有限元方法

Preface to the Series in Information and Computational Science

Preface

Chapter 1Variational Principle

1.1 Sobolev Space

1.2 Poisson Equation

1.2.1 Dirichlet Problem

1.2.2 Neumann Problem

1.3 Biharmonic Equation

1.4 Abstract Variational Problem

1.5 Galerkin Method and Ritz Method

Chapter 2 Finite Element and Finite Element Space

2.1 Triangulation

2.2 Finite Element

2.3 Finite Element Space

2.4 Second Order Problem: Simplex Elements

2.4.1 Simplex Element of Degreek

2.4.2 Linear Simplex Element

2.4.3 Quadric Simplex Element

2.4.4 Cubic Simplex Element

2.4.5 Incomplete Cubic Simplex Element

2.4.6 Crouzeix-Raviart Element

2.4.7 Cubic Hermite Simplex Element

2.4.8 Zienkiewicz Element

2.5 Second Order Problem: Rectangle Elements

2.5.1 Rectangle Element of Type(k)

2.5.2 Incomplete Rectangle Element of Type(2)

2.5.3 Wilson Element

2.5.4 Rectangle C-R Element

2.6 Fourth Order Problem: Simplex Elements

2.6.1 Morley Element

2.6.2 Zienkiewicz Element

2.6.3 Morley-Zienkiewicz Element

2.6.4 Modified Zienkiewicz Element

2.6.5 12-parameter Triangle Plate Element

2.6.6 15-parameter Triangle Plate Element

2.6.7 Argyris Element

2.6.8 Bell Element

2.6.9 Cubic Tetrahedron Element

2.7 Fourth Order Problem: Rectangle Elements

2.7.1 Rectangle Morley Element

2.7.2 Adini Element

2.7.3 Bogner-Fox-Schmit Element

2.8 2m-th Order Problem: MWX Element

Chapter 3 Interpolation Theory of Finite Elements

3.1 Affine Mapping and Affine Family

3.2 Affine Continuity and Scale Invariance

3.3 Interpolation Error

3.4 Inverse Inequality

3.5 Approximate Error of Finite Element Spaces

3.6 Interpolation Error of General Element

Chapter 4 Conforming Finite Element Method

4.1 Poisson Equation

4.2 Plate Bending Problem

4.3 A Posteriori Error Estimate

Chapter 5 Nonconforming Finite Element Methods

5.1 Nonconforming Finite Element

5.2 Weak Continuity

5.3 Second Order Elliptic Problem

5.4 Fourth Order Elliptic Problem

5.5 2m-th Order Elliptic Problem

5.6 A Posteriori Error Estimate

5.7 Error Estimate in L2 Norm

Chapter 6 Convergence of Nonconforming Finite Element

6.1 Generalized Path Test

6.2 Patch Test

6.2.1 Patch Test

6.2.2 Weak Patch Test

6.2.3 Sufficiency of Patch Test

6.2.4 Necessity of Patch Test

6.3 Counter Examples of Patch Test

6.4 F-E-M Test

……

Chapter 7 Quasi-Conforming Element Method

Chapter 8 Unconventional Finite Element Method

Chapter 9 Double Set Parameter Method

Chapter 10 Property of Finite Element Space

Chapter 11 L∞ Error Estimate for Second Order Problem

Chapter 12 L∞ Error Estimate for Plate Bending Problem

Bibliography

Index

《有限元方法（英文版）》系统地论述了有限元方法的数学基础理论。以椭圆偏微分方程边值问题为例，介绍了协调有限元方法以及非协调等非标准有限元方法的数学描述、收敛条件和性质、有限元解的先验和后验误差估计以及有限元空间的基本性质，其中包括作者多年来的部分研究成果。