Penalty finite element method for Stokes problem with nonlinear slip boundary conditions

The penalty finite element method for Stokes problem with nonlinear slip boundary conditions, based on the finite element subspace (Vh,Mh) which satisfies the discrete inf-sup condition, is investigated in this paper. Since this class of nonlinear slip boundary conditions include the subdifferential property, the weak variational formulation associated with Stokes problem is variational inequality. Under some regularity assumptions, we obtain the optimal H1 and L2 error estimates between u and uh, and between u and uhε, where the error orders are ε+h for H1 error and ε+h2 for L2 error.