In this paper we study the convergence of a multigrid method for the solution of a two-dimensional linear second order elliptic equation, discretized by discontinuous Galerkin (DG) methods. For the Baumann-Oden and for the symmetric DG method, we give a detailed analysis of the convergence for cell- and point-wise block-relaxation strategies. We show that, for a suitably constructed two-dimensional polynomial basis, point-wise block partitioning gives much better results than the classical cell-wise partitioning. Independent of the mesh size, for Poisson's equation, simple MG cycles, with block Gauss Seidel and symmetric block Gauss Seidel smoothing, yield a convergence rate of 0.4 - 0.6 per iteration sweep for both DG-methods studied.

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Modelling, Analysis and Simulation [MAS]
Scientific Computing

Hemker, P., & van Raalte, M. (2002). Fourier two-level analysis for higher dimensional discontinuous Galerkin discretisation. Modelling, Analysis and Simulation [MAS]. CWI.