Fourier two-level analysis for higher dimensional discontinuous Galerkin discretisation

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 or symmetric block-Gauss–Seidel smoothing, yield a convergence rate of 0.4–0.6 per iteration sweep for both DG-methods studied.