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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. 169-196, 2024/04/05
In this article a new family of preconditioners is introduced for symmetricpositive definite linear systems. The new preconditioners, called the AWGpreconditioners (for Algebraic-Woodbury-GenEO), are constructedalgebraically. By this we mean that only the knowledge of the matrix $\mathbf{A}$for which the linear system is being solved is required. Thanks to the GenEOspectral coarse space technique, the condition number of the preconditionedoperator is bounded theoretically from above. This upper bound can be madesmaller by enriching the coarse space with more spectral modes.The novelty is that, unlike in previous work on the GenEO coarse spaces, noknowledge of a partially non-assembled form of $\mathbf{A}$ is required. Indeed, thespectral coarse space technique is not applied directly to $\mathbf{A}$ but to alow-rank modification of $\mathbf{A}$ of which a suitable non-assembled form is knownby construction. The extra cost is a second (and to this day rather expensive)coarse solve in the preconditioner. One of the AWG preconditioners has alreadybeen presented in a short preprint by Spillane [DomainDecomposition Methods in Science and EngineeringXXVI, Springer, Cham, 2022, pp. 745–752].This article is the first full presentation of the larger family of AWGpreconditioners. It includes proofs of the spectral bounds as well asnumerical illustrations.
Keywords: preconditioner, domain decomposition, coarse space, algebraic, linear system, Woodbury matrix identity