A robust DPG method for singularly perturbed reaction-diffusion problems

TL;DR

This paper introduces a discontinuous Petrov-Galerkin method with optimal test functions designed for reaction-dominated diffusion problems, ensuring robustness and stability across various small diffusion parameters.

Contribution

The paper develops a new DPG method with an ultra-weak formulation that remains stable and quasi-optimal regardless of the small diffusion parameter, improving robustness for singularly perturbed problems.

Findings

Method achieves error estimates independent of epsilon

Numerical examples confirm stability for very small epsilon

Balanced norm effectively captures boundary layers

Abstract

We present and analyze a discontinuous Petrov-Galerkin method with optimal test functions for a reaction-dominated diffusion problem in two and three space dimensions. We start with an ultra-weak formulation that comprises parameters $\alpha$, $\beta$ to allow for general $\varepsilon$-dependent weightings of three field variables ($\varepsilon$ being the small diffusion parameter). Specific values of $\alpha$ and $\beta$ imply robustness of the method, that is, a quasi-optimal error estimate with a constant that is independent of $\varepsilon$. Moreover, these values lead to a norm for the field variables that is known to be balanced in $\varepsilon$ for model problems with typical boundary layers. Several numerical examples underline our theoretical estimates and reveal stability of approximations even for very small $\varepsilon$.