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 journalArticlepeer-review

4 Scopus citations


We develop the general form of the variational m ultiscale 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 (US)
Pages (from-to)1333-1364
Number of pages32
JournalMultiscale Modeling and Simulation
Issue number3
StatePublished - 2018

Bibliographical note

Funding Information:
∗Received by the editors September 11, 2017; accepted for publication (in revised form) July 3, 2018; published electronically September 4, 2018. Funding: This work was funded by the National Science Foundation through the NSF CAREER Award 1651577. †Department of Civil, Environmental, and Geo-Engineering, University of Minnesota, Minneapolis, MN 55455 (, ‡Faculty of Aerospace Engineering, Delft University of Technology, The Netherlands (,

Publisher Copyright:
© 2018 Society for Industrial and Applied Mathematics.


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


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