Penalty method with Crouzeix-Raviart finite element approximation for the Stokes equations under the slip boundary condition
Penalty method with Crouzeix-Raviart finite element approximation for the Stokes equations under the slip boundary condition
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滑移边界条件下Stokes方程的Crouzeix-Raviart有限元近似罚分法
DOI：
10.1051/m2an/2019008
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
and Guanyu Zhou
中科院分区：
文献类型：
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作者：
Takahito Kashiwabara; Issei Oikawa;and Guanyu Zhou
在光滑域中考虑受非齐次滑移边界条件影响的斯托克斯方程。我们提出了一种基于非相容 P1/P0 近似（Crouzeix-Raviart 近似）并结合惩罚公式和降阶数值积分的有限元方案，以解决基本边界条件 u·n∂Ω=gon∂Ω。由于在应用有限元方法之前，必须将原始域Ω近似为多边形（或多面体）域Ωh，因此需要考虑由于差异Ω≠Ωh而产生的误差，即域扰动问题。特别是，n∂Ωby 的近似使得我们是否有提升定理的离散对应物（即法线迹算子 的连续右逆）变得非常重要； 。在本文中，我们确实利用非一致性近似证明了这样一个离散提升定理，因此我们分别建立了 H1 和 L2 范数中速度的误差估计 和 ，其中 α= 1 ifN= 2 和 α= 1/2 if N= 3。这改进了之前的结果 [T. Kashiwabara等人，数字。 Math.134(2016) 705–740] 获得一致近似，因为在估计中不存在惩罚参数 ϵ 的倒数。
The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domain . 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 conditionu·n∂Ω=gon∂Ω. Because the original domainΩmust be approximated by a polygonal (or polyhedral) domainΩhbefore applying the finite element method, we need to take into account the errors owing to the discrepancyΩ≠Ωh, that is, the issues of domain perturbation. In particular, the approximation ofn∂Ωby makes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., continuous right inverse of the normal trace operator ; . In this paper we indeed prove such a discrete lifting theorem, taking advantage of the nonconforming approximation, and consequently we establish the error estimates and for the velocity in theH1- andL2-norms respectively, whereα= 1 ifN= 2 andα= 1/2 ifN= 3. This improves the previous result [T. Kashiwabaraet al.,Numer. Math.134(2016) 705–740] obtained for the conforming approximation in the sense that there appears no reciprocal of the penalty parameterϵin the estimates.