{\mathcal{F}}(K)} h_F {\lVert \mathtt{s}_F^{1/2}\sigma_K^{-1}\mathbf{w}_K \rVert}^2_{0,F}
\leq  \sum_{K\in {\mathcal{T}}_h^{\Omega_C}}  h_K 
{\lVert \sigma_K^{-1/2}\mathbf{w}_K \rVert}^2_{0,\partial K}.\end{gathered}$$

Similarly, $$\begin{gathered}
\label{transfer}
{\lVert \mathtt{s}_{{\mathcal{E}}}^{1/2} h_{{\mathcal{E}}} {\{\sigma^{-1} \mathbf{w}\}}_{{\mathcal{E}}} \rVert}^2 _{0,{\mathcal{E}}_h} =  
\sum_{e \in {\mathcal{E}}_h} h_e^2 {\lVert\mathtt{s}_{e}^{1/2} {\{\sigma^{-1} \mathbf{w}\}}_e \rVert}^2_{0,e} \\
\leq \sum_{T\in {\mathcal{F}}_h^\Gamma} \sum_{e\in {\mathcal{E}}(T)}  h_e^2 {\lVert\mathtt{s}_{e}^{1/2} \sigma_{K_T}^{-1} 
\mathbf{w}_{K_T}\rVert}^2_{0,e}
\leq \sum_{T\in {\mathcal{F}}_h^\Gamma }
  h_{T}^2 {\lVert\sigma_{K_T}^{-1/2} \mathbf{w}_{K_T}\rVert}^2_{0,\partial T}\, ,\end{gathered}$$ where $K_T\in {\mathcal{T}}_h^{\Omega_C}$ is such that $T=\partial K_T \cap \Gamma$. It follows from that $$\begin{gathered}
{\lVert\mathtt{s}_{{\mathcal{E}}}^{1/2} h_{{\mathcal{E}}} {\{\sigma^{-1} \mathbf{w}\}}_{{\mathcal{E}}}\rVert}^2 _{0,{\mathcal{E}}_h^{\Omega_I}} \leq C^*
\sum_{T \in {\mathcal{F}}_h^\Gamma}  h_{T} {\lVert\sigma_{K_T}^{-1/2} \mathbf{w}_{K_T}\rVert}^2_{0, T} \leq C^*
\sum_{K \in{\mathcal{T}}^{\Omega_C}_h}  h_{K} {\lVert\sigma_K^{-1/2} \mathbf{w}_K\rVert}^2_{0, \partial K} \end{gathered}$$ and follows by applying again the discrete trace inequality in the last estimate and in . Finally, for any $\mathbf{w} \in \prod_{K\in {\mathcal{T}}_h^{\Omega_I}}{\mathcal{P}}_k(K)^3$, $$\label{transfer1}
{\lVerth_{{\mathcal{F}}}^{1/2} {\{\mathbf{w}\}}_{{\mathcal{F}}}\rVert}^2_{0,{\mathcal{F}}_h^{\Omega_I}} = \sum_{F\in {\mathcal{F}}_h^{\Omega_I}} h_F {\lVert{\{\mathbf{w}\}}_F\rVert}_{0,F}^2 
\leq  \sum_{K\in {\mathcal{T}}_h^{\Omega_I}} h_K {\lVert \mathbf{w}_K \rVert}^2_{0,\partial K}$$ and follows again from .

\[boundedness\] There exists a constant $M>0$ independent of $h$ such that $$| A_h(({\boldsymbol{u}}, \phi, c), ({\boldsymbol{v}}, \varphi, m)) | \leq M {\lVert({\boldsymbol{u}}, \phi, c)\rVert}_* {\lVert({\boldsymbol{v}},\varphi, m)\rVert}$$ for all $({\boldsymbol{u}}, \phi,c)$, $({\boldsymbol{v}},\varphi,m)\in \mathbf{X}^s({\mathcal{T}}_h^{\Omega_C})\times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I})\times {\mathbb C}$, with $s>1/2$.

By the Cauchy-Schwarz inequality, we have that $$\begin{array}{l}
|A_h^{\Omega_C}(({\boldsymbol{u}}, \phi,c), ({\boldsymbol{v}}, \varphi,m))| \\[.1cm]
\qquad \leq \omega {\lVert\mu^{1/2}{\boldsymbol{u}}\rVert}_{0,\Omega_C} {\lVert\mu^{1/2}{\boldsymbol{v}}\rVert}_{0,\Omega_C} + 
{\lVert\sigma^{-1/2}{\mathbf{curl}}_h {\boldsymbol{u}}\rVert}_{0,\Omega_C} {\lVert\sigma^{-1/2}{\mathbf{curl}}_h {\boldsymbol{v}}\rVert}_{0,\Omega_C}\\[.1cm] 
\qquad + {\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2} h_{{\mathcal{F}}}^{1/2} {\{\sigma^{-1}{\mathbf{curl}}_h {\boldsymbol{u}}\}}_{{\mathcal{F}}}\rVert}_{0, 