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 efficient estimate of the error and that our atomistic-continuum adaptivity algorithm computes the quantity of interest to within a given tolerance with a nearly minimal atomistic region.