Section: New Results

Asymptotic models for time harmonic wave propagation

Asymptotic models oriented toward more efficient numerical simulation methods have been investigated in three different directions.

In [8] we considered the Poisson equation in a domain with a small hole of size $\delta$, and presented a simple numerical method, based on an asymptotic analysis, which allows to approximate robustly the far field of the solution as $\delta$ goes to zero without meshing the small hole. We proved the stability of the scheme and provide error estimates. This was confirmed with numerous numerical experiments illustrating the efficiency of the technique.

In [11] we considered a Laplace problem with Dirichlet boundary condition in a three dimensional domain containing an inclusion taking the form of a thin tube with small thickness. We proved convergence in operator norm of the resolvent of this problem as the thickness goes to 0, establishing that the perturbation on the resolvent induced by the inclusion is not greater than some (negative) power of the logarithm of the thickness. From this we deduced convergence of the eigenvalues of the perturbed operator toward the limit operator.

In [9] we investigated the eigenvalue problem $-\mathrm{div}\left(\sigma \nabla u\right)=\lambda u\left(𝒫\right)$ in a 2D domain $\Omega$ divided into two regions ${\Omega }_{\pm }$. We were interested in situations where $\sigma$ takes positive values on ${\Omega }_{+}$ and negative ones on ${\Omega }_{-}$. Such problems appear in time harmonic electromagnetics in the modeling of plasmonic technologies. In a recent work $[$ L. Chesnel, X. Claeys, and S.A. Nazarov. A curious instability phenomenon for a rounded corner in presence of a negative material. Asymp. Anal., 88(1):43–74, 2014$]$, we had highlighted an unusual instability phenomenon for the source term problem associated with $\left(𝒫\right)$: for certain configurations, when the interface between the subdomains ${\Omega }_{\pm }$ presents a rounded corner, the solution may depend critically on the value of the rounding parameter. In [9] we explained this property studying the eigenvalue problem $\left(𝒫\right)$. We provided an asymptotic expansion of the eigenvalues and prove error estimates. We established an oscillatory behaviour of the eigenvalues as the rounding parameter of the corner tends to zero. This work was ended with numerical illustrations.