A Taylor-Hood finite element method for the surface Stokes problem without penalization

TL;DR

This paper introduces a novel Taylor-Hood finite element method for the surface Stokes problem that avoids penalization by using edge and vertex degrees of freedom with Gauss-Lobatto nodes, achieving optimal convergence.

Contribution

It extends a recent penalty-free surface Stokes FEM approach to Taylor-Hood elements, enabling optimal convergence without penalization or Lagrange multipliers.

Findings

Achieves optimal-order convergence with Gauss-Lobatto node placement.

Numerical experiments confirm the necessity of nonstandard node placement.

Method circumvents the need for penalization or Lagrange multipliers.

Abstract

Finite element approximation of the velocity-pressure formulation of the surfaces Stokes equations is challenging because it is typically not possible to enforce both tangentiality and $H^1$ conformity of the velocity field. Most previous works concerning finite element methods (FEMs) for these equations thus have weakly enforced one of these two constraints by penalization or a Lagrange multiplier formulation. Recently in [A tangential and penalty-free finite element method for the surface Stokes problem, SINUM 62(1):248-272, 2024], the authors constructed a surface Stokes FEM based on the MINI element which is tangentiality conforming and $H^1$ nonconforming, but possesses sufficient weak continuity properties to circumvent the need for penalization. The key to this method is construction of velocity degrees of freedom lying on element edges and vertices using an auxiliary Piola transform. In this work we extend this methodology to construct Taylor-Hood surface FEMs. The resulting method is shown to achieve optimal-order convergence when the edge degrees of freedom for the velocity space are placed at Gauss-Lobatto nodes. Numerical experiments confirm that this nonstandard placement of nodes is necessary to achieve optimal convergence orders.