In this series of papers 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 required 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：在这一系列论文中,我们展示了一个新的任意命令离散的拉姆(DD)综合体,该综合体基于将多元空间分解成矢量微积分操作员和与Koszul综合体空间相连接的辅助体,对普通多面形色相进行分解。DDR综合体是完全离散的,这意味着空格和离散微积分操作员都由离散的对应方取代。我们证明,这是分析基于这一复杂因素的局部差异方程的离散计划所需的完整结果小组:精确性特性、统一的波因卡尔(Poincar)的不平等,以及原始和联合一致性。我们还表明,DDR综合体如何使得能够设计一个用于磁铁问题的数字方案,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。