# Two-scale FEM for homogenization problems

Ana-Maria Matache; Christoph Schwab

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- 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 36.4 (2010): 537-572. <http://eudml.org/doc/194116>.

@article{Matache2010,

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.
},

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

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

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

language = {eng},

month = {3},

number = {4},

pages = {537-572},

publisher = {EDP Sciences},

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

url = {http://eudml.org/doc/194116},

volume = {36},

year = {2010},

}

TY - JOUR

AU - Matache, Ana-Maria

AU - Schwab, Christoph

TI - Two-scale FEM for homogenization problems

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

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 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.

LA - eng

KW - Homogenization; two-scale regularity; Finite Element Method (FEM); two-scale FEM.; homogenization; Finite Element Method; two-scale FEM; Numerical experiments; error estimates

UR - http://eudml.org/doc/194116

ER -

## References

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- R.C. Morgan and I. Babuska, 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
- C. Schwab, p- and hp- Finite Element Methods. Oxford Science Publications (1998). Zbl0910.73003
- 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). Zbl1007.74079
- 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). Zbl0768.73003

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