+\frac{1}{4}\big(|\psi_h^{n}|^2-1)^2 \bigg)\d x \\
&\quad +
\int_\Omega
\bigg(\frac{1}{2} |\nabla\times {\bf A}_h^{n}-{\bf H}|^2
+\frac{1}{2} |\phi_h^{n}|^2 \bigg)\d x {\nonumber}\end{aligned}$$ for $n=0,1,\cdots,N$. By substituting $\varphi_h=D_\tau\psi_h^{n+1}$, $\chi_h=\phi_h ^{n+1}$ and ${\bf a}_h=D_\tau{\bf A}_h^{n+1}$ into -, we obtain $$\begin{aligned}
\label{FEMEGDecay}
&D_\tau {\cal G}_h^{n+1}
+\int_\Omega 
\big((\eta-\tau/2) |D_\tau\psi_h^{n+1}|^2
+|D_\tau{\bf A}_h^{n+1}|^2 
 \big) \d x  {\nonumber}\\
&\quad 
+\int_\Omega 
\frac{\tau}{2}\bigg( \bigg|D_\tau\bigg(\frac{i}{\kappa} 
\nabla\psi_h^{n+1} + {\bf A}_h^{n}\psi_h^{n+1} \bigg)\bigg|^2
+\big|D_\tau\phi_h^{n+1}|^2 
+\big|D_\tau\nabla\times {\bf A}_h^{n+1}|^2 \bigg)\d x {\nonumber}\\
&\quad 
+\int_\Omega \frac{\tau}{2}\bigg(|\psi_h^{n+1}|^2|D_\tau \psi_h^{n+1}|^2 
+\frac{1}{2}|D_\tau |\psi_h^{n+1}|^2|^2 \bigg)\d x {\nonumber}\\
&= 
-\int_\Omega
i\eta\kappa\Theta(\psi_h^{n})\phi_h^{n}D_\tau\psi_h^{n+1}\d x 
+{\rm Re}\int_\Omega \tau D_\tau\bigg(\frac{i}{\kappa} 
\nabla\psi_h^{n+1} + {\bf A}^{n+1}\psi_h^{n+1}\bigg) 
\overline\psi_h^{n}D_\tau{\bf A}_h^{n+1} \d x {\nonumber}\\
&\leq \frac{\eta}{2}\int_\Omega|D_\tau\psi_h^{n+1}|^2\d x
+ \frac{\eta\kappa^2}{2}\int_\Omega 
|\phi_h^{n}|^2\d x  {\nonumber}\\
&\quad 
+\int_\Omega \frac{\tau}{2} \bigg|D_\tau\bigg(\frac{i}{\kappa} 
\nabla\psi_h^{n+1} + {\bf A}_h^{n}\psi^{n+1}\bigg)\bigg|^2\d x 
+\int_\Omega \frac{\tau}{2}|\psi_h^{n}|^2|D_\tau{\bf A}_h^{n+1}|^2 \d x , 
$$ which reduces to $$\label{FEMEGDecay0}
\begin{aligned}
&D_\tau {\cal G}_h^{n+1}
+\int_\Omega 
\bigg(\frac{\eta-\tau}{2} |D_\tau\psi_h^{n+1}|^2
+\frac{1}{2}|D_\tau{\bf A}_h^{n+1}|^2
+\frac{1-\tau|\psi_h^n|^2}{2}|D_\tau{\bf A}_h^{n+1}|^2
 \bigg) \d x  \\
 &\leq \eta\kappa^2{\cal G}_h^{n}  .
\end{aligned}$$ Unlike the PDE’s solution, it is not obvious whether the finite element solution satisfies $|\psi_h^n|\leq 1$ pointwisely. In Section \[SecUnif\] we shall prove $$\begin{aligned}
\label{BDPsiDecay0}
1-\tau|\psi_h^n|^2\geq 0  \end{aligned}$$ when $\tau<\tau_0$ (for some positive constant $\tau_0$ which is independent of $\tau$ and $h$). Then implies boundedness of the discrete energy via the discrete Gronwall’s inequality. By utilizing the discrete energy, we derive further estimates which 