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This book develops the basic mathematical theory of the finite element method, the most widely used technique for engineering design and analysis. One purpose of this book is to formalize basic tools that are commonly used by researchers in the field but never published. It is intended primarily for mathematics graduate students and mathematically sophisticated engineers and scientists.
The book has been the basis for graduate-level courses at The University of Michigan, Penn State University and the University of Houston. The prerequisite is only a course in real variables, and even this has not been necessary for well-prepared engineers and scientists in many cases.
The book 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.
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.
Basic Concepts
Sobolev Spaces
Variational Formulation of Elliptic Boundary Value Problems
The Construction of a Finite Element Space
Polynomial Approximation Theory in Sobolev Spaces
n-Dimensional Variational Problems
Finite Element Multigrid Methods
Additive Schwarz Preconditioners
Max-norm Estimates
Adaptive Meshes
Variational Crimes
Applications to Planar Elasticity
Mixed Methods
Iterative Techniques for Mixed Methods
Applications of Operator-Interpolation Theory