## Analysis of a quasicontinuum method in one dimension.(English)Zbl 1139.74004

Summary: 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 $$W^{1,\infty}$$-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.

### MSC:

 74A25 Molecular, statistical, and kinetic theories in solid mechanics 74S99 Numerical and other methods in solid mechanics
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### References:

 [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 Series23 (2007) 209-216. · Zbl 1177.65091 [2] A. Braides and M.S. Gelli, Continuum limits of discrete systems without convexity hypotheses. Math. Mech. Solids7 (2002) 41-66. · 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. · 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. Zbl1093.74013 · 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. · Zbl 0488.65021 [6] M. Dobson and M. Luskin, Analysis of a force-based quasicontinuum approximation. ESAIM: M2AN42 (2008) 113-139. Zbl1140.74006 · Zbl 1140.74006 [7] G. Dolzmann, Optimal convergence for the finite element method in Campanato spaces. Math. Comp.68 (1999) 1397-1427. · Zbl 0929.65096 [8] W. E and B. Engquist, The heterogeneous multiscale methods. Commun. Math. Sci.1 (2003) 87-132. · 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. Zbl1104.00302 · Zbl 1104.00302 [10] W. E and P. Ming, Analysis of the local quasicontinuum method, in Frontiers and prospects of contemporary applied mathematics, Ser. Contemp. Appl. Math. CAM6, Higher Ed. Press, Beijing (2005) 18-32. Zbl1188.74019 · Zbl 1188.74019 [11] D.J. Higham, Trust region algorithms and timestep selection. SIAM J. Numer. Anal.37 (1999) 194-210. Zbl0945.65068 · 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. · 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. Zbl1010.74003 · 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). · Zbl 1220.74010 [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. A73 (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). · 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. Zbl1146.65070 · 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. Zbl0973.65100 · 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. Zbl0799.65112 · 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. Zbl0583.47050 · Zbl 0583.47050
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