Static condensation, hybridization, and the devising of the HDG methods

Bernardo Cockburn

Research output: Chapter in Book/Report/Conference proceedingConference contribution

55 Scopus citations

Abstract

In this paper, we review and refine the main ideas for devising the socalled hybridizable discontinuous Galerkin (HDG) methods; we do that in the framework of steady-state diffusion problems. We begin by revisiting the classic techniques of static condensation of continuous finite element methods and that of hybridization of mixed methods, and show that they can be reinterpreted as discrete versions of a characterization of the associated exact solution in terms of solutions of Dirichlet boundary-value problems on each element of the mesh which are then patched together by transmission conditions across interelement boundaries. We then define the HDG methods associated to this characterization as those using discontinuous Galerkin (DG) methods to approximate the local Dirichlet boundaryvalue problems, and using weak impositions of the transmission conditions.We give simple conditions guaranteeing the existence and uniqueness of their approximate solutions, and show that, by their very construction, the HDG methods are amenable to static condensation. We do this assuming that the diffusivity tensor can be inverted; we also briefly discuss the case in which it cannot. We then show how a different characterization of the exact solution, gives rise to a different way of statically condensing an already known HDG method. We devote the rest of the paper to establishing bridges between the HDG methods and other methods (the old DG methods, the mixed methods, the staggered DG method and the so-called Weak Galerkin method) and to describing recent efforts for the construction of HDG methods (one for systematically obtaining superconvergent methods and another, quite different, which gives rise to optimally convergent methods). We end by providing a few bibliographical notes and by briefly describing ongoing work.

Original languageEnglish (US)
Title of host publicationBuilding Bridges
Subtitle of host publicationConnections and Challenges in Modern Approaches to Numerical Partial Differential Equations
EditorsEmmanuil H. Georgoulis, Gabriel R. Barrenechea, Franco Brezzi, Andrea Cangiani, Emmanuil H. Georgoulis
PublisherSpringer- Verlag
Pages129-177
Number of pages49
ISBN (Print)9783319416380
DOIs
StatePublished - Jan 1 2016
EventInternational Conference on Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, 2014 - Durham, United Kingdom
Duration: Jul 8 2014Jul 16 2014

Publication series

NameLecture Notes in Computational Science and Engineering
Volume114
ISSN (Print)1439-7358

Other

OtherInternational Conference on Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, 2014
CountryUnited Kingdom
CityDurham
Period7/8/147/16/14

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