Robust hybrid finite element methods for reaction-dominated diffusion problems

TL;DR

This paper introduces robust hybrid finite element methods for reaction-dominated diffusion problems, employing enriched local spaces and a posteriori error estimators to improve accuracy and stability in the presence of singular perturbations.

Contribution

It develops a new class of hybrid finite element methods with exponential decay enrichment and robust error estimation for reaction-dominated diffusion.

Findings

Methods are robust against singular perturbations.

Numerical experiments show reduced oscillations.

Error estimators are effective and robust.

Abstract

For a reaction-dominated diffusion problem we study a primal and a dual hybrid finite element method where weak continuity conditions are enforced by Lagrange multipliers. Uniform robustness of the discrete methods is achieved by enriching the local discretization spaces with modified face bubble functions which decay exponentially in the interior of an element depending on the ratio of the singular perturbation parameter and the local mesh-size. A posteriori error estimators are derived using Fortin operators. They are robust with respect to the singular perturbation parameter. Numerical experiments are presented that show that oscillations, if present, are significantly smaller then those observed in common finite element methods.