalues 9.1 Introduction 9.1.1 Bilinear element Q1 9.1.2 Rotated Q1 element (Qot) 9.1.3 Exteion of rotated Qz element (EQrzt) 9.1.4 Wilson's element 9.2 Basic Theorems 9.3 Bilinea Elements 9.4 QOt and EQrlt Elements 9.4.1 Proof of Lemma 9.4.1 9.4.2 Proof of Lemma 9.4.2 9.4.3 Proof of Lemma 9.4.3 9.4.4 Proof of Lemma 9.4.4 9.5 Wilson's Element 9.5.1 Proof of Lemma 9.5.1 9.5.2 Proof of Lemma 9.5.2 9.5.3 Proof of Lemma 9.5.3 and Lemma 9.5.4 9.6 Expaio for Eigenfunctie 9.7 Numerical Experiments 9.7.1 Function p=1 9.7.2 Function p=0 9.7.3 Numerical conclusioChapter 10 Eigenvalue Problems with Periodical Boundary Conditio 10.1 Introduction 10.2 Periodic Boundary Conditio 10.3 Adini's Elements for Eigenvalue Problems 10.4 Error Analysis for Poisson's Equation 10.5 Superconvergence for Eigenvalue Problems 10.6 Applicatio to Other Kinds of FEMs 10.6.1 Bi-quadratic Lagrange elements 10.6.2 Triangular elements 10.7 Numerical Results 10.8 Concluding RemarksChapter 11 Semilinear Problems 11.1 Introduction 11.2 Parameter-Dependent Semilinear Problems 11.3 Basic Theorems for Superconvergence of FEMs 11.4 Superconvergence of Bi-p(> 2)-Lagrange Elements 11.5 A Continuation Algorithm Using Adini's Elements 11.6 ConclusioChapter 12 Epilogue 12.1 Basic Framework of Global Superconvergence 12.2 Some Results on Integral Identity Analysis 12.3 Some Results on Global SuperconvergenceBibliographyIndex