In this work we consider the numerical solution of incompressible flows on two-dimensional manifolds. Whereas the compatibility demands of the velocity and the pressure spaces are known from the flat case one further has to deal with the approximation of a velocity field that lies only in the tangential space of the given geometry. Abandoning $H^1$-conformity allows us to construct finite elements which are -- due to an application of the Piola transformation -- exactly tangential. To reintroduce continuity (in a weak sense) we make use of (hybrid) discontinuous Galerkin techniques. To further improve this approach, $H(\operatorname{div}_{\Gamma})$-conforming finite elements can be used to obtain exactly divergence-free velocity solutions. We present several new finite element discretizations. On a number of numerical examples we examine and compare their qualitative properties and accuracy.

中文翻译：

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表面不可压缩流的无散切向有限元方法


在这项工作中，我们考虑二维流形上不可压缩流的数值解。虽然速度和压力空间的兼容性要求是从平面情况中已知的，但还必须处理仅位于给定几何形状的切向空间中的速度场的近似。放弃 $H^1$ 一致性允许我们构造有限元，由于应用了 Piola 变换，这些有限元完全是切向的。为了重新引入连续性（在弱意义上），我们利用（混合）不连续伽辽金技术。为了进一步改进这种方法，可以使用 $H(\operatorname{div}_{\Gamma})$ 符合的有限元来获得精确的无散速度解。我们提出了几种新的有限元离散化。我们通过许多数值例子来检查和比较它们的定性特性和准确性。