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 - <p style="text-align: justify;">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].</p>