Analysis of difference approximations to a singularly perturbed two-point boundary value problem on an adaptively generated grid

Abstract

Over the last few years there has been a significant growth in the use of adaptive grid methods for the numerical solution of differential equations with steep solutions. Little has been done, however, on the error analysis of adaptive methods. In this paper, we present an analysis for an upwind finite difference solution of a singular perturbation problem on a grid that is generated adaptively by equidistributing a monitor function based on the exact solution. It is shown that the discrete solutions converge uniformly with respect to the perturbation parameter, epsilon. This epsilon-uniform convergence is illustrated by numerical computations.