  The efficient and accurate simulation of material systems with defects using atomistic- to-continuum (a/c) coupling methods is a topic of considerable interest in the field of computational materials science. To achieve the desired balance between accuracy and computational efficiency, the use of a posteriori analysis and adaptive algorithms is critical. In this work, we present a rigorous a posteriori error analysis for three typical blended a/c coupling 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 employ first and second-order finite element methods (and potentially higher-order methods) to discretize the Cauchy-Born model in the continuum region. The resulting error estimator provides both an upper bound on the true approximation error and a lower bound up to a theory-based truncation indicator, ensuring its reliability and efficiency. Moreover, we propose an a posteriori analysis for the energy error. We have designed and implemented a corresponding adaptive mesh refinement algorithm for two typical examples of crystalline defects. In both numerical experiments, we observe optimal convergence rates with respect to degrees of freedom when compared to a priori error estimates. 
A Posteriori Analysis and Adaptive Algorithms for Blended Type   Atomistic-to-Continuum Coupling with Higher-Order Finite Elements