  The aim of this article is to derive discontinuous finite elements vector
spaces which can be put in a discrete de-Rham complex for which an harmonic gap
property may be proven. First, discontinuous finite element spaces inspired by
classical N{\'e}d{\'e}lec or Raviart-Thomas conforming space are considered,
and we prove that by relaxing the normal or tangential constraint,
discontinuous spaces ensuring the harmonic gap property can be built. Then the
triangular case is addressed, for which we prove that such a property holds for
the classical discontinuous finite element space for vectors. On Cartesian
meshes, this result does not hold for the classical discontinuous finite
element space for vectors. We then show how to use the de-Rham complex found
for triangular meshes for enriching the finite element space on Cartesian
meshes in order to recover a de-Rham complex, on which the same harmonic gap
property is proven.
