Surface Stokes and Navier-Stokes equations are used to model fluid flow on surfaces. They have attracted significant recent attention in the numerical analysis literature because approximation of their solutions poses significant challenges not encountered in the Euclidean context. One challenge comes from the need to simultaneously enforce tangentiality and $H^1$ conformity (continuity) of discrete vector fields used to approximate solutions in the velocity-pressure formulation. Existing methods in the literature all enforce one of these two constraints weakly either by penalization or by use of Lagrange multipliers. Missing so far is a robust and systematic construction of surface Stokes finite element spaces which employ nodal degrees of freedom, including MINI, Taylor-Hood, Scott-Vogelius, and other composite elements which can lead to divergence-conforming or pressure-robust discretizations. In this paper we construct surface MINI spaces whose velocity fields are tangential. They are not $H^1$-conforming, but do lie in $H({\rm div})$ and do not require penalization to achieve optimal convergence rates. We prove stability and optimal-order energy-norm convergence of the method and demonstrate optimal-order convergence of the velocity field in $L_2$ via numerical experiments. The core advance in the paper is the construction of nodal degrees of freedom for the velocity field. This technique also may be used to construct surface counterparts to many other standard Euclidean Stokes spaces, and we accordingly present numerical experiments indicating optimal-order convergence of nonconforming tangential surface Taylor-Hood $\mathbb{P}^2-\mathbb{P}^1$ elements.

翻译：表面Stokes和Navier-Stokes方程用于模拟曲面上的流体流动。由于求解这些方程的近似解面临着欧几里得背景下未曾遇到的重大挑战，近年来它们在数值分析文献中引起了广泛关注。其中一个挑战来自于需要同时强制离散向量场（用于近似速度-压力公式中的解）满足切向性和$H^1$相容性（连续性）。现有文献中的方法要么通过罚函数，要么通过拉格朗日乘子来弱强制这两个约束之一。目前尚缺乏一种稳健且系统的表面Stokes有限元空间构造方法，该方法采用节点自由度，包括MINI、Taylor-Hood、Scott-Vogelius以及其他可形成散度相容或压力稳健离散化的复合单元。本文构造了速度场具有切向性的表面MINI空间。这些空间不具有$H^1$相容性，但位于$H({\rm div})$中，并且无需罚函数即可获得最优收敛速率。我们证明了该方法的稳定性和最优阶能量范数收敛性，并通过数值实验展示了速度场在$L_2$范数下的最优阶收敛性。本文的核心进展在于速度场节点自由度的构造。该技术还可用于构造许多其他标准欧几里得Stokes空间的曲面版本，因此我们相应地展示了非相容切向表面Taylor-Hood $\mathbb{P}^2-\mathbb{P}^1$单元具有最优阶收敛性的数值实验。