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

<jats:p>The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domain<jats:italic>Ω</jats:italic>⊂ ℝ<jats:sup><jats:italic>N</jats:italic></jats:sup>(<jats:italic>N</jats:italic>= 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<jats:italic>u</jats:italic> · <jats:italic>n</jats:italic><jats:sub>∂Ω</jats:sub> = <jats:italic>g</jats:italic>on<jats:italic>∂Ω</jats:italic>. Because the original domain<jats:italic>Ω</jats:italic>must be approximated by a polygonal (or polyhedral) domain<jats:italic>Ω</jats:italic><jats:sub><jats:italic>h</jats:italic></jats:sub>before applying the finite element method, we need to take into account the errors owing to the discrepancy<jats:italic>Ω</jats:italic> ≠ <jats:italic>Ω</jats:italic><jats:sub><jats:italic>h</jats:italic></jats:sub>, that is, the issues of domain perturbation. In particular, the approximation of<jats:italic>n</jats:italic><jats:sub><jats:italic>∂Ω</jats:italic></jats:sub>by<jats:italic>n</jats:italic><jats:sub>∂Ω<jats:italic>h</jats:italic></jats:sub>makes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., continuous right inverse of the normal trace operator<jats:italic>H</jats:italic><jats:sup>1</jats:sup>(<jats:italic>Ω)<jats:sup>N</jats:sup></jats:italic>→<jats:italic>H</jats:italic><jats:sup>1/2</jats:sup>(<jats:italic>∂Ω</jats:italic>);<jats:italic>u</jats:italic>↦<jats:italic>u</jats:italic>⋅<jats:italic>n<jats:sub>∂Ω</jats:sub></jats:italic>. In this paper we indeed prove such a discrete lifting theorem, taking advantage of the nonconforming approximation, and consequently we establish the error estimates<jats:italic>O</jats:italic>(<jats:italic>h<jats:sup>α</jats:sup></jats:italic>+<jats:italic>ε</jats:italic>) and<jats:italic>O</jats:italic>(<jats:italic>h<jats:sup>2α</jats:sup></jats:italic>+<jats:italic>ε</jats:italic>) for the velocity in the<jats:italic>H</jats:italic><jats:sup>1</jats:sup>- and<jats:italic>L</jats:italic><jats:sup>2</jats:sup>-norms respectively, where<jats:italic>α</jats:italic> = 1 if<jats:italic>N</jats:italic> = 2 and<jats:italic>α</jats:italic> = 1/2 if<jats:italic>N</jats:italic> = 3. This improves the previous result [T. Kashiwabara<jats:italic>et al.</jats:italic>,<jats:italic>Numer. Math.</jats:italic><jats:bold>134</jats:bold>(2016) 705–740] obtained for the conforming approximation in the sense that there appears no reciprocal of the penalty parameter<jats:italic>ϵ</jats:italic>in the estimates.</jats:p>