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

In 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. (C) 2015 Elsevier B.V. All rights reserved.