mathematics of computation volume 56,number 194 april 1991,pages 663-675 AN ANALYSIS OF A SINGULARLY PERTURBED TWO-POINT BOUNDARY VALUE PROBLEM USING ONLY FINITE ELEMENT TECHNIQUES MARTINSTYNES AND EUGENEO'RIORDAN Abstract. We give a new analysis of Petrov-Galerkin finite element methods for solving linear singularly perturbed two-point boundary value problems with- out turning points. No use is made of finite difference methodology such as discrete maximum principles, nor of asymptotic expansions. On meshes which are either arbitrary or slightly restricted, we derive energy norm and L norm error bounds. These bounds are uniform in the perturbation parameter. Our proof uses a variation on the classical Aubin-Nitsche argument, which is novel insofar as the L bound is obtained independently of the energy norm bound. 1. INTRODUCTION We consider the analysis of finite element methods for the singularly per- turbed problem (1.1a) Lu(x) = -eu"(x) + a(x)u'(x) + b(x)u(x) = f(x), 0 < x < 1, (1.1b) u(0) = u(l) = 0, where e G (0, 1] is a parameter, a G C2[0, 1], b G c'[0, I], f e Cx[0, 1], and for x e [0, 1] we have (1.1c) a(x) > a > 0. We assume that problem (1.1) has a unique solution u(x). This is guaranteed if e is sufficiently small (see, e.g., Gartland [4, p. 97]). In general, this solution has a boundary layer at x = 1. It is possible in our analysis to weaken the differentiability assumptions on a, b, and /, but for simplicity of presentation we have not done this. Problem (1.1) may be regarded as a linearized one-dimensional version of a convection-dominated flow problem. Many authors have suggested methods for its numerical solution, and at present it is well understood from a computa- tional point of view. However, the analysis of such methods (i.e., the provision Received April 4, 1989; revised November 22, 1989. 1980 Mathematics Subject Classification(1985 Revision). Primary 65L10, 65L60; Secondary 34E15. The first author's research was partly supported by Arts Faculty Research Fund, University College,Cork. ©1991 American Mathematical Society 0025-5718/91 $1.00+ $.25 per page 663 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use