Error Estimation and Atomistic-Continuum Adaptivity for the Quasicontinuum Approximation of a Frenkel-Kontorova Model

We propose and analyze a goal-oriented a posteriori error estimator for the atomisticcontinuum modeling error in the quasicontinuum method. Based on this error estimator, we develop an algorithm which adaptively determines the atomistic and continuum regions to compute a quantity of interest to within a given tolerance. We apply the algorithm to the computation of the structure of a crystallographic defect described by a Frenkel-Kontorova model and present the results of numerical experiments. The numerical results show that our method gives an e cient estimate of the error and a nearly optimal atomistic-continuum modeling strategy.