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.
C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a
MAS
Modelling, Analysis and Simulation
Modelling, Analysis and SimulationFourier two-level analysis for higher dimensional discontinuous Galerkin discretisation
ABSTRACTIn 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.
C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a
Modelling, Analysis and SimulationFourier two-level analysis for discontinuous Galerkin discretization with linear elements
ABSTRACTIn this paper we study the convergence of 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. In addition to an earlier paper where higher-order methods were studied, here we restrict ourselves to methods using piecewise linear approximations. It is well-known that these methods are unstable if no additional interior penalty is applied.As for the higher order methods, we find that point-wise block-relaxations give much better results than the classical cell-wise relaxations. Both for the Baumann-Oden and for the symmetric DG method, with a sufficient interior penalty, the block relaxation methods studied (Jacobi, Gauss-Seidel and symmetric GaussSeidel) all make excellent smoothing procedures in a classical multigrid setting. Independent of the mesh size, simple MG cycles give convergence factors 0.2 -0.4 per iteration sweep for the different discretizations studied.2000 Mathematics Subject Classification: 65F10, 65N12, 65N15, 65N30, 65N55
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