The convergence of a two-scale FEM for elliptic problems in divergence form with coefficients and geometries oscillating at length scale is analyzed. Full elliptic regularity independent of is shown when the solution is viewed as mapping from the slow into the fast scale. Two-scale FE spaces which are able to resolve the scale of the solution with work independent of and without analytical homogenization are introduced. Robust in error estimates for the two-scale FE spaces are proved. Numerical experiments confirm the theoretical analysis.
Mots clés : homogenization, two-scale regularity, finite element method (FEM), two-scale FEM
@article{M2AN_2002__36_4_537_0,
author = {Matache, Ana-Maria and Schwab, Christoph},
title = {Two-scale {FEM} for homogenization problems},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {537--572},
publisher = {EDP-Sciences},
volume = {36},
number = {4},
year = {2002},
doi = {10.1051/m2an:2002025},
mrnumber = {1932304},
zbl = {1070.65572},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an:2002025/}
}
TY - JOUR AU - Matache, Ana-Maria AU - Schwab, Christoph TI - Two-scale FEM for homogenization problems JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2002 SP - 537 EP - 572 VL - 36 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an:2002025/ DO - 10.1051/m2an:2002025 LA - en ID - M2AN_2002__36_4_537_0 ER -
%0 Journal Article %A Matache, Ana-Maria %A Schwab, Christoph %T Two-scale FEM for homogenization problems %J ESAIM: Modélisation mathématique et analyse numérique %D 2002 %P 537-572 %V 36 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an:2002025/ %R 10.1051/m2an:2002025 %G en %F M2AN_2002__36_4_537_0
Matache, Ana-Maria; Schwab, Christoph. Two-scale FEM for homogenization problems. ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 4, pp. 537-572. doi : 10.1051/m2an:2002025. http://archive.numdam.org/articles/10.1051/m2an:2002025/
[1] and, Survey lectures on the mathematical foundation of the finite element method, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, A.K. Aziz Ed., Academic Press, New York (1973) 5-359. | Zbl
[2] , and, Asymptotic Analysis for Periodic Structures. North Holland, Amsterdam (1978). | MR | Zbl
[3] and, Homogenization of Reticulated Structures. Springer-Verlag, New York (1999). | MR | Zbl
[4] and, A Multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169-189. | Zbl
[5] , and, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comp. 68 (1999) 913-943. | Zbl
[6] , Spectral- and -Finite Elements for problems with microstructure, Ph.D. thesis, ETH Zürich (2000).
[7] , and, Generalized -FEM in Homogenization. Numer. Math. 86 (2000) 319-375. | Zbl
[8] and, Two-scale regularity for homogenization problems with non-smooth fine-scale geometries, submitted. | Zbl
[9] and, Finite dimensional approximations for elliptic problems with rapidly oscillating coefficients, in Multiscale Problems in Science and Technology, N. Antonić, C.J. van Duijn, W. Jäger and A. Mikelić Eds., Springer-Verlag (2002) 203-242. | Zbl
[10] and, An approach for constructing families of homogenized solutions for periodic media, I: An integral representation and its consequences, II: Properties of the kernel. SIAM J. Math. Anal. 22 (1991) 1-33. | Zbl
[11] , - and - Finite Element Methods. Oxford Science Publications (1998). | MR | Zbl
[12] and, High order generalized FEM for lattice materials, in Proc. of the 3rd European Conference on Numerical Mathematics and Advanced Applications, Finland, 1999, P. Neittaanmäki, T. Tiihonen and P. Tarvainen Eds., World Scientific, Singapore (2000). | MR | Zbl
[13] and, Finite Element Analysis1991). | MR | Zbl
[14] , and, Mathematical Problems in Elasticity and Homogenization. North-Holland (1992). | MR | Zbl
Cité par Sources :
