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 the matching between the continuous and discrete cohomology spaces can be proven for periodic meshes. First, the triangular case is addressed, for which we prove that this 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 property is proven.
Vincent Perrier (Thu,) studied this question.