Selfsimilar Perturbation near a Corner: Matching Versus Multiscale Expansions for a Model Problem

Dauge, Monique; Tordeux, Sébastien; Vial, Grégory

doi:10.1007/978-1-4419-1343-2_4

Cited by 26 publications

(45 citation statements)

References 8 publications

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“…In contrast to what happens in the case of holes collapsing at interior points or at smooth boundary points [3], we find that the series expansions in powers of ε that correspond to the asymptotic expansions of [13] are only "stepwise convergent". For corner opening angles ω that are rational multiples of π, the series will be unconditionally convergent, but in general for irrational multiples of π, certain pairs of terms in the series may have to be grouped together in order to achieve convergence.…”

Section: Introductioncontrasting

confidence: 83%

Converging Expansions for Lipschitz Self-Similar Perforations of a Plane Sector

Costabel

Riva

Dauge

et al. 2017

Integr. Equ. Oper. Theory

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Abstract. In contrast with the well-known methods of matching asymptotics and multiscale (or compound) asymptotics, the "functional analytic approach" of Lanza de Cristoforis (Analysis 28, 2008) allows to prove convergence of expansions around interior small holes of size ε for solutions of elliptic boundary value problems. Using the method of layer potentials, the asymptotic behavior of the solution as ε tends to zero is described not only by asymptotic series in powers of ε, but by convergent power series. Here we use this method to investigate the Dirichlet problem for the Laplace operator where holes are collapsing at a polygonal corner of opening ω. Then in addition to the scale ε there appears the scale η = ε π/ω . We prove that when π/ω is irrational, the solution of the Dirichlet problem is given by convergent series in powers of these two small parameters. Due to interference of the two scales, this convergence is obtained, in full generality, by grouping together integer powers of the two scales that are very close to each other. Nevertheless, there exists a dense subset of openings ω (characterized by Diophantine approximation properties), for which real analyticity in the two variables ε and η holds and the power series converge unconditionally. When π/ω is rational, the series are unconditionally convergent, but contain terms in log ε.Mathematics Subject Classification (2010). 35J05, 45A05, 31A10, 35B25, 35C20, 11J99.

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Section: Introductioncontrasting

confidence: 83%

Converging Expansions for Lipschitz Self-Similar Perforations of a Plane Sector

Costabel

Riva

Dauge

et al. 2017

Integr. Equ. Oper. Theory

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“…For such multi-scale problems, two strategies are possible: the matching of expansions, where the solution is described through two expansions which are matched in a transition region, or the multi-scale technique based on superposition via cut-off functions. We refer to [17] and [23] for the presentation of the two methods, respectively -see also [12] for a comparison on a model problem.…”

Section: Rahmani and G Vial / Multi-scale Asymptotic Expansion Fomentioning

confidence: 99%

“…Our aim is to build a solution R λ to problem (P ∞ ), which is nothing but (12) with zero right-hand side, and the additional condition at infinity R λ ∼ s λ * -the extension s λ * of the interior singularity is defined by (3). Of course, the only solution in B of problem (12) with zero right-hand side is Z = 0, and it does not fulfill the asymptotic condition R ∼ s λ * in (P ∞ ).…”

Section: Lemma 41 If the Right-hand Sides Satisfymentioning

confidence: 99%

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Multi-scale asymptotic expansion for a singular problem of a free plate with thin stiffener

Rahmani

Vial

2014

Asymptotic Analysis

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In this paper, we consider a partially clamped plate which is stiffened on a portion of its free boundary. Our aim is to build an asymptotic expansion of the displacement, solution of the Kirchhoff-Love model, with respect to the thickness of the stiffener. Due to the mixed boundary conditions, singularities appear, obstructing the construction of the terms of the asymptotic expansion in the same way as if the plate was surrounded by the stiffener on its whole boundary. Using a splitting into regular and singular parts, we are able to formulate an asymptotic expansion involving profiles which allow to take into account the singularities.

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“…Instead, in Bonnaillie-Noël, Dambrine, Tordeux and Vial [5] the method of multiscale asymptotic expansions is adopted to study the case in which the distance between the holes tends to zero but remains large with respect to their characteristic size. For the comparison of the two methods of multi-scale expansions and matched asymptotic expansions for the analysis of singular perturbation problems in a model example, we refer to Dauge, Tordeux and Vial [13].…”

Section: Introductionmentioning

confidence: 99%

Series expansions for the solution of the Dirichlet problem in a planar domain with a small hole

Riva

Musolino

Rogosin

2015

Asymptotic Analysis

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We consider the Dirichlet problem for the Laplace equation in a planar domain with a small hole. The diameter of the hole is proportional to a real parameter ε and we denote by uε the corresponding solution. If p is a point of the domain, then for ε small we write uε(p) as a convergent power series of ε and of 1/(r 0 + (2π) −1 log |ε|), with r 0 ∈ R. The coefficients of such series are given in terms of solutions of recursive systems of integral equations. We obtain a simplified expression for the series expansion of uε(p) in the case of a ring domain.

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Selfsimilar Perturbation near a Corner: Matching Versus Multiscale Expansions for a Model Problem

Cited by 26 publications

References 8 publications

Converging Expansions for Lipschitz Self-Similar Perforations of a Plane Sector

Converging Expansions for Lipschitz Self-Similar Perforations of a Plane Sector

Multi-scale asymptotic expansion for a singular problem of a free plate with thin stiffener

Series expansions for the solution of the Dirichlet problem in a planar domain with a small hole

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