A-priori analysis of the quasicontinuum method in one dimension

Abstract

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 discrete W1,∞-norm of elastic and fractured atomistic solutions. We then used a fixed point technique to prove the existence of quasicontinuum approximation which satisfies an optimal a-priori error bound.

The first-named author acknowledges the financial support received from the European research project HPRN-CT-2002-00284: New Materials, Adaptive Systems and their Nonlinearities. Modelling, Control and Numerical Simulation, and the kind hospitality of Carlo Lovadina (University of Pavia).