A Spectral Analysis of Subspace Enhanced Preconditioners
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It is well-known that the convergence of Krylov subspace methods for solving linear system of equations depends on the spectrum of the matrix, moreover, it is widely accepted that for both symmetric and unsymmetric systems Krylov subspace methods converge faster if the spectrum of the matrix is clustered. In this paper we investigate the spectrum of the system preconditioned by deflation, coarse correction and adapted deflation preconditioners. Our analysis shows that the spectrum of the preconditioned system is highly impacted by the angle between the coarse space of the three preconditioners and the subspace spanned by the eigenvectors associated with the small eigenvalues of the matrix. Furthermore, we prove that the accuracy of the inverse of the projection matrix also impacts the spectrum of the preconditioned system. Numerical experiments confirm the theoretical analysis.
KeywordsPerturbation analysis Preconditioner Domain decomposition Spectrum Coarse space
The author would like to thank Prof. Frédéric Nataf and Prof. Xiao-Chuan Cai for their supervision and valuable suggestions. The author also thanks the referees for the helpful comments that improve the paper.
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