Solving Discontinuous Galerkin Formulations of Poisson's Equation using Geometric and p Multigrid

Abstract

Methods for solving discontinuous Galerkin formulations of the Poisson equation by coupling p multigrid to geometric multigrid are investigated. The simple approach of performing iterative relaxation on solution approximations of decreasing polynomial degree p down to p = 0 and then applying geometric multigrid is ineffective. The transition from p = 1 top = 0 causes the performance of the entire iteration to degrade because the long wavelength eigenfunctions of the p = 1 discontinuous system are not represented well in thep = 0 space. A new approach is proposed that coarsens from thep = 1 discontinuous space to thep = 1 continuous space. This approach eliminates the problems caused by the p = 1 to p = 0 transition. Furthermore, the p = 1 continuous space is a standard finite element space for which geometric multigrid is well-defined. In addition, when the discontinuous Galerkin equations are restricted to a continuous space, one recovers the Galerkin formulation of continuous finite elements. Thus, applying geometric multigrid to this system is straightforward and effective. Numerical tests agree well with the analysis and confirm that the new approach gives rapid convergence rates that are grid-independent and only weakly sensitive to the polynomial order.