Residual-type A Posteriori Estimators for a Singularly Perturbed Reaction-Diffusion Variational Inequality -- Reliability, Efficiency and Robustness

Abstract

We derive a new residual-type a posteriori estimator for a singularly perturbed reaction-diffusion problem with obstacle constraints. It generalizes robust residual estimators for unconstrained singularly perturbed equations. Upper and lower bounds are derived with respect to an error notion which measures the error of the solution and of a suitable approximation of the constraining force in a $\epsilon$-dependent energy norm and its dual norm. While the robust upper bound is proven for non-discrete obstacle function, the local lower bounds are derived for discrete obstacle functions. For the proof of the local lower bounds we construct special bubble functions which cope with the structure of the approximation of the constraining force and the $\epsilon$-dependency.