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The Fine-scale Field in the Multiscale Discontinuous Galerkin Method: Implications for Stabilized and Variational Multiscale Methods

In Hughes et al. [1], a new procedure was developed with the expressed intent of making the discontinuous Galerkin method (DG) more computationally tractable. Specifically, local, element-wise problems were solved in order to develop a transformation between the degrees-of-freedom of DG and those of a related continuous Galerkin method (CG), for which there were many fewer degrees-of-freedom. The solution of the resulting DG problem was then obtained by local, element-wise post-processing of the solution of the global CG problem. The method was shown numerically to work well and to have a multiscale interpretation, and therefore it was named the multiscale discontinuous Galerkin method (MDG). An error analysis was performed in Buffa, Hughes and Sangalli [2] and generalizations considered in Bochev, Hughes and Scovazzi [3].