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- Title
Penalty method with Crouzeix–Raviart approximation for the Stokes equations under slip boundary condition.
- Authors
Kashiwabara, Takahito; Oikawa, Issei; Zhou, Guanyu
- Abstract
The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domain Ω ⊂ ℝN (N = 2,3) Ω ⊂ R N (N = 2,3) $ \Omega \subset {\mathbb{R}}^N\enspace (N=\mathrm{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 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 n∂Ω by n∂Ωh n ∂ Ω h $ {n}_{\mathrm{\partial }{\mathrm{\Omega }}_h}$ 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(∂Ω) H 1 (Ω) N → H 1 / 2 (∂ Ω) $ {H}^1(\mathrm{\Omega }{)}^N\to {H}^{1/2}(\mathrm{\partial \Omega })$ ; u ↦ u⋅n∂Ω u ↦ u ⋅ n ∂ Ω $ u\mapsto u\cdot {n}_{\mathrm{\partial \Omega }}$. 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) O (h α + ε) $ O({h}^{\alpha }+\epsilon)$ and O(h2α + ε) O (h 2 α + ε) $ O({h}^{2\alpha }+\epsilon)$ 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.
- Subjects
STOKES equations; POLYHEDRAL functions; FINITE element method; NUMERICAL integration
- Publication
ESAIM: Mathematical Modelling & Numerical Analysis (ESAIM: M2AN), 2019, Vol 53, Issue 3, p869
- ISSN
2822-7840
- Publication type
Article
- DOI
10.1051/m2an/2019008