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 our journal 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 proviede 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.
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目录
Preface Acknowledgements Chapter 1 Basic Approaches 1.1 Introduction 1.2 Simplified Hybrid Combined Methods 1.3 Basic Theorem for Global Superconvergence 1.4 Bilinear Elements 1.5 Numerical Experiments 1.6 Concluding Remarks Chapter 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 Remarks Chapter 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 Discussions 3.4.1 Global superconvergence 3.4.2 Special case of h=k and f_xxyy=0 3.4.3 Comparisons 3.4.4 Relation between u_h and ū^*_h 3.5 Concluding Remarks Chapter 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 Remarks Chapter 5 Finite Difference Methods for Singularity Problems 5.1 Introduction 5.2 The Shortley-Weller Difference Approximation 5.3 Analysis for u^D_h with no Error of Divergence Integration 5.4 Analysis for u_h with Approximation of Divergence Integration 5.5 Numerical Verification on Reduced Convergence Rates 5.5.1 The model on stripe domains 5.5.2 The Richardson extrapolation and the least squares method 5.6 Concluding Remarks Chapter 6 Basic Error Estimates for Biharmonic Equations 6.1 Introduction 6.2 Basic Estimates for ∫∫_Ω(u-u_I)_xxv_xxds 6.3 Basic Estimates for ∫∫_Ω(u-u_I)_xyv_xyds 6.4 New Estimates for ∫∫_Ω(u-u_I)_xyv_xyds for Uniform Rectangular Elements 6.5 New Estimates for ∫∫_Ω(u-u_I)_xxv_yyds 6.6 Main Theorem of Global Superconvergence 6.7 Concluding Remarks 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^0_h-elliptic inequality 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 Section 7.4 7.5.2 New superconvergence of average nodal solutions 7.5.3 Superconvergence of L^∞-norm 7.5.4 Global superconvergence of the a posteriori interpolant solutions 7.6 Concluding Remarks Chapter 8 Blending Problems in 3D with Periodical Boundary Conditions 8.1 Introduction 8.2 Biharmonic Equations 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 Superconvergence Chapter 9 Lower Bounds of Leading Eigenvalues 9.1 Introduction 9.1.1 Bilinear element Q_1 9.1.2 Rotated Q_1 element (Q^rot_1) 9.1.3 Extension of rotated Q_1 element (EQ^rot_1) 9.1.4 Wilson's element 9.2 Basic Theorems 9.3 Bilinear Elements 9.4 Q^rot_1 and EQ^rot_1 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 Expansions for Eigenfunctions 9.7 Numerical Experiments 9.7.1 Function ρ=1 9.7.2 Function ρ≠0 9.7.3 Numerical conclusions Chapter 10 Eigenvalue Problems with Periodical Boundary Conditions 10.1 Introduction 10.2 Periodic Boundary Conditions 10.3 Adini's Elements for Eigenvalue Problems 10.4 Error Analysis for Poisson's Equation 10.5 Superconvergence for Eigenvalue Problems 10.6 Applications to Other Kinds of FEMs 10.6.1 Bi-quadratic Lagrange elements 10.6.2 Triangular elements 10.7 Numerical Results 10.8 Concluding Remarks Chapter 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 Conclusions Chapter 12 Epilogue 12.1 Basic Framework of Global Superconvergence 12.2 Some Results on Integral Identity Analysis 12.3 Some Results on Global Superconvergence Bibliography Index
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