A robust DPG method for singularly perturbed reaction-diffusion problems
Norbert Heuer and Michael Karkulik
SIAM J. Numer. Anal. 55 (3), 1218-1242, 2017.
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 ε-dependent weightings of three field variables
(ε 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 ε.
Moreover, these values lead to a norm for the field variables that is known to be balanced
in ε for model problems with typical boundary layers.
Several numerical examples underline our theoretical estimates and reveal
stability of approximations even for very small ε.