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

TL;DR

This work develops a $C^0$ interior penalty method for the surface Stokes problem via the fourth-order stream function formulation, discretizing on an approximate surface with continuous, piecewise polynomial spaces. A key contribution is a curvature-agnostic integration-by-parts framework for the surface biharmonic operator, which yields a positive-definite, single-unknown system and avoids explicit Gauss curvature in the formulation. The authors establish coercivity and continuity, derive geometric-consistency error bounds, and prove convergence in discrete $H^2$ and related norms with rates depending on the FE degree $k$ and geometry-degree $k_g$, complemented by numerical experiments on ellipsoids that corroborate the theoretical predictions. The approach provides a simple, efficiently recoverable velocity field via a tangential curl of the discrete stream function, with robust performance under geometry approximation.

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.

Theorems & Definitions (45)

• Definition 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Lemma 2.8