\psi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}=0$, and ${\llbracket \psi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}=0$, it is straightforward to show that $$\begin{gathered}
\label{diff}
A_h(({\boldsymbol{h}}, \psi, k), ({\boldsymbol{v}}, \varphi, m)) - L_h(({\boldsymbol{v}}, \varphi, m)) = \imath \omega \int_{\Omega_{C}} \mu {\boldsymbol{h}}\cdot {\boldsymbol{v}}+ \int_{\Omega_{C}} {\boldsymbol{e}}\cdot {\mathbf{curl}}_h {\boldsymbol{v}}\\
+ \imath \omega \mu_0 \int_{\Omega_I}(\nabla \psi + k\boldsymbol \rho)\cdot (\nabla_h \varphi + m \boldsymbol \rho)
+ \left({\{{\boldsymbol{e}}\}}_{{\mathcal{F}}}, {\llbracket ({\boldsymbol{v}}, \varphi, m) \rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_{C}}}\\ - \imath \omega \mu_0\left({\{\nabla \psi + k \boldsymbol \rho\}}_{{\mathcal{F}}}, {\llbracket \varphi{\boldsymbol{n}}\rrbracket}_{{\mathcal{F}}}\right)_{{\mathcal{F}}_h^{\Omega_I}}
- \left({\{{\boldsymbol{e}}\}}_{{\mathcal{E}}}, {\llbracket \varphi{\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\right)_{{\mathcal{E}}_h}.\end{gathered}$$ Integrating by parts in each $K\in {\mathcal{T}}_h^{\Omega_{C}}$ and using yield $$\begin{gathered}
\label{GreenOmegaC}
\int_{\Omega_{C}} {\boldsymbol{e}}\cdot {\mathbf{curl}}_h {\boldsymbol{v}}= \sum_{K\in {\mathcal{T}}_h^{\Omega_{C}}} 
\int_K {\mathbf{curl}}\, {\boldsymbol{e}}\cdot {\boldsymbol{v}}-
\sum_{K\in {\mathcal{T}}_h^{\Omega_{C}}} \int_{\partial K} {\boldsymbol{e}}\cdot {\boldsymbol{v}}\times {\boldsymbol{n}}_K \\
= -\imath\omega \int_{\Omega_{C}} \mu{\boldsymbol{h}}\cdot {\boldsymbol{v}}- \sum_{F\in {\mathcal{F}}^0_h(\Omega_{C})}
\int_{F} {\{{\boldsymbol{e}}\}}_F \cdot {\llbracket {\boldsymbol{v}}\times {\boldsymbol{n}}\rrbracket}_F -
\sum_{T\in {\mathcal{F}}_h^\Gamma}
\int_{T} {\boldsymbol{e}}\cdot {\boldsymbol{v}}\times {\boldsymbol{n}}.\end{gathered}$$ Similarly, integrating by parts in each $K\in {\mathcal{T}}_h^{\Omega_I}$ together with and give $$\begin{gathered}
\label{GreenOmegaI}
\imath \omega \mu_{0}\int_{\Omega_I}(\nabla \psi + k\boldsymbol \rho)\cdot (\nabla_h \varphi + m \boldsymbol \rho) = - \imath \omega \mu_{0}\sum_{K\in {\mathcal{T}}_h^{\Omega_I}} \int_K {\mathrm{div}}(\nabla \psi + k \boldsymbol \rho) \varphi \\+\imath \omega \mu_{0}
\sum_{K\in {\mathcal{T}}_h^{\Omega_I}} \int_{\partial K} (\nabla \psi + k\boldsymbol \rho) \cdot  {\boldsymbol{n}}_K \varphi
+ m \int_{\Omega_{I}} (\nabla \psi + k\boldsymbol \rho)\cdot \boldsymbol \rho
= \imath \omega \mu_{0} \sum_{F\in {\mathcal{F}}_{h}^0} \int_{F} {\{\nabla \psi + k \boldsymbol \rho\}}_{F}
\cdot 