2017/05/21 by Chu, Van Tiep, Hoang, Viet Ha
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1705.07407
We study homogenization of multiscale Maxwell wave equation that depends on n separable microscopic scales in a domain D⊂\mathbb Rd on a finite time interval (0,T). Due to the non-compactness of the embedding of H0(\curl,D) in L2(D)d, homogenization of Maxwell wave equation can be significantly more complicated than that of scalar wave equations in the H1(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 L2(D)d norm, which do not appear in the H1(D) setting. For two scale equations, we derive an explicit homogenization error estimate for the case where the solution u0 of the homogenized equation belongs to L^∞((0,T);H1(\curl,D)). When u0 only belongs to a weaker regularity space L^∞((0,T);Hs(\curl,D)) for 0