hphp-FEM convergence for unilateral contact problems with Tresca friction in plane linear elastostatics

This paper is concerned with the convergence of the hphp-version of the finite element method (hphp-FEM) for some nonsmooth unilateral problems in linear elastostatics. We consider in particular the deformation of an elastic body unilaterally supported by a rigid foundation, admitting Tresca friction (given friction) along the rigid foundation, solely subjected to body forces and surface tractions without being fixed along some part of its boundary. For the discretization of the unilateral constraint and the nonsmooth friction functional we employ Gauss–Lobatto quadrature. We show convergence of the hphp-FEM approximations for mechanically definite problems without imposing any regularity assumption. Moreover we treat the coercive case, when the body is fixed along some part of the boundary. Based on an abstract Céa–Falk estimate and operator interpolation arguments, we establish an a priori error estimate in the energy norm under a reasonable regularity assumption.