Numerical analysis of a singularly perturbed convection diffusion problem with shift in space

TL;DR

This paper analyzes a singularly perturbed convection-diffusion problem with a spatial shift, employing asymptotic expansions and layer-adapted meshes to achieve high-order finite element solutions with confirmed numerical stability and efficiency.

Contribution

It introduces a novel numerical analysis approach for a shifted singularly perturbed problem using asymptotic solution decomposition and a new mesh coarsening strategy.

Findings

High-order finite element methods are effective on layer-adapted meshes.

A new mesh coarsening approach reduces computational cost in weak layer regions.

Numerical experiments validate the theoretical stability and accuracy results.

Abstract

We consider a singularly perturbed convection-diffusion problem that has in addition a shift term. We show a solution decomposition using asymptotic expansions and a stability result. Based upon this we provide a numerical analysis of high order finite element method on layer adapted meshes. We also apply a new idea of using a coarser mesh in places where weak layers appear. Numerical experiments confirm our theoretical results.