A-posteriori analysis and adaptive algorithms for the quasicontinuum method in one dimension

The quasicontinuum (QC) 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-posteriori error analysis for the quasi-continuum method in one dimension. We consider atomistic models with Lennard-Jones type finite-range interactions.\ud \ud We prove that, for a stable QC solution with a sufficiently small residual, which is computed in a discrete Sobolev-type norm, there exists an exact solution of the atomistic model problem for which an a-posteriori error estimate holds. We then derive practically computable bounds on the residual and on the inf-sup constants which measure the stability of the QC solution.\ud \ud Finally, we supplement the QC method with a proximal point optimization method with local-error control. We prove that the parameters can be adjusted so that at each step of the optimization algorithm there exists an exact solution to a related atomistic problem whose distance to the numerical solution is smaller than a pre-set tolerance.\ud \ud Key words and phrases: atomistic material models, quasicontinuum method, error analysis, adaptivity, stability\ud \ud The first author acknowledges the financial support received from the European research project HPRB-CT-2002-00284: New Materials, Adaptive Systems and their Nonlinearities. Modelling, Control and Numerical Simulation, and the kind hospitality of Carlo Lovadina and Matteo Negri (University of Pavia).\ud \ud We would like to thank Nick Gould for his advice on practical optimization methods, particularly on proximal point algorithms