Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


A priori and a posteriori analysis of the quasinonlocal quasicontinuum method in 1D
HTML articles powered by AMS MathViewer

by Christoph Ortner PDF
Math. Comp. 80 (2011), 1265-1285 Request permission


For a next-nearest neighbour pair interaction model in a periodic domain, a priori and a posteriori analyses of the quasinonlocal quasicontinuum method (QNL-QC) are presented. The results are valid for large deformations and essentially guarantee a one-to-one correspondence between atomistic solutions and QNL-QC solutions. The analysis is based on consistency error estimates in negative norms, novel a priori and a posteriori stability estimates, and a quantitative inverse function theorem.
  • Marcel Arndt and Mitchell Luskin, Error estimation and atomistic-continuum adaptivity for the quasicontinuum approximation of a Frenkel-Kontorova model, Multiscale Model. Simul. 7 (2008), no. 1, 147–170. MR 2399541, DOI 10.1137/070688559
  • Marcel Arndt and Mitchell Luskin, Goal-oriented adaptive mesh refinement for the quasicontinuum approximation of a Frenkel-Kontorova model, Comput. Methods Appl. Mech. Engrg. 197 (2008), no. 49-50, 4298–4306. MR 2463663, DOI 10.1016/j.cma.2008.05.005
  • Xavier Blanc, Claude Le Bris, and Frédéric Legoll, Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics, M2AN Math. Model. Numer. Anal. 39 (2005), no. 4, 797–826. MR 2165680, DOI 10.1051/m2an:2005035
  • Matthew Dobson and Mitchell Luskin, Analysis of a force-based quasicontinuum approximation, M2AN Math. Model. Numer. Anal. 42 (2008), no. 1, 113–139. MR 2387424, DOI 10.1051/m2an:2007058
  • Matthew Dobson and Mitchell Luskin, An optimal order error analysis of the one-dimensional quasicontinuum approximation, SIAM J. Numer. Anal. 47 (2009), no. 4, 2455–2475. MR 2525607, DOI 10.1137/08073723X
  • M. Dobson, M. Luskin, and C. Ortner, Accuracy of quasicontinuum approximations near instabilities, J. Mech. Phys. Solids 58 (2010), 1741–1757.
  • —, Iterative methods for the force-based quasicontinuum approximation, Comput. Mathods Appl. Mech. Engrg., doi: 10.1016/j.cma.2010.07.008, 2010.
  • —, Sharp stability estimates for the force-based quasicontinuum approximation of homogeneous tensile deformation, Multiscale Model. Simul. 8 (2010), no. 3, 782–802.
  • —, Stability, instability, and error of the force-based quasicontinuum approximation, Arch. Ration. Mech. Anal. 197 (2010), no. 1, 179.
  • W. E, J. Lu, and J.Z. Yang, Uniform accuracy of the quasicontinuum method, Phys. Rev. B 74 (2004), no. 21, 214115.
  • Wei-nan E and Ping-bing Ming, Cauchy-Born rule and the stability of crystalline solids: dynamic problems, Acta Math. Appl. Sin. Engl. Ser. 23 (2007), no. 4, 529–550. MR 2329067, DOI 10.1007/s10255-007-0393
  • Weinan E and Pingbing Ming, Cauchy-Born rule and the stability of crystalline solids: static problems, Arch. Ration. Mech. Anal. 183 (2007), no. 2, 241–297. MR 2278407, DOI 10.1007/s00205-006-0031-7
  • X.H. Li and M. Luskin, A generalized quasi-nonlocal atomistic-to-continuum coupling method with finite range interaction, arXiv:1007.2336v1
  • Ping Lin, Convergence analysis of a quasi-continuum approximation for a two-dimensional material without defects, SIAM J. Numer. Anal. 45 (2007), no. 1, 313–332. MR 2285857, DOI 10.1137/050636772
  • Mitchell Luskin and Christoph Ortner, An analysis of node-based cluster summation rules in the quasicontinuum method, SIAM J. Numer. Anal. 47 (2009), no. 4, 3070–3086. MR 2551158, DOI 10.1137/080743391
  • R.E. Miller and E.B. Tadmor, Benchmarking multiscale methods, Modelling and Simulation in Materials Science and Engineering 17 (2009), p. 053001.
  • —, The Quasicontinuum Method: Overview, Applications and Current Directions, Journal of Computer-Aided Materials Design 9 (2003), 203–239.
  • Pingbing Ming and Jerry Zhijian Yang, Analysis of a one-dimensional nonlocal quasi-continuum method, Multiscale Model. Simul. 7 (2009), no. 4, 1838–1875. MR 2539201, DOI 10.1137/080725842
  • M. Ortiz, R. Phillips, and E.B. Tadmor, Quasicontinuum Analysis of Defects in Solids, Philosophical Magazine A 73 (1996), no. 6, 1529–1563.
  • C. Ortner, A posteriori existence in numerical computations, SIAM J. Numer. Anal. 47 (2009), no. 4, 2550–2577. MR 2525611, DOI 10.1137/060668183
  • Christoph Ortner and Endre Süli, Analysis of a quasicontinuum method in one dimension, M2AN Math. Model. Numer. Anal. 42 (2008), no. 1, 57–91. MR 2387422, DOI 10.1051/m2an:2007057
  • C. Ortner and H. Wang, A priori error analysis for energy-based quasicontinuum approximations of a periodic chain, OxMOS preprint no. 30/2010, Mathematical Institute, University of Oxford.
  • Michael Plum, Computer-assisted enclosure methods for elliptic differential equations, Linear Algebra Appl. 324 (2001), no. 1-3, 147–187. Special issue on linear algebra in self-validating methods. MR 1810529, DOI 10.1016/S0024-3795(00)00273-1
  • S. Prudhomme, P.T. Bauman, and J.T. Oden, Error control for molecular statics problems, International Journal for Multiscale Computational Engineering 4 (2006), no. 5-6, 647–662.
  • A. Shapeev, Consistent energy-based atomistic/continuum coupling for two-body potential: 1d and 2d case, arXiv:1010.0512
  • V. B. Shenoy, R. Miller, E. B. Tadmor, D. Rodney, R. Phillips, and M. Ortiz, An adaptive finite element approach to atomic-scale mechanics—the quasicontinuum method, J. Mech. Phys. Solids 47 (1999), no. 3, 611–642. MR 1675219, DOI 10.1016/S0022-5096(98)00051-9
  • T. Shimokawa, J.J. Mortensen, J. Schiotz, and K.W. Jacobsen, Matching conditions in the quasicontinuum method: Removal of the error introduced at the interface between the coarse-grained and fully atomistic region, Phys. Rev. B 69 (2004), no. 21, 214104.
Similar Articles
  • Retrieve articles in Mathematics of Computation with MSC (2010): 65N12, 65N15, 70C20
  • Retrieve articles in all journals with MSC (2010): 65N12, 65N15, 70C20
Additional Information
  • Christoph Ortner
  • Affiliation: Mathematical Institute, 24-29 St Giles’, Oxford OX1 3LB, United Kingdom
  • MR Author ID: 803698
  • Email:
  • Received by editor(s): November 7, 2009
  • Received by editor(s) in revised form: May 15, 2010
  • Published electronically: December 31, 2010
  • Additional Notes: This work was supported by the EPSRC Critical Mass Programme “New Frontiers in the Mathematics of Solids”.
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 80 (2011), 1265-1285
  • MSC (2010): Primary 65N12, 65N15, 70C20
  • DOI:
  • MathSciNet review: 2785458