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 -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.
Mots clés : 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: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {57--91}, publisher = {EDP-Sciences}, volume = {42}, number = {1}, year = {2008}, doi = {10.1051/m2an:2007057}, mrnumber = {2387422}, zbl = {1139.74004}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an:2007057/} }
TY - JOUR AU - Ortner, Christoph AU - Süli, Endre TI - Analysis of a quasicontinuum method in one dimension JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2008 SP - 57 EP - 91 VL - 42 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2007057/ DO - 10.1051/m2an:2007057 LA - en ID - M2AN_2008__42_1_57_0 ER -
%0 Journal Article %A Ortner, Christoph %A Süli, Endre %T Analysis of a quasicontinuum method in one dimension %J ESAIM: Modélisation mathématique et analyse numérique %D 2008 %P 57-91 %V 42 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2007057/ %R 10.1051/m2an:2007057 %G en %F M2AN_2008__42_1_57_0
Ortner, Christoph; Süli, Endre. Analysis of a quasicontinuum method in one dimension. ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 1, pp. 57-91. doi : 10.1051/m2an:2007057. http://www.numdam.org/articles/10.1051/m2an:2007057/
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