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## Analysis of a quasicontinuum method in one dimension

ESAIM: M2AN, 42 1 (2008) 57-91
Published online: 2008-01-12

## Computational Challenges in the Geosciences

Malgorzata Peszynska
The IMA Volumes in Mathematics and its Applications, Computational Challenges in the Geosciences 156 71 (2013)
DOI: 10.1007/978-1-4614-7434-0_4

## An Analysis of Node-Based Cluster Summation Rules in the Quasicontinuum Method

Mitchell Luskin and Christoph Ortner
SIAM Journal on Numerical Analysis 47 (4) 3070 (2009)
DOI: 10.1137/080743391

## Finite Element Analysis of Cauchy–Born Approximations to Atomistic Models

Charalambos Makridakis and Endre Süli
Archive for Rational Mechanics and Analysis 207 (3) 813 (2013)
DOI: 10.1007/s00205-012-0582-8

## A Priori and A Posteriori $W^{1,\infty}$ Error Analysis of a QC Method for Complex Lattices

Assyr Abdulle, Ping Lin and Alexander V. Shapeev
SIAM Journal on Numerical Analysis 51 (4) 2357 (2013)
DOI: 10.1137/120894300

## Numerical Methods for Multilattices

Assyr Abdulle, Ping Lin and Alexander V. Shapeev
Multiscale Modeling & Simulation 10 (3) 696 (2012)
DOI: 10.1137/110841163

## An analysis of the effect of ghost force oscillation on quasicontinuum error

Matthew Dobson and Mitchell Luskin
ESAIM: Mathematical Modelling and Numerical Analysis 43 (3) 591 (2009)
DOI: 10.1051/m2an/2009007

## An Optimal Order Error Analysis of the One-Dimensional Quasicontinuum Approximation

Matthew Dobson and Mitchell Luskin
SIAM Journal on Numerical Analysis 47 (4) 2455 (2009)
DOI: 10.1137/08073723X

## Error Estimation and Atomistic-Continuum Adaptivity for the Quasicontinuum Approximation of a Frenkel–Kontorova Model

Marcel Arndt and Mitchell Luskin
Multiscale Modeling & Simulation 7 (1) 147 (2008)
DOI: 10.1137/070688559

## Positive Definiteness of the Blended Force-Based Quasicontinuum Method

Xingjie Helen Li, Mitchell Luskin and Christoph Ortner
Multiscale Modeling & Simulation 10 (3) 1023 (2012)
DOI: 10.1137/110859270

## Analysis of the Quasi-Nonlocal Approximation of Linear and Circular Chains in the Plane

Pavel Bělík and Mitchell Luskin
Multiscale Modeling & Simulation 9 (4) 1495 (2011)
DOI: 10.1137/100806242

## On the stability of Bravais lattices and their Cauchy–Born approximations

Thomas Hudson and Christoph Ortner
ESAIM: Mathematical Modelling and Numerical Analysis 46 (1) 81 (2012)
DOI: 10.1051/m2an/2011014

## Atomistic-to-continuum coupling

Mitchell Luskin and Christoph Ortner
Acta Numerica 22 397 (2013)
DOI: 10.1017/S0962492913000068

## Analysis of a One-Dimensional Nonlocal Quasi-Continuum Method

Pingbing Ming and Jerry Zhijian Yang
Multiscale Modeling & Simulation 7 (4) 1838 (2009)
DOI: 10.1137/080725842

## A numerical strategy for coarse-graining two-dimensional atomistic models at finite temperature: The membrane case

X. Blanc and F. Legoll
Computational Materials Science 66 84 (2013)
DOI: 10.1016/j.commatsci.2012.04.045

## Numerical Analysis of Multiscale Computations

Mitchell Luskin and Christoph Ortner
Lecture Notes in Computational Science and Engineering, Numerical Analysis of Multiscale Computations 82 331 (2012)
DOI: 10.1007/978-3-642-21943-6_14

## Analysis of Energy-Based Blended Quasi-Continuum Approximations

Brian Van Koten and Mitchell Luskin
SIAM Journal on Numerical Analysis 49 (5) 2182 (2011)
DOI: 10.1137/10081071X

## The role of the patch test in 2D atomistic-to-continuum coupling methods

Christoph Ortner
ESAIM: Mathematical Modelling and Numerical Analysis 46 (6) 1275 (2012)
DOI: 10.1051/m2an/2012005

## A PRIORI ERROR ESTIMATES FOR ENERGY-BASED QUASICONTINUUM APPROXIMATIONS OF A PERIODIC CHAIN

CHRISTOPH ORTNER and HAO WANG
Mathematical Models and Methods in Applied Sciences 21 (12) 2491 (2011)
DOI: 10.1142/S0218202511005817

