- 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. [CrossRef]
- A. Braides and M.S. Gelli, Continuum limits of discrete systems without convexity hypotheses. Math. Mech. Solids 7 (2002) 41–66. [CrossRef] [MathSciNet]
- 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. [CrossRef] [MathSciNet]
- A. Braides, A.J. Lew and M. Ortiz, Effective cohesive behavior of layers of interatomic planes. Arch. Ration. Mech. Anal. 180 (2006) 151–182. [CrossRef] [MathSciNet]
- 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. [CrossRef] [MathSciNet]
- M. Dobson and M. Luskin, Analysis of a force-based quasicontinuum approximation. ESAIM: M2AN 42 (2008) 113–139. [CrossRef] [EDP Sciences]
- G. Dolzmann, Optimal convergence for the finite element method in Campanato spaces. Math. Comp. 68 (1999) 1397–1427. [CrossRef] [MathSciNet]
- W. E and B. Engquist, The heterogeneous multiscale methods. Commun. Math. Sci. 1 (2003) 87–132. [CrossRef] [MathSciNet]
- 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. [MathSciNet]
- 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.
- D.J. Higham, Trust region algorithms and timestep selection. SIAM J. Numer. Anal. 37 (1999) 194–210. [CrossRef] [MathSciNet]
- 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. [CrossRef]
- 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.
- P. Lin, Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. Math. Comp. 72 (2003) 657–675. [CrossRef] [MathSciNet]
- 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). [CrossRef] [MathSciNet]
- R.E. Miller and E.B. Tadmor, The quasicontinuum method: overview, applications and current directions. J. Computer-Aided Mater. Des. 9 (2003) 203–239. [CrossRef]
- P.M. Morse, Diatomic molecules according to the wave mechanics. II. Vibrational levels. Phys. Rev. 34 (1929) 57–64. [CrossRef]
- M. Ortiz, R. Phillips and E.B. Tadmor, Quasicontinuum analysis of defects in solids. Philos. Mag. A 73 (1996) 1529–1563. [CrossRef]
- C. Ortner, Gradient flows as a selection procedure for equilibria of nonconvex energies. SIAM J. Math. Anal. 38 (2006) 1214–1234 (electronic). [CrossRef] [MathSciNet]
- 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).
- 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. [CrossRef] [MathSciNet]
- 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. [CrossRef] [MathSciNet]
- 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.
- R. Verfürth, A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations. Math. Comp. 62 (1994) 445–475. [MathSciNet]
- 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.
Volume 42, Number 1, January-February 2008
|Page(s)||57 - 91|
|Published online||12 January 2008|