《椭圆方程有限元方法的整体超收敛及其应用》由者Zi-Cai Li、Hung-Tsai Huang、Ningning Yan编著。
《椭圆方程有限元方法的整体超收敛及其应用》内容如下:
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 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.
Preface
Acknowledgements
Chapter 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 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
3.4.3 Comparisons
3.4.4 Relation between Uh and
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 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 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 ..
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-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 Biharmouic 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 Q1
9.1.2 Rotated Q1 element (Qot)
9.1.3 Extension 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 Expansions for Eigenfunctiens
9.7 Numerical Experiments
9.7.1 Function p=1
9.7.2 Function p=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