  Adaptive atomistic/continuum (a/c) coupling method is an important method for
the simulation of material and atomistic systems with defects to achieve the
balance of accuracy and efficiency. Residual based a posteriori error estimator
is often employed in the adaptive algorithm to provide an estimate of the error
of the strain committed by applying the continuum approximation for the
atomistic system and the finite element discretization in the continuum region.
In this work, we propose a theory based approximation for the residual based a
posteriori error estimator which greatly improves the efficiency of the
adaptivity. In particular, the numerically expensive modeling residual is only
computed exactly in a small region around the coupling interface but replaced
by a theoretically justified approximation by the coarsening residual outside
that region. We present a range of adaptive computations based on our modified
a posteriori error estimator and its variants for different types of
crystalline defects some of which are not considered in previous related
literature of the adaptive a/c methods. The numerical results show that,
compared with the original residual based error estimator, the adaptive
algorithm using the modified error estimator with properly chosen parameters
leads to the same optimal convergence rate of the error but reduces the
computational cost by one order with respect to the number of degrees of
freedom.
