  We find new discrete $H^1$- and Poincar\'e-Friedrichs inequalities by
studying the invertibility of the 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 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 2D, general polygonal
elements and, in 3D, general, flat-faced tetrahedral, prismatic, pyramidal and
hexahedral elements.
