Optimal convergence analysis of the unilateral contact problem with and without Tresca friction conditions by the penalty method

We study the linear finite element approximation of the elasticity equations with and without unilateral friction contact (of Tresca type) conditions in a polygonal or polyhedral domain. The unilateral contact condition is weakly imposed by the penalty method. We derive error estimates which depend on the penalty parameter epsilon and the mesh size h. In fact, under the H3/2+nu (Omega), 0 < v <= 1/2, regularity of the solution of the contact problems (with and without friction) and with the requirement epsilon > h, we prove a convergence rate of O (h(1/2+nu) + epsilon(1/2+nu)) in the energy norm. Therefore, if the penalty parameter is taken as epsilon := ch(theta) where 0 < theta <= 1, the convergence rate of O (h(theta(1/2+nu))) is obtained. In particular, we obtain an optimal linear convergence when e behaves like h (i.e. theta = 1) and nu = 1/2. (C) 2018 Elsevier Inc. All rights reserved.