can prove compactness and convergence of a subsequence of $\partial_t\psi_{h,\tau}$, $\psi_{h,\tau}^\pm$, $\partial_t{\bf A}_{h,\tau}$, ${\bf A}_{h,\tau}^\pm$, $\phi_{h,\tau}^\pm$, and prove that the limits of any subsequence coincide with the PDE’s solution, then we can conclude that the sequences $\psi_{h,\tau}^+$, and ${\bf A}_{h,\tau}^+$ converge to the PDE’s solution as $h,\tau\rightarrow 0$.

To estimate the finite element solution (in order to prove the compactness), we introduce a discrete energy function ${\cal G}_{h}^{n} $ and a special time-stepping scheme from which one can derive . By proving , we derive boundedness of the discrete energy from (via the discrete Gronwall’s inequality). Based on the boundedness of the discrete energy, some further estimates need to be derived in order to prove convergence of the finite element solution. For example, in order to prove the weak convergence of a subsequence of $${\rm Re}\bigg[\overline\psi_{h,\tau}^- \bigg(\frac{i}{\kappa} \nabla 
+{\bf A}_{h,\tau}^-\bigg) \psi_{h,\tau}^-\bigg]$$ in $L^2(0,T;L^2)$, we need to prove the following convergence (for a subsequence): $$\begin{aligned}
&\mbox{$\psi_{h,\tau}^-$ converges 
weakly in $L^2(0,T;W^{1,3+\delta})\hookrightarrow L^2(0,T;L^\infty)$
for some $\delta>0$}, \label{psihnL3}\\
&\mbox{$\psi_{h,\tau}^-$\,\, converges strongly in $L^\infty(0,T;L^{6-\epsilon})$ for arbitrarily small $\epsilon>0$}\\
&\mbox{${\bf A}_{h,\tau}^-$\, converges strongly in $L^\infty(0,T;L^{3+\delta})$
for some $\delta>0$} . \label{AAhnL3}\end{aligned}$$ The boundedness of the discrete energy only implies the boundedness of $$\|\psi_{h,\tau}^-\|_{L^\infty(0,T;H^1)} , \quad
\|{\bf A}_{h,\tau}^-\|_{L^\infty(0,T;L^2)}, 