Two-level variational multiscale finite element methods for Navier–Stokes type variational inequality problem

Abstract

Variational inequality problems Variational multiscale method Two-level finite element method a b s t r a c tIn this paper, we present two-level variational multiscale finite element method based on two local Gauss integrations for Navier-Stokes equations with friction boundary conditions which are of the form of Navier-Stokes type variational inequality of the second kind. We solve Navier-Stokes type variational inequality problem on the coarse mesh and solve linearized Navier-Stokes type variational inequality problem corresponding to Newton iteration on the fine mesh. The error estimates in H 1 norm for velocity and L 2 norm for pressure are derived. Meanwhile, Uzawa iteration schemes are constructed to solve the subproblems in this two-level method. Finally, the numerical results are displayed to support the theoretical analysis.

The steady Navier–Stokes and Stokes systems with mixed boundary conditions including one-sided leaks and pressure

“…For the 2-D steady Stokes problems refer to [5], [28], [38], [39], [41] and for the 3-D steady Stokes problems [29]. For the 2-D steady Navier-Stokes problem refer to [2], [35], [36] and [37]. For the 2-D non-steady Navier-Stokes problem refer to [34].…”