Numerical analysis of a bilateral frictional contact problem for linearly elastic materials

We consider a mathematical model which describes the contact between a linearly elastic body and an obstacle, the so-called foundation. The process is quasistatic and the contact is bilateral, i.e. there is no loss of contact during the process. The friction is modelled with Tresca's law. The variational formulation of the problem is a nonlinear evolutionary inequality for the displacement field which has a unique solution under certain assumptions on the given data. We study spatially semi-discrete and fully discrete schemes for the problem with finite-difference discretization in time and finite-element discretization in space. The numerical schemes have unique solutions. We show the convergence of the scheme under the basic solution regularity. Under appropriate regularity assumptions on the solution, we derive optimal order error estimates. Finally, we present numerical results in the study of two-dimensional test problems.