A parametric finite element method for the incompressible Navier--Stokes equations on an evolving surface

TL;DR

This paper introduces a parametric finite element method for simulating incompressible Navier--Stokes equations on evolving surfaces, providing stability analysis and demonstrating practical accuracy through numerical experiments.

Contribution

It develops a stable, high-order finite element approach for surface fluid dynamics with proven stability and validated convergence results.

Findings

The method is stable in the semidiscrete setting.

Numerical experiments confirm the method's convergence and accuracy.

The approach is practical for simulating surface flows on evolving geometries.

Abstract

In this paper we consider the numerical approximation of the incompressible surface Navier--Stokes equations on an evolving surface. For the discrete representation of the moving surface we use parametric finite elements of degree $\ell \geq 2$. In the semidiscrete continuous-in-time setting we are able to prove a stability estimate that mimics a corresponding result for the continuous problem. Some numerical results, including a convergence experiment, demonstrate the practicality and accuracy of the proposed method.