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.

Iterative methods for linear systems (msc 65F10), Stability and convergence of numerical methods (msc 65N12), Error bounds (msc 65N15), Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (msc 65N30), Multigrid methods; domain decomposition (msc 65N55)
Life Sciences (theme 5), Energy (theme 4)
CWI
Modelling, Analysis and Simulation [MAS]
Scientific Computing

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