Two-Level Penalty Finite Element Methods for Navier-Stokes Equations with Nonlinear Slip Boundary Conditions

Abstract

The two-level penalty finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions are investigated in this paper, whose variational formulation is the Navier-Stokes type variational inequality problem of the second kind. The basic idea is to solve the Navier-Stokes type variational inequality problem on a coarse mesh with mesh size $H$ in combining with solving a Stokes type variational inequality problem for simple iteration or solving a Oseen type variational inequality problem for Oseen iteration on a fine mesh with mesh size $h$. The error estimate obtained in this paper shows that if $H = O(h^{5/9})$, then the two-level penalty methods have the same convergence orders as the usual one-level penalty finite element method, which is only solving a large Navier-Stokes type variational inequality problem on the fine mesh. Hence, our methods can save an amount of computational work.