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：本文发展了首个微分形式的多胞体复形。这些复形受离散德·拉姆方法和虚拟元方法的启发，是在由一般多胞体单元构成的网格上构建的微分形式德·拉姆复形的离散版本。两种构造均受益于多胞体方法的高层次途径，在特定网格上可得到比有限元方法更精简的构造。我们建立了插值算子与离散及连续外微分之间的交换性质，证明了复形关键的多项式相容性结果，并指出这些复形的上同调与连续德·拉姆复形的上同调同构。