An arbitrary order and pointwise divergence-free finite element scheme for the incompressible 3D Navier-Stokes equations

TL;DR

This paper introduces a finite element scheme for the 3D incompressible Navier-Stokes equations that ensures pointwise divergence-free velocity fields, enhancing pressure robustness and applicability across various polynomial orders.

Contribution

It develops a novel discretization within finite element exterior calculus that guarantees divergence-free velocity at the discrete level, applicable to arbitrary polynomial orders.

Findings

The scheme is well-posed for linearized equations.

Numerical simulations validate the theoretical error estimates.

The method achieves pressure robustness due to exact divergence-free velocity.

Abstract

In this paper we discretize the incompressible Navier-Stokes equations in the framework of finite element exterior calculus. We make use of the Lamb identity to rewrite the equations into a vorticity-velocity-pressure form which fits into the de Rham complex of minimal regularity. We propose a discretization on a large class of finite elements, including arbitrary order polynomial spaces readily available in many libraries. The main advantage of this discretization is that the divergence of the fluid velocity is pointwise zero at the discrete level. This exactness ensures pressure robustness. We focus the analysis on a class of linearized equations for which we prove well-posedness and provide a priori error estimates. The results are validated with numerical simulations.