This book collects, explains and analyses basic methods and recent results for the successful numerical solution of singularly perturbed differential equations. Such equations model many physical phenomena and their solutions are characterized by the presence of layers. The book is a wide-ranging introduction to the exciting current literature in this area. It concentrates on linear convection-diffusion equations and related nonlinear flow problems, encompassing both ordinary and partial differential equations. While many numerical methods are considered, particular attention is paid to those with realistic error estimates. The book provides a solid and thorough foundation for the numerical analysis and solution of singular perturbation problems.

The analysis of singular perturbed differential equations began early in this century, when approximate solutions were constructed from asymptotic ex­ pansions. (Preliminary attempts appear in the nineteenth century [vD94].) This technique has flourished since the mid-1960s. Its principal ideas and methods are described in several textbooks. Nevertheless, asymptotic ex­ pansions may be impossible to construct or may fail to simplify the given problem; then numerical approximations are often the only option. The systematic study of numerical methods for singular perturbation problems started somewhat later - in the 1970s. While the research frontier has been steadily pushed back, the exposition of new developments in the analysis of numerical methods has been neglected. Perhaps the only example of a textbook that concentrates on this analysis is [DMS80], which collects various results for ordinary differential equations, but many methods and techniques that are relevant today (especially for partial differential equa­ tions) were developed after 1980.Thus contemporary researchers must comb the literature to acquaint themselves with earlier work. Our purposes in writing this introductory book are twofold. First, we aim to present a structured account of recent ideas in the numerical analysis of singularly perturbed differential equations. Second, this important area has many open problems and we hope that our book will stimulate further investigations.Our choice of topics is inevitably personal and reflects our own main interests.

I. Ordinary Differential Equations.- II. Parabolic Initial-Boundary Value Problems in One Space Dimension.- III. Elliptic Boundary Value Problems.- IV. Incompressible Navier-Stokes Equations.- Appendix: Robust Solvers for Linear Systems.- References.