A note on the efficiency of the conjugate gradient method for a class of time‐dependent problems |
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| Authors: | Xing Cai Bjørn Fredrik Nielsen Aslak Tveito |
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| Affiliation: | 1. Simula Research Laboratory, P.O. Box 134, NO‐1325 Lysaker, Norway;2. Department of Informatics, University of Oslo, P.O. Box 1080 Blindern, NO‐0316 Oslo, NorwaySimula Research Laboratory, P.O. Box 134, NO‐1325 Lysaker, Norway===;3. Department of Informatics, University of Oslo, P.O. Box 1080 Blindern, NO‐0316 Oslo, Norway |
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| 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. |
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| Keywords: | the conjugate gradient method parabolic partial differential equations Krylov subspace |
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