This book develops the basic mathematical theory of the finite element  method, the most widely used technique for engineering design and analysis.

The third edition contains four new sections: the BDDC domain decomposition preconditioner, convergence analysis of an adaptive algorithm, interior penalty methods and Poincara\'e-Friedrichs inequalities for piecewise W^1_p functions. New exercises have also been added throughout.

The initial chapter provides an introducton to the entire subject, developed in the one-dimensional case. Four subsequent chapters develop the basic theory in the multidimensional case, and a fifth chapter presents basic applications of this theory. Subsequent chapters provide an introduction to:

 - multigrid methods and domain decomposition methods

 - mixed methods with applications to elasticity and fluid mechanics

 - iterated penalty and augmented Lagrangian methods

 - variational "crimes" including nonconforming andisoparametric  methods, numerical integration and interior penalty methods

 - error estimates in the maximum norm with applications to nonlinear problems

 - error estimators, adaptive meshes and convergence analysis of an adaptive algorithm

- Banach-space operator-interpolation techniques

The book has proved 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. It can also be used for courses that emphasize physical applications or algorithmic efficiency.

Reviews of earlier editions: "This book represents an important contribution to the mathematical literature of finite elements. It is both a well-done text and a good reference." (Mathematical Reviews, 1995)

"This is an excellent, though demanding, introduction to keymathematical topics in the finite element method, and at the same time a valuable reference and source for workers in the area."

 (Zentralblatt,  2002)

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.