Generalized multiscale discontinuous Galerkin method for convection–diffusion equation in perforated media

In this work, we present a Generalized Multiscale Discontinuous Galerkin Method (GMsDGM) for the convection–diffusion equation in perforated media. In order to numerically solve the problem on the fine grid and use the solution as a reference solution, we construct the grid that resolves perforation on the fine grid level and performs finite element approximation using the Interior Penalty Discontinuous Galerkin method (IPDG). To reduce the size of the fine grid system, we present the construction of the coarse grid approximation using a Generalized Multiscale Discontinuous Galerkin Method (GMsDGM). In GMsDGM, we construct the multiscale basis functions in the local domains by solutions of local convection–diffusion problems with different boundary conditions and performing dimension reduction by solutions of local spectral problems. We investigate several types of multiscale space constructions based on the various choices of boundary conditions of the local domains. Numerical results are presented for several test problems with different velocities fields and perforations distributions. We also investigate the influence of various constructions of multiscale basis functions with a different number of basis functions and different model parameters.