We investigate discrete Poincar\'e inequalities on piecewise polynomial subspaces of the Sobolev spaces H(curl) and H(div) in three space dimensions. We characterize the dependence of the constants on the continuous-level constants, the shape regularity and cardinality of the underlying tetrahedral mesh, and the polynomial degree. One important focus is on meshes being local patches (stars) of tetrahedra from a larger tetrahedral mesh. We also review various equivalent results to the discrete Poincar\'e inequalities, namely stability of discrete constrained minimization problems, discrete inf-sup conditions, bounds on operator norms of piecewise polynomial vector potential operators (Poincar\'e maps), and existence of graph-stable commuting projections.

翻译：本文研究三维空间中索伯列夫空间H(curl)与H(div)的分片多项式子空间上的离散庞加莱不等式。我们系统刻画了不等式常数对连续层次常数、底层四面体网格的形状正则性与基数、以及多项式次数的依赖关系。重点探讨了网格作为大型四面体网格局部单元块（星型结构）的情形。同时，我们综述了与离散庞加莱不等式等价的若干重要结果，包括离散约束极小化问题的稳定性、离散inf-sup条件、分片多项式向量势算子（庞加莱映射）的算子范数界，以及图稳定交换投影的存在性。