Uniform Convergence of Multigrid V-cycle Iterations for Indefinite and
Nonsymmetric Problems

James H. Bramble, Do Y. Kwak, and Joseph E. Pasciak

To appear: SIAM Journal of Numerical Analysis

Dedicated to Professor Seymour Parter on the occasion of the sixty fifth
anniversary of his birthday.


Abstract :  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, Jacobi 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 \myrcite{bpnew} 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).