We propose and analyze a generalized finite element method designed for linear quasistatic thermoelastic systems with spatial multiscale coefficients. The method is based on the local orthogonal decomposition technique introduced by Målqvist and Peterseim (Math. Comp. 83 (2014) 2583–2603). We prove convergence of optimal order, independent of the derivatives of the coefficients, in the spatial -norm. The theoretical results are confirmed by numerical examples.
Accepté le :
DOI : 10.1051/m2an/2016054
Mots clés : Linear thermoelasticity, multiscale, generalized finite element, local orthogonal decomposition, a priori analysis
@article{M2AN_2017__51_4_1145_0,
author = {M\r{a}lqvist, Axel and Persson, Anna},
title = {A generalized finite element method for linear thermoelasticity},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1145--1171},
publisher = {EDP-Sciences},
volume = {51},
number = {4},
year = {2017},
doi = {10.1051/m2an/2016054},
mrnumber = {3702408},
zbl = {1397.74191},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2016054/}
}
TY - JOUR AU - Målqvist, Axel AU - Persson, Anna TI - A generalized finite element method for linear thermoelasticity JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 1145 EP - 1171 VL - 51 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2016054/ DO - 10.1051/m2an/2016054 LA - en ID - M2AN_2017__51_4_1145_0 ER -
%0 Journal Article %A Målqvist, Axel %A Persson, Anna %T A generalized finite element method for linear thermoelasticity %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 1145-1171 %V 51 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2016054/ %R 10.1051/m2an/2016054 %G en %F M2AN_2017__51_4_1145_0
Målqvist, Axel; Persson, Anna. A generalized finite element method for linear thermoelasticity. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1145-1171. doi : 10.1051/m2an/2016054. http://archive.numdam.org/articles/10.1051/m2an/2016054/
and , Localized orthogonal decomposition method for the wave equation with a continuum of scales. Math. Comp. 86 (2017) 549–587. | DOI | MR | Zbl
, , and , Thermoelastic contact with Barber’s heat exchange condition. Appl. Math. Optim. 28 (1993) 11–48. | DOI | MR | Zbl
and , Optimal local approximation spaces for generalized finite element methods with application to multiscale problems. Multiscale Model. Simul. 9 (2011) 373–406. | DOI | MR | Zbl
and , Generalized finite element methods: their performance and their relation to mixed methods. SIAM J. Numer. Anal. 20 (1983) 510–536. | DOI | MR | Zbl
, General theory of three-dimensional consolidation. J. Appl. Phys. 18 (1941) 155–164. | DOI | JFM
, Thermoelasticity and irreversible thermodynamics. J. Appl. Phys. 27 (1956) 240–253. | DOI | MR | Zbl
S.C. Brenner and R.L. Scott, The mathematical theory of finite element methods. Vol. 15 of Texts in Applied Mathematics, 3rd edn. Springer, New York (2008). | MR | Zbl
P.G. Ciarlet, Mathematical elasticity. Vol. I. Three-dimensional elasticity. Vol. 20 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam (1988). | MR | Zbl
, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity. Arch. Ration. Mech. Anal. 29 (1968) 241–271. | DOI | MR | Zbl
Ch. Engwer, P. Henning, A. Målqvist and D. Peterseim, Efficient implementation of the localized orthogonal decomposition method. Preprint (2016). | arXiv | MR
and , A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems. ESAIM: M2AN 43 (2009) 353–375. | DOI | Numdam | MR | Zbl
and , Localized orthogonal decomposition techniques for boundary value problems. SIAM J. Sci. Comput. 36 (2014) A1609–A1634. | DOI | MR | Zbl
, and , A localized orthogonal decomposition method for semi-linear elliptic problems. ESAIM: M2AN 48 (2014) 1331–1349. | DOI | Numdam | MR | Zbl
and , A multiscale method for linear elasticity reducing Poisson locking. Comput. Methods Appl. Mech. Engrg. 310 (2016) 156–171. | DOI | MR | Zbl
, , and . The variational multiscale method – a paradigm for computational mechanics. Comput. Methods Appl. Mech. Engrg. 166 (1998) 3–24. | DOI | MR | Zbl
and , Adaptive variational multiscale methods based on a posteriori error estimation: energy norm estimates for elliptic problems. Comput. Methods Appl. Mech. Engrg. 196 (2007) 2313–2324. | DOI | MR | Zbl
A. Målqvist and A. Persson, Multiscale techniques for parabolic equations. Preprint (2015). | arXiv | MR
and , Localization of elliptic multiscale problems. Math. Comp. 83 (2014) 2583–2603. | DOI | MR | Zbl
and , Computation of eigenvalues by numerical upscaling. Numer. Math. 130 (2015) 337–361. | DOI | MR | Zbl
D. Peterseim, Variational multiscale stabilization and the exponential decay of fine-scale correctors. In vol. 114 of Lect. Notes Comput. Sci. Engrg. Springer (2016) 343–369. | MR
and , Overcoming the problem of locking in linear elasticity and poroelasticity: an heuristic approach. Comput. Geosci. 13 (2009) 5–12. | DOI | Zbl
and , A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity. Comput. Geosci. 12 (2008) 417–435. | DOI | MR | Zbl
and , Existence of a solution to the -dimensional problem of thermoelastic contact. Commun. Partial Differ. Eq. 17 (1992) 1597–1618. | DOI | MR | Zbl
, Diffusion in poro-elastic media. J. Math. Anal. Appl. 251 (2000) 310–340. | DOI | MR | Zbl
, The -dimensional quasistatic problem of thermoelastic contact with Barber’s heat exchange conditions. Adv. Math. Sci. Appl. 6 (1996) 559–587. | MR | Zbl
, The existence and uniqueness theorem in Biot’s consolidation theory. Apl. Mat. 29 (1984) 194–211. | MR | Zbl
, Finite element methods for coupled thermoelasticity and coupled consolidation of clay. RAIRO Anal. Numér. 18 (1984) 183–205. | DOI | Numdam | MR | Zbl
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