In this paper we prove Poincar\'e inequalities for the Discrete de Rham (DDR) sequence on a general connected polyhedral domain $\Omega$ of $\mathbb{R}^3$. We unify the ideas behind the inequalities for all three operators in the sequence, deriving new proofs for the Poincar\'e inequalities for the gradient and the divergence, and extending the available Poincar\'e inequality for the curl to domains with arbitrary second Betti numbers. A key preliminary step consists in deriving "mimetic" Poincar\'e inequalities giving the existence and stability of the solutions to topological balance problems useful in general discrete geometric settings. As an example of application, we study the stability of a novel DDR scheme for the magnetostatics problem on domains with general topology.

翻译：本文证明了$\mathbb{R}^3$中一般连通多面体域$\Omega$上离散de Rham（DDR）序列的Poincaré不等式。我们统一了该序列中所有三个算子的不等式思想，给出了梯度和散度Poincaré不等式的新证明，并将已有的旋度Poincaré不等式推广至具有任意第二贝蒂数的域。一个关键预备步骤是推导“模拟”Poincaré不等式，该不等式保证了适用于一般离散几何设定的拓扑平衡问题解的存在性与稳定性。作为应用实例，我们研究了一般拓扑域上静磁问题新型DDR格式的稳定性。