Coupling p-multigrid to geometric multigrid for discontinuous Galerkin formulations of the convection–diffusion equation

An improved p-multigrid algorithm for discontinuous Galerkin (DG) discretizations of convection–diffusion problems is presented. The general p  -multigrid algorithm for DG discretizations involves a restriction from the p=1p=1 to p=0p=0 discontinuous polynomial solution spaces. This restriction is problematic and has limited the efficiency of the p  -multigrid method. For purely diffusive problems, Helenbrook and Atkins have demonstrated rapid convergence using a method that restricts from a discontinuous to continuous polynomial solution space at p=1p=1. It is shown that this method is not directly applicable to the convection–diffusion (CD) equation because it results in a central-difference discretization for the convective term. To remedy this, ideas from the streamwise upwind Petrov–Galerkin (SUPG) formulation are used to devise a transition from the discontinuous to continuous space at p=1p=1 that yields an upwind discretization. The results show that the new method converges rapidly for all Peclet numbers.