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

In this paper we study a multigrid 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 point-wise block-partitioning gives much better results than the classical cell-wise partitioning. Both for the Baumann-Oden and for the symmetric DG method, with and without interior penalty, the block relaxation methods (Jacobi, Gauss-Seidel and symmetric Gauss-Seidel) give excellent smoothing procedures in a classical multigrid setting. Independent of the mesh size, simple MG cycles give convergence factors 0.075 – 0.4 per iteration sweep for the different discretisation methods studied. 2000 Mathematics Subject Classification: 65F10, 65N12, 65N15, 65N30, 65N55