Two-scale FEM for homogenization problems

Matache, Ana-Maria; Schwab, Christoph

doi:10.1051/m2an:2002025

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 4, p. 537-572

Abstract

The convergence of a two-scale FEM for elliptic problems in divergence form with coefficients and geometries oscillating at length scale $ε ≪ 1$ is analyzed. Full elliptic regularity independent of $ε$ 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 $ε$ scale of the solution with work independent of $ε$ and without analytical homogenization are introduced. Robust in $ε$ error estimates for the two-scale FE spaces are proved. Numerical experiments confirm the theoretical analysis.

MR 1932304 | Zbl 1070.65572

DOI : https://doi.org/10.1051/m2an:2002025
Classification: 65N30
Keywords: 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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {36},
     number = {4},
     year = {2002},
     pages = {537-572},
     doi = {10.1051/m2an:2002025},
     zbl = {1070.65572},
     mrnumber = {1932304},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2002__36_4_537_0}
}

Matache, Ana-Maria; Schwab, Christoph. Two-scale FEM for homogenization problems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 4, pp. 537-572. doi : 10.1051/m2an:2002025. http://www.numdam.org/item/M2AN_2002__36_4_537_0/

References

[1] I. Babuška and A.K. Aziz, 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 0268.65052

[2] A. Bensoussan, J.L. Lions and G. Papanicolau, Asymptotic Analysis for Periodic Structures. North Holland, Amsterdam (1978). | MR 503330 | Zbl 0404.35001

[3] D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures. Springer-Verlag, New York (1999). | MR 1676922 | Zbl 0929.35002

[4] T.Y. Hou and X.H. Wu, A Multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169-189. | Zbl 0880.73065

[5] T.Y. Hou, X.H. Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comp. 68 (1999) 913-943. | Zbl 0922.65071

[6] A.-M. Matache, Spectral- and $p$ -Finite Elements for problems with microstructure, Ph.D. thesis, ETH Zürich (2000).

[7] A.-M. Matache, I. Babuška and C. Schwab, Generalized $p$ -FEM in Homogenization. Numer. Math. 86 (2000) 319-375. | Zbl 0964.65125

[8] A.-M. Matache and M.J. Melenk, Two-scale regularity for homogenization problems with non-smooth fine-scale geometries, submitted. | Zbl 1076.35505

[9] A.-M. Matache and C. Schwab, 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 1165.35307 | Zbl pre01827875

[10] R.C. Morgan and I. Babuška, 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 0729.35009

[11] C. Schwab, $p$ - and $h p$ - Finite Element Methods. Oxford Science Publications (1998). | MR 1695813 | Zbl 0910.73003

[12] C. Schwab and A.-M. Matache, 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 1936170 | Zbl 1007.74079

[13] B. Szabó and I. Babuška, Finite Element Analysis1991). | MR 1164869 | Zbl 0792.73003

[14] O.A. Oleinik, A.S. Shamaev and G.A. Yosifian, Mathematical Problems in Elasticity and Homogenization. North-Holland (1992). | MR 1195131 | Zbl 0768.73003