Finite element approximations to a class of viscoelastic problems with short memory under conditions of friction

Abstract

In this paper, we consider the finite element approximations of a class of viscoelastic problems with short memory under frictional conditions. The problem is formulated in the form of a variational inequality and the frictional forces are included in the form of a non differentiable functional. Approximating the friction functional by a convex differentiable functional, we transform the variational inequality formulation into a variational equality form. Based on a priori bounds and compactness arguments, existence, and uniqueness and regularity results are obtained. Then finite element Galerkin method is applied in the spatial direction and error estimates are derived for the semidiscrete scheme. Finally, we discretize in time by replacing the time derivative with the help of difference quotients and discuss the error estimates for the completely discrete scheme.