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Inner-layer asymptotics in partially perforated domains: coupling across flat and oscillating interfaces

Taras Mel'nyk

TL;DR

This work analyzes a Poisson problem in a bounded, partially perforated domain with Neumann boundaries on perforations, addressing both a flat and an $ ext{O}( ext{ε})$-amplitude oscillating interface. Using inner-layer asymptotics together with a two-scale expansion, the authors derive a complete $H^1$-accurate asymptotic expansion for the flat-interface configuration under perforation symmetry and establish a rigorous homogenization with higher-order transmission conditions; for the oscillating interface they obtain a two-term approximation with $H^1$-estimates and show that microstructure effects are captured by the inner layer, while the homogenized problem may remain unaffected. The results provide a principled framework to justify refined coupling conditions across flat and oscillating interfaces, with explicit construction of cell and inner-layer problems and quantitative error bounds. This advances the understanding of transmission across coupled heterogeneous media and offers rigorous tools for high-accuracy multiscale modeling in perforated domains. ${f}$

Abstract

The article examines a boundary-value problem in a domain consisting of perforated and imperforate regions, with Neumann conditions prescribed at the boundaries of the perforations. Assuming the porous medium has symmetric, periodic structure with a small period $\varepsilon,$ we analyse the limit behavior of the problem as $\varepsilon \to 0.$ A crucial aspect of this study is deriving correct coupling conditions at the common interface, which is achieved using inner-layer asymptotics. For the flat interface, we construct and justify a complete asymptotic expansion of the solution in the $H^1$-Sobolev space. Furthermore, for the $\varepsilon$-periodically oscillating interface of amplitude $\mathcal{O}(\varepsilon),$ we provide an approximation to the solution and establish the corresponding asymptotic estimates in $H^1$-Sobolev spaces.

Inner-layer asymptotics in partially perforated domains: coupling across flat and oscillating interfaces

TL;DR

This work analyzes a Poisson problem in a bounded, partially perforated domain with Neumann boundaries on perforations, addressing both a flat and an -amplitude oscillating interface. Using inner-layer asymptotics together with a two-scale expansion, the authors derive a complete -accurate asymptotic expansion for the flat-interface configuration under perforation symmetry and establish a rigorous homogenization with higher-order transmission conditions; for the oscillating interface they obtain a two-term approximation with -estimates and show that microstructure effects are captured by the inner layer, while the homogenized problem may remain unaffected. The results provide a principled framework to justify refined coupling conditions across flat and oscillating interfaces, with explicit construction of cell and inner-layer problems and quantitative error bounds. This advances the understanding of transmission across coupled heterogeneous media and offers rigorous tools for high-accuracy multiscale modeling in perforated domains.

Abstract

The article examines a boundary-value problem in a domain consisting of perforated and imperforate regions, with Neumann conditions prescribed at the boundaries of the perforations. Assuming the porous medium has symmetric, periodic structure with a small period we analyse the limit behavior of the problem as A crucial aspect of this study is deriving correct coupling conditions at the common interface, which is achieved using inner-layer asymptotics. For the flat interface, we construct and justify a complete asymptotic expansion of the solution in the -Sobolev space. Furthermore, for the -periodically oscillating interface of amplitude we provide an approximation to the solution and establish the corresponding asymptotic estimates in -Sobolev spaces.

Paper Structure

This paper contains 11 sections, 19 theorems, 170 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The problem statement
  3. Contraction of formal asymptotics
  4. Analysis in the perforated part
  5. Inner-layer asymptotics
  6. Homogenized problem and problems for $\{v^\pm_k\}$
  7. Justification and the main results
  8. Rapidly oscillating interface
  9. Construction of approximation
  10. Justification
  11. Closing remarks

Key Result

Lemma 3.1

Let $F_0(\xi), \ F_1(\xi), F_2(\xi)$ be $1$-periodic in $\xi$ and smooth functions in $\overline{Y}.$ Then there exists an unique smooth solution $N\in H^1_{per}(Y)$ to the problem if and only if

Figures (5)

  • Figure 1: The periodicity cell $Y$
  • Figure 2: The partial perforated domain $\Omega_\varepsilon$
  • Figure 3: The partial perforated band-cell
  • Figure 4: The partial perforated domain $\Omega_\varepsilon$
  • Figure 5: Partial perforated band-cell

Theorems & Definitions (34)

  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Lemma 3.1
  • Lemma 3.2
  • Remark 3.2
  • Lemma 3.3
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • ...and 24 more