PENALTY METHOD WITH CROUZEIX-RAVIART APPROXIMATION FOR THE STOKES EQUATIONS UNDER SLIP BOUNDARY CONDITION

The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domain Omega subset of R-N (N = 2, 3). We propose a finite element scheme based on the nonconforming P1/P0 approximation (Crouzeix-Raviart approximation) combined with a penalty formulation and with reduced-order numerical integration in order to address the essential boundary condition u . n(partial derivative Omega) = g on partial derivative Omega. Because the original domain Omega must be approximated by a polygonal (or polyhedral) domain Omega(h) before applying the finite element method, we need to take into account the errors owing to the discrepancy Omega not equal Omega(h), that is, the issues of domain perturbation. In particular, the approximation of n(partial derivative Omega) by n(partial derivative Omega h), makes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., continuous right inverse of the normal trace operator H-1(Omega)(N) -> H-1/(2) (partial derivative Omega); u bar right arrow u . n(partial derivative Omega). In this paper we indeed prove such a discrete lifting theorem, taking advantage of the nonconforming approximation, and consequently we establish the error estimates O(h(alpha) + epsilon) and O(h(2 alpha) + epsilon) for the velocity in the H-1 - and L-2-norms respectively, where alpha = 1 if N = 2 and alpha = 1/2 if N = 3. This improves the previous result [T. Kashiwabara et al., Numer. Math. 134 (2016) 705-740] obtained for the conforming approximation in the sense that there appears no reciprocal of the penalty parameter epsilon in the estimates.