Two-Level Fourier Analysis of a Multigrid Approach for Discontinuous Galerkin Discretization

In this paper we study a multigrid (MG) method for the solution of a linear second order elliptic equation, discretized by discontinuous Galerkin (DG) methods, and we give a detailed analysis of the convergence for different block-relaxation strategies. We find that pointwise block-partitioning gives much better results than the classical cellwise partitioning. Both for the Baumann--Oden method and for the symmetric DG method, with and without interior penalty (IP), the block-relaxation methods (Jacobi, Gauss--Seidel, and symmetric Gauss--Seidel) give excellent smoothing procedures in a classical MG setting. Independent of the mesh size, simple MG cycles give convergence factors of 0.075--0.4 per iteration sweep for the different discretization methods studied.