0 Basic Concepts.- 1 Sobolev Spaces.- 2 Variational Formulation of Elliptic Boundary Value Problems.- 3 The Construction of a Finite Element Space.- 4 Polynomial Approximation Theory in Sobolev Spaces.- 5 n-Dimensional Variational Problems.- 6 Finite Element Multigrid Methods.- 7 Max-norm Estimates.- 8 Variational Crimes.- 9 Applications to Planar Elasticity.- 10 Mixed Methods.- 11 Iterative Techniques for Mixed Methods.- 12 Applications of Operator-Interpolation Theory.- References.

Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the clas­ sical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Math­ (AMS) series, which will focus on advanced textbooks ematical Sciences and research level monographs. Preface 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 Uni­ versity of Michigan, Penn State University and the University of Houston.