Mathematical modelling using differential equations is an important part of the modern applied sciences and engineering. The process of mathematical modelling often has two aspects: theoretical investigations of the corresponding differential equations (existence, uniqueness and properties of the solution) and the use of computational tools in order to get this solution in a suitable form. The goal of the book under consideration is to present a unified approach to computational mathematical modelling with differential equations, based on a principle of a fusion of mathematics and computation.

The authors begin the first part of the book by considering the numerical solution of the simplest differential equation by quadrature in order to demonstrate the typical questions that arise while solving the problem numerically. The questions of the convergence and error estimation are discussed. The first part of the book also contains the necessary background material from linear algebra and polynomial approximations. The authors introduce Galerkin's method for more general differential equations and conclude the first part with a survey of methods for the numerical solution of linear algebraic systems.

The second part of the book deals with discretizations by Galerkin's method of two-point boundary value problems, and of scalar and vector initial value problems and with the calculus of variations.

The last part of the book is devoted to problems in two and three dimensions. After introducing some basic results from the calculus of several variables and from piecewise polynomial approximation, the basic

types of linear partial differential equations (the Poisson equation, the heat equation, the wave equation and the mixed parabolic/elliptic-hyperbolic convection-diffusion equation) are considered.

The material is often centered around specific examples, with generalizations coming as additional material and worked out in exercises. Two last sections deal with eigenvalue problems for an elliptic operator and with the abstract theory of the finite element method.

The authors advocate the arrangement of the material in this book on pedagogical grounds. They base their argument on both mathematical and implementational points of view, show that many numerical algorithms have their roots in theorems and proofs and demonstrate how these algorithms work when applied to actual mathematical models. The book includes many exercises of various degrees of complexity and many examples and it is well illustrated. The text is accompanied by very nice historical surveys concerning the subject treated. This book is a substantial revision of Johnson's book [Numerical solution of partial differential equations by the finite element method, Studentlitteratur, Lund, 1987; MR 89b:65003b].