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

We discuss the efficiency of the conjugate gradient (CG) method for solving a sequence of linear systems; Aun+1 = u(n), 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 Au-1 = u(0), contains the solutions (u(n)) for subsequent time steps. The solutions of these equations can therefore be computed by a straightforward projection of the fight-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 (c) 2007 John Wiley & Sons, Ltd.