Two‐level multigrid analysis for the convection–diffusion equation discretized by a discontinuous Galerkin method

Abstract

AbstractIn 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.