A tangential and penalty-free finite element method for the surface Stokes problem

TL;DR

This paper introduces a robust, penalty-free finite element method for the surface Stokes problem, enabling stable and accurate approximation of tangential velocity fields without penalization or Lagrange multipliers.

Contribution

It constructs surface MINI spaces with tangential velocity fields that are not $H^1$-conforming but lie in $H({ m div})$, providing a systematic approach for divergence-conforming surface Stokes discretizations.

Findings

Proves stability and optimal convergence of the method.

Demonstrates optimal-order convergence in numerical experiments.

Provides a new technique for constructing surface Stokes finite element spaces.

Abstract

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.