Stability, Instability, and Error of the Force-based Quasicontinuum Approximation
Stability, Instability, and Error of the Force-based Quasicontinuum Approximation
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DOI：
10.1007/s00205-009-0276-z
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发表时间：
2009-03
影响因子：
2.5
通讯作者：
M. Dobson;M. Luskin;C. Ortner
中科院分区：
文献类型：
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作者：
M. Dobson;M. Luskin;C. Ortner
基于力的模型耦合技术由于其算法简单和高精度，是计算物理中流行的工具。例如，基于力的准连续体（QCF）近似是唯一已知的用于耦合一般原子模型与有限元连续体模型的逐点一致准连续体近似。本文详细分析了该方法的稳定性和误差。我们的最佳顺序误差估计提供了一个理论的理由QCF近似的高精度：他们清楚地表明，连续建模的计算效率可以利用没有显着损失的准确性，如果缺陷被捕获在原子区域。我们需要克服的主要挑战是线性化的QCF算子通常不是正定的。证明了当1 ≤p< ∞时，W1，p-W1，q“对偶对”的离散形式（1/p+ 1/q= 1）不存在一致的inf-sup稳定性条件。然而，我们能够建立一个inf-sup稳定性条件的离散版本的W1，∞-W1，1“对偶配对”，导致最佳阶误差估计在离散W1，∞-范数。
Due to their algorithmic simplicity and high accuracy, force-based model coupling techniques are popular tools in computational physics. For example, the force-based quasicontinuum (QCF) approximation is the only known pointwise consistent quasicontinuum approximation for coupling a general atomistic model with a finite element continuum model. In this paper, we present a detailed stability and error analysis of this method. Our optimal order error estimates provide a theoretical justification for the high accuracy of the QCF approximation: they clearly demonstrate that the computational efficiency of continuum modeling can be utilized without a significant loss of accuracy if defects are captured in the atomistic region. The main challenge we need to overcome is the fact that the linearized QCF operator is typicallynotpositive definite. Moreover, we prove that no uniform inf-sup stability condition holds for discrete versions of theW1,p-W1,q“duality pairing” with 1/p+ 1/q= 1, if 1 ≤p< ∞. However, we were able to establish an inf-sup stability condition for a discrete version of theW1,∞-W1,1“duality pairing” which leads to optimal order error estimates in a discreteW1,∞-norm.