A convergence analysis of the quasicontinuum method

Malena I. Espãnol, Dennis M. Kochmann, Sergio Contixz, Michael Ortiz

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


We present a ł-convergence analysis of the quasicontinuum method focused on the behavior of the approximate energy functionals in the continuum limit of a harmonic and defectfree crystal. The analysis shows that, under general conditions of stability and boundedness of the energy, the continuum limit is attained provided that the continuum-e.g., finite-element-Approximation spaces are strongly dense in an appropriate topology and provided that the lattice size converges to zero more rapidly than the mesh size. The equicoercivity of the quasicontinuum energy functionals is likewise established with broad generality, which, in conjunction with -convergence, ensures the convergence of the minimizers. We also show under rather general conditions that, for interatomic energies having a clusterwise additive structure, summation or quadrature rules that suitably approximate the local element energies do not affect the continuum limit. Finally, we propose a discrete patch test that provides a practical means of assessing the convergence of quasicontinuum approximations. We demonstrate the utility of the discrete patch test by means of selected examples of application.

Original languageEnglish (US)
Pages (from-to)766-794
Number of pages29
JournalMultiscale Modeling and Simulation
Issue number3
StatePublished - 2013
Externally publishedYes


  • Atomistic-To-continuum modelsł
  • Convergence
  • Quasicontinuum method

ASJC Scopus subject areas

  • Chemistry(all)
  • Modeling and Simulation
  • Ecological Modeling
  • Physics and Astronomy(all)
  • Computer Science Applications


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