  QM (quantum mechenics) and MM (molecular mechenics) coupling methods are
widely used in simulations of crystalline defects. In this paper, we construct
a residual based a posteriori error indicator for QM/MM coupling
approximations. We prove the reliability of the error indicator (upper bound of
the true approximation error) and develop some sampling techniques for its
efficient calculation. Based on the error indicator and D\"{o}rfler marking
strategy, we design an adaptive QM/MM algorithm for crystalline defects and
demonstrate the efficiency with some numerical experiments.
