N}_h^k$ the nonlinear operator ${\mathcal M}:  {\mathbb S}_{h}^r
\rightarrow  {\mathbb S}_{h}^r$ defined via duality by $$\begin{aligned}
({\mathcal M}{\mathscr S}_h,\varphi_h)
:&= \frac{\eta}{\tau}({\mathscr S}_h, \varphi_h)  
+ \bigg( \bigg(\frac{i}{\kappa}\nabla+{\bf A}_h^{n+1}\bigg) {\mathscr S}_h \,, 
\bigg(\frac{i}{\kappa}\nabla+{\bf A}_h^{n+1}\bigg) \varphi_h\bigg) {\nonumber}\\
&\quad 
+((|{\mathscr S}_h|^{2}-1){\mathscr S}_h, \varphi_h) , 
&&\forall\, \varphi_h \in {\mathbb S}_{h}^r ,\end{aligned}$$ is continuous and monotone, i.e.[^4] $$\begin{aligned}
({\mathcal M}{\mathscr S}_h
-{\mathcal M}\widetilde{\mathscr S}_h,{\mathscr S}_h-\widetilde{\mathscr S}_h)
\ge \bigg(\frac{\eta}{\tau}-1\bigg)\|{\mathscr S}_h-\widetilde{\mathscr S}_h\|_{L^2}^2 \, , 
&&\forall\, {\mathscr S}_h,\widetilde{\mathscr S}_h \in {\mathbb S}_{h}^r .\end{aligned}$$ Hence, [@Showalter Lemma 2.1 and Corollary 2.2 of Chapter 2] implies that for any given $f_h\in {\mathbb S}_{h}^r$ the equation ${\mathcal M}{\mathscr S}_h=f_h$ has a solution ${\mathscr S}_h\in {\mathbb S}_{h}^r$. In other words, equation has a solution $\psi_h^{n+1}\in {\mathbb S}_{h}^r$. The uniqueness of the solution $\psi_h^{n+1}\in {\mathbb S}_{h}^r$ is an obvious consequence of the monotonicity of the operator ${\mathcal M}$.

Overall, for any given $(\psi_h^{n},{\bf A}_h^{n})\in {\mathbb S}_{h}^{r} \times {\mathbb N}_{h}^{k}$, the system - has a unique solution $(\psi_h^{n+1},\phi_h^{n+1},{\bf A}_h^{n+1})\in {\mathbb S}_{h}^r\times {\mathbb V}_{h}^{k+1} \times {\mathbb N}_{h}^{k}$ when $\tau<\eta$. In the rest part of this paper, we prove the convergence of the finite element solution. Some frequently used basic lemmas are listed in Section \[PreLem\].

Preliminary lemmas {#PreLem}
------------------

The following lemma is concerned with the 