A Gamma convergence analysis of the quasicontinuum method
M. I. Espanol, D. M. Kochmann, S. Conti and M. Ortiz
Multiscale Modeling & Simulation, 11, 766-794 (2013).

ABSTRACT
We present a Gamma-convergence analysis of the quasicontinuum method focused on
the behavior of the approximate energy functionals in the continuum limit of a
harmonic and defect-free crystal. The analysis shows that, under general
conditions of stability and boundedness of the energy, the continuum limit is
attained provided that the continuum-e. g., finite-element-approximation spaces
are strongly dense in an appropriate topology and provided that the lattice size
converges to zero more rapidly than the mesh size. The equicoercivity of the
quasicontinuum energy functionals is likewise established with broad generality,
which, in conjunction with Gamma-convergence, ensures the convergence of the
minimizers. We also show under rather general conditions that, for interatomic
energies having a clusterwise additive structure, summation or quadrature rules
that suitably approximate the local element energies do not affect the continuum
limit. Finally, we propose a discrete patch test that provides a practical means
of assessing the convergence of quasicontinuum approximations. We demonstrate
the utility of the discrete patch test by means of selected examples of
application.