Mass, momentum and energy preserving FEEC and broken-FEEC schemes for the incompressible Navier-Stokes equations

TL;DR

This paper introduces two finite element schemes for the incompressible Navier-Stokes equations that preserve mass, momentum, and energy, utilizing FEEC and broken-FEEC approaches with high-order accuracy and stability.

Contribution

The paper presents novel FEEC-based and broken-FEEC-based finite element schemes that preserve key physical invariants for Navier-Stokes, with analysis and numerical validation.

Findings

Both schemes preserve divergence-free velocity and energy.

Numerical tests confirm high-order accuracy and robustness.

Schemes handle general boundary conditions effectively.

Abstract

In this article we propose two finite element schemes for the Navier-Stokes equations, based on a reformulation that involves differential operators from the de Rham sequence and an advection operator with explicit skew-symmetry in weak form. Our first scheme is obtained by discretizing this formulation with conforming FEEC (Finite Element Exterior Calculus) spaces: it preserves the pointwise divergence free constraint of the velocity, its total momentum and its energy, in addition to being pressure robust. Following the broken-FEEC approach, our second scheme uses fully discontinuous spaces and local conforming projections to define the discrete differential operators. It preserves the same invariants up to a dissipation of energy to stabilize numerical discontinuities. For both schemes we use a middle point time discretization which preserve these invariants at the fully discrete level and we analyse its well-posedness in terms of a CFL condition. Numerical test cases performed with spline finite elements allow us to verify the high order accuracy of the resulting numerical methods, as well as their ability to handle general boundary conditions.