Robust exponential convergence of hp-FEM in balanced norms for singularly perturbed reaction-diffusion problems: corner domains

TL;DR

This paper demonstrates that the hp-FEM achieves robust exponential convergence in balanced norms for reaction-diffusion problems with boundary and corner layers on polygonal domains, using specially refined meshes.

Contribution

It proves robust exponential convergence of hp-FEM in balanced norms for singularly perturbed reaction-diffusion problems on polygonal domains with corner and boundary layers.

Findings

Achieves exponential convergence in balanced norms

Effective mesh refinement near corners and boundaries

Applicable to singularly perturbed reaction-diffusion equations

Abstract

The hp-version of the finite element method is applied to singularly perturbed reaction-diffusion type equations on polygonal domains. The solution exhibits boundary layers as well as corner layers. On a class of meshes that are suitable refined near the boundary and the corners, robust exponential convergence (in the polynomial degree) is shown in both a balanced norm and the maximum norm.