\\
&{\bf B}_{h,\tau}^+(t)={\bf B}_h^{n+1}:=
\nabla\times{\bf A}_{h}^{n+1},
&&{\bf E}_{h,\tau}^+(t)={\bf E}_h^{n+1}
:=-D_\tau{\bf A}_h^{n+1}-\nabla\phi_h^{n+1}  .
\label{Eht1} \end{aligned}$$

In this paper we prove the following theorem.

\[MainTHM\] [*Under Assumption [2.1]{}, for any given $\psi_0\in {\cal H}^1$ and ${\bf A}_0\in {\bf H}({\rm curl,div})$ such that $|\psi_0|\leq 1$, the system - has a unique finite element solution when $\tau<\eta$ $(\eta$ is the parameter in , which converges to the unique solution of - as $\tau,h\rightarrow 0$ in the following sense: $$\begin{aligned}
&\psi_{h,\tau}^+\rightarrow \psi 
&&\mbox{strongly in\, $L^\infty(0,T;{\cal L}^2)$
and weakly$^*$\, in\, $L^\infty(0,T;{\cal H}^1)$ }, \\ 
&{\bf A}_{h,\tau}^+\rightarrow {\bf A}
&& \mbox{strongly in\, $L^\infty(0,T;{\bf L}^2)$ 
and weakly$^*$ in\, $L^\infty(0,T;{\bf H}({\rm curl}))$} , \\
&\phi_{h,\tau}^+\rightarrow \phi 
&&\mbox{strongly\,\,\, in\, $L^2(0,T;L^2)$ 
and weakly$^*$ in\, $L^\infty(0,T;L^2)$}, \\ 
&{\bf B}_{h,\tau}^+\rightarrow {\bf B}
&&\mbox{weakly$^*$\, in\, $L^\infty(0,T;{\bf L}^2)$},\\
&{\bf E}_{h,\tau}^+\rightarrow {\bf E}
&&\mbox{weakly\,\,\, in\, $L^2(0,T;{\bf L}^2)$} . \end{aligned}$$* ]{}

In the meantime of proving Theorem \[MainTHM\], we also obtain global well-posedness of the PDE problem - (see Appendix).

If $\Omega$ is a curved polygon in $\R^2$ and the external magnetic field ${\bf H}$ is perpendicular to the domain, i.e. ${\bf H}=(0,0,H)$, then - hold when ${\bf H}$ is replaced by $H$, with the following two-dimensional notations: $$\begin{aligned}
&\nabla\times {\bf A}
=\frac{\partial A_2}{\partial x_1}-\frac{\partial A_1}{\partial x_2},
