Averaging techniques yield reliable a posteriori finite element error control for obstacle problems

The reliability of frequently applied averaging techniques for a posteriori error control has recently been established for a series of finite element methods in the context of second-order partial differential equations. This paper establishes related reliable and efficient a posteriori error estimates for the energy-norm error of an obstacle problem on unstructured grids as a model example for variational inequalities. The surprising main result asserts that the distance of the piecewise constant discrete gradient to any continuous piecewise affine approximation is a reliable upper error bound up to known higher order terms, consistency terms, and a multiplicative constant.