Parametric Finite Element Discretization of the Surface Stokes Equations

TL;DR

This paper introduces a higher-order surface finite element method for the surface Stokes equations, analyzing its stability, accuracy, and convergence, with a focus on discretization choices and implementation benefits.

Contribution

It presents a novel discretization approach for surface Stokes equations, including stability analysis and optimal convergence results for an isogeometric setting.

Findings

Optimal order convergence in tangential norms

Stable discretization schemes with divergence and diffusion terms

Implementation advantages of the proposed method

Abstract

We study a higher-order surface finite element (SFEM) penalty-based discretization of the tangential surface Stokes problem. Several discrete formulations are investigated which are equivalent in the continuous setting. The impact of the choice of discretization of the diffusion term and of the divergence term on numerical accuracy and convergence, as well as on implementation advantages, is discussed. We analyze the inf-sup stability of the discrete scheme in a generic approach by lifting stable finite element pairs known from the literature. A discretization error analysis in tangential norms then shows optimal order convergence of an isogeometric setting that requires only geometric knowledge of the discrete surface.