Journal on Fluid Mechanics, Vol. 368, 1998.

Reviewer: C. Pozrikidis

This unconventional book emerged from the extensive rewriting of its precursor entitled Numerical Solution of Partial Differential Equations, single-authored by C. Johnson. It is the second part of a trilogy consisting of an introductory book entitled Introduction to Computational Differential Equations where fundamental calculus and linear algebra are discussed, and an advanced book entitled Advanced Computational Differential Equations where applications of the methods to the physical and Engineering sciences are discussed and specific numerical implementations are outlined. The goal of this book, as stated at the preface, is to provide a student with the essential theoretical and computational tools that make it possible to use differential equations in science and engineering effectively. Furthermore, the authors state, and a quick glance through the table of contents confirms, that the backbone of this book is a unified presentation of numerical solution techniques for differential equations based on Galerkin methods. Accordingly, several classes of other methods, including finite-difference, finite-volume, and spectral methods, are not discussed.

I used the word unconventional in the first paragraph for several reasons. Most important, the authors introduce the subject from a historical point of view, and motivate the reading of their book by arguing that to interface calculus and numerical computation is both useful and necessary. To the unsuspecting reader, this book may be erroneously mistaken for a tribute to Leibniz, for every chapter contains extensive discussions of, drawings and excerpts from, the work and life of the co-founder of differential and integral calculus. Parts of the book may also be mistaken for a brief history of mathematical modelling and numerical simulation based on differential equations. Indeed, the first two chapters are devoted entirely to historical retrospectives.

The book is divided into three parts: the first part is an introduction; the second part discusses ordinary differential equations; and the third part discusses partial differential equations. The authors' enthusiasm in writing this text is apparent, and is clearly reflected in the manner by which they introduce new concepts; intellectual curiosity takes priority over the need to build an apparatus. An example is the ingenious introduction of the Cauchy sequence in Chapter 3 entitled a Review of Calculus. This chapter contains a summary of key concepts and theorems, often accompanied by rigorous mathematical proofs. Among other things, one learns that Leibniz conceived the idea of integral calculus on October 29, which also happens to be the birthday of this reviewer. Each subsequent chapter concisely but clearly introduces and explains a key idea or class of methods, often by example and always on a need-to-know basis.

Chapters 4 and 5 introduce selected concepts of linear algebra, polynomial and trigonometric interpolation and integration. The level of the presentation is typical of an intermediate-level text on numerical analysis. The purpose is to prepare the ground for the finite-element method, and the material is screened carefully for that purpose. Chapter 6 introduces the Galerkin projection and the finite-element method for problems in one dimension. Part I concludes with Chapter 7 which reviews the solution of systems of linear algebraic equations by direct and iterative methods.

Chapters 8-12, comprising Part II, discuss the solution of several classes of problems involving linear ordinary differential equations by finite-element methiods. This includes two-point boundary value problems, and initial-value problems for a single, or a system of, first-order equations. The intentional exclusion of nonlinear equations explains the absence of the shooting and related methods, reminds us that the authors' interest lies in the Galerkin projection method, and emphasizes, that this text should be regarded as an introduction to finite-element methods from the perspective of the applied mathematician.

Part III extends established concepts and procedures to two and higher dimensions. Chapter 13 is an introduction to differential and integral calculus for functions of more than one variable; Chapter 14 discusses polynomial interpolation in two dimensions, with an informative section on triangulation Chapter 15 focuses on the finite-element solution of the Poisson equation; Chapters 16, 17, 18 and 19 concentrate, respectively, on the one-dimensional unsteady heat conduction equation, the two-dimensional wave equation, the steady two-dimensional convection-diffusion equation, and the unsteady two-dimensional convection-diffusion equation. The penultimate chapter, 20, discusses eigenvalue problems. The final chapter, 21, entitled the power of abstraction, explains how a generalized formalism for strongly elliptic equation may be used to derive useful results for equations commonly encountered in mathematical physics.

This book is not about practical numerical computation but is intended to explain the framework upon which the computation relies. It is addressed to students of, or researchers with a sound background on, applied mathematics. Frequent error analyses and proofs of convergence are reminders of this focus. The careful reader will learn the fundamentals of Galerkin and finite-element methods, and should then be in a position to use them rigorously and skilfully to solve particular problems, carrying out error and convergence analyses if so desired. A software package called Femlab accompanies the book and is freely available through the internet, although I was unable to locate the distribution URL or e-mail addresses anywhere in the text. The programs reportedly solve several classes of differential equations with adaptive error control. I suspect that the software packa2e is under production and will be available upon publication of the introductory volume.

One cannot find the heart to criticize any aspect of this delightful and illuminating book which I recommend for general reading and one's continuing education. The book could be used as a text in an advanced undergraduate course or introductory graduate course in curricula of applied mathematics. It could also be used as a text in corresponding courses in a number of disciplines of applied sciences and engineering, but the students must have a strong mathematics back round as well as carry a certain amount of intellectual curiosity, or they may complain about the authors' rightful demand for attention to the fundamentals.