  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.
