  In this paper, we mainly employed the idea of the previous paper to study the
sharp uniform $W^{1,p}$ estimates with $1<p\leq \infty$ for more general
elliptic systems with the Neumann boundary condition on a bounded $C^{1,\eta}$
domain, arising in homogenization theory. Based on the skills developed by Z.
Shen and by T. Suslina for different purposes, we also established the $L^2$
convergence rates on a bounded $C^{1,1}$ domain and a Lipschitz domain,
respectively. Here we found a "rough" version of the first order correctors
(see Theorem 1.3), It allows us to skip the corresponding convergence results
on $\mathbb{R}^d$ that are the preconditions in T. Suslina's papers.
