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方程用于模拟表面上的流体流动。这些方程近年来在数值分析文献中引起了广泛关注，因为其解的近似面临欧几里得背景下未遇到的显著挑战。一个挑战来自需要同时满足离散向量场的切向性和$H^1$相容性（连续性），这些向量场用于速度-压力公式中的近似解。现有文献中的方法均通过惩罚或使用拉格朗日乘子，弱满足这两个约束条件之一。目前缺乏一种稳健且系统化的表面Stokes有限元空间构造方法，该方法应使用节点自由度，包括MINI、Taylor-Hood、Scott-Vogelius以及其他复合单元，这些单元可以导致散度相容或压力鲁棒的离散化。本文构造了表面MINI空间，其速度场是切向的。这些空间不满足$H^1$相容性，但属于$H({\rm div})$空间，且不需要惩罚即可达到最优收敛速率。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$L_2$中的最优阶收敛。本文的核心进展是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空间的表面对应物，因此我们给出了数值实验，表明非相容切向表面Taylor-Hood $\mathbb{P}^2-\mathbb{P}^1$单元具有最优阶收敛性。