$\mathbf{X}^s({\mathcal{T}}_h^{\Omega_C})\times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I})\times {\mathbb C}$ by $$\begin{aligned}
{\lVert({\boldsymbol{v}}, \varphi,m)\rVert}^2 := & {\lVert(\omega\mu)^{1/2}{\boldsymbol{v}}\rVert}^2_{0,\Omega_C} + {\lVert\sigma^{-1/2}{\mathbf{curl}}_h {\boldsymbol{v}}\rVert}^2_{0,\Omega_C} + \omega\mu_0 {\lVert\nabla_h \varphi+ m \boldsymbol \rho\rVert}^2_{0,\Omega_I} \\
+ & {\lVert\mathtt{s}^{-1/2}_{{\mathcal{F}}}h_{{\mathcal{F}}}^{-1/2}{\llbracket ({\boldsymbol{v}},\varphi,m)  \rrbracket}_{{\mathcal{F}}}\rVert}^2_{0,{\mathcal{F}}_h^{\Omega_C}} + \omega\mu_0 {\lVerth_{{\mathcal{F}}}^{-1/2}{\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\rVert}^2_{0,{\mathcal{F}}_h^{\Omega_I}}\\
+ &{\lVert\mathtt{s}^{-1/2}_{{\mathcal{E}}} h_{{\mathcal{E}}}^{-1}{\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}^2_{0,{\mathcal{E}}_h}.\end{aligned}$$ We also need to introduce $$\begin{gathered}
{\lVert({\boldsymbol{v}}, \varphi,m)\rVert}_{\ast}^2 :=  {\lVert({\boldsymbol{v}}, \varphi,m)\rVert}^2 + 
{\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2} h_{{\mathcal{F}}}^{1/2} {\{\sigma^{-1}{\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{F}}}\rVert}^2_{0,{\mathcal{F}}_h^{\Omega_C}}\\
+ {\lVert\mathtt{s}_{{\mathcal{E}}}^{1/2} h_{{\mathcal{E}}} {\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{E}}}\rVert}^2_{0,{\mathcal{E}}_h}
+{\lVerth_{{\mathcal{F}}}^{1/2} {\{\nabla_h \varphi + m \boldsymbol \rho\}}_{{\mathcal{F}}}\rVert}^2_{0,{\mathcal{F}}_h^{\Omega_I}}.\end{gathered}$$

The following discrete trace inequality is standard, (see, e.g. [@DiPietroErn Lemma 1.46]).

For all integer $k\geq 0$ there exists a constant $C^*>0$ independent of $h$ such that, $$\label{discreteTrace3D} 
  h_Q {\lVert v\rVert}^2_{0,\partial Q} \leq C^* {\lVert v\rVert}^2_{0,Q} \quad \forall \, v\in {\mathcal{P}}_k(Q),\quad 
  \forall Q\in \{{\mathcal{T}}_h,{\mathcal{F}}_h^\Gamma\}.$$

It is used to prove the following auxiliary result.

\[equivalence\] For all $k\geq 0$, there exist constants $C_{\Omega_C}>0$ and $C_{\Omega_I}>0$ independent of the mesh size and the coefficients such that $$\label{discIneq1}
{\lVert\mathtt{s}_{{\mathcal{E}}}^{1/2} h_{{\mathcal{E}}} {\{\sigma^{-1} \mathbf{w}\}}_{{\mathcal{E}}}\rVert} _{0,{\mathcal{E}}_h} +
{\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2} h_{{\mathcal{F}}}^{1/2} {\{\sigma^{-1}\mathbf{w}\}}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}} \leq C_{\Omega_C} {\lVert\sigma^{-1/2} \mathbf{w}\rVert}_{0,{\Omega_C}}\, ,$$ for all $\mathbf{w} \in \prod_{K\in {\mathcal{T}}_h^{\Omega_C}}{\mathcal{P}}_k(K)^3$, and $$\label{discIneq2}
{\lVerth_{{\mathcal{F}}}^{1/2} {\{\mathbf{w}\}}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_I}} \leq C_{\Omega_I} {\lVert\mathbf{w}\rVert}_{0,\Omega_I} \, ,$$ for all $\mathbf{w} \in \prod_{K\in {\mathcal{T}}_h^{\Omega_I}}{\mathcal{P}}_k(K)^3$.

By definition of $\mathtt{s}_{{\mathcal{F}}}$, for any $\mathbf{w} \in \prod_{K\in {\mathcal{T}}_h^{\Omega_C}}{\mathcal{P}}_k(K)^3$, $$\begin{gathered}
\label{transfer0}
{\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2} h_{{\mathcal{F}}}^{1/2} {\{\sigma^{-1}\mathbf{w}\}}_{{\mathcal{F}}}\rVert}^2_{0,{\mathcal{F}}_h^{\Omega_C}} =  
\sum_{F\in {\mathcal{F}}_h^{\Omega_C}} h_F {\lVert \mathtt{s}_F^{1/2}{\{\sigma^{-1}\mathbf{w}\}}_F \rVert}^2_{0,F}\\ 
\leq \sum_{K\in {\mathcal{T}}_h^{\Omega_C}} \sum_{F\in 