Hybridizable discontinuous Galerkin methods for second-order elliptic problems: overview, a new result and open problems

We describe, in the framework of steady-state diffusion problems, the history of the development of the so-called hybridizable discontinuous Galerkin (HDG) methods, since their inception in 2009 until now. We show how it runs parallel to the development of the so-called hybridized mixed (HM) methods and how, a few years ago, it prompted the introduction of the M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{M}$$\end{document}-decompositions as a novel tool for the construction of superconvergent HM and HDG methods for elements of quite general shapes. We then uncover a new link between HM and HDG methods, namely, that any HM method can be rewritten as an HDG method by a suitable transformation of a subspace of the approximate fluxes of the HM method into a stabilization function. We end by listing several open problems which are a direct consequence of this result.