Numerical approximation of the singularly perturbed heat equation in a circle

TL;DR

This paper develops a finite element method with an enriched Galerkin space to numerically approximate the singularly perturbed heat equation in a circular domain, avoiding boundary mesh refinement.

Contribution

It introduces a boundary layer element based on asymptotic analysis and integrates it into the finite element space for improved numerical approximation.

Findings

The method achieves uniform convergence without boundary mesh refinement.

Numerical simulations validate the effectiveness of the enriched Galerkin approach.

The approach simplifies boundary layer representation in singularly perturbed problems.

Abstract

In this article we study the two dimensional singularly perturbed heat equation in a circular domain. The aim is to develop a numerical method with a uniform mesh, avoiding mesh refinement at the boundary thanks to the use of a relatively simple representation of the boundary layer. We provide the asymptotic expansion of the solution at first order and derive the boundary layer element resulting from the boundary layer analysis. We then perform the convergence analysis introducing the boundary layer element in the finite element space thus obtaining what is called an "enriched Galerkin space". Finally we present and comment on numerical simulations using a quasi-uniform grid and the modified finite element method.