椭圆方程有限元方法的整体超收敛及其应用
2012-3
科学出版社
Zi-Cai Li，Hung-Tsai Huang，Ningning Yan 编
324
This book cove the advanced study on the global
superconvegence of elliptic equatio in both theory and computation,
where the main materials are adapted from our journal pape
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 equatio 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
publicatio. This book has also summarized some important results of
Lin, his colleagues and othe. This book is written for researche
and graduate students of mathematics and engineering to study and
apply the global superconvergence for numerical PDE.
Zi-Cai Li graduated in 1963 from Tsinghua University， and received the Ph. D. degree in 1986 from the University of Toronto. Since 1993， he has been a Professor in Department of Applied .Mathematics，Sun Yat-sen University， Kaohsiung， Taiwan. His research areas are numerical analysis， scientific computing， image processing and pattern recognition. Hung-Tsai Huang is a Professor at Department of Applied Mathematics， I-Shou University， Kaohsiung， Taiwan. He received the Ph.D. degree in 2003 from the Department of Applied Mathematics， Sun Yat-sen University， Kaohsiung， Taiwan. His research areas are numericalanalysis and scientific computing. Ningning Yan earned her Ph.D. from the Institute of Computational Mathematics， Chinese Academy of Sciences in 1990. She is currently the professor of Academy of Mathematics and Systems Science， Chinese Academy of Sciences. Her research interests are numerical methods for partial differential equations and optimal control problems.
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 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 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 domai
5.5.2 The Richardson extrapolation and the least squares
method
5.6 Concluding Remarks
Chapter 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-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 solutio
7.5.3 Superconvergence of L-norm
7.5.4 Global superconvergence of the a posteriori interpolant
solutio
7.6 Concluding Remarks
Chapter 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 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 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 conclusio
Chapter 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 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 Conclusio
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
