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**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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\textit{C. Ortner} and \textit{E. Süli}, ESAIM, Math. Model. Numer. Anal. 42, No. 1, 57--91 (2008; Zbl 1139.74004)

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