Penalty method with Crouzeix–Raviart approximation for the Stokes equations under slip boundary condition

Abstract

The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domainΩ⊂ ℝN(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 conditionu · n∂Ω = gon∂Ω. Because the original domainΩmust be approximated by a polygonal (or polyhedral) domainΩhbefore applying the finite element method, we need to take into account the errors owing to the discrepancyΩ ≠ Ωh, that is, the issues of domain perturbation. In particular, the approximation ofn∂Ωbyn∂Ωhmakes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., continuous right inverse of the normal trace operatorH1(Ω)N→H1/2(∂Ω);u↦u⋅n∂Ω. In this paper we indeed prove such a discrete lifting theorem, taking advantage of the nonconforming approximation, and consequently we establish the error estimatesO(hα+ε) andO(h2α+ε) for the velocity in theH1- andL2-norms respectively, whereα = 1 ifN = 2 andα = 1/2 ifN = 3. This improves the previous result [T. Kashiwabaraet al.,Numer. Math.134(2016) 705–740] obtained for the conforming approximation in the sense that there appears no reciprocal of the penalty parameterϵin the estimates.