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 $$u\cdot n = g$$u·n=g on $$\partial \Omega $$∂Ω, which avoids a variational crime and simultaneously facilitates the numerical implementation. We give $$O(h^{1/2} + \epsilon ^{1/2} + h/\epsilon ^{1/2})$$O(h1/2+ϵ1/2+h/ϵ1/2)-error estimate for velocity and pressure in the energy norm, where h and $$\epsilon $$ϵ denote the discretization parameter and the penalty parameter, respectively. In the two-dimensional case, it is improved to $$O(h + \epsilon ^{1/2} + h^2/\epsilon ^{1/2})$$O(h+ϵ1/2+h2/ϵ1/2) by applying reduced-order numerical integration to the penalty term. The theoretical results are confirmed by numerical experiments.