Error control in h- and hp-adaptive FEM for Signorini's problem

This paper presents a posteriori finite element error estimates for Signorini’s problem. The discretization is based on a mixed variational formulation proposed by Haslinger et al. which is extended to higher-order finite elements. The a posteriori error control relies on estimating the discretization error of an auxiliary problem which is given as a variational equation. The estimation consists of error bounds for the discretization error of the auxiliary problem and some further terms which capture the geometrical error and the error in the complementary condition. The derived estimates are applied to hand hp-adaptive refinement and enrichment strategies. Numerical results confirm the applicability of the theoretical findings. In particular, optimal algebraic and almost exponential convergence rates are obtained.