solvability of \[eq:robin variation\]. We have following lemma, the proof is postponed in appendix.

\[lem:coercive for Robin\] Let $\Omega$ be a Lipschitz domain and $\Gamma$ its boundary. Then there exists a constant $C(\Omega)$, such that $\forall \psi \in H^1(\Omega)$: $$C(\Omega) {\left\lVert \psi \right\rVert}_{1,\Omega}^2 \leq \int_{\Omega} {\left\lvert \nabla \psi \right\rvert}^2 + \int_{\Gamma} \psi^2 .$$

Similarly, we can prove

The solutions $u_\epsilon$ of Robin problems \[eq:robin variation\] converge weakly to $u^0$, while $u^0$ is the solution of homogenized Robin problem: $$\left\{
    \begin{aligned}
    -{\mathrm{div}}(\hat{A}\nabla u_0) = f ~~~~ &\text{ in } L^2(\Omega) \\
    \bm{n}\cdot \hat{A}\nabla u_0+\alpha(\bm{x})u_0=g ~~~~ &\text{ in } L^2(\Gamma)
    \end{aligned} .
    \right.$$ And the associated variation form is: $$\label{eq:homogenized robin problem variation form}
    \left\{
    \begin{aligned}
    &\text{Find} ~~ u_0 \in H^1(\Omega) \text{ s.t. } \forall v \in H^1(\Omega)\\
    &\int_{\Omega}\hat{A} \nabla u_0 \cdot \nabla v + \int_{\Gamma} \alpha u_0 v = \int_{\Omega}fv + \int_{\Gamma} gv
    \end{aligned} .
    \right.$$

Estimation \[lem:boundary estimation\] is cited from Thm 1.5.1.10 in [@Grisvard2011], we use this lemma to prove $O(\epsilon^{1/2})$ convergence rate in $H^1(\Omega)$ norm.

\[lem:boundary estimation\] Let $\Omega$ be a Lipschitz domain in ${\mathbb{R}}^d$, then for $u \in H^1(\Omega)$, $$\int_{\Gamma} u^2 \leq C(\Omega) {\left(  t\int_{\Omega} {\left\lvert \nabla u \right\rvert}^2 + t^{-1} \int_{\Omega} u^2 \right)} .$$ Here $t$ 