5R3. Finite Element Solution of Boundary Value Problems: Theory and Computation. Classics in Applied Math, Vol 35. - O Axelsson (Univ of Nijmegen, Nijmegen, Netherlands) and VA Barker (Tech Univ, Lyngby, Denmark). SIAM, Philadelphia. 2001. 432 pp. Softcover. ISBN 0-89871-499-0. $50.00.
Reviewed by DJ Benson (Dept of Appl Mech and Eng Sci, UCSD, 9500 Gilman, La Jolla CA 92093-0411).
Quoting from the preface, “The purpose of this book is to provide an introduction to both the theoretical and computational aspects of the finite element method for solving boundary value problems for partial differential equations.” Its intended readers are advanced undergraduates and graduate students in numerical analysis, mathematics, computer science, and the “theoretically inclined workers” in engineering and the physical sciences. In comparison to some recent texts, the book is short at 432 pages, but the topics the authors decided to treat are treated thoroughly.
The first chapter is devoted to quadratic functionals on finite-dimensional vector spaces. Chapters 2 and 3 discuss the variational formulation of boundary value problems, and Chapters 4 and 5 discuss the Ritz-Galerkin and the finite element method. Some readers may be disappointed to find that the text does not present a catalog of all the common interpolation functions that are currently in popular use, or address methods, but these are minor quibbles. The final two chapters cover direct and iterative methods for solving the linear equations generated by the finite element method. The last chapter also includes an introduction to the multigrid method. Exercises and references are provided at the end of each chapter. The text is independent of any particular programing language, which is a nice feature. Some of the algorithms are written out in a form of pseudo-code that anyone familiar with any computer language should be able to read.
Because the book suits the interests of students in numerical analysis and mathematics so well, the majority of mechanical engineers will probably not be interested in this text. Most of the partial differential equations are model linear ones, and the challenges associated with beam, plate, and shell elements are largely ignored as are all nonlinear problems. In addition, it does not address the issues of selective reduced integration and other variational crimes.
In terms of its stated aims, the book is a success. It is mathematically rigorous, yet does not become bogged down in either ponderous notation or the details of arcane points that are of little interest to the reader new to finite element methods. The explanations are clear and, for its intended audience, it should be a good read. This book, Finite Element Solution of Boundary Value Problems: Theory and Computation, will appeal to readers who have enjoyed the books by Ciarlet, Claes Johnson, and Oden.