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

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 point-wise 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 that preserve these invariants at the fully discrete level and we analyze its well-posedness in terms of a CFL condition. While our theoretical results hold for general finite elements preserving the de Rham structure, we consider one application to tensor-product spline spaces. Specifically, we conduct several numerical test cases to verify the high order accuracy of the resulting numerical methods, as well as their ability to handle general boundary conditions.