The Spectrum of the Force-Based Quasicontinuum Operator for a Homogeneous Periodic Chain
The Spectrum of the Force-Based Quasicontinuum Operator for a Homogeneous Periodic Chain
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DOI：
10.1137/110825704
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发表时间：
2010-04
期刊：
影响因子：
--
通讯作者：
M. Dobson;C. Ortner;A. Shapeev
中科院分区：
文献类型：
--
作者：
M. Dobson;C. Ortner;A. Shapeev
我们证明，在一般条件下，线性化的基于力的拟连续谱 (QCF) 算子具有真实的正谱。在第二邻域相互作用的情况下，该谱与准非局域准连续谱 (QNL) 算子的谱相同。我们为线性化 QCF 算子构造了一个特征基，其条件数在原子数量和原子区域大小上是一致的。这些结果确立了最近数值观测的有效性并对其进行了改进[M。 Dobson、M. Luskin 和 C. Ortner，计算机。方法应用机甲。工程，200 (2011)，第 2697--2709 页，多尺度模型。模拟，8(2010)，第782--802页]。作为我们结果的直接结果，我们获得了（预处理的）GMRES 算法的收敛速度的严格估计以及 QCF 方法的新稳定性估计。
We show under general conditions that the linearized force-based quasicontinuum (QCF) operator has a real, positive spectrum. The spectrum is identical to that of the quasinonlocal quasicontinuum (QNL) operator in the case of second-neighbor interactions. We construct an eigenbasis for the linearized QCF operator whose condition number is uniform in the number of atoms and the size of the atomistic region. These results establish the validity of and improve upon recent numerical observations [M. Dobson, M. Luskin, and C. Ortner, Comput. Methods Appl. Mech. Engrg., 200 (2011), pp. 2697--2709, Multiscale Model. Simul., 8 (2010), pp. 782--802]. As immediate consequences of our results we obtain rigorous estimates for convergence rates of (preconditioned) GMRES algorithms as well as a new stability estimate for the QCF method.