Discrete $H^1$-inequalities for spaces admitting M-decompositions

TL;DR

This paper develops new discrete $H^1$-inequalities for spaces with M-decompositions, enabling the design of superconvergent HDG and mixed methods with simplified error analysis for incompressible Navier-Stokes equations.

Contribution

It introduces novel discrete inequalities and stabilization strategies that lead to superconvergent, energy-bounded HDG and mixed methods applicable on general unstructured meshes.

Findings

New discrete $H^1$-inequalities for M-decompositions.

Superconvergent HDG and mixed methods with simplified error analysis.

Applicable to various polygonal and polyhedral meshes in 2D and 3D.

Abstract

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.