Analysis of a quasicontinuum method in one dimension
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 1, p. 57-91

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 W 1, -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.

DOI : https://doi.org/10.1051/m2an:2007057
Classification:  70C20,  70-08,  65N15
Keywords: atomistic material models, quasicontinuum method, error analysis, stability
@article{M2AN_2008__42_1_57_0,
     author = {Ortner, Christoph and S\"uli, Endre},
     title = {Analysis of a quasicontinuum method in one dimension},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {1},
     year = {2008},
     pages = {57-91},
     doi = {10.1051/m2an:2007057},
     zbl = {1139.74004},
     mrnumber = {2387422},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2008__42_1_57_0}
}
Ortner, Christoph; Süli, Endre. Analysis of a quasicontinuum method in one dimension. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 1, pp. 57-91. doi : 10.1051/m2an:2007057. http://www.numdam.org/item/M2AN_2008__42_1_57_0/

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