# Two-scale FEM for homogenization problems

Ana-Maria Matache; Christoph Schwab

- Volume: 36, Issue: 4, page 537-572
- ISSN: 0764-583X

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topMatache, Ana-Maria, and Schwab, Christoph. "Two-scale FEM for homogenization problems." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 36.4 (2002): 537-572. <http://eudml.org/doc/245742>.

@article{Matache2002,

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

author = {Matache, Ana-Maria, Schwab, Christoph},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {homogenization; two-scale regularity; finite element method (FEM); two-scale FEM; Finite Element Method; Numerical experiments; error estimates},

language = {eng},

number = {4},

pages = {537-572},

publisher = {EDP-Sciences},

title = {Two-scale FEM for homogenization problems},

url = {http://eudml.org/doc/245742},

volume = {36},

year = {2002},

}

TY - JOUR

AU - Matache, Ana-Maria

AU - Schwab, Christoph

TI - Two-scale FEM for homogenization problems

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2002

PB - EDP-Sciences

VL - 36

IS - 4

SP - 537

EP - 572

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

LA - eng

KW - homogenization; two-scale regularity; finite element method (FEM); two-scale FEM; Finite Element Method; Numerical experiments; error estimates

UR - http://eudml.org/doc/245742

ER -

## References

top- [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. Zbl0268.65052
- [2] A. Bensoussan, J.L. Lions and G. Papanicolau, Asymptotic Analysis for Periodic Structures. North Holland, Amsterdam (1978). Zbl0404.35001MR503330
- [3] D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures. Springer-Verlag, New York (1999). Zbl0929.35002MR1676922
- [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. Zbl0880.73065
- [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. Zbl0922.65071
- [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. Zbl0964.65125
- [8] A.-M. Matache and M.J. Melenk, Two-scale regularity for homogenization problems with non-smooth fine-scale geometries, submitted. Zbl1076.35505
- [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. Zbl1165.35307
- [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. Zbl0729.35009
- [11] C. Schwab, $p$- and $hp$- Finite Element Methods. Oxford Science Publications (1998). Zbl0910.73003MR1695813
- [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). Zbl1007.74079MR1936170
- [13] B. Szabó and I. Babuška, Finite Element Analysis. John Wiley $\&$ Sons, Inc. (1991). Zbl0792.73003MR1164869
- [14] O.A. Oleinik, A.S. Shamaev and G.A. Yosifian, Mathematical Problems in Elasticity and Homogenization. North-Holland (1992). Zbl0768.73003MR1195131

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