The analysis of singular perturbed differential equations began early in this century, when approximate solutions were constructed from asymptotic expansions. (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 expansions 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 equations) were developed after 1980. Thus contemporary researches 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 singular 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.

We have learned a great deal about singular perturbation problems from other researchers. We thank those colleagues who helped and influenced us, including A.E. Berger, P.A. Farrell. A. Felgenhauer, E.C. Gartland, Ch. Großmann, A.F. Hegarty, R.B. Kellogg, J.J.H. Miller, K.W. Morton, G.I. Shishkin, E. Süli, and R. Vulanovic, and especially Herbert Goering and Eugene O´Riordan.

Our work was supported by the Deutsche Forschungsgemeinschaft and by University College Cork Arts Faculty. We are grateful to them, to the Mathematisches Forschungsinstitut in Oberwolfach for its hospitality, and to Springer-Verlag for its cooperation.