N_l^\epsilon \epsilon \partial_i \theta_\epsilon {\mathbf{S}}_\epsilon(\partial_l {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0) \right\rVert}_{0,\Omega}\\
    &+{\left\lVert \epsilon N_l \theta_\epsilon {\mathbf{S}}_\epsilon(\partial_{il}^2 {\mkern 1.5mu\overline{\mkern-1.5muu\mkern-1.5mu}\mkern 1.5mu}_0) \right\rVert}_{0,\Omega} \\
    \leq& C(\Omega, \kappa_1, \kappa_2){\left( \epsilon^{1/2}{\left\lVert u_0 \right\rVert}_{2,\Omega}+\epsilon {\left\lVert u_0 \right\rVert}_{2,\Omega} \right)} \leq C(\Omega, \kappa_1, \kappa_2) \epsilon^{1/2} {\left\lVert u_0 \right\rVert}_{2,\Omega}
    \end{aligned}.$$ Substitute $v=u_\epsilon-\tilde{u}_\epsilon \in V$ into \[lem:key\], and recall $\gamma_j u_0 = \gamma_j \tilde{u_\epsilon}$, we have: $$\begin{aligned}
    &\int_{\Omega} A^\epsilon \nabla(u_\epsilon - \tilde{u}_\epsilon) \cdot \nabla (u_\epsilon -\tilde{u}_\epsilon) \\
    =& \int_\Omega A^\epsilon \nabla w_\epsilon \cdot \nabla (u_\epsilon -\tilde{u}_\epsilon) + \int_{\Omega} A^\epsilon \nabla r_\epsilon \cdot \nabla (u_\epsilon -\tilde{u}_\epsilon) \\
    \leq& j^0(\gamma_j u_0; \gamma_j (u_\epsilon-\tilde{u}_\epsilon))+j^0(\gamma_j u_\epsilon; \gamma_j(\tilde{u}_\epsilon-u_\epsilon))\\
    &+C\epsilon^{1/2}{\left\lVert u_0 \right\rVert}_{2,\Omega} {\left\lVert \nabla(u_\epsilon -\tilde{u}_\epsilon) \right\rVert}_{0, \Omega}+\kappa_2 {\left\lVert \nabla r_\epsilon \right\rVert}_{0,\Omega} {\left\lVert \nabla (u_\epsilon-\tilde{u}_\epsilon) \right\rVert}_{0,\Omega} \\       
    \leq& j^0(\gamma_j \tilde{u}_\epsilon; \gamma_j (u_\epsilon-\tilde{u}_\epsilon))+j^0(\gamma_j u_\epsilon; \gamma_j(\tilde{u}_\epsilon-u_\epsilon))+C\epsilon^{1/2}{\left\lVert u_0 \right\rVert}_{2,\Omega} {\left\lVert \nabla(u_\epsilon -\tilde{u}_\epsilon) \right\rVert}_{0,\Omega}\\
    \leq& \alpha_j c_j^2 {\left\lVert u_\epsilon-\tilde{u}_\epsilon \right\rVert}_V^2 + C\epsilon^{1/2} {\left\lVert u_0 \right\rVert}_{2,\Omega} {\left\lVert u_\epsilon-\tilde{u}_\epsilon \right\rVert}_V
    \end{aligned}.$$ By a direct calculation, we have ${\left\lVert u_\epsilon-\tilde{u}_{\epsilon} \right\rVert}_V \leq 1/\Delta C\epsilon^{1/2} {\left\lVert u_0 \right\rVert}_{2,\Omega}$. Then ${\left\lVert \nabla w_\epsilon \right\rVert}_{0,\Omega} \leq C\epsilon^{1/2} {\left\lVert u_0 \right\rVert}_{2,\Omega}$, $C$ depends on $\Omega, \kappa_1, \kappa_2, \Delta$, since ${\left\lVert \nabla w_\epsilon \right\rVert}_{0,\Omega} \leq {\left\lVert u_\epsilon-\tilde{u}_\epsilon \right\rVert}_V+ {\left\lVert \nabla r_\epsilon \right\rVert}_{0,\Omega}$.

We have following corollary to quantify the $L^2$ convergence rate:

Let the assumptions \[ass:A\]-\[ass:C\] be satisfied, and $u_0 \in 