arXiv 1809.094642018-09-25math.NAcs.NA

带有Crouzeix-Raviart逼近的惩罚法用于具有滑边界条件的Stokes方程

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

The University of Tokyo（东京大学）；Waseda University（早稻田大学）；Tokyo University of Science（东京科学大学）

Takahito Kashiwabara, Issei Oikawa, Guanyu Zhou

备注：: 21 pages

AI总结：

本文研究了在滑边界条件下非齐次Stokes方程的有限元方案，结合非配准P1/P0逼近（Crouzeix-Raviart逼近）和惩罚法，以处理本质边界条件u·n_∂Ω = g在∂Ω上的问题，并证明了离散提升定理，从而建立了速度在H^1和L^2范数下的误差估计。

摘要：

在光滑域Ω⊂R^N (N=2,3)中考虑具有非齐次滑边界条件的Stokes方程。我们提出了一种基于非配准P1/P0逼近（Crouzeix-Raviart逼近）结合惩罚法和降阶数值积分的有限元方案，以处理边界条件u·n_∂Ω = g在∂Ω上的问题。由于在应用有限元方法之前必须用多边形（或多面体）域Ω_h近似原域Ω，因此需要考虑由于Ω≠Ω_h带来的误差，即域扰动问题。特别是，n_∂Ω由n_∂Ω_h近似时，是否具有离散提升定理的离散对应物，即右连续逆的法向迹算子H^1(Ω)^N→H^{1/2}(∂Ω)；u↦u·n_∂Ω，这是一个非平凡的问题。在本文中，我们确实证明了这样的离散提升定理，利用非配准逼近，并因此建立了速度在H^1和L^2范数下的误差估计O(h^α+ε)和O(h^{2α}+ε)，其中α=1如果N=2，α=1/2如果N=3。这改进了之前的结果[T. Kashiwabara等，Numer. Math. 134 (2016)，pp. 705--740]，该结果针对配准逼近，即误差估计中没有惩罚参数ε的倒数。

英文摘要：

The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domain $Ω\subset \mathbb R^N \, (N=2,3)$. 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 $u \cdot n_{\partialΩ} = g$ on $\partialΩ$. 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 $Ω\neq Ω_h$, that is, the issues of domain perturbation. In particular, the approximation of $n_{\partialΩ}$ by $n_{\partialΩ_h}$ makes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., right-continuous inverse of the normal trace operator $H^1(Ω)^N \to H^{1/2}(\partialΩ)$; $u \mapsto u\cdot n_{\partialΩ}$. 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), pp. 705--740] obtained for the conforming approximation in the sense that there appears no reciprocal of the penalty parameter $ε$ in the estimates.

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