Parameter uniform essentially first order convergence of a fitted mesh method for a class of parabolic singularly perturbed Robin problem for a system of reaction-diffusion equations

TL;DR

This paper introduces a finite difference method on a Shishkin mesh for reaction-diffusion systems with singular perturbations, achieving uniform first order convergence in space and time.

Contribution

The paper develops and proves the uniform convergence of a fitted mesh method for a class of parabolic singularly perturbed reaction-diffusion systems, addressing boundary layers.

Findings

First order convergence in time

Essentially first order convergence in space

Uniform convergence independent of perturbation parameters

Abstract

In this paper, a class of linear parabolic systems of singularly perturbed second order differential equations of reaction-diffusion type with initial and Robin boundary conditions is considered. The components of the solution $\vec u$ of this system exhibit parabolic boundary layers with sublayers. A numerical method composed of a classical finite difference scheme on a piecewise uniform Shishkin mesh is suggested. This method is proved to be first order convergent in time and essentially first order convergent in the space variable in the maximum norm uniformly in the perturbation parameters