A numerical method for singularly perturbed convection-diffusion problems posed on smooth domains

TL;DR

This paper introduces a finite difference method tailored for singularly perturbed convection-diffusion problems on smooth domains, combining domain decomposition with specialized meshes to effectively handle boundary layers.

Contribution

It presents a novel numerical approach that integrates a Shishkin mesh aligned with boundary curvature and a domain decomposition technique for improved accuracy.

Findings

Numerical results confirm the method's effectiveness.

The approach accurately captures boundary layers.

The method demonstrates stability and convergence.

Abstract

A finite difference method is constructed to solve singularly perturbed convection-diffusion problems posed on smooth domains. Constraints are imposed on the data so that only regular exponential boundary layers appear in the solution. A domain decomposition method is used, which uses a rectangular grid outside the boundary layer and a Shishkin mesh, aligned to the curvature of the outflow boundary, near the boundary layer. Numerical results are presented to demonstrate the effectiveness of the proposed numerical algorithm.