Analysis of a Time Implicit Scheme for the Oseen Model Driven by Nonlinear Slip Boundary Conditions

Abstract

This work is concerned with the time discrete analysis of the Oseen system of equations driven by nonlinear slip boundary conditions of friction type. We study the existence of solutions of the time discrete model and derive several a priori estimates needed to recover the solution of the continuous problem by means of weak compactness. Moreover, for the difference between the exact and approximate solutions, we obtain the rate of convergence of order one with respect to the time step without imposing extra regularity on the weak solution.

Evolutionary Oseen Model for Generalized Newtonian Fluid with Multivalued Nonmonotone Friction Law

“…The standard argument shows that the entire sequence {Av n } converges weakly in V * to Av. This concludes the proof of property (5).+…”

“…Such an approximation is used in implicit time discretization method applied together with a fixed point strategy, see e.g. [5,12] and the references therein.…”

“…If the potential generating the slip condition is a convex function, then the variational formulation of the problem is a variational inequality, see e.g. [5,9,10,14]. The stationary and non-stationary Oseen equations with homogeneous Dirichlet boundary condition were studied by the Galerkin method in [8] while stationary flow of non-Newtonian fluid with frictional boundary conditions have been recently treated in [23].…”

“…x ∈ Γ C . This choice of j leads to a threshold slip condition considered in [5,9,10]. It is easy to calculate that…”

A new class of variational‐hemivariational inequalities for steady Oseen flow with unilateral and frictional type boundary conditions

“…Such an approximation is used in implicit time discretization method applied together with a fixed point strategy, see e.g. [] and the references therein.…”