The Mathematical Theory of Finite Element Methods

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. The results contained here have never appeared before, and should prove interesting to researchers. It is intended primarily for 1st year mathematics graduate students and mathematically sophisticated engineers and scientists. The book has been the basis for courses at the graduate level at The University of Michigan, at Penn State University and at the University of Houston. The only prerequisite is a course in real variables, and even this has not been necessary for well prepared engineers and scientists in many cases. The material in this book would be appropriate for a second semester course on basic functional analysis taught to 1st year math graduate students. 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. The book contains such topics as Sobolev spaces, construction of a finite element space, n-dimensional variational problems, finite element multigrid methods, applications to planar elasticity. Each chapter ends with a series of exercises.