Fourier two-level analysis for discontinuous Galerkin discretization with linear elements
In 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 fordifferent 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-relaxationsgive 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 Gauss-Seidel) 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.
|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)|
|Modelling, Analysis and Simulation [MAS]|
Hemker, P.W, Hoffmann, W, & van Raalte, M.H. (2002). Fourier two-level analysis for discontinuous Galerkin discretization with linear elements. Modelling, Analysis and Simulation [MAS]. CWI.