On Homogenization of Problems in Domains of the “Infusorium” Type

Chechkin, Gregory A.; Chechkina, T. P.

doi:10.1023/b:joth.0000016062.22939.73

Cited by 15 publications

(16 citation statements)

References 7 publications

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“…We refer the reader to [13] for a discussion on uniform descriptions of heterogeneous media, while the working technique is detailed for instance in [14, pp. 11-22] [18][19][20] contain more theoretical approaches able to justify the asymptotics at least for simpler PDE models.…”

Section: A Few Comments On Related Literaturementioning

confidence: 99%

“…Homogenization problems in locally periodic perforated domains have been dealt with in, for instance, [18], [21][22][23]; see [24] for a more recent account of bibliographic information. At the technical level, we rely on the analysis reported in [23] for the case of a Poisson problem with a linear Fourier condition imposed at the boundary of the perforation.…”

Section: A Few Comments On Related Literaturementioning

confidence: 99%

“…In the locally periodic setting, one represents the normal vector n ε to the "oscillating" internal boundaries of the perforations in the form suggested, for instance, in [18,22] as follows:…”

Section: Locally Periodic Array Of Perforationsmentioning

confidence: 99%

See 2 more Smart Citations

Homogenization of a reaction–diffusion system modeling sulfate corrosion of concrete in locally periodic perforated domains

Fatima¹,

Arab²,

Zemskov

et al. 2010

J Eng Math

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A reaction-diffusion system modeling concrete corrosion in sewer pipes is discussed. The system is coupled, semi-linear, and partially dissipative. It is defined on a locally periodic perforated domain with nonlinear Robin-type boundary conditions at water-air and solid-water interfaces. Asymptotic homogenization techniques are applied to obtain upscaled reaction-diffusion models together with explicit formulae for the effective transport and reaction coefficients. It is shown that the averaged system contains additional terms appearing due to the deviation of the assumed geometry from a purely periodic distribution of perforations for two relevant parameter regimes: (a) all diffusion coefficients are of order of O(1) and (b) all diffusion coefficients are of order of O(ε 2 ) except the one for H 2 S(g) which is of order of O(1). In case (a) a set of macroscopic equations is obtained, while in case (b) a two-scale reaction-diffusion system is derived that captures the interplay between microstructural reaction effects and the macroscopic transport.

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Section: A Few Comments On Related Literaturementioning

confidence: 99%

Section: A Few Comments On Related Literaturementioning

confidence: 99%

See 1 more Smart Citation

Homogenization of a reaction–diffusion system modeling sulfate corrosion of concrete in locally periodic perforated domains

Fatima¹,

Arab²,

Zemskov

et al. 2010

J Eng Math

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show abstract

“…, p be the eigenvalues of problem (1), which converge to 0 , as → 0. Then where (12) and is the outward unit normal to + .…”

Section: Theorem 23mentioning

confidence: 99%

Asymptotic approximation of eigenelements of the Dirichlet problem for the Laplacian in a domain with shoots

Amirat

Chechkin

Gadyl’shin

2010

Math Methods in App Sciences

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Communicated by E. Sánchez-PalenciaWe study the asymptotic behavior of the eigenelements of the Dirichlet problem for the Laplacian in a two-dimensional bounded domain with thin shoots, depending on a small parameter e. Under the assumption that the width of the shoots goes to zero, as e tends to zero, we construct the limit (homogenized) problem and prove the convergence of the eigenvalues and eigenfunctions to the eigenvalues and eigenfunctions of the limit problem, respectively. Under the additional assumption that the shoots, in a fixed vicinity of the basis, are straight and periodic, and their width and the distance between the neighboring shoots are of order e, we construct the two-term asymptotics of the eigenvalues of the problem, as e → 0.

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“…In some cases, the estimates for the convergence rate were proven. It was also shown that when constructing the next terms of the asymptotics for the perturbed solutions, one can get estimates for the convergence rate or improve it [1][2][3][4][5][6][7][8]. In some cases, complete asymptotic expansions were constructed [9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Waveguides with fast oscillating boundary

Cardone¹

2017

Nanosystems: Phys. Chem. Math.

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We consider an elliptic operator in a planar waveguide with a fast oscillating boundary where we impose Dirichlet, Neumann or Robin boundary conditions assuming that both the period and the amplitude of the oscillations are small. We describe the homogenized operator, establish the norm resolvent convergence of the perturbed resolvent to the homogenized one, and prove the estimates for the rate of convergence. It is shown that under the homogenization, the type of the boundary condition can change.

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On Homogenization of Problems in Domains of the “Infusorium” Type

Cited by 15 publications

References 7 publications

Homogenization of a reaction–diffusion system modeling sulfate corrosion of concrete in locally periodic perforated domains

Homogenization of a reaction–diffusion system modeling sulfate corrosion of concrete in locally periodic perforated domains

Asymptotic approximation of eigenelements of the Dirichlet problem for the Laplacian in a domain with shoots

Waveguides with fast oscillating boundary

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