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

¹ The Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan
² Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan
³ Department of Applied Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku, Tokyo 162-8601, Japan

^** Corresponding author: tkashiwa@ms.u-tokyo.ac.jp

Received: 25 September 2018
Accepted: 26 January 2018

Abstract

The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domain Ω ⊂ 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_∂Ω = g on ∂Ω. 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 ≠ Ω_h, that is, the issues of domain perturbation. In particular, the approximation of n_∂Ω by n_∂Ωh makes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., continuous right inverse of the normal trace operator H¹ (Ω)^N → H^1/2(∂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 estimates O(h^α + ε) and O(h^2α + ε) for the velocity in the H¹- and L²-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) 705–740] obtained for the conforming approximation in the sense that there appears no reciprocal of the penalty parameter ϵ in the estimates.