Uniform convergence of multigrid V-cycle iterations for indefinite and nonsymmetric problems

J. H. Bramble

Department of Mathematics, Cornell University, Ithaca, NY 14853

D. Y. Kwak

Department of Mathematics, Korea Advanced Institute of Science and Technology, Taejon, Korea 305-701

J. E. Pasciak

Department of Aplied Science, Brookhaven National Laboratory, Upton, NY 11973


In this paper, we present an analysis of a multigrid method for nonsymmetric and/or indefinite elliptic problems. In this multigrid method various types of smoothers may be used. One type of smoother which we consider is defined in terms of an associated symmetric problem and includes point and line, Jaoci and Gauss-Seidel iterations. We also study smoothers based entirely on the original operator. One is based on the normal form, that is, the product of the operator and its transpose. Other smoothers studied include point and line, Jacobi and Gauss-Seidel (with certain orderings). We show that the uniform estimates of [6] for symmetric positive definite problems carry over to these algorithms. More precisely, the multigrid iteration for the nonsymmetric and/or indefinite problem is shown to converge at a uniform rate provided that the coarsest grid in the multilevel iteration is sufficiently fine (but not depending on the number of multigrid levels).