Stability, Instability, and Error of the Force-based Quasicontinuum Approximation

被引:50
作者
Dobson, Matthew [1 ,2 ]
Luskin, Mitchell [2 ]
Ortner, Christoph [3 ]
机构
[1] CERMICS, ENPC, F-77455 Marne La Vallee 2, France
[2] Univ Minnesota, Sch Math, Minneapolis, MN 55455 USA
[3] Math Inst, Oxford OX1 3LB, England
基金
美国国家科学基金会; 英国工程与自然科学研究理事会;
关键词
FINITE-ELEMENT; CONVERGENCE; EQUATIONS;
D O I
10.1007/s00205-009-0276-z
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
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 typically not positive definite. Moreover, we prove that no uniform inf-sup stability condition holds for discrete versions of the W-1,W-p-W-1,W-q "duality pairing" with 1/p + 1/q = 1, if 1 <= p < infinity. However, we were able to establish an inf-sup stability condition for a discrete version of the W-1,W-infinity-W-1,W-1 "duality pairing" which leads to optimal order error estimates in a discrete W-1,W-infinity-norm.
引用
收藏
页码:179 / 202
页数:24
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