We prove discrete versions of the first and second Weber inequalities on $\boldsymbol{H}(\mathbf{curl})\cap\boldsymbol{H}(\mathrm{div}_{\eta})$-like hybrid spaces spanned by polynomials attached to the faces and to the cells of a polyhedral mesh. The proven hybrid Weber inequalities are optimal in the sense that (i) they are formulated in terms of $\boldsymbol{H}(\mathbf{curl})$- and $\boldsymbol{H}(\mathrm{div}_{\eta})$-like hybrid semi-norms designed so as to embed optimally (polynomially) consistent face penalty terms, and (ii) they are valid for face polynomials in the smallest possible stability-compatible spaces. Our results are valid on domains with general, possibly non-trivial topology. In a second part we also prove, within a general topological setting, related discrete Maxwell compactness properties.

翻译：我们证明了第一类与第二类Weber不等式在$\boldsymbol{H}(\mathbf{curl})\cap\boldsymbol{H}(\mathrm{div}_{\eta})$型杂化空间上的离散版本，该空间由附着于多面体网格的面与胞腔上的多项式张成。所证明的杂化Weber不等式在以下意义下是最优的：(i) 它们以$\boldsymbol{H}(\mathbf{curl})$型和$\boldsymbol{H}(\mathrm{div}_{\eta})$型杂化半范数表述，这些半范数经过专门设计以最优（多项式）一致地嵌入面惩罚项；(ii) 它们适用于最小可能的稳定性相容空间中的面多项式。我们的结果对具有一般（可能非平凡）拓扑的域成立。在第二部分中，我们还在一般拓扑框架下证明了相关的离散Maxwell紧性性质。