Dobson, Matthew, Ortner, Christoph and Shapeev, A. V. (2010) The spectrum of the force-based quasicontinuum operator for a homogenous periodic chain. Working Paper. ArXiv e-prints. ArXiv e-prints (No.1004.3435v1).

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Abstract

We show under general conditions that the linearized force-based quasi-continuum (QCF) operator has a positive spectrum, which is identical to the spectrum of the quasinonlocal quasicontinuum (QNL) operator in the case of second-neighbour interactions. Moreover, we establish a bound on the condition number of a matrix of eigenvectors that is uniform in the number of atoms and the size of the atomistic region. These results establish the validity of and improve upon recent conjectures ([7, Conjecture 2] and [6, Conjecture 8]) which were based on numerical experiments. 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.

Item Type:	Working or Discussion Paper (Working Paper)
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Series Name:	ArXiv e-prints
Publisher:	ArXiv e-prints
Official Date:	20 April 2010
Dates:	Date Event 20 April 2010 ["eprint_fieldopt_dates_date_type_available" not defined]
Number:	No.1004.3435v1
Number of Pages:	27
Institution:	University of Warwick
Status:	Not Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access (Creative Commons)
Funder:	EPSRC, NSF

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