Shift-Invert Arnoldi's Method with Preconditioned Iterative Solves
SIAM Journal on Matrix Analysis and Applications
A Coarse Space Construction Based on Local Dirichlet-to-Neumann Maps
SIAM Journal on Scientific Computing
Journal of Computational and Applied Mathematics
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For quite some time, the deflation preconditioner has been proposed and used to accelerate the convergence of Krylov subspace methods. For symmetric positive definite linear systems, the convergence of conjugate gradient methods combined with deflation has been analyzed and compared with other preconditioners, e.g., with the abstract balancing preconditioner [R. Nabben and C. Vuik, SIAM J. Sci. Comput., 27 (2006), pp. 1742-1759]. In this paper, we extend the convergence analysis to nonsymmetric linear systems in the context of GMRES iteration and compare it with the abstract nonsymmetric balancing preconditioner. We are able to show that many results for symmetric positive definite matrices carry over to arbitrary nonsymmetric matrices. First we establish that the spectra of the preconditioned systems are similar. Moreover, we show that under certain conditions, the 2-norm of residuals produced by GMRES combined with deflation is never larger than the 2-norm of residuals produced by GMRES combined with the abstract balancing preconditioner. Numerical experiments are done to nonsymmetric linear systems arising from a finite volume discretization of the convection-diffusion equation, and the numerical results confirm our theoretical results.