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arXiv · 1808.05709

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

Bernardo Cockburn·Guosheng Fu·Weifeng Qiu
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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.

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Bernardo Cockburn, Guosheng Fu, Weifeng Qiu. 2018-08-17. Discrete $H^1$-inequalities for spaces admitting M-decompositions. https://arxiv.org/abs/1808.05709

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