Numerical approximations to the scaled first derivatives of a two parameter singularly perturbed problem

TL;DR

This paper develops a finite difference method on a special mesh to accurately approximate scaled derivatives in a two-parameter singularly perturbed problem, ensuring uniform convergence across parameters.

Contribution

It introduces a discretization approach using a Shishkin mesh that achieves parameter-uniform convergence of scaled derivatives in a two-parameter singularly perturbed problem.

Findings

Scaled discrete derivatives converge uniformly to continuous derivatives.

Method effectively handles two singular perturbation parameters.

Provides a reliable numerical scheme for complex layered problems.

Abstract

A singularly perturbed problem involving two singular perturbation parameters is discretized using the classical upwinded finite difference scheme on an appropriate piecewise-uniform Shishkin mesh. Scaled discrete derivatives (with scaling only used within the layers) are shown to be parameter uniformly convergent to the scaled first derivatives of the continuous solution.