Static condensation, hybridization and the devising of HDG methods

Bernardo Cockburn
University of Minnesota

We give an overview of the evolution of the so-called hybridizable discontinuous Galerkin (HDG) methods. Working in the framework of steady-state diffusion, we show how the application of the ideas of static condensation and hybridization, proposed back in the 60's, to discontinuous Galerkin methods produced the appearance of the HDG methods. We then show that the development of the HDG methods has followed two main tracks. One stresses the development of their "local spaces" whereas the other, more recent, focuses on the development of their "stabilization functions". The former is strongly related to the theory of hybridized mixed methods and leads to the theory of M-decompositions. The latter is associated to the introduction of new remarkable stabilization functions, namely, the one proposed by Lehrenfeld-Schoeberl and the one devised for the Hybrid High-Order methods. Special emphasis will be given to the contributions that Francisco-Javier Sayas made.