Eigenvalues of the Neumann-Poincaré Operator for Two Inclusions with Contact of Order $m$: A Numerical Study (Journal of Computational Mathematics, 2018)
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.