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 \(\varvec{H}({{\,\mathrm{{\textbf {curl}}}\,}})\cap \varvec{H}({{\,\textrm{div}\,}}_{\eta })\)-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.