Two-level pressure projection finite element methods for Navier–Stokes equations with nonlinear slip boundary conditions

The two-level pressure projection stabilized 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. Based on the P1–P1 triangular element and using the pressure projection stabilized finite element method, we solve a small Navier–Stokes type variational inequality problem on the coarse mesh with mesh size H and solve a large Stokes type variational inequality problem for simple iteration or a large Oseen type variational inequality problem for Oseen iteration on the fine mesh with mesh size h. The error analysis obtained in this paper shows that if h=O(H2), the two-level stabilized methods have the same convergence orders as the usual one-level stabilized finite element methods, which is only solving a large Navier–Stokes type variational inequality problem on the fine mesh. Finally, numerical results are given to verify the theoretical analysis.