A higher order numerical method for singularly perturbed elliptic problems with characteristic boundary layers

TL;DR

This paper introduces a Petrov-Galerkin finite element method using exponential splines and Shishkin meshes to effectively solve singularly perturbed elliptic problems with boundary layers, achieving higher convergence rates.

Contribution

The paper presents a novel higher order Petrov-Galerkin finite element method combining exponential splines and Shishkin meshes for improved accuracy in singularly perturbed elliptic problems.

Findings

The method is stable and parameter-uniform.

Achieves higher order convergence than upwinding.

Effectively captures boundary layers.

Abstract

A Petrov-Galerkin finite element method is constructed for a singularly perturbed elliptic problem in two space dimensions. The solution contains a regular boundary layer and two characteristic boundary layers. Exponential splines are used as test functions in one coordinate direction and are combined with bilinear trial functions defined on a Shishkin mesh. The resulting numerical method is shown to be a stable parameter-uniform numerical method that achieves a higher order of convergence compared to upwinding on the same mesh.