We find new discrete H 1 - and Poincaré-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.
Bibliographical noteFunding Information:
The work of the first author was supported in part by the National Science Foundation (grant DMS-1522657) and by the University of Minnesota Supercomputing Institute. The work of the third author was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project CityU 11302014).
© 2018 Society for Industrial and Applied Mathematics.
- Discontinuous galerkin