Theory and Practice of Finite Elements

This book presents the mathematical theory of finite elements, starting from basic results on approximation theory and finite element interpolation and building up to more recent research topics, such as and Discontinuous Galerkin, subgrid viscosity stabilization, and a posteriori error estimation.

The body of the text is organized into three parts plus two appendices collecting the functional analysis  results used in the book. The first part develops the theoretical basis for the finite element method and emphasizes the fundamental role of inf-sup conditions. The second party addresses various applications encompassing elliptic PDE's, mixed formulations, first-order PDEs, and the time-dependent versions of these problems. The third part covers implementation issues and should provide readers with  most of  the practical details needed to write or understand a finite element code.

Written at the graduate level, the text contains numerous examples and exercises and is intended to serve as a graduate textbook. Depending on one's interests, several reading paths can be followed, emphasizing either theoretical results, numerical algorithms, code efficiency, or applications in the engineering sciences.

The book will be useful to researchers and graduate students in mathematics, computer science and engineering.

From the reviews:

"This book represents an excellent compendium of information about the mathematics and numerical analysis of the finite … . Its user-ability, for the writing … is guaranteed through the inclusion of a discussion … . will be a useful text for advanced mathematics/engineering graduates who wish to learn about … good background information for the engineer who will eventually apply the finite element method to practical real-world problems. In addition, it will be an excellent text for the mathematics graduate … . (R.S.Anderssen, Mathematical Reviews, 2005)

"This book is an expanded version of Lecture Notes published by the authors in French … . It has been used as a textbook for graduate finite element courses … . The book can be used in several courses in Mathematics, Computer Science, and Engineering programs. The authors offer suggestions for course titles and syllabi. … The book is organized into three parts. … Many bibliographic entries to the extensive literature on finite elements are given throughout the book." (I.N. Katz, Zentralblatt Math, Vol. 1059 (10), 2005)

"A relative complete coverage of issues concerning finite element methodology based soundly on theory. … this is a self-contained presentation which goes relatively far regarding the questions of stability, approximation and error estimation and demonstrates the use of the rather abstract concepts in concrete situations. The presentation is suited for the mathematician and also for applied scientists should they be ready to digest the more abstract concepts which, however, prove very useful understanding how to obtain a working code for the problem at hand." (H. Muthsam, Monatshefte für Mathematik, Vol. 148 (2), 2006)

"Overall, the authors have largery succeeded in giving a rather comprehensive exposition of the finite element method and its challenges, up to some current research topics, in less than 500 pages, yet without taking any unreasonable shortcut, by no means an easy task."

(SIAM Review)