with its usual Hilbertian norm ${\lVert{\boldsymbol{v}}\rVert}_{\mathbf{H}({\mathbf{curl}}, \mathcal O)}^2:=
{\lVert{\boldsymbol{v}}\rVert}_{0, \mathcal O}^2 + {\lVert{\mathbf{curl}}\, {\boldsymbol{v}}\rVert}_{0, \mathcal O}^2$.

We consider a sequence $\{{\mathcal{T}}_h\}_h$ of conforming and shape-regular triangulations of $\overline \Omega_C \cup \overline \Omega_I$. We assume that each partition ${\mathcal{T}}_h$ consists of tetrahedra $K$ of diameter $h_K$ and unit outward normal to $\partial K$ denoted ${\boldsymbol{n}}_K$. We also assume that for all $K\in {\mathcal{T}}_h$ we have either $K\subset \overline\Omega_C$ or $K\subset \overline \Omega_I$ and denote $${\mathcal{T}}_h^{\Omega_C}:= {\left\{K\in {\mathcal{T}}_h;\quad K\subset \overline\Omega_C\right\}},\qquad {\mathcal{T}}_h^{\Omega_I}:= {\left\{K\in {\mathcal{T}}_h;\quad K\subset \overline\Omega_I\right\}}.$$ We also assume that the meshes $\{{\mathcal{T}}_h^{\Omega_C}\}_h$ are aligned with the discontinuities of the coefficients $\sigma$ and $\mu$. The parameter $h:= \max_{K\in {\mathcal{T}}_h} \{h_K\}$ represents the mesh size.

We denote by ${\mathcal{F}}_h^0(\Omega_C)$ and ${\mathcal{F}}_h^0(\Omega_I)$ the sets of interior faces of the triangulations ${\mathcal{T}}_h^{\Omega_C}$ and ${\mathcal{T}}_h^{\Omega_I}$ respectively. We also introduce the sets of boundary faces $${\mathcal{F}}_h^\Gamma:= {\left\{F = \overline K\cap\overline{K'};\quad  K\in {\mathcal{T}}_h^{\Omega_C},\,\, K'\in {\mathcal{T}}_h^{\Omega_I}\right\}}
\quad 
\text{and}
\quad
{\mathcal{F}}_h^\Sigma:= {\left\{F = \partial K \cap \Sigma;\quad K\in {\mathcal{T}}_h^{\Omega_I}\right\}}$$ and consider $${\mathcal{F}}_h^{\Omega_C}:={\mathcal{F}}_h^0(\Omega_C) \cup  {\mathcal{F}}_h^\Gamma, \quad {\mathcal{F}}_h^{\Omega_I}:={\mathcal{F}}_h^0(\Omega_I) \cup  {\mathcal{F}}_h^\Sigma
\quad \text{and} \quad {\mathcal{F}}_h := {\mathcal{F}}_h^{\Omega_C}\cup {\mathcal{F}}_h^{\Omega_I}.$$ We notice that ${\left\{{\mathcal{F}}_h^\Gamma\right\}}_h$ is a shape regular family of triangulations of $\Gamma$ into triangles $T$ of diameter $h_T$. Finally, we consider the set ${\mathcal{E}}_h$ of 