In this work, we uncover hidden geometric aspect of low-order compatible numerical schemes. First, we rewrite standard mimetic reconstruction operators defined by Stokes theorem using geometric elements of the barycentric dual grid, providing the equivalence between mimetic numerical schemes and discrete geometric approaches. Second, we introduce a novel global property of the reconstruction operators, called $P_0$-consistency, which extends the standard consistency requirement of the mimetic framework. This concept characterizes the whole class of reconstruction operators that can be used to construct a global mass matrix in such a way that a global patch test is passed. Given the geometric description of the scheme, we can set up a correspondence between entries of reconstruction operators and geometric elements of a secondary grid, which is built by duality from the primary grid used in the scheme formulation. Finally, we show the that the geometric interpretation is necessary for the correct evaluation of certain physical variables in the post-processing stage. A discussion on how the geometric viewpoint allows to optimize reconstruction operators completes the exposition.

在这项工作中，我们发现了低阶兼容数值方案的隐藏几何方面。首先，我们使用重心双网格的几何元素重写由斯托克斯定理定义的标准模拟重构算符，从而提供模拟数值方案和离散几何方法之间的等效性。其次，我们介绍了重构算子的一种新的全局性质，即$P_0$-consistency，扩展了模拟框架的标准一致性要求。这个概念表征了整类重构算子，这些算子可以用来构建全局质量矩阵，从而通过全局补丁测试。给定方案的几何描述，我们可以在重建算子的条目与辅助网格的几何元素之间建立对应关系，该辅助网格是通过从方案制定中使用的主要网格的对偶性构建的。最后，我们证明了几何解释对于后期处理阶段中某些物理变量的正确评估是必要的。关于几何视点如何允许优化重构算子的讨论完成了本说明。