A multiscale formulation of the Discontinuous Petrov–Galerkin method for advective–diffusive problems

We consider the Discontinuous Petrov–Galerkin method for the advection–diffusion model problem, and we investigate the application of the variational multiscale method to this formulation. We show the exact modeling of the fine scale modes at the element level for the linear case, and we discuss the approximate modeling both in the linear and in the non-linear cases. Furthermore, we highlight the existing link between this multiscale formulation and the p-version of the finite element method. Numerical examples illustrate the behavior of the proposed scheme.