%0 Journal Article
%T A robust DPG method for singularly perturbed reaction-diffusion problems
%A Norbert Heuer
%A Michael Karkulik
%J Mathematics
%D 2015
%I arXiv
%X 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 $\varepsilon$.
%U http://arxiv.org/abs/1509.07560v1