Two-level multigrid analysis for the convection-diffusion equation discretized by a discontinuous Galerkin method.

Keywords
continuous Galerkin method ? multigrid iteration ? two-level Fourier analysis ? point-wise block-relaxation

Abstract
In this paper, we study a multigrid (MG) method for the solution of a linear one-dimensional convection-diffusion equation that is discretized by a discontinuous Galerkin method. In particular we study the convection-dominated case when the perturbation parameter, i.e. the inverse cell-Reynolds-number, is smaller than the finest mesh size.
We show that, if the diffusion term is discretized by the non-symmetric interior penalty method (NIPG) with feasible penalty term, multigrid is sufficient to solve the convection-diffusion or the convection-dominated equation. Then, independent of the mesh-size, simple MG cycles with symmetric Gauss-Seidel smoothing give an error reduction factor of 0.2-0.3 per iteration sweep.
Without penalty term, for the Baumann-Oden (BO) method we find that only a robust (i.e. cell-Reynolds-number uniform) two-level error-reduction factor (0.4) is found if the point-wise block-Jacobi smoother is used. Copyright ? 2005 John Wiley & Sons, Ltd.