Numerical solution of a singularly perturbed two-point boundary value problem using equidistribution: analysis of convergence

Abstract

Adaptive grid methods are becoming established as valuable computational techniques for the numerical solution of differential equations with near-singular solutions. Adaptive methods are equally effective in approximating solutions of problems with boundary layers or interior layers (see, for example, Mulholland et al., SIAM J. Sci. Comput. 19(4) (1998) 1261–1289). Much is now being done in developing error analyes for methods that are based on adaptivity. In this paper, we present a rigorous error analysis for the solution of a singularly perturbed two-point boundary value problem on a grid that is constructed adaptively from a knowledge of the exact solution. The discrete solutions are generated by an upwind finite difference scheme and the grid is formed by equidistributing a monitor function based on arc-length. An error analysis shows that the discrete solutions are uniformly convergent with respect to the perturbation parameter, epsilon. The epsilon-uniform convergence is confirmed by numerical computations.