  Due to their algorithmic simplicity and high accuracy, force-based model
coupling techniques are an exciting development in computational physics. For
example, the force-based quasicontinuum approximation is the only known
pointwise consistent quasicontinuum approximation for coupling a general
atomistic model with a finite element continuum model. In this paper, we
analyze the stability of the force-based quasicontinuum approximation. We then
use our stability result to obtain an optimal order error analysis of this
coupling method that provides theoretical justification for the high accuracy
of the force-based quasicontinuum approximation -- the computational efficiency
of continuum modeling can be utilized without the loss of significant accuracy
if defects are captured in the atomistic region. The main challenge we need to
overcome is the fact (which we prove) that the linearized quasicontinuum
operator is typically not positive definite. Moreover, we prove that no uniform
inf-sup stability condition holds for discrete versions of the
$W^{1,p}$-$W^{1,q}$ "duality pairing" with $1/p+1/q=1$, if $1 \leq p < \infty$.
We must therefore derive an inf-sup stability condition for a discrete version
of the $W^{1,\infty}$-$W^{1,1}$ "duality pairing" which then leads to optimal
order error estimates in a discrete $W^{1,\infty}$-norm.
