A posteriori analysis and adaptive algorithms for blended type atomistic-to-continuum coupling with higher-order finite elements
Y. S. Wang
Computer Physics Communications, 310, 109533 (2025).

ABSTRACT
The accurate and efficient simulation of material systems with defects using
atomistic-to-continuum (a/c) coupling methods is a significant focus in
computational materials science. Achieving a balance between accuracy and
computational cost requires the application of a posteriori error analysis and
adaptive algorithms. In this paper, we provide a rigorous a posteriori error
analysis for three common blended a/c methods: the blended energy-based
quasi-continuum (BQCE) method, the blended force-based quasi-continuum (BQCF)
method, and the atomistic/continuum blending with ghost force correction (BGFC)
method. We discretize the Cauchy-Born model in the continuum region using first-
and second-order finite element methods, with the potential for extending to
higher-order schemes. The resulting error estimator provides both an upper bound
on the true error and a reliable lower bound, subject to a controllable
truncation term. Furthermore, we offer an a posteriori analysis of the energy
error. We develop and implement an adaptive mesh refinement algorithm applied to
two typical defect scenarios: a micro-crack and a Frenkel defect. In both cases,
our numerical experiments demonstrate optimal convergence rates with respect to
degrees of freedom, in agreement with a priori error estimates.