Smoothed Aggregation Multigrid for the Discontinuous Galerkin Method

The aim of this paper is to investigate theoretically as well as experimentally an algebraic multilevel algorithm for the solution of the linear systems arising from the discontinuous Galerkin method. The smoothed aggregation multigrid, introduced by Vaněk for the conforming finite element method, is applied to low-order discretizations of convection-diffusion equations. For the elliptic model problem the algorithm is shown to be optimal. Adjustments for the case of nonvanishing advection, such as directionally implicit smoothing and a suitable splitting of the operator are discussed. Several numerical experiments are presented, including a Newton type linearization of the compressible Navier-Stokes equations.