Homogenization of the Poisson Equation in a Thick Periodic Junction

Mel'nyk, Taras A.

doi:10.4171/zaa/923

Cited by 65 publications

(22 citation statements)

References 12 publications

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“…-a diffusion equation with respect to x N , having Neumann boundary conditions, and involving an integral operator with respect to the rapid variable y (see the first three equations in (18)), coupled with -a partial differential equation with respect to y , having homogeneous Neumann boundary condition on ∂ ω and replacing the algebraic system of [6] (see the last two equations in (18)). Moreover, the asymptotic analysis in Ω − extends the one of [6].…”

Section: Resultsmentioning

confidence: 99%

“…The function u ∈ V p (Ω + ) solving Problem (18) is unique. Furthermore, the energies converge in the sense that, for the previously selected subsequence:…”

Section: Definition 1 (Two Scale Convergence) Formentioning

confidence: 99%

“…Moreover, there exist a subsequence of {ε }, still denoted by {ε }, and u 1 ∈ L p (Ω + ,W 1,p (ω)), depending possibly on the selected subsequence, satisfying (14) and such that (u, u 1 ) is a weak solution of (18). Furthermore, the energies converge in the sense that, for the previously selected subsequence:…”

Section: Definition 1 (Two Scale Convergence) Formentioning

confidence: 99%

“…For the study of highly oscillating boundaries we refer to [2], [3], [4], [7], [9], [10], [11], [14], [16], [18], [19] and [21]. For the junction of a plate with a beam we refer to [12], [13] and [15] and for the study of heterogeneous fibers we refer to [22].…”

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confidence: 99%

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Limit behavior of thin heterogeneous domain with rapidly oscillating boundary

Mossino

Sili

2007

Ricerche mat.

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We give a generalization of the work presented in [6] where the asymptotic behaviour, as ε → 0, of a monotone nonlinear problem in a bounded multidomain of R N depending on ε was addressed. We extend the previous results to the case where the nonlinear operator depends both on the slow and rapid variable and we prove that, due to the presence of the rapid variable, the algebraic equation contained in the limit problem obtained in [6] must be replaced by a partial differential equation with respect to the microscopic variable y .

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Section: Resultsmentioning

confidence: 99%

“…The function u ∈ V p (Ω + ) solving Problem (18) is unique. Furthermore, the energies converge in the sense that, for the previously selected subsequence:…”

Section: Definition 1 (Two Scale Convergence) Formentioning

confidence: 99%

Section: Definition 1 (Two Scale Convergence) Formentioning

confidence: 99%

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confidence: 99%

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Limit behavior of thin heterogeneous domain with rapidly oscillating boundary

Mossino

Sili

2007

Ricerche mat.

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“…In [14], R. Brizzi and J. P. Chalot derive the limit problem for the Laplace equation with the homogeneous Neumann boundary condition and with the right-hand side in L 2 . For the same problem, a nonoscillating approximation at order O ε 1−δ , δ > 0, for the H 1 -norm is obtained by T. A. Mel'nyk in [21], under an additional assumption on the right-hand side. A monotone problem is considered by D. Blanchard, L. Carbone and A. Gaudiello in [9] (see also [10] and [13] for thin multidomains with rough boundaries).…”

Section: Introductionmentioning

confidence: 98%

Asymptotic approximation of the solution of Stokes equations in a domain with highly oscillating boundary

Amirat¹,

Bodart²,

Maio³

et al. 2007

Ann. Univ. Ferrara

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We consider a viscous incompressible flow in an infinite horizontal domain bounded at the bottom by a smooth wall and at the top by a rough wall. The latter is assumed to consist in a plane wall covered with periodically distributed asperities which size depends on a small parameter, and with a fixed height. We assume that the flow is governed by the stationary Stokes equations. Using a boundary layer corrector we derive and analyze a first order asymptotic approximation of the flow.

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Homogenization of the Signorini boundary-value problem in a thick junction and boundary integral equations for the homogenized problem

Mel'nyk¹,

Nakvasiuk²,

Wendland³

2010

Math. Meth. Appl. Sci.

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We consider a mixed boundary-value problem for the Poisson equation in a thick junction X e which is the union of a domain X 0 and a large number of e-periodically situated thin cylinders. The non-uniform Signorini conditions are given on the lateral surfaces of the cylinders. The asymptotic analysis of this problem is done as e → 0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove a convergence theorem and show that the non-uniform Signorini boundary conditions are transformed in the limiting variational inequalities in the region that is filled up by the thin cylinders as e → 0. The convergence of the energy integrals is proved as well. The existence and uniqueness of the solution to this non-standard limit problem is established. This solution can be constructed by using a penalty formulation and successive iteration. For some subclass, these problems can be reduced to an obstacle problem in X 0 and an appropriate postprocessing. The equations in X 0 finally are also treated with boundary integral equations.

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Homogenization of the Poisson Equation in a Thick Periodic Junction

Cited by 65 publications

References 12 publications

Limit behavior of thin heterogeneous domain with rapidly oscillating boundary

Limit behavior of thin heterogeneous domain with rapidly oscillating boundary

Asymptotic approximation of the solution of Stokes equations in a domain with highly oscillating boundary

Homogenization of the Signorini boundary-value problem in a thick junction and boundary integral equations for the homogenized problem

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