## Iterative methods for the force-based quasicontinuum approximation: Analysis of a 1D model problem

M. Dobson, M. Luskin and C. Ortner
Computer Methods in Applied Mechanics and Engineering 200 (37-40) 2697 (2011)
DOI: 10.1016/j.cma.2010.07.008

## Accuracy of quasicontinuum approximations near instabilities

M. Dobson, M. Luskin and C. Ortner
Journal of the Mechanics and Physics of Solids 58 (10) 1741 (2010)
DOI: 10.1016/j.jmps.2010.06.011

## Goal-oriented adaptive mesh refinement for the quasicontinuum approximation of a Frenkel-Kontorova model

Marcel Arndt and Mitchell Luskin
Computer Methods in Applied Mechanics and Engineering 197 (49-50) 4298 (2008)
DOI: 10.1016/j.cma.2008.05.005

## Multiresolution molecular mechanics: Convergence and error structure analysis

Qingcheng Yang, Emre Biyikli and Albert C. To
Computer Methods in Applied Mechanics and Engineering 269 20 (2014)
DOI: 10.1016/j.cma.2013.10.012

## A priori error analysis of two force-based atomistic/continuum models of a periodic chain

Charalambos Makridakis, Christoph Ortner and Endre Süli
Numerische Mathematik 119 (1) 83 (2011)
DOI: 10.1007/s00211-011-0380-5

## On a $\Gamma$-Convergence Analysis of a Quasicontinuum Method

Mathias Schäffner and Anja Schlömerkemper
Multiscale Modeling & Simulation 13 (1) 132 (2015)
DOI: 10.1137/140971439

## Analysis of the Residual-Type and the Gradient Recovery-Type a Posteriori Error Estimators for a Consistent Atomistic-to-Continuum Coupling Method in One-Dimension

Hao Wang and Siyao Yang
Multiscale Modeling & Simulation 16 (2) 679 (2018)
DOI: 10.1137/17M1118579

## A Study on the Quasi-continuum Approximations of a One-Dimensional Fracture Model

Xiantao Li and Pingbing Ming
Multiscale Modeling & Simulation 12 (3) 1379 (2014)
DOI: 10.1137/130939547

## Consistent Energy-Based Atomistic/Continuum Coupling for Two-Body Potentials in One and Two Dimensions

Alexander V. Shapeev
Multiscale Modeling & Simulation 9 (3) 905 (2011)
DOI: 10.1137/100792421

## Sharp Stability Estimates for the Force-Based Quasicontinuum Approximation of Homogeneous Tensile Deformation

M. Dobson, M. Luskin and C. Ortner
Multiscale Modeling & Simulation 8 (3) 782 (2010)
DOI: 10.1137/090767005

## A Posteriori Error Estimation and Adaptive Algorithm for Atomistic/Continuum Coupling in Two Dimensions

Hao Wang, Mingjie Liao, Ping Lin and Lei Zhang
SIAM Journal on Scientific Computing 40 (4) A2087 (2018)
DOI: 10.1137/17M1131106

## N -break states in a chain of nonlinear oscillators

A. S. Rodrigues, P. G. Kevrekidis and M. Dobson
Physical Review E 99 (2) (2019)
DOI: 10.1103/PhysRevE.99.022201

## Iterative Solution of the Quasicontinuum Equilibrium Equations with Continuation

Matthew Dobson and Mitchell Luskin
Journal of Scientific Computing 37 (1) 19 (2008)
DOI: 10.1007/s10915-008-9208-6

## Stability, Instability, and Error of the Force-based Quasicontinuum Approximation

Matthew Dobson, Mitchell Luskin and Christoph Ortner
Archive for Rational Mechanics and Analysis 197 (1) 179 (2010)
DOI: 10.1007/s00205-009-0276-z

## Finite-Temperature Coarse-Graining of One-Dimensional Models: Mathematical Analysis and Computational Approaches

X. Blanc, C. Le Bris, F. Legoll and C. Patz
Journal of Nonlinear Science 20 (2) 241 (2010)
DOI: 10.1007/s00332-009-9057-y

## Numerical Analysis of Multiscale Problems

Brian Van Koten, Xingjie Helen Li, Mitchell Luskin and Christoph Ortner
Lecture Notes in Computational Science and Engineering, Numerical Analysis of Multiscale Problems 83 67 (2012)
DOI: 10.1007/978-3-642-22061-6_3

## Sharp Uniform Convergence Rate of the Supercell Approximation of a Crystalline Defect

Julian Braun and Christoph Ortner
SIAM Journal on Numerical Analysis 58 (1) 279 (2020)
DOI: 10.1137/18M122830X