Deflation and Balancing Preconditioners for Krylov Subspace Methods Applied to Nonsymmetric Matrices

For quite some times, deflation preconditioner has been proposed and used to accelerate the convergence of Krylov subspace methods. For symmetric positive definite linear systems, convergence of CG combined with deflation has been analyzed and compared with other preconditioners, e.g. with the abstract balancing preconditioner [Nabben and 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 show that under certain conditions, the 2norm of residuals produced by GMRES combined with deflation is never larger than the 2-norm of residuals produced by GMRES combined with the 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.