Convergence analysis of a quasi-continuum approximation for a two-dimensional material without defects
P. Lin
SIAM Journal on Numerical Analysis, 45, 313-332 (2007).

ABSTRACT
In many applications, materials are modeled by a large number of particles ( or
atoms), where any particle can interact with any other. The computational cost
is very high since the number of atoms is huge. Recently much attention has been
paid to a so-called quasi-continuum (QC) method, which is a mixed
atomistic/continuum model. The QC method uses an adaptive finite element
framework to effectively integrate the majority of the atomistic degrees of
freedom in regions where there is no serious defect. However, numerical analysis
of this method is still in its infancy. In this paper we will conduct a
convergence analysis of the QC method in the case when there is no defect. We
will also remark on the case when the defect region is small. The difference
between our analysis and conventional analysis is that our exact atomistic
solution is not a solution of a continuous partial differential equation, but a
discrete lattice scale solution which is not approximately related to any
conventional partial differential equation.