Solving the Convection-Diffusion Equations via a Multiscale and Discontinuous Galerkin Approach

We present a multiscale and discontinuous Galerkin approach for solving the convection-dominated diffusion problems. The unresolved fine-scales are discretized element-wise using bubble functions. A discontinuous Galerkin discretization is employed in the resolved macro scales. To enhance stability to the numerical formulation, we add a nonlinear artificial diffusion operator inside of the elements, acting on both scales, and an extra stabilization on the interelement edges. Numerical tests demonstrate that the method is stable and accurate in solving transport problems with dominant convection.