\qquad 
\nabla\cdot {\bf A}=\frac{\partial A_1}{\partial x_1}
+\frac{\partial A_2}{\partial x_2},\\
&\nabla\times H=\bigg(\frac{\partial H}{\partial x_2},\,
-\frac{\partial H}{\partial x_1}\bigg),\quad 
\nabla\psi=\bigg(\frac{\partial \psi}{\partial x_1},\,
\frac{\partial \psi}{\partial x_2}\bigg).\end{aligned}$$ With these notations, - can also be used for solving the two-dimensional problem, and Theorem \[MainTHM\] can also be proved in the similar way.

An overview of the proof {#OverviewP}
------------------------

Our basic idea is to introduce $\psi_{h,\tau}$ and ${\bf A}_{h,\tau}$ ($\psi_{h,\tau}^+$ and ${\bf A}_{h,\tau}^+$) as the piecewise linear (piecewise constant) interpolation of the finite element solutions $\psi_h^{n+1}$ and ${\bf A}_h^{n+1}$ in the time direction, respectively, and denote by $\psi_{h,\tau}^-$ and ${\bf A}_{h,\tau}^-$ the piecewise constant interpolation of $\psi_h^{n}$ and ${\bf A}_h^{n}$, respectively. Rewrite the finite element equations as the following equations defined continuously in time: $$\begin{aligned}
&\int_0^T\bigg[(\eta \partial_t\psi_{h,\tau}, \varphi_{h,\tau}) 
+\bigg( \bigg(\frac{i}{\kappa}\nabla+{\bf A}_{h,\tau}^+\bigg)\psi_{h,\tau}^+ \,, 
\bigg(\frac{i}{\kappa}\nabla+{\bf A}_{h,\tau}^+\bigg)\varphi_{h,\tau}\bigg)
 \bigg]\d t {\nonumber}\\
&\quad 
+((|\psi_{h,\tau}^+|^{2}-1)\psi_{h,\tau}^+,\varphi_{h,\tau})\bigg]\d t=
\int_0^T(i\eta \kappa
\Theta(\psi_{h,\tau}^-)\phi_{h,\tau}^- ,\varphi_{h,\tau})\d t   . {\nonumber}\end{aligned}$$ $$\begin{aligned}
&\int_0^T\bigg[ (\phi_{h,\tau}^+, \chi_{h,\tau}) 
- ({\bf A}_{h,\tau}^+,\nabla \chi_{h,\tau} ) \bigg]\d t= 0 \, , \\[10pt]
&\int_0^T\bigg[ (\partial_t{\bf A}_{h,\tau},{\bf a}_{h,\tau}) 
+ (\nabla\phi_{h,\tau}^+ \, ,{\bf a}_{h,\tau})
+ (\nabla\times{\bf A}_{h,\tau}^+  \, , \nabla\times {\bf a}_{h,\tau})
\bigg]\d t{\nonumber}\\
&\quad +\int_0^T\bigg[ {\rm Re}\bigg(\overline\psi_{h,\tau}^- \bigg(\frac{i}{\kappa} \nabla 
+{\bf A}_{h,\tau}^-\bigg) \psi_{h,\tau}^-  , {\bf a}_{h,\tau}\bigg)\bigg]\d t
=\int_0^T\bigg[ (\nabla\times{\bf H} \, ,{\bf a}_{h,\tau})\bigg]\d t  \, . 
$$ If we 