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Christoph Ortner, Endre Süli:

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. 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. 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.

 Bibliographical note: submitted to ESAIM : M2AN