Luskin, Mitchell; Ortner, Christoph An analysis of node-based cluster summation rules in the quasicontinuum method. (English) Zbl 1196.82122 SIAM J. Numer. Anal. 47, No. 4, 3070-3086 (2009). Summary: We investigate two examples of node-based cluster summation rules that have been proposed for the quasicontinuum (QC) method: a force-based approach and an energy-based approach which is a generalization of the nonlocal QC method. We show that, even for the case of nearest-neighbor interaction in a one-dimensional periodic chain, both of these approaches create large errors that cannot be removed by increasing the cluster size when used with graded and, more generally, nonsmooth meshes. We offer some suggestions for how the accuracy of (cluster) summation rules may be improved. Cited in 20 Documents MSC: 82D25 Statistical mechanics of crystals 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs Keywords:atomistic-to-continuum coupling; coarse-graining; quasicontinuum method; cluster summation rule PDFBibTeX XMLCite \textit{M. Luskin} and \textit{C. Ortner}, SIAM J. Numer. Anal. 47, No. 4, 3070--3086 (2009; Zbl 1196.82122) Full Text: DOI arXiv