and Eigenvalue Problems

Department of Mathematics

University of Southern California

1042 W. 36th Place

DRB-155

Los Angeles, CA 90089-1113

Center for Computational Mathematics

University of Colorado at Denver

Denver, CO 80217-3364

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.