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 H1 norm for velocity and L2L2 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.