The needle problem approach to non-periodic homogenization

Ben Schweizer; Marco Veneroni; Ben Schweizer; Marco Veneroni

doi:10.3934/nhm.2011.6.755

Networks and Heterogeneous Media

2011, Volume 6, Issue 4: 755-781. doi: 10.3934/nhm.2011.6.755

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The needle problem approach to non-periodic homogenization

Ben Schweizer ^{1
,},
Marco Veneroni ^{2
,}

1.
Technische Universität Dortmund, Fakultät für Mathematik, Vogelpothsweg 87, D-44227 Dortmund
2.
McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street West, H3A 2K6 Montreal QC

Received: 01 January 2011 Revised: 01 July 2011
Primary: 35J15, 35B27; Secondary: 65N30.

We introduce a new method to homogenization of non-periodic problems and illustrate the approach with the elliptic equation $-\nabla\cdot (a^\epsilon\nabla u^\epsilon) = f$. On the coefficients $a^\epsilon$ we assume that solutions $u^\epsilon$ of homogeneous $\epsilon$-problems on simplices with average slope $\xi\in \mathbb{R}^n$ have the property that flux-averages $f a^\epsilon\nabla u^\epsilon\in \mathbb{R}^n$ converge, for $\epsilon\to 0$, to some limit $a^\star(\xi)$, independent of the simplex. Under this assumption, which is comparable to H-convergence, we show the homogenization result for general domains and arbitrary right hand side. The proof uses a new auxiliary problem, the needle problem. Solutions of the needle problem depend on a triangulation of the domain, they solve an $\epsilon$-problem in each simplex and are affine on faces.
- Non-periodic homogenization,
- elliptic boundary-value problems,
- representative volume elements,
- div-curl lemma,
- adapted grids
Citation: Ben Schweizer, Marco Veneroni. The needle problem approach to non-periodic homogenization[J]. Networks and Heterogeneous Media, 2011, 6(4): 755-781. doi: 10.3934/nhm.2011.6.755

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Abstract

We introduce a new method to homogenization of non-periodic problems and illustrate the approach with the elliptic equation $-\nabla\cdot (a^\epsilon\nabla u^\epsilon) = f$. On the coefficients $a^\epsilon$ we assume that solutions $u^\epsilon$ of homogeneous $\epsilon$-problems on simplices with average slope $\xi\in \mathbb{R}^n$ have the property that flux-averages $f a^\epsilon\nabla u^\epsilon\in \mathbb{R}^n$ converge, for $\epsilon\to 0$, to some limit $a^\star(\xi)$, independent of the simplex. Under this assumption, which is comparable to H-convergence, we show the homogenization result for general domains and arbitrary right hand side. The proof uses a new auxiliary problem, the needle problem. Solutions of the needle problem depend on a triangulation of the domain, they solve an $\epsilon$-problem in each simplex and are affine on faces.

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