The quasicontinuum method is a coarse-graining technique for
reducing the complexity of atomistic simulations in a static and
quasistatic setting. In this paper we aim to give a detailed a
priori and a posteriori error analysis for a quasicontinuum
method in one dimension. We consider atomistic models with
Lennard–Jones type long-range interactions and a QC formulation
which incorporates several important aspects of practical QC
methods. First, we prove the existence, the local uniqueness and the
stability with respect to a discrete W1,∞-norm of
elastic and fractured atomistic solutions. We use a fixed point
argument to prove the existence of a quasicontinuum approximation
which satisfies a quasi-optimal a priori error bound. We then
reverse the role of exact and approximate solution and prove that,
if a computed quasicontinuum solution is stable in a sense that we
make precise and has a sufficiently small residual, there exists a
`nearby' exact solution which it approximates, and we give an a
posteriori error bound. We stress that, despite the fact that we
use linearization techniques in the analysis, our results apply to
genuinely nonlinear situations.