are used to prove compactness and convergence of the finite element solution.

Main theorem
------------

Let $W^{s,p}$, $s\geq 0$ and $1\leq p\leq\infty$, be the conventional Sobolev spaces of real-valued functions defined on $\Omega$, and let ${\bf W}^{s,p}=W^{s,p}\times W^{s,p}\times W^{s,p}$ be the corresponding Sobolev space of vector fields. The case of integer $s$ can be found in [@Adams], and the characterization of more general function spaces with fractional $s$ can be found in [@Rychkov]. Let ${\cal W}^{s,p}:=W^{s,p}+i\,W^{s,p}$ denote the complex-valued Sobolev space and define the abbreviations $$\begin{aligned}
&L^p:=W^{0,p},\quad\,  
{\cal L}^p:={\cal W}^{0,p},\quad\, 
{\bf L}^p:={\bf W}^{0,p} ,
\quad\mbox{for $1\leq p\leq\infty$,}\\
&H^s:=W^{s,2},\quad 
{\cal H}^s:={\cal W}^{s,2},\quad 
{\bf H}^s:={\bf W}^{s,2},
\quad\mbox{for $s\geq 0$} . \end{aligned}$$ Moreover, we define $$\begin{aligned}
&{\bf H}({\rm curl}):=
\{{\bf u}\in {\bf L}^2:\,
\nabla\times {\bf u}\in {\bf L}^2\}  , \\
& {\bf H}({\rm div})\,\,:=
\{{\bf u}\in {\bf L}^2:\,
\nabla\cdot {\bf u}\in L^2 \}  ,\\
&{\bf H}({\rm curl,div}):=
\{{\bf u}\in {\bf L}^2:\,
\nabla\times {\bf u}\in {\bf L}^2,
\,\,\,\nabla\cdot {\bf u}\in L^2 \,\,\,\mbox{and}\,\,\, 
{\bf u}\cdot{\bf n}=0\,\,\mbox{on}\,\,\partial\Omega\} . \label{Def_Hcd_bd1}
$$

Let $\psi_{h,\tau}^+$, $\psi_{h,\tau}^-$, ${\bf A}_{h,\tau}^+$, ${\bf A}_{h,\tau}^-$, ${\bf B}_{h,\tau}^+$ and ${\bf E}_{h,\tau}^+$ be the piecewise constant functions on $(0,T]$ such that on each subinterval $(t_n,t_{n+1}]$ $$\begin{aligned}
&\psi_{h,\tau}^+(t)=\psi_h^{n+1},  
&&\psi_{h,\tau}^-(t)=\psi_h^{n},  \\
&{\bf A}_{h,\tau}^+(t)={\bf A}_h^{n+1},   
&&{\bf A}_{h,\tau}^-(t)={\bf A}_h^{n} ,  