Analysis of a quasicontinuum method in one dimension

Mathematical Modelling and Numerical Analysis

Volumn 42, Issue 1, 2008, Pages 57-91

  Ortner, Christoph ^a   Süli, Endre ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Atomistic material models; Error analysis; Quasicontinuum method; Stability

Indexed keywords

EID: 38849126369     PISSN: 0764583X     EISSN: 12903841     Source Type: Journal    
DOI: 10.1051/m2an:2007057     Document Type: Article

References (25)

1. X. Blanc, C. Le Bris and F. Legoll, Analysis of a prototypical multiscale method coupling atomistic and continuum mechanics: the convex case. Acta Math. Appl. Sinica English Series 23 (2007) 209-216.
2. A. Braides and M.S. Gelli, Continuum limits of discrete systems without convexity hypotheses. Math. Mech. Solids 7 (2002) 41-66.
3. A. Braides, G. Dal Maso and A. Garroni, Variational formulation of softening phenomena in fracture mechanics: the one-dimensional case. Arch. Ration. Mech. Anal. 146 (1999) 23-58.
4. A. Braides, A.J. Lew and M. Ortiz, Effective cohesive behavior of layers of interatomic planes. Arch. Ration. Mech. Anal. 180 (2006) 151-182.
5. F. Brezzi, J. Rappaz and P.-A. Raviart, Finite-dimensional approximation of nonlinear problems. I. Branches of nonsingular solutions. Numer. Math. 36 (1980) 1-25.
6. M. Dobson and M. Luskin, Analysis of a force-based quasicontinuum approximation. ESAIM: M2AN 42 (2008) 113-139.
7. G. Dolzmann, Optimal convergence for the finite element method in Campanato spaces. Math. Comp. 68 (1999) 1397-1427.
8. W. E and B. Engquist, The heterogeneous multiscale methods. Commun. Math. Sci. 1 (2003) 87-132.
9. W. E and P. Ming, Analysis of multiscale methods. J. Comput. Math. 22 (2004) 210-219. Special issue dedicated to the 70th birthday of Professor Zhong-Ci Shi.
10. W. E and P. Ming, Analysis of the local quasicontinuum method, in Frontiers and prospects of contemporary applied mathematics, Ser. Contemp. Appl. Math. CAM 6, Higher Ed. Press, Beijing (2005) 18-32.
11. D.J. Higham, Trust region algorithms and timestep selection. SIAM J. Numer. Anal. 37 (1999) 194-210.
12. J.E. Jones, On the Determination of Molecular Fields. III. From Crystal Measurements and Kinetic Theory Data. Proc. Roy. Soc. London A. 106 (1924) 709-718.
13. B. Lemaire, The proximal algorithm, in New methods in optimization and their industrial uses (Pau/Paris, 1987), of Internat. Schriftenreihe Numer. Math. 87, Birkhäuser, Basel (1989) 73-87.
14. P. Lin, Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. Math. Comp. 72 (2003) 657-675.
15. P. Lin, Convergence analysis of a quasi-continuum approximation for a two-dimensional material without defects. SIAM J. Numer. Anal. 45 (2007) 313-332 (electronic).
16. R.E. Miller and E.B. Tadmor, The quasicontinuum method: overview, applications and current directions. J. Computer-Aided Mater. Des. 9 (2003) 203-239.
17. P.M. Morse, Diatomic molecules according to the wave mechanics. II. Vibrational levels. Phys. Rev. 34 (1929) 57-64.
18. M. Ortiz, R. Phillips and E.B. Tadmor, Quasicontinuum analysis of defects in solids. Philos. Mag. A 73 (1996) 1529-1563.
19. C. Ortner, Gradient flows as a selection procedure for equilibria of nonconvex energies. SIAM J. Math. Anal. 38 (2006) 1214-1234 (electronic).
20. C. Ortner and E. Süli, A posteriori analysis and adaptive algorithms for the quasicontinuum method in one dimension. Technical Report NA06/13, Oxford University Computing Laboratory (2006).
21. C. Ortner and E. Süli, Discontinuous Galerkin finite element approximation of nonlinear second-order elliptic and hyperbolic systems. SIAM J. Numer. Anal. 45 (2007) 1370-1397.
22. M. Plum, Computer-assisted enclosure methods for elliptic differential equations. Linear Algebra Appl. 324 (2001) 147-187. Special issue on linear algebra in self-validating methods.
23. L. Truskinovsky, Fracture as a phase transformation, in Contemporary research in mechanics and mathematics of materials, R.C. Batra and M.F. Beatty Eds., CIMNE (1996) 322-332.
24. R. Verfürth, A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations. Math. Comp. 62 (1994) 445-475.
25. E. Zeidler, Nonlinear functional analysis and its applications. I Fixed-point theorems. Springer-Verlag, New York (1986). Translated from the German by Peter R. Wadsack.