Convergence of a Force-Based Hybrid Method in Three Dimensions 
J. F. Lu and P. B. Ming
Communications on Pure and Applied Mathematics, 66, 83-108 (2013).

ABSTRACT
We study a force-based hybrid method that couples an atomistic model with the
Cauchy-Born elasticity model. We show that the proposed scheme converges to the
solution of the atomistic model with second-order accuracy, since the ratio
between lattice parameter and the characteristic length scale of the deformation
tends to 0. Convergence is established for the three-dimensional system without
defects, with general finite-range atomistic potential and simple lattice
structure. The proof is based on consistency and stability analysis. General
tools for stability analysis are developed in the framework opseudodifference
operators in arbitrary dimensions.