Modelling, Analysis and Simulation 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 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