A parameter uniform numerical method for 2D singularly perturbed elliptic differential-difference equations

Abstract

The objective of this article is to develop and analyze a parameter uniform numerical method for solving 2D singularly perturbed elliptic convection-diffusion differential-difference equations. The previous studies in this area have mainly focused on cases involving small shifts, but practical scenarios may involve shifts of arbitrary sizes, both larger and smaller. Despite this, the 2D singularly perturbed elliptic differential-difference equations involving shifts of arbitrary sizes have remained unsolved. To address this challenge, a fitted operator finite difference method is developed on a special equidistant mesh that ensures the nodal point coincides with the shifts after discretization.. The scheme is rigorously analyzed to prove parameter uniform a priori error estimate in \(L_{\infty }\) norm. To illustrate the theoretical findings, we provide numerical results and implement numerical experiments to investigate the effect of shifts on the solution.