The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations - 1st Edition - ISBN: 9780120686506, 9781483267982

The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations

1st Edition

Editors: A. K. Aziz
eBook ISBN: 9781483267982
Imprint: Academic Press
Published Date: 1st January 1972
Page Count: 796
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The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations is a collection of papers presented at the 1972 Symposium by the same title, held at the University of Maryland, Baltimore County Campus. This symposium relates considerable numerical analysis involved in research in both theoretical and practical aspects of the finite element method.

This text is organized into three parts encompassing 34 chapters. Part I focuses on the mathematical foundations of the finite element method, including papers on theory of approximation, variational principles, the problems of perturbations, and the eigenvalue problem. Part II covers a large number of important results of both a theoretical and a practical nature. This part discusses the piecewise analytic interpolation and approximation of triangulated polygons; the Patch test for convergence of finite elements; solutions for Dirichlet problems; variational crimes in the field; and superconvergence result for the approximate solution of the heat equation by a collocation method. Part III explores the many practical aspects of finite element method.

This book will be of great value to mathematicians, engineers, and physicists.

Table of Contents



Part I. Survey Lectures on the Mathematical Foundations of the Finite Element Method


1. Preliminary Remarks

2. The Fundamental Notions

3. Properties of Solutions of Elliptic Boundary Value Problems

4. Theory of Approximation

5. Variational Principles

6. Rate of Convergence of the Finite Element Method

7. One Parameter Families of Variational Principles

8. Finite Element Method for Non-Smooth Domains and Coefficients

9. The Problems of Perturbations in the Finite Element Method

10. The Eigenvalue Problem

11. The Finite Element Method for Time Dependent Problems

Part II. Invited Hour Lectures

Piecewise Analytic Interpolation and Approximation in Triangulated Polygons

Approximation of Steklov Eigenvalues of Non-Selfadjoint Second Order Elliptic Operators

The Combined Effect of Curved Boundaries and Numerical Integration in Isoparametric Finite Element Methods

A Superconvergence Result for the Approximate Solution of the Heat Equation by a Collocation Method

Some L2 Error Estimates for Parabolic Galerkin Methods

Computational Aspects of the Finite Element Method

Effects of Quadrature Errors in Finite Element Approximation of Steady State, Eigenvalue, and Parabolic Problems

Experience with the Patch Test for Convergence of Finite Elements

Higher Order Singularities for Interface Problems

On Dirichlet Problems Using Subspaces with Nearly Zero Boundary Conditions

Generalized Conjugate Functions for Mixed Finite Element Approximations of Boundary Value Problems

Finite Element Formulation by Variational Principles with Relaxed Continuity Requirements

Variational Crimes in the Finite Element Method

Spline Approximation and Difference Schemes for the Heat Equation

Part III. Short Communications

The Extension and Application of Sard Kernel Theorems to Compute Finite Element Error Bounds

Two Types of Piecewise Quadratic Spaces and Their Order of Accuracy for Poisson’s Equation

A Method of Galerkin Type Achieving Optimum L2 Accuracy for First Order Hyperbolics and Equations of Schrödinger Type

Richardson Extrapolation for Parabolic Galerkin Methods

Geometric Aspects of the Finite Element Method

The Use of Interpolatory Polynomials for a Finite Element Solution of the Multigroup Diffusion Equation

A “Local” Basis of Generalized Splines over Right Triangles Determined from a Nonuniform Partitioning of the Plane

Least Square Polynomial Spline Approximation

Subspaces with Accurately Interpolated Boundary Conditions


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© Academic Press 1972
Academic Press
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About the Editor

A. K. Aziz

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