\quad
\|\nabla\times {\bf A}_{h,\tau}^-\|_{L^\infty(0,T;L^2)} 
\quad
\mbox{and}\quad 
\|\phi_{h,\tau}^-\|_{L^\infty(0,T;L^2)} ,$$ which are not enough for $\psi_{h,\tau}^-$ and ${\bf A}_{h,\tau}^-$ to be compact and converge in the sense of -.

We shall prove by establishing a discrete Sobolev embedding inequality (Lemma \[SobolevD\]): $$\begin{aligned}
\label{AhL33}
\|{\bf A}_{h}^n\|_{L^{3+\delta}}
\leq C(\|{\bf A}_{h}^n\|_{L^2}
+\|\nabla\times {\bf A}_{h}^n\|_{L^2}+\|\phi_{h}^n\|_{L^2}) ,\end{aligned}$$ and we also need to show that this embedding is compact. Since we allow the domain to be multi-connected, in order to prove , we need to use the discrete Hodge decomposition $$\begin{aligned}
{\bf A}_h^n =
{\bf c}_h + \nabla\theta_h 
+ \sum_{j=1}^{\frak M}\alpha_{j,h}{\bf w}_{j,h}  \end{aligned}$$ and show that the divergence-free part ${\bf c}_h$, the curl-free part $\nabla\theta_h $ and the discrete harmonic part $ \sum_{j=1}^{\frak M}\alpha_{j,h}{\bf w}_{j,h}$ are all bounded in ${\bf L}^{3+\delta}$. For this purpose, we need to construct the basis functions ${\bf w}_{j,h}$, $j=1,\cdots,\frak M$, of the discrete harmonic vector fields and prove that they are bounded in ${\bf L}^{3+\delta}$ (Lemma \[RegDHarV\]).

In order to prove , we rewrite the finite element equation of $\psi_h^{n+1}$ in the form of $$\begin{aligned}
\eta D_\tau\psi_h^{n+1}
-\frac{1}{\kappa^2}\Delta_h\psi_h^{n+1}
=f_h^{n+1}\end{aligned}$$ and prove the following inequality (Lemma 3.8): $$\begin{aligned}
\label{LemW1qDa}
\sum_{n=0}^{N-1}\tau\|\psi_h^{n+1}\|_{W^{1,q+\delta}}^2\leq 
C\sum_{n=0}^{N-1}\tau\|f_h^{n+1}\|_{L^{q/2}}^2
+C\|\psi_h^0\|_{H^1}^2
\quad\mbox{ 
for some $q>3$ and $\delta>0$. }\end{aligned}$$ Then we prove $$\begin{aligned}
\sum_{n=0}^{N-1}\tau\|f_h^{n+1}\|_{L^{q/2}}^2
