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.

翻译：本文旨在构造可嵌入离散de-Rham复形的不连续有限元向量空间，并证明其具有调和间隙性质。首先，借鉴经典Nédélec或Raviart-Thomas协调空间，考虑不连续有限元空间，并证明通过放松法向或切向约束，可构造出满足调和间隙性质的不连续空间。随后针对三角网格情形，证明经典向量不连续有限元空间具备该性质。对于笛卡尔网格，经典向量不连续有限元空间不满足此性质。进而展示如何利用三角网格下的de-Rham复形，对笛卡尔网格上的有限元空间进行扩充，以恢复de-Rham复形，并证明在该复形上成立的相同调和间隙性质。