Two-scale FEM for homogenization problems

Ana-Maria Matache; Christoph Schwab

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2002)

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

Abstract

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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.

How to cite

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Matache, 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

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  2. [2] A. Bensoussan, J.L. Lions and G. Papanicolau, Asymptotic Analysis for Periodic Structures. North Holland, Amsterdam (1978). Zbl0404.35001MR503330
  3. [3] D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures. Springer-Verlag, New York (1999). Zbl0929.35002MR1676922
  4. [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. [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. [6] A.-M. Matache, Spectral- and p -Finite Elements for problems with microstructure, Ph.D. thesis, ETH Zürich (2000). 
  7. [7] A.-M. Matache, I. Babuška and C. Schwab, Generalized p -FEM in Homogenization. Numer. Math. 86 (2000) 319–375. Zbl0964.65125
  8. [8] A.-M. Matache and M.J. Melenk, Two-scale regularity for homogenization problems with non-smooth fine-scale geometries, submitted. Zbl1076.35505
  9. [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. [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. [11] C. Schwab, p - and h p - Finite Element Methods. Oxford Science Publications (1998). Zbl0910.73003MR1695813
  12. [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. [13] B. Szabó and I. Babuška, Finite Element Analysis. John Wiley & Sons, Inc. (1991). Zbl0792.73003MR1164869
  14. [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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