Penalty method with P1/P1 finite element approximation for the Stokes equations under the slip boundary condition

We consider the P1/P1 or P1b/P1 finite element approximations to the Stokes equations in a bounded smooth domain subject to the slip boundary condition. A penalty method is applied to address the essential boundary condition (Formula presented.) on (Formula presented.), which avoids a variational crime and simultaneously facilitates the numerical implementation. We give (Formula presented.)-error estimate for velocity and pressure in the energy norm, where h and (Formula presented.) denote the discretization parameter and the penalty parameter, respectively. In the two-dimensional case, it is improved to (Formula presented.) by applying reduced-order numerical integration to the penalty term. The theoretical results are confirmed by numerical experiments.