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