A new class of history-dependent quasivariational inequalities was recently studied in
[M. Sofonea and A. Matei, History-dependent quasivariational inequalities arising in
contact mechanics. Eur. J. Appl. Math. 22 (2011) 471–491].
Existence, uniqueness and regularity results were proved and used in the study of several
mathematical models which describe the contact between a deformable body and an obstacle.
The aim of this paper is to provide numerical analysis of the quasivariational
inequalities introduced in the aforementioned paper. To this end we introduce temporally
semi-discrete and fully discrete schemes for the numerical approximation of the
inequalities, show their unique solvability, and derive error estimates. We then apply
these results to a quasistatic frictional contact problem in which the material’s behavior
is modeled with a viscoelastic constitutive law, the contact is bilateral, and friction is
described with a slip-rate version of Coulomb’s law. We discuss implementation of the
corresponding fully-discrete scheme and present numerical simulation results on a
two-dimensional example.