Homogenization of Differential Operators and Integral Functionals

Jikov, V. V.; Козлов, С. М.; Oleinik, O. A.

doi:10.1007/978-3-642-84659-5

Cited by 1,821 publications

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“…The homogenization theory was discussed in many papers and books, see [5,9,12,14] and references therein. On the other hand, our theory differs from the earlier ones in several respects: we do not assume periodic structure in the medium, which is often assumed in the cited literature, our differential expressions are non-selfadjoint, in contrast to the usual assumptions, and our boundary conditions are non-selfadjoint as well, so we treat non-selfadjoint operators.…”

Section: Introductionmentioning

confidence: 99%

Electromagnetic wave scattering by many small perfectly conducting particles of an arbitrary shape

Ramm

2012

Optics Communications

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In this paper the author's invited plenary talk at the 7-th PACOM (PanAfrican Congress of Mathematicians), is presented.Asymptotic solution to many-body wave scattering problem is given in the case of many small scatterers. The small scatterers can be particles whose physical properties are described by the boundary impedances, or they can be small inhomogeneities, whose physical properties are described by their refraction coefficients. Equations for the effective field in the limiting medium are derived. The limit is considered as the size a of the particles or inhomogeneities tends to zero while their number M (a) tends to infinity. These results are applied to the problem of creating materials with a desired refraction coefficient. For example, the refraction coefficient may have wave-focusing property, or it may have negative refraction, i.e., the group velocity may be directed opposite to the phase velocity. This paper is a review of the author's results presented in

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Section: Introductionmentioning

confidence: 99%

Electromagnetic wave scattering by many small perfectly conducting particles of an arbitrary shape

Ramm

2012

Optics Communications

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“…Many important physical problems are of this nature. In particular, they arise in the design and study of composite materials built from essentially different components; see, e.g., [9,38,7,34,24].…”

Section: Introductionmentioning

confidence: 99%

Lavrentiev regularization + Ritz approximation = uniform finite element error estimates for differential equations with rough coefficients

Knyazev

Widlund

2001

Math. Comp.

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Abstract. We consider a parametric family of boundary value problems for a diffusion equation with a diffusion coefficient equal to a small constant in a subdomain. Such problems are not uniformly well-posed when the constant gets small. However, in a series of papers, Bakhvalov and Knyazev have suggested a natural splitting of the problem into two well-posed problems. Using this idea, we prove a uniform finite element error estimate for our model problem in the standard parameter-independent Sobolev norm. We also study uniform regularity of the transmission problem, needed for approximation. A traditional finite element method with only one additional assumption, namely, that the boundary of the subdomain with the small coefficient does not cut any finite element, is considered.One interpretation of our main theorem is in terms of regularization. Our FEM problem can be viewed as resulting from a Lavrentiev regularization and a Ritz-Galerkin approximation of a symmetric ill-posed problem. Our error estimate can then be used to find an optimal regularization parameter together with the optimal dimension of the approximation subspace.

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“…Various kinds of simpli cations of this proof were done in the 70's by Murat, Tartar, Bakhvalov, Bensoussan, Lions and Papanicolaou (for more information, see e.g. the book [16]). …”

Section: The Periodic Casementioning

confidence: 99%