On the Homogenization of a Scalar Scattering Problem for Highly Oscillating Anisotropic Media

Source abstract

We study the homogenization of a transmission problem arising in the scattering theory for bounded inhomogeneities with periodic coefficients modeled by the anisotropic Helmholtz equation. The coefficients are assumed to be periodic functions of the fast variable, specified over the unit cell with characteristic size $\epsilon$. By way of multiple scales expansion, we focus on the $O(\epsilon^{k})$, $k=1,2$, bulk and boundary corrections of the leading-order $(O(1))$ homogenized transmission problem. The analysis in particular provides the $H^1$ and $L^2$ estimates of the error committed by the first-order-corrected solution considering (i) bulk correction only and (ii) boundary and bulk correction. We treat explicitly the $O(\epsilon)$ boundary correction for the transmission problem when the scatterer is a unit square and show it has an $L^2$-limit as $\epsilon\to 0$, provided that the boundary cutoff of cells is fixed. We also establish the $O(\epsilon^{2})$ bulk correction describing the mean wave motion inside the scatterer. The analysis also highlights a previously established, yet scarcely recognized, fact that the $O(\epsilon)$ bulk correction of the mean motion vanishes identically. (A correction is attached.)