Homogenization error for two scale Maxwell equations

Abstract

For two scale elliptic equations in a domain $D$, standard homogenization errors are deduced with the assumption that the solution $u_0$ of the homogenized equation belongs to $H^2(D)$. For two scale Maxwell equations, the corresponding required regularity is $u_0\in H^1({\rm curl}, D)$. These regularity conditions normally do not hold in general polygonal domains, which are of interests for finite element discretization. The paper establishes homogenization errors when $u_0$ belongs to a weaker regularity space $H^{1+s}(D)$ for elliptic problems and $H^s({\rm curl},D)$ for Maxwell problems where $0<s<1$. Though we only present the results for two scale Maxwell equations when $u_0\in H^s({\rm curl},D)$ with $0<s<1$, the procedure works verbatim for elliptic equations when $u_0$ belongs to $H^{1+s}(D)$ with $0<s<1$.