+ A_h^{\Omega_I}(({\boldsymbol{u}}, \phi, c), ({\boldsymbol{v}}, \varphi, m))\, .$$

Let us assume that $\sigma^{-1} {\boldsymbol{j}}\in \mathbf{H}^{1/2+s}({\mathcal{T}}_h^{\Omega_C})$ with $s>1/2$. Then we can define the linear form $L_h(\cdot)$ on $\mathbf{X}^s({\mathcal{T}}_h^{\Omega_C}) \times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I}) \times {\mathbb C}$ by $$L_h(({\boldsymbol{v}}, \varphi)) := (\sigma^{-1} {\boldsymbol{j}}, {\mathbf{curl}}_h {\boldsymbol{v}})_{{\mathcal{T}}_h^{\Omega_C}} + \left( {\{\sigma^{-1} {\boldsymbol{j}}\}}_{{\mathcal{F}}}, {\llbracket ({\boldsymbol{v}}, \varphi, m) \rrbracket}_{{\mathcal{F}}} \right)_{{\mathcal{F}}_h^{\Omega_C}}
- \left({\{\sigma^{-1} {\boldsymbol{j}}\}}_{{\mathcal{E}}}, {\llbracket \varphi {\boldsymbol{t}}\rrbracket}_{{\mathcal{E}}}\right)_{{\mathcal{E}}_h}.$$

We propose the following DG formulation of problem -: $$\label{DG-FEM}
\begin{array}{l}
\text{Find $({\boldsymbol{h}}_{h}, \psi_h,k_h)\in \mathbf{X}_h\times V_h\times {\mathbb C}$ such that,}\\[2ex]
A_h(({\boldsymbol{h}}_{h}, \psi_h,k_h), ({\boldsymbol{v}}, \varphi, m)) = L_h(({\boldsymbol{v}}, \varphi, m))\quad \forall \,  ({\boldsymbol{v}}, \varphi, m)\in \mathbf{X}_h\times V_h\times {\mathbb C}\, .
\end{array}$$

The existence and uniqueness of the solution of this problem is proved in Theorem \[LM\]

We end this section by showing that the DG scheme is consistent.

\[consistency0\] Let $({\boldsymbol{h}}, \psi, k)\in \mathbf{H}({\mathbf{curl}}, {\Omega_C})\times {\mathrm{H}}^1(\Omega_I)\times {\mathbb C}$ be the solution of -. Under the assumption $\sigma^{-1} {\boldsymbol{j}}\in \mathbf{H}^{1/2+s}({\mathcal{T}}_h^{\Omega_C})$ and the regularity conditions $({\boldsymbol{h}},\psi, k) \in \mathbf{X}^s({\mathcal{T}}_h^{\Omega_C})\times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I})\times {\mathbb C}$, with $s>1/2$, we have that $$A_h(({\boldsymbol{h}}, \psi, k), ({\boldsymbol{v}}, \varphi, m)) = L_h(({\boldsymbol{v}}, \varphi, m)) \quad
    \forall \, ({\boldsymbol{v}},\varphi, m)\in \mathbf{X}_h\times V_h \times {\mathbb C}.$$

Using again the notation ${\boldsymbol{e}}= \sigma^{-1} ( {\mathbf{curl}}\, {\boldsymbol{h}}- {\boldsymbol{j}})$ and taking into account that ${\llbracket ({\boldsymbol{h}},\psi,k) \rrbracket}_{{\mathcal{F}}} =0$, ${\llbracket 