{\mathcal{F}}_h^{\Omega_C}}
 {\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2} h_{{\mathcal{F}}}^{-1/2} {\llbracket ({\boldsymbol{v}}, \varphi,m) \rrbracket}_{{\mathcal{F}}}\rVert}_{0, {\mathcal{F}}_h^{\Omega_C}}\\[.1cm]
\qquad + {\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2} h_{{\mathcal{F}}}^{1/2} {\{\sigma^{-1}{\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{F}}}\rVert}_{0, {\mathcal{F}}_h^{\Omega_C}}
 {\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2} h_{{\mathcal{F}}}^{-1/2} {\llbracket ({\boldsymbol{u}}, \phi,c) \rrbracket}_{{\mathcal{F}}}\rVert}_{0, {\mathcal{F}}_h^{\Omega_C}}\\[.1cm]
 \qquad + \mathtt{a}^{\Omega_C} {\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2}h_{{\mathcal{F}}}^{-1/2} {\llbracket ({\boldsymbol{u}}, \phi,c) \rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}} {\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2}h_{{\mathcal{F}}}^{-1/2} {\llbracket ({\boldsymbol{v}}, \varphi,m) \rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_C}}.
\end{array}$$ Applying with $\mathbf{w} = {\mathbf{curl}}_h {\boldsymbol{v}}$ we obtain $$|A_h^{\Omega_C}(({\boldsymbol{u}}, \phi,c), ({\boldsymbol{v}}, \varphi, m))| \leq (1+ C_\Omega + \mathtt{a}^{\Omega_C})\,  {\lVert({\boldsymbol{u}}, \phi,c)\rVert}_* {\lVert({\boldsymbol{v}},\varphi,m)\rVert}$$ for all $({\boldsymbol{u}}, \phi,c)$ and $({\boldsymbol{v}},\varphi,m)\in \mathbf{X}^s({\mathcal{T}}_h^\Omega)\times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I}) \times {\mathbb C}$. On the other hand, $$\begin{array}{l}
|A_h^{\Omega_I}(({\boldsymbol{u}}, \phi, c), ({\boldsymbol{v}}, \varphi,m))| \leq \omega \mu_0 {\lVert\nabla_h \phi+c \boldsymbol \rho\rVert}_{0,\Omega_I} {\lVert\nabla_h \varphi + m \boldsymbol \rho\rVert}_{0,\Omega_I} \\[.1cm]
\quad\qquad+ \omega \mu_0 {\lVerth_{{\mathcal{F}}}^{1/2} {\{\nabla_h \varphi+m \boldsymbol \rho\}}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_I}} {\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket \phi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_I}} \\[.1cm]
\quad\qquad+ \omega \mu_0 {\lVerth_{{\mathcal{F}}}^{1/2} {\{\nabla_h \phi+c \boldsymbol \rho\}}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_I}} {\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_I}}\\[.1cm]
\quad\qquad+ \alpha {\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2}h_{{\mathcal{E}}}^{-1} {\llbracket \phi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h} {\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2}h_{{\mathcal{E}}}^{-1} {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h}\\[.1cm]
\quad\qquad+  {\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2}h_{{\mathcal{E}}}{\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{v}}\}}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h} {\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2}h_{{\mathcal{E}}}^{-1} {\llbracket \phi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h}\\[.1cm]
\quad\qquad+ {\lVert\mathtt{s}_{{\mathcal{F}}}^{1/2}h_{{\mathcal{E}}}{\{\sigma^{-1} {\mathbf{curl}}_h {\boldsymbol{u}}\}}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h} {\lVert\mathtt{s}_{{\mathcal{F}}}^{-1/2}h_{{\mathcal{E}}}^{-1} {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\rVert}_{0,{\mathcal{E}}_h}\\[.1cm]
\quad\qquad+\mathtt{a}^{\Omega_I}{\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket \phi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h^{\Omega_I}} {\lVerth_{{\mathcal{F}}}^{-1/2} {\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\rVert}_{0,{\mathcal{F}}_h}
\end{array}$$ and it follows from (applied with $\mathbf{w}= \nabla_h \varphi+ m \boldsymbol \rho$ ) and (applied with $\mathbf{w}= {\mathbf{curl}}_h {\boldsymbol{v}}$) that $$|A_h^{\Omega_I}(({\boldsymbol{u}}, \phi,c), ({\boldsymbol{v}}, \varphi, m))| \leq (1 + C_{\Omega_I} + C_{\Omega}+ \mathtt{a}^{\Omega_I}+ \alpha) \,  {\lVert({\boldsymbol{u}}, \phi,c)\rVert}_* {\lVert({\boldsymbol{v}},\varphi,m)\rVert},$$ which gives the result.

\[discElip\] There exists a constant $\alpha_0>0$ independent of the mesh size and the coefficients such 