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 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, and satisfies suitable exactness properties depending on the topology of the domain. In conjunction with bespoke discrete counterparts of \(\text {L}^2\)-products, it can be used to design schemes for partial differential equations that benefit from the exactness of the sequence but, unlike classical (e.g., Raviart–Thomas–Nédélec) finite elements, are nonconforming. We prove a complete panel of results for the analysis of such schemes: exactness properties, uniform Poincaré 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.

在本文中，我们基于将多项式空间分解为与 Koszul 复数中的空间相关联的向量微积分算子和补的范围，在一般多面体网格上提出了一种新颖的任意阶离散 de Rham (DDR) 复数。DDR 复合体是完全离散的，这意味着空间和离散微积分算子都被离散的对应物所取代，并且根据域的拓扑满足合适的精确性属性。结合\(\text {L}^2\) 的定制离散对应物-products，它可用于设计偏微分方程的方案，该方案受益于序列的精确性，但与经典（例如，Raviart-Thomas-Nédélec）有限元不同，它是不一致的。我们证明了用于分析此类方案的完整结果面板：精确性属性、均匀庞加莱不等式以及原始和伴随一致性。我们还展示了这种 DDR 复合体如何能够设计用于静磁问题的数值方案，并使用上述结果来证明该方案的稳定性和最佳误差估计。