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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.
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
- No. of pages:
- © Academic Press 1972
- 1st January 1972
- Academic Press
- eBook ISBN:
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