Residual-type a posteriori error estimator for a quasi-static Signorini contact problem

Abstract

Abstract We present a new residual-type a posteriori estimator for a quasi-static Signorini problem. The theoretical results are derived for two- and three-dimensional domains and the case of nondiscrete gap functions is addressed. We derive global upper and lower bounds with respect to an error notion, which measures the error in the displacements, the velocities and a suitable approximation of the contact forces. Further, local lower bounds for the spatial error at each discrete time point are given. The estimator splits in temporal and spatial contributions, which can be used for the adaptation of the time step as well as the mesh size. In the derivation of the estimator the local properties of the solution are exploited such that the spatial estimator has no contributions related to the nonlinearities in the interior of the actual time-dependent contact zone, but gives rise to an appropriate refinement of the free boundary zone.