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

In this paper we introduce a new discretization of the incompressible Navier-Stokes equations. We use the Lamb identity for the advection term $(u \cdot \nabla)u$ and the general idea allows a lot of freedom in the treatment of the non-linear term. The main advantage of this scheme is that the divergence of the fluid velocity is pointwise zero at the discrete level. This exactness allows for exactly conserved quantities and pressure robustness. Discrete spaces consist of piecewise polynomials, they may be taken of arbitrary order and are already implemented in most libraries. Although the nonlinear term may be implemented as is, most proofs here are done for the linearized equation. The whole problem is expressed in the finite element exterior calculus framework. We also present numerical simulations, our codes are written with the FEniCS computing platform, version 2019.1.0.