The atomistic to continuum interface for quasicontinuum energiesexhibits nonzero forces under uniform strain that have beencalled ghost forces.In this paper,we prove for a linearization of a one-dimensional quasicontinuum energyaround a uniform strainthat the effect of the ghost forces on the displacementnearly cancels and has a small effect on the error away from the interface.We give optimal order error estimatesthat show that the quasicontinuum displacementconverges to the atomistic displacement at the rate O(h)in the discrete $\ell^\infty$ andw 1,1 norms where h is the interatomic spacing.We also give a proof that the error in the displacement gradientdecays away from the interface to O(h) at distance O(h|logh|)in the atomistic region and distance O(h) in the continuum region.Our work gives an explicit and simplified form for the decay of the effect of theatomistic to continuum coupling error in terms of a general underlying interatomic potential and gives the estimates described above in the discrete $\ell^\infty$ and w1,p norms.

We analyze a force-based quasicontinuum approximation to aone-dimensional system of atoms that interact by a classicalatomistic potential. This force-based quasicontinuum approximationcan be derived as the modification of an energy-basedquasicontinuum approximation by the addition of nonconservativeforces to correct nonphysical “ghost” forces that occur in theatomistic to continuum interface during constant strain. The algorithmicsimplicity and consistency with the purely atomistic modelat constant strain has made the force-basedquasicontinuum approximation popular for large-scalequasicontinuum computations.We prove that the force-based quasicontinuum equations havea unique solution when the magnitude of the external forces satisfyexplicit bounds. For Lennard-Jones next-nearest-neighborinteractions, we show that unique solutions existfor external forces that extend the system nearly to its tensile limit.We give an analysis of the convergence of the ghost force iteration methodto solve the equilibrium equations for the force-based quasicontinuum approximation.We show that the ghost force iteration is a contraction and give an analysis for itsconvergence rate.