Scattering by a bounded highly oscillating periodic medium and the effect of boundary correctors

Fioralba Cakoni; Bojan B. Guzina; Shari Moskow; Tayler Pangburn

doi:10.1137/19M1237089

Scattering by a bounded highly oscillating periodic medium and the effect of boundary correctors

Fioralba Cakoni, Bojan B. Guzina, Shari Moskow, Tayler Pangburn

Civil, Environmental, and Geo- Engineering

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8 Scopus citations

Abstract

We study the homogenization of a transmission problem arising in the scattering theory for bounded inhomogeneities with periodic coefficient in the lower-order term of the Helmholtz equation. The squared index of refraction is assumed to be a periodic function of the fast variable, specified over the unit cell with characteristic size ϵ. We obtain improved convergence results that assume lower regularity than previous estimates (which also allow for periodicity in the second-order operator), and we describe the asymptotic behavior of boundary correctors for general domains at all orders. In particular we show that, in contrast to Dirichlet problems, the O(ϵ) boundary corrector is nontrivial and can be observed in the far field. We further demonstrate the latter far field effect is larger than that of the \bulk" corrector|the so-called periodic drift, which is found to emerge only at O(ϵ2). We illustrate the analysis by examples in one and two spatial dimensions.

Original language	English (US)
Pages (from-to)	1448-1474
Number of pages	27
Journal	SIAM Journal on Applied Mathematics
Volume	79
Issue number	4
DOIs	https://doi.org/10.1137/19M1237089
State	Published - 2019

Bibliographical note

Funding Information:
∗Received by the editors January 8, 2019; accepted for publication (in revised form) June 14, 2019; published electronically July 30, 2019. https://doi.org/10.1137/19M1237089 Funding: The work of the first author was partially supported by NSF grant DMS-1813492 and by AFOSR grant FA9550-13-1-0199. The work of the second author was partially supported by the DOE NEUP through grant 10-862. The work of the third author was partially supported by NSF grant DMS-1715425. †Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019 (fiora.cakoni@ rutgers.edu). ‡Civil, Environmental, and Geo-Engineering, University of Minnesota, Minneapolis, MN 55455 (guzin001@umn.edu). §Drexel University, Philadelphia, PA 19104 (slm84@drexel.edu). ¶Mathematics, Drexel University, Philadelphia PA 19104 (tayler.anne.pangburn@drexel.edu).

Publisher Copyright:
© 2019 Society for Industrial and Applied Mathematics.

Keywords

Boundary layers
Higher-order expansion
Periodic inhomogeneities
Scattering

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title = "Scattering by a bounded highly oscillating periodic medium and the effect of boundary correctors",

abstract = "We study the homogenization of a transmission problem arising in the scattering theory for bounded inhomogeneities with periodic coefficient in the lower-order term of the Helmholtz equation. The squared index of refraction is assumed to be a periodic function of the fast variable, specified over the unit cell with characteristic size ϵ. We obtain improved convergence results that assume lower regularity than previous estimates (which also allow for periodicity in the second-order operator), and we describe the asymptotic behavior of boundary correctors for general domains at all orders. In particular we show that, in contrast to Dirichlet problems, the O(ϵ) boundary corrector is nontrivial and can be observed in the far field. We further demonstrate the latter far field effect is larger than that of the \bulk{"} corrector|the so-called periodic drift, which is found to emerge only at O(ϵ2). We illustrate the analysis by examples in one and two spatial dimensions.",

keywords = "Boundary layers, Higher-order expansion, Periodic inhomogeneities, Scattering",

author = "Fioralba Cakoni and Guzina, {Bojan B.} and Shari Moskow and Tayler Pangburn",

note = "Funding Information: ∗Received by the editors January 8, 2019; accepted for publication (in revised form) June 14, 2019; published electronically July 30, 2019. https://doi.org/10.1137/19M1237089 Funding: The work of the first author was partially supported by NSF grant DMS-1813492 and by AFOSR grant FA9550-13-1-0199. The work of the second author was partially supported by the DOE NEUP through grant 10-862. The work of the third author was partially supported by NSF grant DMS-1715425. †Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019 (fiora.cakoni@ rutgers.edu). ‡Civil, Environmental, and Geo-Engineering, University of Minnesota, Minneapolis, MN 55455 (guzin001@umn.edu). §Drexel University, Philadelphia, PA 19104 (slm84@drexel.edu). ¶Mathematics, Drexel University, Philadelphia PA 19104 (tayler.anne.pangburn@drexel.edu). Publisher Copyright: {\textcopyright} 2019 Society for Industrial and Applied Mathematics.",

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N1 - Funding Information: ∗Received by the editors January 8, 2019; accepted for publication (in revised form) June 14, 2019; published electronically July 30, 2019. https://doi.org/10.1137/19M1237089 Funding: The work of the first author was partially supported by NSF grant DMS-1813492 and by AFOSR grant FA9550-13-1-0199. The work of the second author was partially supported by the DOE NEUP through grant 10-862. The work of the third author was partially supported by NSF grant DMS-1715425. †Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019 (fiora.cakoni@ rutgers.edu). ‡Civil, Environmental, and Geo-Engineering, University of Minnesota, Minneapolis, MN 55455 (guzin001@umn.edu). §Drexel University, Philadelphia, PA 19104 (slm84@drexel.edu). ¶Mathematics, Drexel University, Philadelphia PA 19104 (tayler.anne.pangburn@drexel.edu). Publisher Copyright: © 2019 Society for Industrial and Applied Mathematics.

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N2 - We study the homogenization of a transmission problem arising in the scattering theory for bounded inhomogeneities with periodic coefficient in the lower-order term of the Helmholtz equation. The squared index of refraction is assumed to be a periodic function of the fast variable, specified over the unit cell with characteristic size ϵ. We obtain improved convergence results that assume lower regularity than previous estimates (which also allow for periodicity in the second-order operator), and we describe the asymptotic behavior of boundary correctors for general domains at all orders. In particular we show that, in contrast to Dirichlet problems, the O(ϵ) boundary corrector is nontrivial and can be observed in the far field. We further demonstrate the latter far field effect is larger than that of the \bulk" corrector|the so-called periodic drift, which is found to emerge only at O(ϵ2). We illustrate the analysis by examples in one and two spatial dimensions.

AB - We study the homogenization of a transmission problem arising in the scattering theory for bounded inhomogeneities with periodic coefficient in the lower-order term of the Helmholtz equation. The squared index of refraction is assumed to be a periodic function of the fast variable, specified over the unit cell with characteristic size ϵ. We obtain improved convergence results that assume lower regularity than previous estimates (which also allow for periodicity in the second-order operator), and we describe the asymptotic behavior of boundary correctors for general domains at all orders. In particular we show that, in contrast to Dirichlet problems, the O(ϵ) boundary corrector is nontrivial and can be observed in the far field. We further demonstrate the latter far field effect is larger than that of the \bulk" corrector|the so-called periodic drift, which is found to emerge only at O(ϵ2). We illustrate the analysis by examples in one and two spatial dimensions.

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Scattering by a bounded highly oscillating periodic medium and the effect of boundary correctors

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