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 $Ω\subset \mathbb R^N \, (N=2,3)$. 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 $u \cdot n_{\partialΩ} = g$ on $\partialΩ$. Because the original domain $Ω$ must be approximated by a polygonal (or polyhedral) domain $Ω_h$ before applying the finite element method, we need to take into account the errors owing to the discrepancy $Ω\neq Ω_h$, that is, the issues of domain perturbation. In particular, the approximation of $n_{\partialΩ}$ by $n_{\partialΩ_h}$ makes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., right-continuous inverse of the normal trace operator $H^1(Ω)^N \to H^{1/2}(\partialΩ)$; $u \mapsto u\cdot n_{\partialΩ}$. In this paper we indeed prove such a discrete lifting theorem, taking advantage of the nonconforming approximation, and consequently we establish the error estimates $O(h^α+ ε)$ and $O(h^{2α} + ε)$ for the velocity in the $H^1$- and $L^2$-norms respectively, where $α= 1$ if $N=2$ and $α= 1/2$ if $N=3$. This improves the previous result [T. Kashiwabara et al., Numer. Math. 134 (2016), pp. 705--740] obtained for the conforming approximation in the sense that there appears no reciprocal of the penalty parameter $ε$ in the estimates.