An ?-uniform method on equidistant meshes

For a singularly-perturbed two-point boundary value problem, we propose an?-uniform finite difference method on an equidistant mesh which requires no exactsolution of a differential equation. We start with a full-fitted operator methodreflecting the singular perturbation nature of the problem through a local boundaryvalue problem. However, to solve the local boundary value problem, we employ anupwind method on a Shishkin mesh in local domain, instead of solving it exactly.We further study the convergence properties of the numerical method proposed andprove it nodally converges to the true solution for any ?.