Solving the incompressible surface Navier-Stokes equation by surface finite elements

TL;DR

This paper introduces a surface finite element method for solving the incompressible surface Navier-Stokes equations on complex surfaces, enabling efficient numerical simulations that respect the topology of the surface.

Contribution

It presents a novel finite element approach reformulating the equations in Cartesian coordinates, compatible with standard finite element tools, and demonstrates its effectiveness through computational experiments.

Findings

Successful simulation of flow on a torus demonstrating topological effects

Comparison with discrete exterior calculus validates the method

Flow topology interactions illustrated via Poincaré-Hopf theorem

Abstract

We consider a numerical approach for the incompressible surface Navier-Stokes equation on surfaces with arbitrary genus $g(\mathcal{S})$. The approach is based on a reformulation of the equation in Cartesian coordinates of the embedding $\mathbb{R}^3$, penalization of the normal component, a Chorin projection method and discretization in space by surface finite elements for each component. The approach thus requires only standard ingredients which most finite element implementations can offer. We compare computational results with discrete exterior calculus (DEC) simulations on a torus and demonstrate the interplay of the flow field with the topology by showing realizations of the Poincar\'e-Hopf theorem on $n$-tori.