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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
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Official URL: http://dx.doi.org/10.1137/110825704
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 | ||||||||
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| Subjects: | Q Science > QA Mathematics T Technology > TA Engineering (General). Civil engineering (General) |
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| Divisions: | Faculty of 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: |
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| 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 | ||||||||
| 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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