An efficient block preconditioner for Jacobian‐free global–local multiscale methods

Abstract In this paper, we develop a block preconditioner for Jacobian‐free global–local multiscale methods, in which the explicit computation of the Jacobian may be circumvented at the macroscale by using a Newton–Krylov process. Effective preconditioning is necessary for the Krylov subspace iterations (e.g. GMRES) to enhance computational efficiency. This is, however, challenging since no explicit information regarding the Jacobian matrix is available. The block preconditioning technique developed in this paper circumvents this problem by effectively deflating the spectrum of the Jacobian matrix at the current Newton step using information about only the Krylov subspaces corresponding to the Jacobian matrices in the previous Newton steps and their representations on those subspaces. This approach is optimal and results in exponential convergence of the GMRES iterations within each Newton step, thus minimizing expensive microscale computations without requiring explicit Jacobian formation in any step. In terms of both computational cost and storage requirements, the action of a single block of the preconditioner per GMRES step scales linearly as the number of degrees of freedom of the macroscale problem as well as the dimension of the invariant subspace of the preconditioned Jacobian matrix. Copyright © 2011 John Wiley & Sons, Ltd.