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.

翻译：利用原子到连续介质（a/c）耦合方法高效准确地模拟含缺陷材料系统是计算材料科学领域备受关注的研究课题。为了在精度与计算效率之间达到理想平衡，采用后验分析与自适应算法至关重要。本研究针对三种典型的混合型a/c耦合方法——混合能量型准连续体（BQCE）方法、混合力型准连续体（BQCF）方法以及具有伪力校正的原子/连续介质混合（BGFC）方法——提出了严格的后验误差分析。我们采用一阶和二阶有限元方法（并可能扩展至更高阶方法）对连续介质区域中的Cauchy-Born模型进行离散。所构建的误差估计量既提供了真实逼近误差的上界，又给出了基于理论截断指标的下界，确保了其可靠性与有效性。此外，我们还提出了能量误差的后验分析，并针对两类典型晶体缺陷设计实现了相应的自适应网格细化算法。在两项数值实验中，与先验误差估计相比，我们观察到关于自由度的最优收敛速率。