椭圆方程有限元方法的整体超收敛及其应用

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

《椭圆方程有限元方法的整体超收敛及其应用》总结了作者Zi-Cai Li、Hung-Tsai Huang、Ningning Yan近十几年来在有限元高精度算法（主要是整体超收敛分析）方面的主要结果，其中包括许多已发表或尚未发表的成果。本书采用统一的分析方法，即中国学者独创的积分恒等式方法，对常见的椭圆型偏微分方程的各种有限元方法进行了深入、系统的分析，给出了相应的整体超收敛结果及高精度有限元算法。该书还讨论了非线性问题、特征值问题及差分方法等的整体超收敛，研究了相应的稳定性分析和奇异问题的特殊处理技术，介绍了大量实际应用问题的超收敛分析和数值计算结果，以验证整体超收敛分析的有效性。《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.