Conjugate gradient for nonsingular saddle-point systems with a maximally rank-deficient leading block

Abstract

We consider iterative solvers for large, sparse, symmetric linear systems with a saddle-point structure. Since such systems are indefinite, the conjugate gradient (CG) method is not naturally designed for solving them. However, in the case of a maximally rank-deficient leading block, we prove that there are two sufficient conditions that allow for CG to be used. We show that the conditions are satisfied for a model time-harmonic Maxwell problem. To support our analysis, we present several numerical experiments for three-dimensional problems on complicated computational domains with constant and variable coefficients.