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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 $$\\partial \\Omega $$źΩ, 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+∈1/2+h/∈1/2)-error estimate for velocity and pressure in the energy norm, where h and $$\\epsilon $$∈ 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+∈1/2+h2/∈1/2) by applying reduced-order numerical integration to the penalty term. The theoretical results are confirmed by numerical experiments.

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