We develop in this work the first polytopal complexes of differential forms. These complexes, inspired by the Discrete De Rham and the Virtual Element approaches, are discrete versions of the de Rham complex of differential forms built on meshes made of general polytopal elements. Both constructions benefit from the high-level approach of polytopal methods, which leads, on certain meshes, to leaner constructions than the finite element method. We establish commutation properties between the interpolators and the discrete and continuous exterior derivatives, prove key polynomial consistency results for the complexes, and show that their cohomologies are isomorphic to the cohomology of the continuous de Rham complex.

翻译：本文开发了第一种离散形式的微分形式多面体复合体，这些复合体受到了离散德拉姆和虚拟元素方法的启发，是建立在通用多面体网格上的微分形式的 de Rham 复合体的离散版本。这两种构造都受益于多面体方法的高级方法，某些网格上的构造比有限元法更为简洁。我们证明了插值器与离散和连续外微分之间的交换性质，证明了复合体的关键多项式一致性结果，并展示了它们的上同调同构于连续德拉姆复合体的上同调。