The Stokes equation posed on surfaces is important in some physical models, but its numerical solution poses several challenges not encountered in the corresponding Euclidean setting. These include the fact that the velocity vector should be tangent to the given surface and the possible presence of degenerate modes (Killing fields) in the solution. We analyze a surface finite element method which provides solutions to these challenges. We consider an interior penalty method based on the well-known Brezzi-Douglas-Marini $H({\rm div})$-conforming finite element space. The resulting spaces are tangential to the surface, but require penalization of jumps across element interfaces in order to weakly maintain $H^1$ conformity of the velocity field. In addition our method exactly satisfies the incompressibility constraint in the surface Stokes problem. Secondly, we give a method which robustly filters Killing fields out of the solution. This problem is complicated by the fact that the dimension of the space of Killing fields may change with small perturbations of the surface. We first approximate the Killing fields via a Stokes eigenvalue problem and then give a method which is asymptotically guaranteed to correctly exclude them from the solution. The properties of our method are rigorously established via an error analysis and illustrated via numerical experiments.

中文翻译：

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表面斯托克斯方程的符合发散性的有限元方法

表面上提出的斯托克斯方程在某些物理模型中很重要，但其数值解提出了一些在相应的欧几里德设置中没有遇到的挑战。其中包括速度矢量应与给定表面相切的事实以及解中可能存在的简并模式（Killing 场）。我们分析了为这些挑战提供解决方案的表面有限元方法。我们考虑一种基于著名的 Brezzi-Douglas-Marini $H({\rm div})$-符合有限元空间的内部惩罚方法。由此产生的空间与表面相切，但需要对跨元素界面的跳跃进行惩罚，以便微弱地保持速度场的 $H^1$ 一致性。此外，我们的方法完全满足表面斯托克斯问题中的不可压缩性约束。其次，我们给出了一种从解决方案中稳健地过滤掉 Killing 场的方法。由于Killing 场空间的维度可能会随着表面的小扰动而变化，这个问题变得复杂了。我们首先通过斯托克斯特征值问题来近似杀死域，然后给出一种渐近保证将它们从解决方案中正确排除的方法。我们方法的特性是通过误差分析严格建立的，并通过数值实验进行说明。我们首先通过斯托克斯特征值问题来近似杀死域，然后给出一种渐近保证将它们从解决方案中正确排除的方法。我们方法的特性是通过误差分析严格建立的，并通过数值实验进行说明。我们首先通过斯托克斯特征值问题来近似杀死域，然后给出一种渐近保证将它们从解决方案中正确排除的方法。我们方法的特性是通过误差分析严格建立的，并通过数值实验进行说明。