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Analysis of a quasicontinuum method in one dimension
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Ortner, Christoph and Süli, Endre (2008) Analysis of a quasicontinuum method in one dimension. ESAIM: Mathematical Modelling and Numerical Analysis, Volume 42 (Number 1). pp. 57-91. doi:10.1051/m2an:2007057 ISSN 0764-583X.
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Official URL: http://dx.doi.org/10.1051/m2an:2007057
Abstract
The quasicontinuum method is a coarse-graining technique for reducing the complexity of atomistic simulations in a static and quasistatic setting. In this paper we aim to give a detailed a priori and a posteriori error analysis for a quasicontinuum method in one dimension. We consider atomistic models with Lennard–Jones type long-range interactions and a QC formulation which incorporates several important aspects of practical QC methods. First, we prove the existence, the local uniqueness and the stability with respect to a discrete W1,∞-norm of elastic and fractured atomistic solutions. We use a fixed point argument to prove the existence of a quasicontinuum approximation which satisfies a quasi-optimal a priori error bound. We then reverse the role of exact and approximate solution and prove that, if a computed quasicontinuum solution is stable in a sense that we make precise and has a sufficiently small residual, there exists a `nearby' exact solution which it approximates, and we give an a posteriori error bound. We stress that, despite the fact that we use linearization techniques in the analysis, our results apply to genuinely nonlinear situations.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Multiscale modeling, Continuum mechanics | ||||
Journal or Publication Title: | ESAIM: Mathematical Modelling and Numerical Analysis | ||||
Publisher: | EDP Sciences | ||||
ISSN: | 0764-583X | ||||
Official Date: | January 2008 | ||||
Dates: |
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Volume: | Volume 42 | ||||
Number: | Number 1 | ||||
Page Range: | pp. 57-91 | ||||
DOI: | 10.1051/m2an:2007057 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access | ||||
Date of first compliant deposit: | 20 December 2015 | ||||
Date of first compliant Open Access: | 20 December 2015 |
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