Robust exponential convergence of $hp$-FEM in balanced norms for singularly perturbed reaction-diffusion equations

TL;DR

This paper proves that the $hp$-finite element method achieves robust exponential convergence in balanced and maximum norms for singularly perturbed reaction-diffusion equations on spectral boundary layer meshes, supported by numerical experiments.

Contribution

It introduces a new analysis demonstrating robust exponential convergence of $hp$-FEM in balanced norms for reaction-diffusion problems with boundary layers.

Findings

Robust exponential convergence in balanced norms.

Robust exponential convergence in maximum norm.

Numerical experiments confirm theoretical results.

Abstract

The $hp$-version of the finite element method is applied to a singularly perturbed reaction-diffusion equation posed in one- and two-dimensional domains with analytic boundary. On suitably designed \emph{Spectral Boundary Layer meshes}, robust exponential convergence in a balanced norm is shown. This balanced norm is stronger than the energy norm in that the boundary layers are $O(1)$ uniformly in the singular perturbation parameter. Robust exponential convergence in the maximum norm is also established. The theoretical findings are illustrated with two numerical experiments.