A robust DPG method for singularly perturbed reaction-diffusion problems
For computational scientists solving singularly perturbed reaction-diffusion problems, this method provides a robust numerical scheme with epsilon-independent error bounds.
The paper presents a discontinuous Petrov-Galerkin method for reaction-dominated diffusion problems that achieves robust error estimates independent of the small diffusion parameter, with numerical examples confirming stability for very small epsilon.
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 $α$, $β$ to allow for general $\varepsilon$-dependent weightings of three field variables ($\varepsilon$ being the small diffusion parameter). Specific values of $α$ and $β$ 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 $\varepsilon$.