Multigrid Methods for Nearly Singular Linear Equations and Eigenvalue Problems

Abstract

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. One of the important aspects of this theory is that it allows such shifts to vary anywhere in the multigrid scheme, enabling its application to a wider class of eigenproblem solvers. 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 cycle. It is then applied to the Rayleigh quotient multigrid (RQMG) method, which is a more direct multigrid procedure for solving eigenproblems. Local convergence of the multigrid V-cycle and global convergence for a full multigrid version of both methods is obtained.