A Robust Dpg Method For Singularly Perturbed Reaction-Diffusion Problems

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 ultraweak formulation that comprises parameters alpha, beta to allow for general epsilon-dependent weightings of three field variables (epsilon 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 epsilon. Moreover, these values lead to a norm for the field variables that is known to be balanced in epsilon for model problems with typical boundary layers. Several numerical examples underline our theoretical estimates and reveal stability of approximations even for very small epsilon. Keywords. Author Keywords:reaction-dominated diffusion

singularly perturbed problem

boundary layers

discontinuous Petrov-Galerkin method