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We develop the general form of the variational multiscale method in adiscontinuous Galerkin framework. Our method is based on the decomposition ofthe true solution into discontinuous coarse-scale and discontinuous fine-scaleparts. The obtained coarse-scale weak formulation includes two types offine-scale contributions. The first type corresponds to a fine-scale volumetricterm, which we formulate in terms of a residual-based model that also takesinto account fine-scale effects at element interfaces. The second type consistsof independent fine-scale terms at element interfaces, which we formulate interms of a new fine-scale "interface model". We demonstrate for theone-dimensional Poisson problem that existing discontinuous Galerkinformulations, such as the interior penalty method, can be rederived by choosingparticular fine-scale interface models. The multiscale formulation thus opensthe door for a new perspective on discontinuous Galerkin methods and theirnumerical properties. This is demonstrated for the one-dimensionaladvection-diffusion problem, where we show that upwind numerical fluxes can beinterpreted as an ad hoc remedy for missing volumetric fine-scale terms.

Residual-based variational multiscale modeling in a discontinuous Galerkin framework