有限元素方法的数学理论

Jillian

The mathematical theory of finite element methods
作者Susanne C. Brenner等

This book develops the basic mathematical theory of the finite element method, the most widely used technique for engineering design and analysis. This expanded second edition contains new chapters on additive Schwarz preconditioners and adaptive meshes. New exercises have also been added throughout. The book will be useful to mathematicians as well as engineers and physical scientists. It can be used for a course that provides an introduction to basic functional analysis, approximation theory, and numerical analysis, while building upon and applying basic techniques of real variable theory. Different course paths can be chosen, allowing the book to be used for courses designed for students with different interests.

目录
Series Preface
Preface to the Second Edition
Preface to the First Edition
0 Basic Concepts
0.1 Weak Formulation of Boundary Value Problems
0.2 Ritz-Galerkin Approximation
0.3 Error Estimates
0.4 Piecewise Polynomial Spaces - The Finite Element Method ...
0.5 Relationship to Difference Methods
0.6 Computer Implementation of Finite Element Methods
0.7 Local Estimates
0.8 Adaptive Approximation
0.9 Weighted Norm Estimates
0.x Exercises
1 Sobolev Spaces
1.1 Review of Lebesgue Integration Theory
1.2 Generalized (Weak) Derivatives
1.3 Sobolev Norms and Associated Spaces
1.4 Inclusion Relations and Sobolev's Inequality
1.5 Review of Chapter o
1.6 Trace Theorems
1.7 Negative Norms and Duality
1.x Exercises
2 Variational Formulation of Elliptic Boundary Value Problems
2.1 Inner-Product Spaces
2.2 Hilbert Spaces
2.3 Projections onto Subspaces
　2.4 Riesz Representation Theorem
　2.5 Formulation of Symmetric Variational Problems
　2.6 Formulation of Nonsymmetric Variational Problems
　2.7 Tile Lax-Milgram Theorem
　2.8 Estimates for General Finite Element Approximation
　2.9 Higher-dimensional Examples
　2.10 Exercises
3 The Construction of a Finite Element Space
3.1 The Finite Element
3.2 Triangular Finite Elements
The Lagrange Element
The Hermite Element
The Argyris Element
3.3 The Interpolant
3.4 Equivalence of Elements
3.5 Rectangular Elements
Tensor Product Elements
The Serendipity Element
　3.6 Higher-dimensional Elements
　3.7 Exotic Elements
　3.8 Exercises
4 Polynomial Approximation Theory in Sobolev Spaces
4.1 Averaged Taylor Polynomials
4.2 Error Representation
4.3 Bounds for Riesz Potentials
4.4 Bounds for the Interpolation Error
4.5 Inverse Estimates
4.6 Tensor-product Polynomial Approximation
4.7 Isoparanmtric Polynomial Approximation
4.8 Interpolation of Non-smooth Functions
4.9 A Discrete Sobolev Inequality
4.x Exercises
5 n-Dimensional Variational Problems
6 Finite Element Multigrid Methods
7 Additive Schwarz Preconditioners
8 Max-norm Estimates
9 Adaptive Meshes
10 Variational Crimes
11 Applications to Planar Elasticity
12 Mixed Methods
13 Iterative Techniques for Mixed Methods
14 Applications of Operator-Interpolation Theory
References
Index