  We derive a model problem for quasicontinuum approximations that allows a
simple, yet insightful, analysis of the optimal-order convergence rate in the
continuum limit for both the energy-based quasicontinuum approximation and the
quasi-nonlocal quasicontinuum approximation. The optimal-order error estimates
for the quasi-nonlocal quasicontinuum approximation are given for all strains
up to the continuum limit strain for fracture. The analysis is based on an
explicit treatment of the coupling error at the atomistic to continuum
interface, combined with an analysis of the error due to atomistic and
continuum schemes using the stability of the quasicontinuum approximation.
