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The atomistic to continuum interface for quasicontinuum energies exhibitsnonzero forces under uniform strain that have been called ghost forces. In thispaper, we prove for a linearization of a one-dimensional quasicontinuum energyaround a uniform strain that the effect of the ghost forces on the displacementnearly cancels and has a small effect on the error away from the interface. Wegive optimal order error estimates that show that the quasicontinuumdisplacement converges to the atomistic displacement at the optimal rate O($h$)in the discrete $\ell^\infty$ norm and O($h^{1/p}$) in the $w^{1,p}$ norm for$1 \leq p < \infty.$ where $h$ is the interatomic spacing. We also give a proofthat the error in the displacement gradient decays away from the interface toO($h$) at distance O($h|\log h|$) in the atomistic region and distance O($h$)in the continuum region. E, Ming, and Yang previously gave a counterexample toconvergence in the $w^{1,\infty}$ norm for a harmonic interatomic potential.Our work gives an explicit and simplified form for the decay of the effect ofthe atomistic to continuum coupling error in terms of a general underlyinginteratomic potential and gives the estimates described above in the discrete$\ell^\infty$ and $w^{1,p}$ norms.

An analysis of the effect of ghost force oscillation on quasicontinuum error