introduce notations, review some preliminary materials including generalized directional derivative, and state the model problem and assumptions for later proof. In \[sec:homogenization\], firstly, we prove a uniform bound for solutions which is missing in [@Liu2008], and we think it is indispensable. Then we apply div-curl lemma to prove the homogenization result, the proof will also be provided for the self-containing. We give an $O(\epsilon^{1/2})$ estimation for first order asymptotic expansion in \[sec:estimation\], and the insight most comes from [@Shen2016]. We discuss Robin problem in \[sec:robin\], and show that with the duality technique from [@Shen2016] the difference between original and homogenized solutions in $L^2$ norm is $O(\epsilon)$. A computational framework based on finite element methods will be presented in \[sec:computation\] together with its numerical analysis. Experiments are reported in \[sec:experiments\], and the results are in good agreement with predicted estimation.

Preliminaries {#sec:preliminaries}
=============

Generally, when $X$ is used, it denotes a real Banach space with its norm as ${\left\lVert \cdot \right\rVert}_X$, $X^*$ as its topological dual, ${\left\langle \cdot,\cdot \right\rangle}_{X^*\times X}$ as duality pairing. Without confusion, we omit the subscript and simply write ${\left\langle \cdot,\cdot \right\rangle}$. Weak convergence is indicated by $\rightharpoonup$. Given two normed spaces $X$ and $Y$, $\mathcal{L}(X, Y)$ is 