An $hp$ finite element method for a singularly perturbed reaction-convection-diffusion boundary value problem with two small parameters

TL;DR

This paper develops an $hp$ finite element method on a spectral boundary layer mesh for a reaction-convection-diffusion problem with two small parameters, achieving exponential convergence uniformly across parameters.

Contribution

It introduces a novel $hp$ finite element approach on spectral boundary layer meshes for problems with two small parameters, ensuring uniform exponential convergence.

Findings

Method converges exponentially in energy norm

Convergence is uniform with respect to both small parameters

Numerical examples confirm theoretical results

Abstract

We consider a second order singularly perturbed boundary value problem, of reaction-convection-diffusion type with two small parameters, and the approximation of its solution by the $hp$ version of the Finite Element Method on the so-called {\emph{Spectral Boundary Layer}} mesh. We show that the method converges uniformly, with respect to both singular perturbation parameters, at an exponential rate when the error is measured in the energy norm. Numerical examples are also presented, which illustrate our theoretical findings.