We introduce a decoupled backward Euler scheme for solving -: $$\begin{aligned}
&\eta D_\tau \psi^{n+1} 
-i\eta\kappa\Theta(\psi^{n})\nabla\cdot{\bf A}^n
+ \bigg(\frac{i}{\kappa} \nabla + {\bf A}^{n+1}\bigg)^{2} \psi^{n+1}
 + (|\psi^{n+1}|^{2}-1) \psi^{n+1}= 0, 
\label{TD1}\\[5pt] 
& D_\tau {\bf A}^{n+1} 
-\nabla(\nabla\cdot{\bf A}^{n+1})
+ \nabla\times(\nabla\times{\bf A}^{n+1})
+  {\rm Re}\bigg[\overline\psi^{n}\bigg(\frac{i}{\kappa} \nabla 
+{\bf A}^{n}\bigg) \psi^{n}\bigg]
=  \nabla\times {\bf H}  , 
\label{TD2}  \end{aligned}$$ where we have used a cut-off function $$\begin{aligned}
\Theta(z):=z/\max(|z|,1) ,\quad\forall~ z\in\C ,\end{aligned}$$ which satisfies $\Theta(z)=z$ if $|z|\leq 1$. For any given integers $r$ and $k$ which satisfy the condition , we solve by the Galerkin FEM and solve by a mixed FEM. Let $(\psi_{h}^{0}, {\bf A}_h^0):=(\psi_{0},
{\bf A}_0)$ at the initial time step and define $\phi_h^0:=-\nabla\cdot{\bf A}_0$. We look for $\psi_{h}^{n+1} \in {\mathbb S}_{h}^r$ and $(\phi_{h}^{n+1},{\bf A}_h ^{n+1} ) 
\in {\mathbb V}_{h}^{k+1} \times {\mathbb N}_{h}^{k}$, $n=0,1,\cdots,N-1$, satisfying the equations $$\begin{aligned}
&(\eta D_{\tau}\psi_{h}^{n+1}, \varphi_h)  
+ \bigg( \bigg(\frac{i}{\kappa}\nabla+{\bf A}_h^{n+1}\bigg) {\psi}_{h}^{n+1} \,, 
\bigg(\frac{i}{\kappa}\nabla+{\bf A}_h^{n+1}\bigg) \varphi_h\bigg) {\nonumber}\\
&\quad 
+((|\psi_{h}^{n+1}|^{2}-1)\psi_{h}^{n+1}, \varphi_h) 
=- (i\eta \kappa\Theta(\psi_{h}^{n})\phi_{h}^{n} ,\varphi_h) \, , 
&&\forall\, \varphi_h \in {\mathbb S}_{h}^r ,
\label{FEM1} \\[10pt]
& (\phi_h ^{n+1}, \chi_h) 
- ({\bf A}_h^{n+1},\nabla \chi_h ) = 0 \, , 
&&\forall\, \chi_h\in {\mathbb V}_{h}^{k+1}, 
\label{FEM3} \\[10pt]
&(D_{\tau}{\bf A}_{h}^{n+1},{\bf a}_{h}) 
+(\nabla\phi_{h}^{n+1} \, , {\bf a}_{h} )
+ (\nabla\times{\bf A}_h ^{n+1} \, , \nabla\times {\bf a}_{h}) {\nonumber}\\
& 
