This book develops the basic mathematical theory of the finite element method, the most widely used technique for engineering design and analysis. It formalizes basic tools that are commonly used by researchers in the field but not previously published.

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.

The initial chapter provides an introduction 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 other iterative solution techniques, mixed methods with applications to elasticity and fluid flow models, variational "crimes" including nonconforming and isoparametric methods and numerical integration, error estimates in the maximum norm with applications to nonlinear problems, and Banach-space operator-interpolation techniques.

Different course paths can be chosen, allowing the book to be used for courses designed for students with different interests. For example, courses can emphasize physical applications, or algorithmic efficiency and code development issues, or the more difficult convergence theorems of the subject.