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

This paper deals with the non-conservative coupling of two one-dimensional barotropic Euler systems at an interface at x = 0. The closure pressure laws differ in the domains x < 0 and x > 0, and a Dirac source term concentrated at x = 0 models singular pressure losses. We propose two numerical methods. The first one relies on ghost state reconstructions at the interface while the second is based on a suitable relaxation framework. Both methods satisfy a well-balanced property for stationary solutions. In addition, the second method preserves mass conservation and exactly restores the prescribed singular pressure drops for both unsteady and steady solutions.

The instationary Stokes and Navier−Stokes equations are considered in a simultaneously space-timevariational saddle point formulation, so involving both velocities u and pressure p. For the instationaryStokes problem, it is shown that the corresponding operator is a boundedlyinvertible linear mapping between H1 and H'2, both Hilbertspaces H1 and H2 beingCartesian products of (intersections of) Bochner spaces, or duals of those. Based on theseresults, the operator that corresponds to the Navier−Stokes equations is shown to mapH1 into H'2, with a Fréchetderivative that, at any (u,p) ∈H1, is boundedly invertible. These resultsare essential for the numerical solution of the combined pair of velocities and pressureas function of simultaneously space and time. Such a numerical approach allows for theapplication of (adaptive) approximation from tensor products of spatial and temporal trialspaces, with which the instationary problem can be solved at a computational complexitythat is of the order as for a corresponding stationary problem.