  Force-based atomistic-continuum hybrid methods are the only known pointwise
consistent methods for coupling a general atomistic model to a finite element
continuum model. For this reason, and due to their algorithmic simplicity,
force-based coupling methods have become a popular class of atomistic-continuum
hybrid models as well as other types of multiphysics models. However, the
recently discovered unusual stability properties of the linearized force-based
quasicontinuum (QCF) approximation, especially its indefiniteness, present a
challenge to the development of efficient and reliable iterative methods. We
present analytic and computational results for the generalized minimal residual
(GMRES) solution of the linearized QCF equilibrium equations. We show that the
GMRES method accurately reproduces the stability of the force-based
approximation and conclude that an appropriately preconditioned GMRES method
results in a reliable and efficient solution method.

Numerical Analysis