Analysis of a quasicontinuum method in one dimension
ESAIM: Mathematical Modelling and Numerical Analysis - 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 ${\mathrm{W}}^{1,\infty }$-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 : https://doi.org/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: Mathematical Modelling and Numerical Analysis - 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},
zbl = {1139.74004},
mrnumber = {2387422},
language = {en},
url = {archive.numdam.org/item/M2AN_2008__42_1_57_0/}
}
Ortner, Christoph; Süli, Endre. Analysis of a quasicontinuum method in one dimension. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 42 (2008) no. 1, pp. 57-91. doi : 10.1051/m2an:2007057. http://archive.numdam.org/item/M2AN_2008__42_1_57_0/

[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 2300225

[2] A. Braides and M.S. Gelli, Continuum limits of discrete systems without convexity hypotheses. Math. Mech. Solids 7 (2002) 41-66. | MR 1900933 | Zbl 1024.74004

[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 1682660 | Zbl 0945.74006

[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 2210908 | Zbl 1093.74013

[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 595803 | Zbl 0488.65021

[6] M. Dobson and M. Luskin, Analysis of a force-based quasicontinuum approximation. ESAIM: M2AN 42 (2008) 113-139. | Numdam | MR 2387424 | Zbl 1140.74006

[7] G. Dolzmann, Optimal convergence for the finite element method in Campanato spaces. Math. Comp. 68 (1999) 1397-1427. | MR 1677478 | Zbl 0929.65096

[8] W. E and B. Engquist, The heterogeneous multiscale methods. Commun. Math. Sci. 1 (2003) 87-132. | MR 1979846 | Zbl 1093.35012

[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 2058933 | Zbl 1046.65108

[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 2249291 | Zbl pre05050158

[11] D.J. Higham, Trust region algorithms and timestep selection. SIAM J. Numer. Anal. 37 (1999) 194-210. | MR 1742752 | Zbl 0945.65068

[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 1001168 | Zbl 0692.90079

[14] P. Lin, Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model. Math. Comp. 72 (2003) 657-675. | MR 1954960 | Zbl 1010.74003

[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 2285857 | Zbl pre05246529

[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 55.0539.02

[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 2274480 | Zbl 1117.35004

[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 2338392 | Zbl 1146.65070

[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 1810529 | Zbl 0973.65100

[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 1213837 | Zbl 0799.65112

[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 816732 | Zbl 0583.47050