A posteriori error control of hp-finite elements for variational inequalities of the first and second kind

In this paper, residual‐based a posteriori error estimates for variational inequalities, including those of the second kind, are proposed. The variational formulation and its discretizations are considered in an abstract framework of general Banach spaces. The main idea is to express the residual in terms of some Lagrange multipliers which can be obtained, for instance, by some post‐processing. The residual itself is defined for an arbitrary element of the trial space. The error of this element and of the solution of the variational inequality is then estimated by the dual norm of the residual plus some computable remainder terms. This concept is applied to a variety of (frictional) contact problems, such as Signorini and obstacle problems as well as to the Bingham fluid problem. The applicability of the estimates is confirmed by several numerical experiments. In particular, the general framework allows for the discretization with $hp$ ‐adaptivity which leads to nearly exponential convergence rates in most cases.