The spectrum of the force-based quasicontinuum operator for a homogeneous periodic chain
M. Dobson, C. Ortner and A. V. Shapeev
Multiscale Modeling and Simulation, 10, 744-765 (2012).
ABSTRACT
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.