Finite Element Error Analysis of Surface Stokes Equations in Stream Function Formulation

TL;DR

This paper presents an error analysis of a trace finite element method for solving surface Stokes equations in stream function form on curved surfaces, providing optimal error bounds and numerical validation.

Contribution

It offers the first rigorous error analysis for a trace finite element discretization of surface Stokes equations in stream function formulation, applicable to existing surface FEM methods.

Findings

Optimal order discretization error bounds established.

Numerical experiments confirm theoretical error estimates.

Methods for reconstructing velocity and pressure are demonstrated.

Abstract

We consider a surface Stokes problem in stream function formulation on a simply connected oriented surface $\Gamma \subset \mathbb{R}^3$ without boundary. This formulation leads to a coupled system of two second order scalar surface partial differential equations (for the stream function and an auxiliary variable). To this coupled system a trace finite element discretization method is applied. The main topic of the paper is an error analysis of this discretization method, resulting in optimal order discretization error bounds. The analysis applies to the surface finite element method of Dziuk-Elliott, too. We also investigate methods for reconstructing velocity and pressure from the stream function approximation. Results of numerical experiments are included.