In this paper we present a novel arbitrary-order discrete de Rham (DDR) complex on general polyhedral meshes based on the decomposition of polynomial spaces into the ranges of vector calculus operators and complements linked to the spaces in the Koszul complex. The DDR complex is fully discrete, meaning that both the spaces and discrete calculus operators are replaced by discrete counterparts. We prove a complete panel of results for the analysis of discretisation schemes for partial differential equations based on this complex: exactness properties, uniform Poincar\'e inequalities, as well as primal and adjoint consistency. We also show how this DDR complex enables the design of a numerical scheme for a magnetostatics problem, and use the aforementioned results to prove stability and optimal error estimates for this scheme.

翻译：在本文中,我们展示了一个关于普通多面体的独立命令(DDD)综合体,其基础是将多元空间分解成矢量微积分操作员的范围,并与Koszul综合体的空间相连接。DDR综合体是完全独立的,这意味着空间和离散微积分操作员都由离散对应方取代。我们证明这是一个完整的分析结果小组,用于分析基于这一复杂因素的局部差异方程的离散计划:精确性、统一的Poincar'e不平等以及原始和联合一致性。我们还表明,DDR综合体如何使设计一个用于磁层问题的数字计划成为可能,并利用上述结果来证明这一计划的稳定性和最佳误差估计。