有限元法被广泛用于工程设计和工程分析。本书是Springer出版的《应用数学教材》丛书之15。全书分成15章，在第1版的基础上增加了加性Schwarz预条件和自适应格；书中不但提供有限元法系统的数学理论。还兼重在工程设计和分析中的应用算法效率、程序开发和较难的收敛问题。

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 0

 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 The Lax-Milgram Theorem

 2.8 Estimates for General Finite Element Approximation

 2.9 Higher-dimensional Examples

 2.x 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.x 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 Isoparametric Polynomial Approximation

 4.8 Interpolation of Non-smooth Functions

 4.9 A Discrete Sobolev Inequality

 4.x Exercises

5 n-Dimensional Variational Problems

 5.1 Variational Formulation of Poisson's Equation .

 5.2 Variational Formulation of the Pure Neumann Problem .

 5.3 Coercivity of the Variational Problem

 5.4 Variational Approximation of Poisson's Equation

 5.5 Elliptic Regularity Estimates

 5.6 General Second-Order Elliptic Operators

 5.7 Variational Approximation of General Elliptic Problems .

 5.8 Negative-Norm Estimates

 5.9 The Plate-Bending Biharmonic Problem

 5.x Exercises

6 Finite Element Multigrid Methods

 6.1 A Model Problem

 6.2 Mesh-Dependent Norms

 6.3 The Multigrid Algorithm

 6.4 Approximation Property

 6.5 W-cycle Convergence for the kth Level Iteration

 6.6 V-cycle Convergence for the kth Level Iteration

 6.7 Full Multigrid Convergence Analysis and Work Estimates

 6.x Exercises

7 Additive Schwarz Preconditioners

 7.1 Abstract Additive Schwarz Framework

 7.2 The Hierarchical Basis Preconditioner

 7.3 The BPX Preconditioner

 7.4 The Two-level Additive Schwarz Preconditioner

 7.5 Nonoverlapping Domain Decomposition Methods

 7.6 The BPS Preconditioner

 7.7 The Neumann-Neumann Preconditioner

 7.x Exercises

8 Max-norm Estimates

 8.1 Main Theorem

 8.2 Reduction to Weighted Estimates

 8.3 Proof of Lemma 8.2.6

 8.4 Proofs of Lemmas 8.3.7 and 8.3.11

 8.5 Lp Estimates (Regular Coefficients)

 8.6 Lp Estimates (Irregular Coefficients)

 8.7 A Nonlinear Example

 8.x Exercises

9 Adaptive Meshes

 9.1 A priori Estimates

 9.2 Error Estimators

 9.3 Local Error Estimates

 9.4 Estimators for Linear Forms and Other Norms

 9.5 Conditioning of Finite Element Equations

 9.6 Bounds on the Condition Number

 9.7 Applications to the Conjugate-Gradient Method

 9.x Exercises

10 Variational Crimes

 10.1 Departure from the Framework

 10.2 Finite Elements with Interpolated Boundary Conditions .

 10.3 Nonconforming Finite Elements

 10.4 Isoparametric Finite Elements

 10.x Exercises

11 Applications to Planar Elasticity

 11.1 The Boundary Value Problems

 11.2 Weak Formulation and Korn's Inequality

 11.3 Finite Element Approximation and Locking

 11.4 A Robust Method for the Pure Displacement Problem ..

 11.x Exercises

12 Mixed Methods

 12.1 Examples of Mixed Variational Formulations

 12.2 Abstract Mixed Formulation

 12.3 Discrete Mixed Formulation

 12.4 Convergence Results for Velocity Approximation

 12.5 The Discrete Inf-Sup Condition

 12.6 Verification of the Inf-Sup Condition

 12.x Exercises

13 Iterative Techniques for Mixed Methods

 13.1 Iterated Penalty Method

 13.2 Stopping Criteria

 13.3 Augmented Lagrangian Method

 13.4 Application to the Navier-Stokes Equations

 13.5 Computational Examples

 13.x Exercises

14 Applications of Operator-Interpolation Theory

 14.1 The Real Method of Interpolation

 14.2 Real Interpolation of Sobolev Spaces

 14.3 Finite Element Convergence Estimates

 14.4 The Simultaneous Approximation Theorem

 14.5 Precise Characterizations of Regularity

 14.x Exercises

References

Index