Homogenization error for two scale Maxwell equations

TL;DR

This paper extends homogenization error estimates for two scale Maxwell equations to cases where the homogenized solution has weaker regularity, specifically in fractional Sobolev spaces, which is relevant for polygonal domains.

Contribution

It establishes homogenization error bounds for two scale Maxwell equations under weaker regularity assumptions on the solution, applicable to non-smooth domains.

Findings

Homogenization errors are derived for solutions in fractional Sobolev spaces.

Results apply to Maxwell equations with regularity $H^s({ m curl}, D)$ for $0<s<1$.

Procedure also applicable to elliptic equations with similar regularity conditions.

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 $0<s<1$.