Commun. Comput. Phys., 5 (2009), pp. 683-693.


Bilinear Forms for the Recovery-Based Discontinuous Galerkin Method for Diffusion

Marc van Raalte 1, Bram van Leer 2*

1 Centrum voor Wiskunde en Informatica, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands.
2 W.M. Keck Foundation Laboratory for Computational Fluid Dynamics, Department of Aerospace Engineering, The University of Michigan, Ann Arbor, MI 48109-2140, USA.

Received 9 October 2007; Accepted (in revised version) 28 March 2008
Available online 1 August 2008

Abstract

The present paper introduces bilinear forms that are equivalent to the recovery-based discontinuous Galerkin formulation introduced by Van Leer in 2005. The recovery method approximates the solution of the diffusion equation in a discontinuous function space, while inter-element coupling is achieved by a local L_2 projection that recovers a smooth continuous function underlying the discontinuous approximation. Here we introduce the concept of a local "recovery polynomial basis" - smooth polynomials that are in the weak sense indistinguishable from the discontinuous basis polynomials - and show it allows us to eliminate the recovery procedure. The recovery method reproduces the symmetric discontinuous Galerkin formulation with additional penalty-like terms depending on the targeted accuracy of the method. We present the unique link between the recovery method and discontinuous Galerkin bilinear forms.

AMS subject classifications: 35K05, 65M60

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Key words: Diffusion equation, discontinuous Galerkin methods, recovery basis, L_2 projection.

*Corresponding author.
Email: marc.van.raalte@planet.nl (M. van Raalte), bram@umich.edu (B. van Leer)
 

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