A note on the efficiency of the conjugate gradient method for a class of time‐dependent problems

Abstract We discuss the efficiency of the conjugate gradient (CG) method for solving a sequence of linear systems; Aun+1 = un, where A is assumed to be sparse, symmetric, and positive definite. We show that under certain conditions the Krylov subspace, which is generated when solving the first linear system Au1 = u0, contains the solutions {un} for subsequent time steps. The solutions of these equations can therefore be computed by a straightforward projection of the right‐hand side onto the already computed Krylov subspace. Our theoretical considerations are illustrated by numerical experiments that compare this method with the order‐optimal scheme obtained by applying the multigrid method as a preconditioner for the CG‐method at each time step. Copyright © 2007 John Wiley & Sons, Ltd.