Residual-based variational multiscale modeling in a discontinuous Galerkin framework

Stein K.F. Stoter, Sergio R. Turteltaub, Steven J. Hulshoff, Dominik Schillinger

Research output: Contribution to journalArticleScientificpeer-review

4 Citations (Scopus)
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Abstract

We develop the general form of the variational multiscale method in a discontinuous Galerkin framework. Our method is based on the decomposition of the true solution into discontinuous coarse-scale and discontinuous fine-scale parts. The obtained coarse-scale weak formulation includes two types of fine-scale contributions. The first type corresponds to a fine-scale volumetric term, which we formulate in terms of a residual-based model that also takes into account fine-scale effects at element interfaces. The second type consists of independent fine-scale terms at element interfaces, which we formulate in terms of a new fine-scale "interface model." We demonstrate for the one-dimensional Poisson problem that existing discontinuous Galerkin formulations, such as the interior penalty method, can be rederived by choosing particular fine-scale interface models. The multiscale formulation thus opens the door for a new perspective on discontinuous Galerkin methods and their numerical properties. This is demonstrated for the one-dimensional advection-diffusion problem, where we show that upwind numerical fluxes can be interpreted as an ad hoc remedy for missing volumetric fine-scale terms.
Original languageEnglish
Pages (from-to)1333-1364
Number of pages32
JournalMultiscale Modeling and Simulation
Volume16
Issue number3
DOIs
Publication statusPublished - 1 Jan 2018

Bibliographical note

Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

Keywords

  • Multiscale discontinuous Galerkin methods
  • Residual-based multiscale modeling
  • Upwinding
  • Variational multiscale method

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