  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
$0<s<1$.
