Semi-discrete stabilized finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions based on regularization procedure

Based on the pressure projection stabilized methods, the semi-discrete finite element approximation to the time-dependent Navier-Stokes equations with nonlinear slip boundary conditions is considered in this paper. Because this class of boundary condition includes the subdifferential property, then the variational formulation is the Navier-Stokes type variational inequality problem. Using the regularization procedure, we obtain a regularized problem and give the error estimate between the solutions of the variational inequality problem and the regularized problem with respect to the regularized parameter e, which means that the solution of the regularized problem converges to the solution of the Navier-Stokes type variational inequality problem as the parameter epsilon --> 0. Moreover, some regularized estimates about the solution of the regularized problem are also derived under some assumptions about the physical data. The pressure projection stabilized finite element methods are used to the regularized problem and some optimal error estimates of the finite element approximation solutions are derived.