An optimal order error analysis of the one-dimensional quasicontinuum approximation

Matthew Dobson, Mitchell Luskin

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41 Scopus citations


We derive a model problem for quasicontinuum approximations that allows a simple, yet insightful, analysis of the optimal-order convergence rate in the continuum limit for both the energy-based quasicontinuum approximation and the quas i-nonlocal quasicontinuum approximation. For simplicity, the analysis is restricted to the case of second-neighbor interactions and is linearized about a uniformly stretched reference lattice. The o ptimal-order error estimates for the quasi-nonlocal quasicontinuum approximation are given for al l strains up to the continuum limit strain for fracture. The analysis is based on an explicit treatment of the coupling error at the atomistic-to-continuum interface, combined with an analysis of the error due to the atomistic and continuum schemes using the stability of the quasicontinuum approximation.

Original languageEnglish (US)
Pages (from-to)2455-2475
Number of pages21
JournalSIAM Journal on Numerical Analysis
Issue number4
StatePublished - 2009


  • Atomistic-to-continuum
  • Error analysis
  • Quasicontinuum


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