Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T20:01:49.791Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-scale FEM for homogenization problems

Published online by Cambridge University Press:  15 September 2002

Ana-Maria Matache  and
Christoph Schwab
Ana-Maria Matache
Affiliation:
Seminar for Applied Mathematics, ETH-Zentrum, CH-8092 Zürich, Switzerland. [email protected].
Christoph Schwab
Affiliation:
Seminar for Applied Mathematics, ETH-Zentrum, CH-8092 Zürich, Switzerland. [email protected].
Get access

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 shownwhen 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 thesolution with work independent of ε and withoutanalytical homogenization are introduced. Robustin ε error estimates for the two-scale FE spaces are proved. Numerical experiments confirm thetheoretical analysis.

Keywords

Homogenizationtwo-scale regularityFinite Element Method (FEM)two-scale FEM.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 4 , July 2002 , pp. 537 - 572
Copyright
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

I. Babuska 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.
A. Bensoussan, J.L. Lions and G. Papanicolau, Asymptotic Analysis for Periodic Structures. North Holland, Amsterdam (1978).
D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures. Springer-Verlag, New York (1999).
Hou, T.Y. and Wu, X.H., Multiscale, A finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169-189. CrossRef
Hou, T.Y., Wu, X.H. and Cai, Z., Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comp. 68 (1999) 913-943. CrossRef
A.-M. Matache, Spectral- and p -Finite Elements for problems with microstructure, Ph.D. thesis, ETH Zürich (2000).
Matache, A.-M., Babuska, I. and Schwab, C., Generalized p-FEM in Homogenization. Numer. Math. 86 (2000) 319-375. CrossRef
A.-M. Matache and M.J. Melenk, Two-scale regularity for homogenization problems with non-smooth fine-scale geometries, submitted.
A.-M. Matache and C. Schwab, Finite dimensional approximations for elliptic problems with rapidly oscillating coefficients, in Multiscale Problems in Science and Technology, N. Antonic, C.J. van Duijn, W. Jäger and A. Mikelic Eds., Springer-Verlag (2002) 203-242.
Morgan, R.C. and Babuska, I., 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. CrossRef
C. Schwab, p - and hp - Finite Element Methods. Oxford Science Publications (1998).
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).
B. Szab ó and I. Babuska, Finite Element Analysis. John Wiley & Sons, Inc. (1991).
O.A. Oleinik, A.S. Shamaev and G.A. Yosifian, Mathematical Problems in Elasticity and Homogenization. North-Holland (1992).