A posteriori analysis and adaptive algorithms for blended type atomistic-to-continuum coupling with higher-order finite elements

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.