Singularly perturbed reaction-diffusion problems as first order systems

TL;DR

This paper reformulates singularly perturbed reaction-diffusion problems as first order systems, employing specialized finite element methods to achieve optimal convergence on layer-adapted meshes.

Contribution

It introduces a novel first order system approach with tailored finite element discretizations for improved convergence analysis.

Findings

Optimal convergence order achieved in a balanced norm.

Effective handling of layer phenomena on adapted meshes.

Comparable accuracy to second order formulations.

Abstract

We consider a singularly perturbed reaction diffusion problem as a first order two-by-two system. Using piecewise discontinuous polynomials for the first component and $H_{div}$-conforming elements for the second component we provide a convergence analysis on layer adapted meshes and an optimal convergence order in a balanced norm that is comparable with a balanced $H^2$-norm for the second order formulation.