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ETNA - Electronic Transactions on Numerical Analysis
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |
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DATUM, UNTERSCHRIFT / DATE, SIGNATURE
BANK AUSTRIA CREDITANSTALT, WIEN (IBAN AT04 1100 0006 2280 0100, BIC BKAUATWW), DEUTSCHE BANK MÜNCHEN (IBAN DE16 7007 0024 0238 8270 00, BIC DEUTDEDBMUC)
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ETNA - Electronic Transactions on Numerical Analysis, pp. 221-237, 2024/04/29
We consider the iterative solution of symmetric saddle-point matrices with a singular leading block. We develop a new ideal positive definite block-diagonal preconditioner that yields a preconditioned operator with four distinct eigenvalues. We offer a few techniques for making the preconditioner practical and illustrate the effectiveness of our approach with numerical experiments. The novelty of the paper lies in the generality of the assumptions made: as long as the saddle-point matrix is nonsingular, there is no assumption on the specific rank of the leading block. Current ideal preconditioners typically rely either on invertibility or a high nullity of the leading block, and the new technique aims to bridge this gap. A spectral analysis is offered, accompanied by numerical experiments.
Keywords: saddle-point systems, preconditioning, augmentation, Schur complement