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

Takahito Kashiwabara, Issei Oikawa, Guanyu Zhou

Research output: Contribution to journalArticle

### 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 · nδ ω = g on δ ω. Because the original domain ω must be approximated by a polygonal (or polyhedral) domain ωh before applying the finite element method, we need to take into account the errors owing to the discrepancy ω = ωh, that is, the issues of domain perturbation. In particular, the approximation of nδ ω by nδ 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 → H1/2(δ ω); u → u? nδ 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(hα + ϵ) and O(h + 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.

Original language English 869-891 23 ESAIM: Mathematical Modelling and Numerical Analysis 53 3 https://doi.org/10.1051/m2an/2019008 Published - 1 May 2019

### Keywords

• Discrete H1/2-norm
• Domain perturbation
• Nonconforming FEM
• Slip boundary condition
• Stokes equations