In this work we prove that, for a general polyhedral domain of $\Real^3$, the cohomology spaces of the discrete de Rham complex of [Di Pietro and Droniou, An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness, Poincar\'e inequalities, and consistency, Found. Comput. Math., 2021, DOI: \href{https://dx.doi.org/10.1007/s10208-021-09542-8}{10.1007/s10208-021-09542-8}] are isomorphic to those of the continuous de Rham complex. This is, to the best of our knowledge, the first result of this kind for an arbitrary-order complex built from a general polyhedral mesh.

翻译：在这项工作中，我们证明了对于$\Real^3$ 的一般多面体域，根据[Di Pietro和Droniou，An arbitrary-order discrete de Rham complex on polyhedral meshes：Exactness，Poincaré inequalities，and consistency，Found. Comput. Math.，2021，DOI：\href {https://dx.doi.org/10.1007/s10208-021-09542-8} {10.1007/s10208-021-09542-8}]构建的任意阶复形的上同调空间与连续deRham复形相同。据我们所知，这是从一般的多面体网格构建任意阶复形的这种结果中的首个成果。