An Analysis of the Finite Element Method

Gilbert Strang and George Fix
Wellesley-Cambridge Press
(This book was originally published by Prentice-Hall, 1973)

TABLE OF CONTENTS

   1. An Introduction to the Theory

1.1  The Basic Ideas
1.2  A Two-point Boundary Value Problem
1.3  The Variational Form of the Problem
1.4  Finite Difference Approximations
1.5  The Ritz Method and Linear Elements
1.6  The Error with Linear Elements
1.7  The Finite Element Method in One Dimension
1.8  Boundary Value Problems in Two Dimensions
1.9  Triangular and Rectangular Elements
1.10 Element Matrices in Two-dimensional Problems

   2. A Summary of the Theory

2.1  Basis Functions for the Finite Element Spaces
2.2  Rates of Convergence
2.3  Galerkin's Method, Collocation, and the Mixed Method
2.4  Systems; Shell Problems; Variations on the FEM

   3. Approximation

3.1  Pointwise Approximation
3.2  Mean-square Approximation
3.3  Curved Elements and Isoparametric Transformations
3.4  Error Estimates

   4. Variational Crimes

4.1  Violations of the Rayleigh-Ritz Code
4.2  Non-conforming Elements
4.3  Numerical Integration
4.4  Approximation of Domain and Boundary Conditions

   5. Stability

5.1  Independence of the Basis
5.2  The Condition Number

   6. Eigenvalue Problems

6.1  Variational Formulation and the Min-max Principle
6.2  Some Elementary Examples
6.3  Eigenvalue and Eigenfunction Errors
6.4  Computational Techniques

   7. Initial-Value Problems

7.1  Galerkin-Crank-Nicolson Method for the Heat Equation
7.2  Stability and Convergence in Parabolic Problems
7.3  Hyperbolic Equations

   8. Singularities

8.1  Corners and Interfaces
8.2  Singular Functions
8.3  Errors in the Presence of Singularities
8.4  Experimental Results

BIBLIOGRAPHY

INDEX OF NOTATIONS

INDEX

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