Volume 42, Number 1, January-February 2008
|Page(s)||57 - 91|
|Published online||12 January 2008|
Analysis of a quasicontinuum method in one dimension
Oxford University Computing Laboratory, Wolfson Building,
Parks Road, Oxford OX1 3QD, UK. firstname.lastname@example.org; email@example.com
Revised: 17 April 2007
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.
Mathematics Subject Classification: 70C20 / 70-08 / 65N15
Key words: Atomistic material models / quasicontinuum method / error analysis / stability.
© EDP Sciences, SMAI, 2008
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