SUMMARY The penalty finite element method for Navier–Stokes equations with nonlinear slip boundary conditions is investigated in this paper. Since this class of nonlinear slip boundary conditions include the subdifferential property, the weak variational formulation is a variational inequality problem of the second kind. Using the penalty finite element approximation, we obtain optimal error estimates between the exact solution and the finite element approximation solution. Finally, we show the numerical results which are in full agreement with the theoretical results. Copyright © 2011 John Wiley & Sons, Ltd.

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 ε, which means that the solution of the regularized problem converges to the solution of the Navier-Stokes type variational inequality problem as the parameter ε −→ 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.

The time-dependent Navier-Stokes equations with leak boundary conditions are investigated in this paper. The equivalent variational inequality is derived, and the weak and strong solvabilities of this variational inequality are obtained by the Galerkin approximation method and the regularized method. In addition, the continuous dependence property of solutions on given initial data is established, from which the strong solution is unique.

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