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：我们证明了第一类和第二类Weber不等式的离散形式，这些不等式定义在由多面体网格的面和胞元上的多项式张成的$\boldsymbol{H}(\mathbf{curl})\cap\boldsymbol{H}(\mathrm{div}_{\eta})$型混合空间上。本文证明的混合Weber不等式具有最优性，体现在：(i) 它们通过$\boldsymbol{H}(\mathbf{curl})$型和$\boldsymbol{H}(\mathrm{div}_{\eta})$型混合半范数表述，这些范数经过精心设计以嵌入最优（多项式）一致的面罚项；(ii) 它们适用于最小可能稳定兼容空间中的面多项式。我们的结果适用于具有一般拓扑（可能包含非平凡拓扑）的域。在第二部分中，我们还在一股拓扑框架下证明了相关的离散Maxwell紧致性质。