The Spectrum Of The Force-Based Quasicontinuum Operator For A Homogeneous Periodic Chain

We show under general conditions that the linearized force-based quasicontinuum (QCF) operator has a real, positive spectrum. The spectrum is identical to that of the quasinon-local quasicontinuum (QNL) operator in the case of second-neighbor interactions. We construct an eigenbasis for the linearized QCF operator whose condition number is uniform in the number of atoms and the size of the atomistic region. These results establish the validity of and improve upon recent numerical observations [M. Dobson, M. Luskin, and C. Ortner, Comput. Methods Appl. Mech. Engrg., 200 (2011), pp. 2697-2709, Multiscale Model. Simul., 8 (2010), pp. 782-802]. As immediate consequences of our results we obtain rigorous estimates for convergence rates of (preconditioned) GMRES algorithms as well as a new stability estimate for the QCF method.