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Analysis of a quasicontinuum method in one dimension

Published online by Cambridge University Press:  12 January 2008

Christoph Ortner  and
Endre Süli
Christoph Ortner
Affiliation:
Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, UK. christoph.ortner@comlab.ox.ac.uk; endre.suli@comlab.ox.ac.uk
Endre Süli
Affiliation:
Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, UK. christoph.ortner@comlab.ox.ac.uk; endre.suli@comlab.ox.ac.uk
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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.

Keywords

Atomistic material modelsquasicontinuum methoderror analysisstability.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 1 , January 2008 , pp. 57 - 91
Copyright
© EDP Sciences, SMAI, 2008

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