A $C^0$ interior penalty method for the stream function formulation of the surface Stokes problem

TL;DR

This paper introduces a novel $C^0$ interior penalty finite element method for solving the surface Stokes problem using a fourth-order stream function formulation, with error estimates independent of surface curvature.

Contribution

It develops a curvature-independent $C^0$ interior penalty scheme for surface Stokes, utilizing a new integration-by-parts formula for the surface biharmonic operator.

Findings

The scheme is positive definite.

Error estimates are derived in various norms.

The method does not explicitly depend on surface curvature.

Abstract

We propose a $C^0$ interior penalty method for the fourth-order stream function formulation of the surface Stokes problem. The scheme utilizes continuous, piecewise polynomial spaces defined on an approximate surface. We show that the resulting discretization is positive definite and derive error estimates in various norms in terms of the polynomial degree of the finite element space as well as the polynomial degree to define the geometry approximation. A notable feature of the scheme is that it does not explicitly depend on the Gauss curvature of the surface. This is achieved via a novel integration-by-parts formula for the surface biharmonic operator.