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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. 210-243, 2018/10/15
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