Analysis of a quasicontinuum method in one dimension 
C. Ortner and E. Suli
ESAIM-Mathematical Modelling and Numerical Analysis, 42, 57-91 (2008).

ABSTRACT
he 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
W-1,W-infinity-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.