Analysis of patch-test consistent atomistic-to-continuum coupling with higher-order finite elements
Dedner, Andreas1 ; Ortner, Chistoph1 ; Wu, Huan1
1 Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 6, pp. 2263-2288.

We formulate a patch test consistent atomistic-to-continuum coupling (a/c) scheme that employs a second-order (potentially higher-order) finite element method in the material bulk. We prove a sharp error estimate in the energy-norm, which demonstrates that this scheme is (quasi-)optimal amongst energy-based sharp-interface a/c schemes that employ the Cauchy–Born continuum model. Our analysis also shows that employing a higher-order continuum discretisation does not yield qualitative improvements to the rate of convergence.

DOI : 10.1051/m2an/2017013
Classification : 65N12, 65N15, 70C20
Mots-clés : Atomistic models, coarse graining, atomistic-to-continuum coupling, quasicontinuum method, error analysis
Dedner, Andreas 1 ; Ortner, Chistoph 1 ; Wu, Huan 1

1 Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK. 
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     author = {Dedner, Andreas and Ortner, Chistoph and Wu, Huan},
     title = {Analysis of patch-test consistent atomistic-to-continuum coupling with higher-order finite elements},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {2263--2288},
     publisher = {EDP-Sciences},
     volume = {51},
     number = {6},
     year = {2017},
     doi = {10.1051/m2an/2017013},
     zbl = {1383.82062},
     mrnumber = {3745172},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2017013/}
}
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Dedner, Andreas; Ortner, Chistoph; Wu, Huan. Analysis of patch-test consistent atomistic-to-continuum coupling with higher-order finite elements. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 6, pp. 2263-2288. doi : 10.1051/m2an/2017013. http://archive.numdam.org/articles/10.1051/m2an/2017013/

G. Acosta and R.G. Duran, An optimal Poincaré inequality in l1 for convex domains. Proc. Amer. Math. Soc. 132 (2003) 195–202. | DOI | MR | Zbl

C. Carstensen, Quasi-interpolation and a posteriori error analysis in finite element methods. ESAIM: M2AN 33 (1999) 1187–1202. | DOI | Numdam | MR | Zbl

W. E, J. Lu and J.Z. Yang, Uniform accuracy of the quasicontinuum method. Phys. Rev. B 74 (2006) 214–115.

V. Ehrlacher, C. Ortner and A.V. Shapeev, Analysis of boundary conditions for crystal defect atomistic simulations (2013).

X. H. Li, C. Ortner, A. Shapeev and B. Van Koten. Analysis of blended atomistic/continuum hybrid methods. Numer. Math. 134 (2016) 275–326. | DOI | MR | Zbl

M. Luskin and C. Ortner. Atomstic-to-continuum coupling. Acta Numer. 22 (2013) 397–508. | DOI | MR | Zbl

C. Ortner, The role of the patch test in 2D atomistic-to-continuum coupling methods. ESAIM: M2AN 46 (2012) 1275–1319. | DOI | Numdam | MR | Zbl

C. Ortner and A. Shapeev, Interpolation of lattice functions and applications to atomistic/continuum multiscale methods. Preprint (2012). | arXiv

C. Ortner, A. Shapeev and L. Zhang, (in-)stability and stabilisation of QNL-type atomistic-to-continuum coupling methods. Preprint (2013). | arXiv | MR

C. Ortner and E. Süli, A note on linear elliptic systems on R d . Preprint (2012). | arXiv

C. Ortner and L. Zhang, Atomistic/continuum blending with ghost force correction. SIAM: J. Sci. Comput. 38 (2016) A346–A375. | MR | Zbl

C. Ortner and L. Zhang, Construction and sharp consistency estimates for atomistic/continuum coupling methods with general interfaces: a 2D model problem. SIAM J. Numer. Anal. (2012) 50. | MR | Zbl

Christoph Ortner, A priori and a posteriori analysis of the quasinonlocal quasicontinuum method in 1D. Math. Comput. 80 (2011) 1265–1285. | DOI | MR | Zbl

C. Schwab, p- and hp- finite element methods: theory and applications in solid and fluid mechanics. Oxford Universiy Press, Oxford (1998). | MR | Zbl

T. Shimokawa, J.J. Mortensen, J. Schiotz and K.W. Jacobsen, Matching conditions in the quasicontinuum method: Removal of the error introduced at the interface between the coarse-grained and fully atomistic region. Phys. Rev. B 69 (2004) 214104.. | DOI

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