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 ISBN 978-3-7001-8258-0 Online Edition Research Article
Gabriele Ciaramella,
Martin J. Gander
S. 210 - 243 doi:10.1553/etna_vol49s210 Verlag der Österreichischen Akademie der Wissenschaften doi:10.1553/etna_vol49s210
Abstract: In the ddCOSMO solvation model for the numerical simulation ofmolecules (chains of atoms), the unusual observation was made thatthe associated Schwarz domain-decomposition method converges independentlyof the number of subdomains (atoms) and this without coarsecorrection, i.e., the one-level Schwarz method is scalable.We analyzed this unusual property for the simplified caseof a rectangular molecule and square subdomains using Fourier analysis,leading to robust convergence estimates in the $L^2$-norm and lateralso for chains of subdomains represented by disks using maximumprinciple arguments, leading to robust convergence estimates in$L^{\\infty}$. A convergence analysis in the more natural$H^1$-setting proving convergence independently of the number ofsubdomains was, however, missing. We close this gap in this paperusing tools from the theory of alternating projection methodsand estimates introduced by P.-L. Lions for the study of domaindecomposition methods. We prove that robust convergenceindependently of the number of subdomains is possible also in $H^1$and show furthermore that even for certain two-dimensional domainswith holes, Schwarz methods can be scalable without coarse-space corrections.As a by-product, we review some of the results of P.-L. Lions[On the Schwarz alternating method. I, in DomainDecomposition Methods for Partial Differential Equations, SIAM,Philadelphia, 1988, pp. 1-42]and in some cases provide simpler proofs. Keywords: domain decomposition methods, Schwarz methods, chain of subdomains, elliptic PDE, Laplace equation, COSMO solvation model Published Online: 2018/10/15 12:23:28 Object Identifier: 0xc1aa5576 0x0039f90e Rights: . Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613. …
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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 |