A Unified Homogenization Approach for the Dirichlet Problem in Perforated Domains

Abstract

We revisit the periodic homogenization of Dirichlet problems for the Laplace operator in perforated domains, and establish a unified proof that works for different regimes of hole-cell ratios, that is the ratio between the scaling factor of the holes and that of the periodic cells. The approach is then made quantitative and it yields correctors and error estimates for vanishing holecell ratios. For positive volume fraction of holes, the approach is just the standard oscillating test function method; for vanishing volume fraction of holes, we study asymptotic behaviors of a properly rescaled cell problems and use them to build oscillating test functions. Our method reveals how the different regimes are intrinsically connected through the cell problems and the connection with periodic layer potentials.

High Order Homogenization of the Stokes System in a Periodic Porous Medium

“…Similar results are expected to hold in dimension d = 2 but would require a different treatment, as e.g. in [6,41]. The hole ηT is assumed to be an non-empty open subset strictly included in the unit cell for any η ≤ 1 (it does not touch the boundary):…”

“…The asymptotic behaviors of the tensors X k * , M k are obtained by following the methodology of [6,41,34], which relies on several technical results stated in this part.…”

“…This matter is to be adressed in a future work through a more accurate analysis of the rate of convergence of the coefficients X 2k * and X 2k+1 * in the asymptotic (5.12) and (5.13). To date, let us note that Jing obtained recently X 0 * = F/η d−2 + O(1) in the scalar case (proof of the Lemma 5.1 in [41]) by using layer potential techniques.…”

Convergence rate for the homogenization of stationary diffusions in dilutely perforated domains with reflecting boundaries

“…In this paper, we focus on the case with zero Neumann data at the boundary of the holes, and investigate the continuous transition of the effective models with respect to the relative smallness of the holes, and we establish quantitative convergence results for homogenization when the holes are dilute. The proofs are based on a careful study of the rescaled cell-problems using layer potential techniques, a method started in [25,26,27] for the case of Dirichlet data in the holes, and exhibit the effectiveness of this approach. Before we refocus on the problem (1.1), let us mention that homogenizations in perforated domains and in similar geometric settings have gained many attentions recently with new convergence rates and uniform regularity [36,38,39,35,40,30] following the framework of [10,11,31,29], derivations in the non-periodic settings [23,22,21,20], derivation of higher order models [17,16,18] and so on.…”

“…σ ε is the bounding constant of a version of Poincaré inequality for H 1 functions in B ε with zero value in B ηε ; see e.g. [2,25]. Note that η cr,2 ∼ ε 2 d−2 for d ≥ 3 and η cr,2 ∼ exp(− C ε 2 ) for d = 2.…”