In this paper, we are concerned with the nonconforming finite element discretization of geometric partial differential equations. We construct a surface Crouzeix–Raviart element on the linear approximated surface, analogous to a flat surface. The optimal convergence theory for the new nonconforming surface finite element method is developed even though the geometric error exists. By taking an intrinsic viewpoint of manifolds, we introduce a new superconvergent gradient recovery method for the surface Crouzeix–Raviart element using only the information of discretized surface. The potential of serving as an asymptotically exact a posteriori error estimator is also exploited. A series of benchmark numerical examples are presented to validate the theoretical results and numerically demonstrate the superconvergence of the gradient recovery method.

中文翻译：

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拉普拉斯-贝尔特拉米方程的表面 Crouzeix-Raviart 元素

在本文中，我们关注几何偏微分方程的非一致性有限元离散化。我们在线性近似表面上构造一个表面 Crouzeix-Raviart 元素，类似于平面。即使存在几何误差，也为新的非一致曲面有限元方法开发了最优收敛理论。通过采取流形的内在观点，我们引入了一种新的超收敛梯度恢复方法，用于仅使用离散表面信息的表面 Crouzeix-Raviart 元。还利用了作为渐近精确后验误差估计器的潜力。提出了一系列基准数值例子来验证理论结果并数值证明梯度恢复方法的超收敛性。