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 -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.
@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},
volume = {42},
year = {2008},
pages = {57-91},
doi = {10.1051/m2an:2007057},
mrnumber = {2387422},
zbl = {1139.74004},
language = {en},
url = {http://dml.mathdoc.fr/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) pp. 57-91. doi : 10.1051/m2an:2007057. http://gdmltest.u-ga.fr/item/M2AN_2008__42_1_57_0/
[1] , and , 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] and , Continuum limits of discrete systems without convexity hypotheses. Math. Mech. Solids 7 (2002) 41-66. | MR 1900933 | Zbl 1024.74004
[3] , and , 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] , and , Effective cohesive behavior of layers of interatomic planes. Arch. Ration. Mech. Anal. 180 (2006) 151-182. | MR 2210908 | Zbl 1093.74013
[5] , and , Finite-dimensional approximation of nonlinear problems. I. Branches of nonsingular solutions. Numer. Math. 36 (1980) 1-25. | MR 595803 | Zbl 0488.65021
[6] and , Analysis of a force-based quasicontinuum approximation. ESAIM: M2AN 42 (2008) 113-139. | Numdam | MR 2387424 | Zbl 1140.74006
[7] , 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] , Trust region algorithms and timestep selection. SIAM J. Numer. Anal. 37 (1999) 194-210. | MR 1742752 | Zbl 0945.65068
[12] , On the Determination of Molecular Fields. III. From Crystal Measurements and Kinetic Theory Data. Proc. Roy. Soc. London A. 106 (1924) 709-718.
[13] , 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] , 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] , 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] and , The quasicontinuum method: overview, applications and current directions. J. Computer-Aided Mater. Des. 9 (2003) 203-239.
[17] , Diatomic molecules according to the wave mechanics. II. Vibrational levels. Phys. Rev. 34 (1929) 57-64. | JFM 55.0539.02
[18] , and , Quasicontinuum analysis of defects in solids. Philos. Mag. A 73 (1996) 1529-1563.
[19] , 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] and , A posteriori analysis and adaptive algorithms for the quasicontinuum method in one dimension. Technical Report NA06/13, Oxford University Computing Laboratory (2006).
[21] and , 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] , 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] , 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] , 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] , 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