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.
Bibliographical noteFunding Information:
∗Received by the editors September 11, 2017; accepted for publication (in revised form) July 3, 2018; published electronically September 4, 2018. http://www.siam.org/journals/mms/16-3/M114704.html 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 (Stote031@umn.edu, Dominik@umn.edu). ‡Faculty of Aerospace Engineering, Delft University of Technology, The Netherlands (S.R.Turteltaub@tudelft.nl, S.J.Hulshoff@tudelft.nl).
© 2018 Society for Industrial and Applied Mathematics.
- Multiscale discontinuous Galerkin methods
- Residual-based multiscale modeling
- Variational multiscale method