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

Dobson, Matthew; Luskin, Mitchell

ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 3, pp. 591-604.

The atomistic to continuum interface for quasicontinuum energies exhibits nonzero forces under uniform strain that have been called ghost forces. In this paper, we prove for a linearization of a one-dimensional quasicontinuum energy around a uniform strain that the effect of the ghost forces on the displacement nearly cancels and has a small effect on the error away from the interface. We give optimal order error estimates that show that the quasicontinuum displacement converges to the atomistic displacement at the rate O($h$) in the discrete ${\ell }^{\infty }$ and $w^{1,1}$ norms where $h$ is the interatomic spacing. We also give a proof that the error in the displacement gradient decays 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 the atomistic to continuum coupling error in terms of a general underlying interatomic potential and gives the estimates described above in the discrete ${\ell }^{\infty }$ and $w^{1,p}$ norms.

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