Penalty method with P1/P1 finite element approximation for the Stokes equations under 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 $u\cdot n=g$ on $\mathrm{\partial }\mathrm{\Omega }$, which avoids a variational crime and simultaneously facilitates the numerical implementation. We give $O(h^{1\mathrm{/}2}+{ϵ}^{1\mathrm{/}2}+h\mathrm{/}{ϵ}^{1\mathrm{/}2})$-error estimate for velocity and pressure in the energy norm, where $h$ and $ϵ$ denote the discretization parameter and the penalty parameter, respectively. In the two-dimensional case, it is improved to $O(h+{ϵ}^{1\mathrm{/}2}+h^2\mathrm{/}{ϵ}^{1\mathrm{/}2})$ by applying reduced-order numerical integration to the penalty term. The theoretical results are confirmed by numerical experiments.