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

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. The robust upper and local 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. For the proof of efficiency we construct special bubble functions which cope with the structure of the approximation of the constraining force and the $\epsilon$-dependency.