### 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_{δ ω} = g on δ 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_{δ ω} by n_{δ 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 H^{1} (ω)^{N} → H^{1/2}(δ ω); u → u? n_{δ 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 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) 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 |
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Pages (from-to) | 869-891 |

Number of pages | 23 |

Journal | ESAIM: Mathematical Modelling and Numerical Analysis |

Volume | 53 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 May 2019 |

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### Keywords

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

### Cite this

*ESAIM: Mathematical Modelling and Numerical Analysis*,

*53*(3), 869-891. https://doi.org/10.1051/m2an/2019008