Analysis of a quasicontinuum method in one dimension
ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 1, pp. 57-91.

The quasicontinuum method is a coarse-graining technique for reducing the complexity of atomistic simulations in a static and quasistatic setting. In this paper we aim to give a detailed a priori and a posteriori error analysis for a quasicontinuum method in one dimension. We consider atomistic models with Lennard-Jones type long-range interactions and a QC formulation which incorporates several important aspects of practical QC methods. First, we prove the existence, the local uniqueness and the stability with respect to a discrete W 1, -norm of elastic and fractured atomistic solutions. We use a fixed point argument to prove the existence of a quasicontinuum approximation which satisfies a quasi-optimal a priori error bound. We then reverse the role of exact and approximate solution and prove that, if a computed quasicontinuum solution is stable in a sense that we make precise and has a sufficiently small residual, there exists a ‘nearby’ exact solution which it approximates, and we give an a posteriori error bound. We stress that, despite the fact that we use linearization techniques in the analysis, our results apply to genuinely nonlinear situations.

DOI : 10.1051/m2an:2007057
Classification : 70C20, 70-08, 65N15
Mots clés : atomistic material models, quasicontinuum method, error analysis, stability
@article{M2AN_2008__42_1_57_0,
     author = {Ortner, Christoph and S\"uli, Endre},
     title = {Analysis of a quasicontinuum method in one dimension},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {57--91},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {1},
     year = {2008},
     doi = {10.1051/m2an:2007057},
     mrnumber = {2387422},
     zbl = {1139.74004},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2007057/}
}
TY  - JOUR
AU  - Ortner, Christoph
AU  - Süli, Endre
TI  - Analysis of a quasicontinuum method in one dimension
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2008
SP  - 57
EP  - 91
VL  - 42
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2007057/
DO  - 10.1051/m2an:2007057
LA  - en
ID  - M2AN_2008__42_1_57_0
ER  - 
%0 Journal Article
%A Ortner, Christoph
%A Süli, Endre
%T Analysis of a quasicontinuum method in one dimension
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2008
%P 57-91
%V 42
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2007057/
%R 10.1051/m2an:2007057
%G en
%F M2AN_2008__42_1_57_0
Ortner, Christoph; Süli, Endre. Analysis of a quasicontinuum method in one dimension. ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 1, pp. 57-91. doi : 10.1051/m2an:2007057. http://www.numdam.org/articles/10.1051/m2an:2007057/

[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. | MR

[2] A. Braides and M.S. Gelli, Continuum limits of discrete systems without convexity hypotheses. Math. Mech. Solids 7 (2002) 41-66. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[6] M. Dobson and M. Luskin, Analysis of a force-based quasicontinuum approximation. ESAIM: M2AN 42 (2008) 113-139. | Numdam | MR | Zbl

[7] G. Dolzmann, Optimal convergence for the finite element method in Campanato spaces. Math. Comp. 68 (1999) 1397-1427. | MR | Zbl

[8] W. E and B. Engquist, The heterogeneous multiscale methods. Commun. Math. Sci. 1 (2003) 87-132. | MR | Zbl

[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. | MR | Zbl

[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. | MR

[11] D.J. Higham, Trust region algorithms and timestep selection. SIAM J. Numer. Anal. 37 (1999) 194-210. | MR | Zbl

[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. | MR | Zbl

[14] P. Lin, Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. Math. Comp. 72 (2003) 657-675. | MR | Zbl

[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). | MR

[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. | JFM

[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). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

Cité par Sources :