An analysis of the effect of ghost force oscillation on quasicontinuum error
M. Dobson and M. Luskin
ESAIM-Mathematical Modelling and Numerical Analysis, 43, 591-604 (2009).
ABSTRACT
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 l(infinity) 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 vertical bar log h vertical bar) 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 l(infinity) and w(1,p)
norms.