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. To complement 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 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. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]