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The spectrum of the force-based quasicontinuum operator for a homogeneous periodic chain

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Dobson, Matthew, Ortner, Christoph and Shapeev, A. V. (2012) The spectrum of the force-based quasicontinuum operator for a homogeneous periodic chain. Multiscale Modeling & Simulation, Volume 10 (Number 3). pp. 744-765. doi:10.1137/110825704 ISSN 1540-3459.

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Official URL: http://dx.doi.org/10.1137/110825704

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Abstract

We show under general conditions that the linearized force-based quasicontinuum (QCF) operator has a real, positive spectrum. The spectrum is identical to that of the quasinonlocal quasicontinuum (QNL) operator in the case of second-neighbor interactions. We construct an eigenbasis for the linearized QCF operator whose condition number is uniform in the number of atoms and the size of the atomistic region. These results establish the validity of and improve upon recent numerical observations [M. Dobson, M. Luskin, and C. Ortner, Comput. Methods Appl. Mech. Engrg., 200 (2011), pp. 2697--2709, Multiscale Model. Simul., 8 (2010), pp. 782--802]. As immediate consequences of our results we obtain rigorous estimates for convergence rates of (preconditioned) GMRES algorithms as well as a new stability estimate for the QCF method.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
T Technology > TA Engineering (General). Civil engineering (General)
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH): Multiscale modeling, Continuum mechanics
Journal or Publication Title: Multiscale Modeling & Simulation
Publisher: World Scientific Publishing Co. Pte. Ltd.
ISSN: 1540-3459
Official Date: 17 July 2012
Dates:
DateEvent
17 July 2012Available
3 April 2012Accepted
24 February 2011Submitted
Volume: Volume 10
Number: Number 3
Page Range: pp. 744-765
DOI: 10.1137/110825704
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Date of first compliant deposit: 27 December 2015
Date of first compliant Open Access: 27 December 2015
Funder: Engineering and Physical Sciences Research Council (EPSRC), National Science Foundation (U.S.) (NSF), France. Agence nationale de la recherche (ANR)
Grant number: ANR-06-CIS6-006 (ANR)
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