An adaptation of Nitscheʼs method to the Tresca friction problem

Abstract

We propose a simple adaptation to the Tresca friction case of the Nitsche-based finite element method introduced previously for frictionless unilateral contact. Both cases of unilateral and bilateral contact with friction are taken into account, with emphasis on frictional unilateral contact for the numerical analysis. We manage to prove theoretically the fully optimal convergence rate of the method in the H1(Ω)-norm which is O(h12+ν) when the solution lies in H32+ν(Ω), 0<ν⩽k−1/2, in two dimensions and three dimensions, for Lagrange piecewise linear (k=1) and quadratic (k=2) finite elements. No additional assumption on the friction set is needed to obtain this proof.