Parameter-uniform numerical methods for singularly perturbed parabolic problems with incompatible boundary-initial data

TL;DR

This paper introduces a numerical method combining finite difference schemes on specialized meshes with analytical functions to effectively approximate solutions of singularly perturbed parabolic problems with incompatible boundary-initial data, achieving near first-order uniform convergence.

Contribution

The paper presents a novel parameter-uniform numerical method that handles incompatible boundary-initial data in singularly perturbed parabolic problems, with proven convergence and numerical validation.

Findings

Almost first-order parameter-uniform convergence achieved.

Numerical results confirm theoretical error bounds.

Method effectively manages incompatible boundary-initial data.

Abstract

Numerical approximations to the solution of a linear singularly perturbed parabolic reaction-diffusion problem with incompatible bound\-ary-initial data are generated, The method involves combining the computational solution of a classical finite difference operator on a tensor product of two piecewise-uniform Shishkin meshes with an analytical function that captures the local nature of the incompatibility. A proof is given to show almost first order parameter-uniform convergence of these numerical/analytical approximations. Numerical results are given to illustrate the theoretical error bounds.