Discrete $H^1$-inequalities for spaces admitting M-decompositions
Provides a theoretical framework for designing stable and superconvergent numerical methods for fluid dynamics on complex meshes, but is incremental as it extends existing M-decomposition theory.
The paper derives new discrete H^1 and Poincaré-Friedrichs inequalities for spaces with M-decompositions, enabling simplified error analysis and construction of superconvergent HDG and mixed methods for incompressible Navier-Stokes equations on unstructured meshes with general polygonal/polyhedral elements.
We find new discrete $H^1$- and Poincaré-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.