Multigrid Methods for Nearly Singular Linear Equations
and Eigenvalue Problems

Z. Cai
Department of Mathematics
University of Southern California
1042 W. 36th Place
Los Angeles, CA 90089-1113

J. Mandel
Center for Computational Mathematics
University of Colorado at Denver
Denver, CO 80217-3364

S. F. McCormick
Program in Applied Mathematics
Campus Box 526
University of Colorado at Boulder
Boulder, CO 80309-0526


The purpose of this paper is to develop a convergence theory for multigrid methods applied to nearly singular linear elliptic partial differential equations, of the type produced from a positive definite system by a shift with the identity. The theory is first applied to a method for computing eigenvalues and eigenvectors that consists of multigrid iterations with zero right-hand side and updating the shift from the Rayleigh quotient before every iteration. It is then applied to the Rayleigh quotient multigrid method (RQMG), which is a more direct multigrid procedure for solving eigenproblems. Local convergence of the multigrid V-cycle and global convergence of a full multigrid version of both is obtained.

Contributed December 16, 1992.