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.