©Society for Industrial and Applied Mathematics, 1998.

Properly used, this book would be an excellent text for a lower level, mathematically oriented, introductory course on finite element methods. The book is a substantial revision of [Johnson's text] and provides a readable, up-to-date, easily accessible, and mathematically sound introduction to finite element methods. The book is divided into three parts. Part I reviews some necessary background material and introduces the Galerkin method. Part II considers finite element methods for two-point boundary value problems and initial value problems, and Part III applies finite elements to the Poisson, heat, wave, and convection-diffusion equations. Eigenvalue problems are also briefly considered.

With some exceptions, the level of mathematical technicalities is kept low to make the book accessible for a wide audience of mathematics, science, and engineering students. It is confined to linear problems, and the finite element methods are often studied only in their lower order versions (e.g., piecewise linear). The Lax-Milgram theorem and a heuristic definition of Sobolev spaces are relegated to the last chapter. A forthcoming companion book, Advanced Computational Differential Equations, is to provide further theory and applications.

Although the review chapters in Part I make this book, in principle, very self-contained (even a review of calculus is included), in practice suitable prerequisites would be a standard two year calculus sequence, some linear algebra, and knowledge of basic numerical analysis (Gaussian elimination, polynomial interpolation, quadrature). Prior knowledge of differential equations could be minimal: as each new problem-class is introduced, there is a brief discussion of applications, qualitative properties, and analytic solution techniques. Separation of variables, Green's Functions, d'Alembert's formula, and related topics are all discussed. The heat and wave equations are derived from physical principles.

Indeed the authors advocate a unified approach wherein the analytic and numerical study of differential equations proceeds step-in-step, and the present book is written with this viewpoint in mind. However, the main focus of the book remains on numerical computations, and with a proper selection of chapters, it is certainly feasible to use this book as the text in a more traditional one-semester course focused on particular applications (e.g., finite element methods for elliptic problems).

Two particular features distinguish this book. From the beginning equal emphasis is placed on a priori and a posteriori error estimates, the latter being motivated by their use in error control and adaptive mesh refinement. (The duality arguments used in the proofs of these a posteriori estimates will likely be the material that readers find most mathematically challenging.) Also, equal emphasis is placed on all types of differential equations, whereas a more traditional approach would concentrate on elliptic boundary value problems and only then consider extensions to time-dependent problems. These emphases reflect advances in finite element methods made over the last two decades and in particular much of the authors' own research.

However, the authors' enthusiasm has resulted in some choices of questionable pedagogical merit. The very first example of a finite element calculation for a boundary value problem that readers see (section 6.2.4) involves an adaptively refined mesh - certainly for the first example a uniform mesh would suffice! And the methods presented for time-dependent partial differential equations all utilize finite elements in both space and time (e.g., the discontinuous Galerkin method); connections with the more traditional paradigm of obtaining a semidiscrete formulation using finite elements in space only, and then applying finite difference methods in time, are only briefly mentioned. Readers new to the subject may be left with an incomplete understanding of its historical development and of common contemporary practice. In fairness to students, especially since the book's title seems to imply a comprehensive treatment, instructors using this book as a text should make students aware of alternate methods.

The desire for a complete unified treatment from first principles also results in some oddities. In Chapter 3, A Review of Calculus, the fundamental theorem of calculus is proved. The proof is used to introduce the ideas of piecewise-constant approximations and convergence. However, the discussion requires the concepts of uniform continuity and Cauchy sequences, which is more mathematical sophistication than typically assumed in the rest of the book. And halfway through the book (section 9.6), the existence of the exponential function is established by proving the convergence of the piecewise-constant discontinuous Galerkin approximations to the equation u' = u. This seems a bit esoteric for the book's likely audience. But again, instructors wishing to do so may simply bypass these sections.

The book makes occasional reference to Femlab, a collection of software freely available through the internet (http://www.md.chalmers.se/Math/Research/ Femlab/), but is not dependent on Femlab, and instructors could easily complement the book with other packages (such as Matlab's pdetool) if desired. Instructors who wish students to write their own codes (especially two-dimensional) will need to provide some additional details on data structures, loop ordering, implementation of boundary conditions, etc. This book provides a clear elucidation of the fundamentals of the finite element method at an elementary level. There are many examples and numerical results, throughout, and there is a good selection of exercises The emphasis on error control and adaptivity is both important in practice and provides students with a good example of the practical application of theoretical analysis. I would only caution that instructors using this as a text need to consider carefully, taking into account their audience, their viewpoints and objectives, what material to cover, and whether to provide any supplemental material.