Analysis of the Taylor-Hood surface finite element method for the surface Stokes equation

Abstract

We consider the surface Stokes equation on a smooth closed hypersurface in R 3 \mathbb {R}^3 . 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.