2 research outputs found
Discrete -inequalities for spaces admitting M-decompositions
We find new discrete - 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 -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.Comment: 22 page
Superconvergent HDG methods for Maxwell's equations via the -decomposition
The concept of the -decomposition was introduced by Cockburn et al.\ in
Math. Comp.\ vol.\ 86 (2017), pp.\ 1609-1641 {to provide criteria to guarantee
optimal convergence rates for the Hybridizable Discontinuous Galerkin (HDG)
method for coercive elliptic problems}. In that paper they systematically
constructed superconvergent hybridizable discontinuous Galerkin (HDG) methods
to approximate the solutions of elliptic PDEs on unstructured meshes. In this
paper, we use the -decomposition to construct HDG methods for the Maxwell's
equations on unstructured meshes in two dimension. In particular, we show the
any choice of spaces having an -decomposition, together with sufficiently
rich auxiliary spaces, has an optimal error estimate and superconvergence even
though the problem is not in general coercive. Unlike the elliptic case, we
obtain a superconvergent rate for the curl of the solution, not the solution,
and this is confirmed by our numerical experiments