Recently a nonconforming surface finite element was developed to discretize 2D vector-valued compressible flow problems in a 3D domain. In this contribution we derive an error analysis for this approach on a vector-valued Laplace problem, which is an important operator for fluid-equations on the surface. In our setup, the problem is approximated via edge-integration on local flat triangles using the nonconforming linear Crouzeix-Raviart element. The flat planes coincide with the surface at the edge midpoints. This is also the place, where the Crouzeix-Raviart element requires continuity between two neighbouring elements. The developed Crouzeix-Raviart approximation is a non-parametric approach that works on local coordinate systems, established in each triangle. This setup is numerically efficient and straightforward to implement. For this Crouzeix-Raviart discretization we derive optimal error bounds in the $H^1$-norm and $L^2$-norm and present an estimate for the geometric error. Numerical experiments validate the theoretical results.

翻译：最近，一种非协调曲面有限元被提出，用于离散三维区域中的二维向量值可压缩流动问题。本文针对向量值拉普拉斯问题，推导了该方法的误差分析，该算子是曲面上流体方程的一个重要算子。在我们的设定中，问题通过使用非协调线性 Crouzeix-Raviart 元在局部平坦三角形上进行边积分来近似。这些平坦平面在边的中点处与曲面重合。此处也正是 Crouzeix-Raviart 元要求相邻单元间保持连续性的位置。所发展的 Crouzeix-Raviart 近似是一种非参数方法，它在每个三角形上建立的局部坐标系中工作。这种设定数值效率高且易于实现。针对此 Crouzeix-Raviart 离散化，我们推导了在 $H^1$ 范数和 $L^2$ 范数下的最优误差界，并给出了几何误差的估计。数值实验验证了理论结果。