Linear stationary iterative methods for the force-based quasicontinuum approximation

Mitchell Luskin, Christoph Ortner

Research output: Chapter in Book/Report/Conference proceedingConference contribution

2 Scopus citations


Force-based multiphysics coupling methods have become popular since they provide a simple and efficient coupling mechanism, avoiding the difficulties in formulating and implementing a consistent coupling energy. They are also the only known pointwise consistent methods for coupling a general atomistic model to a finite element continuum model. However, the development of efficient and reliable iterative solution methods for the force-based approximation presents a challenge due to the non-symmetric and indefinite structure of the linearized force-based quasicontinuum approximation, as well as to its unusual stability properties. In this paper, we present rigorous numerical analysis and computational experiments to systematically study the stability and convergence rate for a variety of linear stationary iterative methods.

Original languageEnglish (US)
Title of host publicationNumerical Analysis of Multiscale Computations - Proceedings of a Winter Workshop at the Banff International Research Station 2009
EditorsOlof Runborg, Bjorn Engquist, Yen-Hsi R. Tsai
PublisherSpringer Verlag
Number of pages38
ISBN (Print)9783642219429
StatePublished - 2012
EventWorkshop on Numerical Analysis and Multiscale Computations, 2009 - Banff, Canada
Duration: Dec 6 2009Dec 11 2009

Publication series

NameLecture Notes in Computational Science and Engineering
ISSN (Print)1439-7358


OtherWorkshop on Numerical Analysis and Multiscale Computations, 2009

Bibliographical note

Funding Information:
Acknowledgements This work was supported in part by the National Science Foundation under DMS-0757355, DMS-0811039, the PIRE Grant OISE-0967140, the Institute for Mathematics and Its Applications, and the University of Minnesota Supercomputing Institute. This work was also supported by the Department of Energy under Award Number DE-SC0002085. CO was supported by the EPSRC grant EP/H003096/1 “Analysis of Atomistic-to-Continuum Coupling Methods.” Fig. 4 Graphs of the operator norm kGqcekUk;p, .k;p/ 2 f.1;1/;.2;1/g, for f ixed N D 256;K D 15, F00 D 1, plotted against AF. For the case U1;1 only estimates are available and upper and lower bounds are shown instead (cf. Appendix 7). The graphs confirm the result of Corollary 2 that kGqcekUk;p ! 1 as AF C K 2F00 ! 0. Moreover, they clearly indicate that kGqcekUk;p > 1 already for strains F in the region AF 0:5, which are much lower than the critical strain at which LFqce becomes singular

Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2012.


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