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A fully discrete ε-uniform method for singular perturbation problems on equidistant meshes

We propose a fully discrete ε-uniform finite-difference method on an equidistant mesh for a singularly perturbed two-point boundary-value problem (BVP). We start with a fitted operator method reflecting the singular perturbation nature of the problem through a local BVP. However, to solve the local BVP, we employ an upwind method on a Shishkin mesh in local domain, instead of solving it exactly. Thus, we show that it is possible to develop a ε-uniform method, totally in the context of finite differences, without solving any differential equation exactly. We further study the convergence properties of the numerical method proposed and prove that it nodally converges to the true solution for any ε. Finally, a set of numerical experiments is carried out to validate the theoretical results computationally.