曲面Stokes问题的一种Crouzeix-Raviart元分析

中文摘要 AI 辅助

近年来，定义在曲面上的向量值流动问题的有限元离散化受到越来越多的关注。本文研究定义在嵌入三维空间的二维流形上的曲面Stokes系统。曲面速度场在曲面的多面体逼近上使用非协调Crouzeix-Raviart有限元逼近，而压力用分片常数函数离散。控制方程涉及对称应变率张量，用Crouzeix-Raviart有限元逼近该张量会在速度场中产生伪振荡，因为离散Korn不等式在非协调空间上不成立。因此，我们通过添加边跳跃惩罚项来稳定动量方程，该惩罚项恢复了双线性项的强制性，因为离散Korn不等式对稳定化后的版本成立。此外，我们证明了该速度-压力有限元对满足离散inf-sup相容性条件。进一步，我们导出了能量范数和$L^2$范数下的最优先验误差估计。这些理论结果得到了数值实验的验证。

英文摘要

Recently, increasing attention has been paid to finite element discretizations of vector-valued flow problems posed on curved surfaces. In this work, we study a surface Stokes system defined on a two-dimensional manifold embedded in three-dimensional space. The surface velocity field is approximated using a nonconforming Crouzeix-Raviart finite element on a polyhedral approximation of the surface, while the pressure is discretized by piecewise constant functions. The governing equations involve the symmetric strain-rate tensor, whose approximation with Crouzeix-Raviart finite elements leads to spurious oscillations in the velocity field because the discrete Korn inequality fails on the nonconforming space. We therefore stabilize the momentum equation by adding an edge-jump penalty term, which restores the coervity of the bilinear term as the discrete Korn inequality holds for this stabilized version. Additionally, we establish that this finite element pair of velocity and pressure spaces satisfies the discrete inf-sup compatibility condition. Furthermore, we derive optimal a-priori error estimates in both the energy norm and the $L^2$-norm. These theoretical results are corroborated by numerical experiments.

发表机构

• Otto-von-Guericke University Magdeburg（马格德堡奥托·冯·古里克大学）

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