Discrete Weber inequalities and related Maxwell compactness for hybrid spaces over polyhedral partitions of domains with general topology

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.