Pointwise A Posteriori Error Control of Discontinuous Galerkin Methods for Unilateral Contact Problems

In this article, we derive a reliable and efficient a posteriori error estimator in the supremum norm for a class of discontinuous Galerkin (DG) methods for the frictionless unilateral contact problem between two elastic bodies. The proposed error estimator generalizes the basic residual type estimators for the linear problems in linear elasticity taking into account the nonlinearity on a part of the boundary. The analysis hinges on the super- and sub-solutions constructed by modifying the discrete solution appropriately, and it is carried out in a unified manner which holds for several DG methods. The terms arising from the contact stresses in the error estimator vanish on the discrete full contact set. We illustrate the performance of the proposed error estimator via several numerical experiments in two dimensions.