Homogenization of multiscale Maxwell wave equations

Abstract: We study homogenization of multiscale Maxwell wave equation that depends on $n$ separable microscopic scales in a domain $D\subset{\mathbb R}<sup>d$ on a finite time interval $(0,T)$. Due to the non-compactness of the embedding of $H_0(\curl,D)$ in $L<sup>2(D)<sup>d$, homogenization of Maxwell wave equation can be significantly more complicated than that of scalar wave equations in the $H<sup>1(D)$ setting, and requires analysis uniquely for Maxwell wave equations. We employ multiscale convergence. The homogenized Maxwell wave equation and the initial condition are deduced from the multiscale homogenized equation. When the coefficient of the second order time derivative in the multiscale equation depends on the microscopic scales, the derivation is significantly more complicated, comparing to scalar wave equations, due to the corrector terms for the solution $\ue$ of the multiscale equation in the $L<sup>2(D)<sup>d$ norm, which do not appear in the $H<sup>1(D)$ setting. For two scale equations, we derive an explicit homogenization error estimate for the case where the solution $u_0$ of the homogenized equation belongs to $L<sup>\infty((0,T);H<sup>1(\curl,D))$. When $u_0$ only belongs to a weaker regularity space $L<sup>\infty((0,T);H<sup>s(\curl,D))$ for $0<s<1$, we contribute an approach to deduce a new homogenization error in this case, which depends on $s$. For general multiscale problems, a corrector is derived albeit without an explicit homogenization error estimate. These correctors and homogenization errors play an essential role in deriving numerical correctors for approximating the solutions to the multiscale problems numerically, as considered in our forthcoming publication.