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Volume 36, Issue 1
Eigenvalues of the Neumann-Poincaré Operator for Two Inclusions with Contact of Order $m$: A Numerical Study

Eric Bonnetier, Faouzi Triki & Chun-Hsiang Tsou

DOI: 10.4208/jcm.1607-m2016-0543

J. Comp. Math., 36 (2018), pp. 17-28.

Published online: 2018-02

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  • Abstract

In a composite medium that contains close-to-touching inclusions, the pointwise values of the gradient of the voltage potential may blow up as the distance $δ$ between some inclusions tends to 0 and as the conductivity contrast degenerates. In a recent paper [9], we showed that the blow-up rate of the gradient is related to how the eigenvalues of the associated Neumann-Poincaré operator converge to $±\frac{1}{2}$ as $δ$ → 0, and on the regularity of the contact. Here, we consider two connected 2-D inclusions, at a distance $δ$ › 0 from each other. When $δ$ = 0, the contact between the inclusions is of order $m ≥ 2$. We numerically determine the asymptotic behavior of the first eigenvalue of the Neumann-Poincaré operator, in terms of $δ$ and $m$, and we check that we recover the estimates obtained in [10].

  • Keywords

Elliptic equations, Eigenvalues, Numerical approximation.

  • AMS Subject Headings

Primary 35J25, 73C40.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

Eric.Bonnetier@univ-grenoble-alpes.fr (Eric Bonnetier)

Faouzi.Triki@univ-grenoble-alpes.fr (Faouzi Triki)

Chun-Hsiang.Tsou@imag.fr (Chun-Hsiang Tsou)

  • BibTex
  • RIS
  • TXT
@Article{JCM-36-17, author = {Bonnetier , EricTriki , Faouzi and Tsou , Chun-Hsiang}, title = {Eigenvalues of the Neumann-Poincaré Operator for Two Inclusions with Contact of Order $m$: A Numerical Study}, journal = {Journal of Computational Mathematics}, year = {2018}, volume = {36}, number = {1}, pages = {17--28}, abstract = {

In a composite medium that contains close-to-touching inclusions, the pointwise values of the gradient of the voltage potential may blow up as the distance $δ$ between some inclusions tends to 0 and as the conductivity contrast degenerates. In a recent paper [9], we showed that the blow-up rate of the gradient is related to how the eigenvalues of the associated Neumann-Poincaré operator converge to $±\frac{1}{2}$ as $δ$ → 0, and on the regularity of the contact. Here, we consider two connected 2-D inclusions, at a distance $δ$ › 0 from each other. When $δ$ = 0, the contact between the inclusions is of order $m ≥ 2$. We numerically determine the asymptotic behavior of the first eigenvalue of the Neumann-Poincaré operator, in terms of $δ$ and $m$, and we check that we recover the estimates obtained in [10].

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1607-m2016-0543}, url = {http://global-sci.org/intro/article_detail/jcm/10580.html} }
TY - JOUR T1 - Eigenvalues of the Neumann-Poincaré Operator for Two Inclusions with Contact of Order $m$: A Numerical Study AU - Bonnetier , Eric AU - Triki , Faouzi AU - Tsou , Chun-Hsiang JO - Journal of Computational Mathematics VL - 1 SP - 17 EP - 28 PY - 2018 DA - 2018/02 SN - 36 DO - http://doi.org/10.4208/jcm.1607-m2016-0543 UR - https://global-sci.org/intro/article_detail/jcm/10580.html KW - Elliptic equations, Eigenvalues, Numerical approximation. AB -

In a composite medium that contains close-to-touching inclusions, the pointwise values of the gradient of the voltage potential may blow up as the distance $δ$ between some inclusions tends to 0 and as the conductivity contrast degenerates. In a recent paper [9], we showed that the blow-up rate of the gradient is related to how the eigenvalues of the associated Neumann-Poincaré operator converge to $±\frac{1}{2}$ as $δ$ → 0, and on the regularity of the contact. Here, we consider two connected 2-D inclusions, at a distance $δ$ › 0 from each other. When $δ$ = 0, the contact between the inclusions is of order $m ≥ 2$. We numerically determine the asymptotic behavior of the first eigenvalue of the Neumann-Poincaré operator, in terms of $δ$ and $m$, and we check that we recover the estimates obtained in [10].

Bonnetier , EricTriki , Faouzi and Tsou , Chun-Hsiang. (2018). Eigenvalues of the Neumann-Poincaré Operator for Two Inclusions with Contact of Order $m$: A Numerical Study. Journal of Computational Mathematics. 36 (1). 17-28. doi:10.4208/jcm.1607-m2016-0543
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