We propose a $C^0$ interior penalty method for the fourth-order stream function formulation of the surface Stokes problem. The scheme utilizes continuous, piecewise polynomial spaces defined on an approximate surface. We show that the resulting discretization is positive definite and derive error estimates in various norms in terms of the polynomial degree of the finite element space as well as the polynomial degree to define the geometry approximation. A notable feature of the scheme is that it does not explicitly depend on the Gauss curvature of the surface. This is achieved via a novel integration-by-parts formula for the surface biharmonic operator.

翻译：本文针对表面Stokes问题的四阶流函数公式，提出了一种$C^0$内罚方法。该方案采用在近似表面上定义的连续分片多项式空间。我们证明了所得离散化是正定的，并依据有限元空间的多项式次数以及定义几何近似的多项式次数，推导了多种范数下的误差估计。该方案的一个显著特点是其不显式依赖于表面的高斯曲率。这是通过一种新颖的表面双调和算子分部积分公式实现的。