{\lVert{\boldsymbol{u}}\rVert}_{s, {\mathcal{T}}_h^{\Omega_C}} + {\lVert{\mathbf{curl}}_h {\boldsymbol{u}}\rVert}_{s, {\mathcal{T}}_h^{\Omega_C}} \big).$$

We introduce $
\mathbf{L}^2_t(\Gamma) = {\left\{\boldsymbol{\varphi}\in \mathbf{L}^2(\Gamma);\,\,  \boldsymbol{\varphi}\cdot {\boldsymbol{n}}= 0\right\}}
$ and consider the $m$-order order Brezzi-Douglas-Marini (BDM) finite element approximation of the space $$\mathbf{H}(\text{div}_\Gamma, \Gamma) := {\left\{ \boldsymbol{\varphi}\in \mathbf{L}_t^2(\Gamma); \quad 
\text{div}_\Gamma \boldsymbol{\varphi}\in L^2(\Gamma) \right\}}$$ relatively to the mesh ${\mathcal{F}}_h^\Gamma$ (see, e.g. [@Boffi]). It is given by $$\mathcal{\mathbf{BDM}}({\mathcal{F}}_h^\Gamma) = {\left\{\boldsymbol{\varphi}\in  \mathbf{H}(\text{div}_\Gamma, \Gamma); \quad 
\boldsymbol{\varphi}|_T \in {\mathcal{P}}_m(T)^2,\quad \forall T\in {\mathcal{F}}_h^\Gamma\right\}}.$$ The corresponding interpolation operator $\Pi_{h,m}^{\text{BDM}}$ is bounded on $
\mathbf{H}(\text{div}_\Gamma, \Gamma)\cap \prod_{T\in {\mathcal{F}}_h^\Gamma} {\mathrm{H}}^\delta(T)^2$ for all $\delta>0$ and we recall that it is uniquely characterized on each $T\in {\mathcal{F}}_h^\Gamma$ by the conditions $$\label{BDMfreedom1}
\int_e \Pi_{h,m}^{\text{BDM}} \boldsymbol{\varphi}\cdot {\boldsymbol{n}}_T q = \int_e \boldsymbol{\varphi}\cdot {\boldsymbol{n}}_T q\quad 
\forall q \in {\mathcal{P}}_m(e),\quad \forall e\in {\mathcal{E}}(T),$$ $$\label{BDMfreedom2}
\int_T \Pi_{h,m}^{\text{BDM}} \boldsymbol{\varphi}\cdot  \mathbf{q} = \int_T \boldsymbol{\varphi}\cdot  \mathbf{q}\quad 
\forall \mathbf{q} \in {\mathcal{P}}_{m-2}(T)^2 + \mathbf{S}_{m-1}(T),$$ where $\mathbf{S}_{m-1}(T):= {\left\{\mathbf{q}\in \tilde{\mathcal{P}}_{m-1}(T)^2;\quad \mathbf{q}\cdot \begin{pmatrix}x_1\\x_2
\end{pmatrix} = 0\right\}}$ with $\tilde{\mathcal{P}}_{m-1}(T)$ representing the set of homogeneous polynomials of degree $m-1$ and $\begin{pmatrix}x_1\\x_2\end{pmatrix}$ being the local variable on the plane containing $T$.

The commuting diagram property $$\begin{aligned}
\label{commuting2} ({\boldsymbol{\Pi}}_{h,m}^{\text{curl}} {\boldsymbol{u}}) \times {\boldsymbol{n}}_\Gamma &= \Pi_{h,m}^{\text{BDM}} ({\boldsymbol{u}}\times {\boldsymbol{n}}_\Gamma)\end{aligned}$$ holds true for all ${\boldsymbol{u}}\in \mathbf{H}({\mathbf{curl}}, \Omega_C)\cap \mathbf{H}^s({\mathbf{curl}}, {\mathcal{T}}_h^{\Omega_C})$, $s>1/2$, see [@Hiptmair1 section 9] for more details.

For all $K\in {\mathcal{T}}_h^{\Omega_I}$ we define the 