Residual-based variational multiscale modeling in a discontinuous Galerkin framework

摘要

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.