### Abstract

We study the behavior of the spectrum of a family of one-dimensional operators with periodic high-contrast coefficients as the period goes to zero, which may represent, e.g., the elastic or electromagnetic response of a two-component composite medium. Compared to the standard operators with moderate contrast, they exhibit a number of new effects due to the underlying nonuniform ellipticity of the family. The effective behavior of such media in the vanishing period limit also differs notably from that of multidimensional models investigated thus far by other authors, due to the fact that neither component of the composite forms a connected set. We then discuss a modified problem, where the equation coefficient is set to a positive constant on an interval that is independent of the period. Formal asymptotic analysis and numerical tests with finite elements suggest the existence of localized eigenfunctions (``defect modes''), whose eigenvalues are situated in the gaps of the limit spectrum for the unperturbed problem.

Read More: http://epubs.siam.org/doi/10.1137/130947106

Language | English |
---|---|

Pages | 72-98 |

Number of pages | 17 |

Journal | Multiscale Modeling and Simulation |

Volume | 13 |

Issue number | 1 |

Early online date | 8 Jan 2015 |

DOIs | |

Status | Published - 2015 |

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### Cite this

*Multiscale Modeling and Simulation*,

*13*(1), 72-98. https://doi.org/10.1137/130947106

**Spectral analysis of one-dimensional high-contrast elliptic problems with periodic coefficients.** / Cherednichenko, Kirill; Cooper, Shane; Guenneau, Sebastien.

Research output: Contribution to journal › Article

*Multiscale Modeling and Simulation*, vol. 13, no. 1, pp. 72-98. https://doi.org/10.1137/130947106

}

TY - JOUR

T1 - Spectral analysis of one-dimensional high-contrast elliptic problems with periodic coefficients

AU - Cherednichenko, Kirill

AU - Cooper, Shane

AU - Guenneau, Sebastien

PY - 2015

Y1 - 2015

N2 - We study the behavior of the spectrum of a family of one-dimensional operators with periodic high-contrast coefficients as the period goes to zero, which may represent, e.g., the elastic or electromagnetic response of a two-component composite medium. Compared to the standard operators with moderate contrast, they exhibit a number of new effects due to the underlying nonuniform ellipticity of the family. The effective behavior of such media in the vanishing period limit also differs notably from that of multidimensional models investigated thus far by other authors, due to the fact that neither component of the composite forms a connected set. We then discuss a modified problem, where the equation coefficient is set to a positive constant on an interval that is independent of the period. Formal asymptotic analysis and numerical tests with finite elements suggest the existence of localized eigenfunctions (``defect modes''), whose eigenvalues are situated in the gaps of the limit spectrum for the unperturbed problem.Read More: http://epubs.siam.org/doi/10.1137/130947106

AB - We study the behavior of the spectrum of a family of one-dimensional operators with periodic high-contrast coefficients as the period goes to zero, which may represent, e.g., the elastic or electromagnetic response of a two-component composite medium. Compared to the standard operators with moderate contrast, they exhibit a number of new effects due to the underlying nonuniform ellipticity of the family. The effective behavior of such media in the vanishing period limit also differs notably from that of multidimensional models investigated thus far by other authors, due to the fact that neither component of the composite forms a connected set. We then discuss a modified problem, where the equation coefficient is set to a positive constant on an interval that is independent of the period. Formal asymptotic analysis and numerical tests with finite elements suggest the existence of localized eigenfunctions (``defect modes''), whose eigenvalues are situated in the gaps of the limit spectrum for the unperturbed problem.Read More: http://epubs.siam.org/doi/10.1137/130947106

UR - http://epubs.siam.org/journal/mmsubt

UR - http://dx.doi.org/10.1137/130947106

U2 - 10.1137/130947106

DO - 10.1137/130947106

M3 - Article

VL - 13

SP - 72

EP - 98

JO - Multiscale Modeling and Simulation

T2 - Multiscale Modeling and Simulation

JF - Multiscale Modeling and Simulation

SN - 1540-3459

IS - 1

ER -