Research reports

Years: 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

High order topological asymptotics: reconciling layer potentials and compound asymptotic expansions

by F. Feppon and H. Ammari

(Report number 2021-38)

Abstract
A systematic two-step procedure is proposed for the derivation of full asymptotic expansions of the solution of elliptic partial differential equations set on a domain perforated with a small hole on which a Dirichlet boundary condition is applied. First, an integral representation of the solution is sought, which enables to exploit the explicit dependence with respect to the small parameter to predict the correct form of a two-scale ansatz. Second, the terms of the ansatz are characterized by the method of matched asymptotic expansions as the solutions of a cascade of successive interior and exterior domains. This allows to interpret them as high-order correctors, for which error bounds can be proved using variational estimates. The methodology is illustrated on two different problems: we start by revisiting the perforated Poisson problem with Dirichlet boundary conditions on both the hole and the outer boundary, where we highlight how the method enables to obtain very naturally the correct ansatz in the most delicate two-dimensional setting. Then, we provide original and complete asymptotic expansions for a perforated cell-problem featuring periodic conditions.

Keywords: Topological asymptotics, layer potentials, matched asymptotic expansions, Deny-Lions spaces, exterior Dirichlet problem, periodic cell problem.

@Techreport{FA21_980,
  author = {F. Feppon and H. Ammari},
  title = {High order topological asymptotics: reconciling layer potentials and compound asymptotic expansions},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2021-38},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2021/2021-38.pdf },
  year = {2021}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).