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We find new discrete $H^{1}\!$- and Poincare--Friedrichs inequalities by studying the invertibility of the discontinuous Galkerkin (DG) approximation of the flux for local spaces admitting M-decompositions. We then show how to use these inequalities to define and analyze new, superconvergent hybridizable DG (HDG) and mixed methods for which the stabilization function is defined in such a way that the approximations satisfy new $H^1$-stability results with which their error analysis is greatly simplified. We apply this approach to define a wide class of energy-bounded, superconvergent HDG and mixed methods for the incompressible Navier--Stokes equations defined on unstructured meshes using, in two dimensions, general polygonal elements and, in three dimensions, general, flat-faced tetrahedral, prismatic, pyramidal, and hexahedral elements.

Discrete $H^1$-Inequalities for Spaces Admitting M-Decompositions