# Analysis of a quasicontinuum method in one dimension

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

- Volume: 42, Issue: 1, page 57-91
- ISSN: 0764-583X

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topOrtner, Christoph, and Süli, Endre. "Analysis of a quasicontinuum method in one dimension." ESAIM: Mathematical Modelling and Numerical Analysis 42.1 (2008): 57-91. <http://eudml.org/doc/250345>.

@article{Ortner2008,

abstract = {
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 W1,∞-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.
},

author = {Ortner, Christoph, Süli, Endre},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Atomistic material models; quasicontinuum method; error
analysis; stability.; atomistic material models; error analysis; stability; linearization},

language = {eng},

month = {1},

number = {1},

pages = {57-91},

publisher = {EDP Sciences},

title = {Analysis of a quasicontinuum method in one dimension},

url = {http://eudml.org/doc/250345},

volume = {42},

year = {2008},

}

TY - JOUR

AU - Ortner, Christoph

AU - Süli, Endre

TI - Analysis of a quasicontinuum method in one dimension

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2008/1//

PB - EDP Sciences

VL - 42

IS - 1

SP - 57

EP - 91

AB -
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 W1,∞-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.

LA - eng

KW - Atomistic material models; quasicontinuum method; error
analysis; stability.; atomistic material models; error analysis; stability; linearization

UR - http://eudml.org/doc/250345

ER -

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## Citations in EuDML Documents

top- Thomas Hudson, Christoph Ortner, On the stability of Bravais lattices and their Cauchy–Born approximations
- Thomas Hudson, Christoph Ortner, On the stability of Bravais lattices and their Cauchy–Born approximations
- Christoph Ortner, The role of the patch test in 2D atomistic-to-continuum coupling methods
- Christoph Ortner, The role of the patch test in 2D atomistic-to-continuum coupling methods
- Matthew Dobson, Mitchell Luskin, An analysis of the effect of ghost force oscillation on quasicontinuum error

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