Discrete de-Rham complex involving a discontinuous finite element space for velocities: the case of periodic straight triangular and Cartesian meshes

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.