LAUSR

Abstract 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 e and the mesh size h. In fact, under the H 3 2 + ν ( Ω ) , 0 ν ≤ 1 2 , regularity of the solution of the contact problems (with and without friction) and with the requirement e > h , we prove a convergence rate of O ( h 1 2 + ν + e 1 2 + ν ) in the energy norm. Therefore, if the penalty parameter is taken as e : = c h θ where 0 θ ≤ 1 , the convergence rate of O ( h θ ( 1 2 + ν ) ) is obtained. In particular, we obtain an optimal linear convergence when e behaves like h (i.e. θ = 1 ) and ν = 1 2 .