A-Priori Analysis of the Quasicontinuum Method in One Dimension

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 give an a-priori error analysis for the quasicontinuum method in one dimension. We consider atomistic models with Lennard–Jones type long range interactions and a practical QC formulation. 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 then use a fixed point technique to prove the existence of a quasicontinuum approximation which satisfies an optimal a-priori error bound.