Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

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Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model
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by Ping Lin;
Math. Comp. 72 (2003), 657-675
DOI: https://doi.org/10.1090/S0025-5718-02-01456-4
Published electronically: June 4, 2002
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Abstract:

In many applications materials are modeled by a large number of particles (or atoms) where any one of particles interacts with all others. Near or nearest neighbor interaction is expected to be a good simplification of the full interaction in the engineering community. In this paper we shall analyze the approximate error between the solution of the simplified problem and that of the full-interaction problem so as to answer the question mathematically for a one-dimensional model. A few numerical methods have been designed in the engineering literature for the simplified model. Recently much attention has been paid to a finite-element-like quasicontinuum (QC) method which utilizes a mixed atomistic/continuum approximation model. No numerical analysis has been done yet. In the paper we shall estimate the error of the QC method for this one-dimensional model. Possible ill-posedness of the method and its modification are discussed as well.
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Bibliographic Information
  • Ping Lin
  • Affiliation: Department of Mathematics, The National University of Singapore, 2 Science Drive 2, Singapore 117543
  • Email: matlinp@math.nus.edu.sg
  • Received by editor(s): June 9, 1998
  • Received by editor(s) in revised form: May 29, 2001
  • Published electronically: June 4, 2002
  • © Copyright 2002 American Mathematical Society
  • Journal: Math. Comp. 72 (2003), 657-675
  • MSC (2000): Primary 65C20, 65K10, 65M15, 65M60, 74N15, 74G65
  • DOI: https://doi.org/10.1090/S0025-5718-02-01456-4
  • MathSciNet review: 1954960