SFEM for the unsteady Navier-Stokes Equations on a stationary surface

TL;DR

This paper develops and analyzes a fully discrete surface finite element method for solving the unsteady Navier-Stokes equations on a stationary surface, providing stability, error bounds, and numerical validation.

Contribution

It introduces a novel SFEM with a Lagrange multiplier for tangential velocity enforcement and offers comprehensive stability and error analysis including optimal convergence results.

Findings

Optimal velocity error bounds in energy norm for certain finite element spaces.

Optimal pressure error bounds under regularity assumptions.

Numerical simulations confirm theoretical results and compare with penalty methods.

Abstract

In this paper we consider a fully discrete numerical method for the unsteady Navier-Stokes equations on a smooth closed stationary surface in $\mathbb{R}^3$. We use the surface finite element method (SFEM) with a generalized Taylor-Hood finite element pair $\mathrm{\mathbf{P}}_{k_u}$-- $\mathrm{P}_{k_{pr}}$-- $\mathrm{P}_{k_{\lambda}}$, where we enforce the tangential condition of the velocity field weakly, by introducing an extra Lagrange multiplier $\lambda$. Depending on the richness of the finite element space involving this extra Lagrange multiplier we present a fully discrete stability and error analysis. For the velocity, we establish optimal $L^{2}(a_h)$-norm bounds ($a_h$ - an energy norm) when $k_\lambda=k_u$ and suboptimal with respect to the geometric approximation error when $k_{\lambda} = k_u-1$ (optimal when \emph{super-parametric finite elements} are used). For the pressure, optimal $L^2(L^2)$-norm error bounds are established when $k_\lambda=k_u$. Assuming further regularity assumptions for our continuous problem, we are also able to show optimal convergence (using \emph{super-parametric finite elements} again) when $k_\lambda=k_u-1$. Numerical simulations that confirm the established theory are provided, along with a comparative analysis against a penalty approach.