edges $e = \overline T\cap \overline{T'}$ (where $T$ and $T'$ are two adjacent triangles from ${\mathcal{F}}_h^\Gamma$).

Let $\mathcal{O}_h$ be anyone of the previously introduced partitions of $\overline\Omega_C\cup \overline \Omega_I$, $\overline\Omega_C$, $\overline \Omega_I$ or $\Gamma$ and let $E$ be a generic element of the given partition. We introduce for any $s\geq 0$ the broken Sobolev spaces $${\mathrm{H}}^s(\mathcal{O}_h) := \prod_{E\in \mathcal{O}_h} {\mathrm{H}}^s(E)\quad \text{and} \quad  \mathbf{H}^s(\mathcal{O}_h) := \prod_{E\in \mathcal{O}_h} \mathbf{H}^s(E)^3\, .$$

For each $w:= {\left\{w_E\right\}}\in {\mathrm{H}}^s(\mathcal{O}_h)$, the components $w_E$ represents the restriction $w|_E$. When no confusion arises, the restrictions will be written without any subscript.

The space ${\mathrm{H}}^s(\mathcal{O}_h)$ is endowed with the Hilbertian norm $${\lVertw\rVert}_{s,\mathcal{O}_h}^2 := \sum_{E\in \mathcal{O}_h} {\lVertw_E\rVert}^2_{s,E}.$$

We consider identical definitions for the norm and the seminorm on the vectorial version $\mathbf{H}^s(\mathcal{O}_h)$. We use the standard conventions ${\mathrm{L}}^2(\mathcal{O}_h):={\mathrm{H}}^0(\mathcal{O}_h)$ and $\mathbf{L}^2(\mathcal{O}_h):=\mathbf{H}^0(\mathcal{O}_h)$ and introduce the bilinear forms $$(w, z)_{\mathcal{O}_h} = \sum_{E\in \mathcal{O}_h} \int_E w_E z_E, \quad \forall w, z \in {\mathrm{L}}^2(\mathcal{O}_h)$$ and $$(\boldsymbol{w}, \boldsymbol{z})_{\mathcal{O}_h} = \sum_{E\in \mathcal{O}_h} \int_E \boldsymbol{w}_E\cdot \boldsymbol{z}_E, \quad \forall \boldsymbol{w}, \boldsymbol{z}\in \mathbf{L}^2(\mathcal{O}_h).$$

Assume that $({\boldsymbol{v}},\varphi,m)\in \mathbf{H}^{1+s}({\mathcal{T}}_h^{\Omega_C})\times {\mathrm{H}}^{1+s}({\mathcal{T}}_h^{\Omega_I})\times {\mathbb C}$, with $s>1/2$. Moreover, let us recall that $\boldsymbol \rho$ has been constructed as a piecewise-linear vector field, therefore its restriction to any face $F$ has a meaning. We define ${\mathbf{curl}}_h{\boldsymbol{v}}\in \mathbf{H}^s({\mathcal{T}}_h^{\Omega_C})$ 