An analysis of the effect of ghost force oscillation on quasicontinuum error. (English) Zbl 1165.81414

Dobson, Matthew; Luskin, Mitchell

ESAIM, Math. Model. Numer. Anal. 43, No. 3, 591 (2009).

Summary: 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 \(O(h|\log h|)\) 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.

MSC:

81V70	Many-body theory; quantum Hall effect
82B30	Statistical thermodynamics
65Z05	Applications to the sciences
70C20	Statics

Keywords:

quasicontinuum; atomistic to continuum; ghost force

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