Penalty Method for the Stationary Navier–Stokes Problems Under the Slip Boundary Condition

We consider the penalty method for the stationary Navier–Stokes equations with the slip boundary condition. The well-posedness and the regularity theorem of the penalty problem are investigated, and we obtain the optimal error estimate (Formula presented.) in (Formula presented.)-norm, where (Formula presented.) is the penalty parameter. We are concerned with the finite element approximation with the P1b / P1 element to the penalty problem. The well-posedness of discrete problem is proved. We obtain the error estimate (Formula presented.) for the non-reduced-integration scheme with (Formula presented.), and the reduced-integration scheme with (Formula presented.), where h is the discretization parameter and d is the spatial dimension. For the reduced-integration scheme with (Formula presented.), we prove the convergence order (Formula presented.). The theoretical results are verified by numerical experiments.