Analysis of the Taylor-Hood Surface Finite Element Method for the surface Stokes equation
Arnold Reusken
TL;DR
This work provides a rigorous discretization error analysis for the Taylor-Hood SFEM applied to the surface Stokes equations on smooth closed surfaces, using a penalty to enforce tangentiality. It establishes well-posedness of an extended variational formulation, proves discrete inf-sup stability, and derives optimal energy-norm error bounds that account for geometry approximation via a parametric surface mapping. The analysis shows that the penalty formulation does not adversely affect conditioning and identifies spectrally optimal preconditioning strategies for the resulting saddle-point system. Together, these results give a solid theoretical foundation for the practical Hood-Taylor-SFEM on surfaces and guide efficient linear-algebra solutions. Numerical experiments and a detailed study are reported in a related FOR joint paper.
Abstract
We consider the surface Stokes equation on a smooth closed hypersurface in three-dimensional space. For discretization of this problem a generalization of the surface finite element method (SFEM) of Dziuk-Elliott combined with a Hood-Taylor pair of finite element spaces has been used in the literature. We call this method Hood-Taylor-SFEM. This method uses a penalty technique to weakly satisfy the tangentiality constraint. In this paper we present a discretization error analysis of this method resulting in optimal discretization error bounds in an energy norm. We also address linear algebra aspects related to (pre)conditioning of the system matrix.