UNIFORM-CONVERGENCE OF MULTIGRID V-CYCLE ITERATIONS FOR INDEFINITE AND NONSYMMETRIC PROBLEMS

Abstract

In this paper, an analysis of a multigrid method for nonsymmetric and/or indefinite elliptic problems is presented. In this multigrid method various types of smoothers may be used. One type of smoother considered is defined in terms of an associated symmetric problem and includes point and line, Jacobi, and Gauss-Seidel iterations. Smoothers based entirely on the original operator are also considered. One smoother 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. It is shown that the uniform estimates of [J.H. Bramble and J. E. Pasciak, Math. Comp., 60 (1993), pp. 447-471] 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 dependent on the number of multigrid levels).