Dobson, Matthew, Luskin, Mitchell Barry and Ortner, Christoph (2010) Stability, instability, and error of the force-based quasicontinuum approximation. Archive for Rational Mechanics and Analysis, Volume 197 (Number 1). pp. 179-202. doi:10.1007/s00205-009-0276-z

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Abstract

Due to their algorithmic simplicity and high accuracy, force-based model coupling techniques are popular tools in computational physics. For example, the force-based quasicontinuum (QCF) approximation is the only known pointwise consistent quasicontinuum approximation for coupling a general atomistic model with a finite element continuum model. In this paper, we present a detailed stability and error analysis of this method. Our optimal order error estimates provide a theoretical justification for the high accuracy of the QCF approximation: they clearly demonstrate that the computational efficiency of continuum modeling can be utilized without a significant loss of accuracy if defects are captured in the atomistic region. The main challenge we need to overcome is the fact that the linearized QCF operator is typically not positive definite. Moreover, we prove that no uniform inf-sup stability condition holds for discrete versions of the W 1,p -W 1,q "duality pairing" with 1/p + 1/q = 1, if 1 ≤ p < ∞. However, we were able to establish an inf-sup stability condition for a discrete version of the W 1,∞-W 1,1 "duality pairing" which leads to optimal order error estimates in a discrete W 1,∞-norm.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics T Technology > TA Engineering (General). Civil engineering (General)
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Multiscale modeling, Continuum mechanics
Journal or Publication Title:	Archive for Rational Mechanics and Analysis
Publisher:	Springer
ISSN:	0003-9527
Official Date:	July 2010
Dates:	Date Event July 2010 Published
Date of first compliant deposit:	20 December 2015
Volume:	Volume 197
Number:	Number 1
Page Range:	pp. 179-202
DOI:	10.1007/s00205-009-0276-z
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	National Science Foundation (U.S.) (NSF), United States. Department of Energy, Institute for Mathematics and Its Applications (IMA), University of Minnesota. Supercomputer Institute, University of Minnesota, Engineering and Physical Sciences Research Council (EPSRC)
Grant number:	DMS-0757355 (NSF), DMS-0811039 (NSF), DE-FG02-05ER25706 (DOE)

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