We prove discrete versions of the first and second Weber inequalities on \(\varvec{H}({{\,\mathrm{{\textbf {curl}}}\,}})\cap \varvec{H}({{\,\textrm{div}\,}}_{\eta })\) -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 \(\varvec{H}({{\,\mathrm{{\textbf {curl}}}\,}})\) - and \(\varvec{H}({{\,\textrm{div}\,}}_{\eta })\) -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.
