All references refer to "Finite Element Methods for Maxwell's Equations" by Monk.
Preliminaries: Let $\Omega\subset \mathbb{R}^3$ be a bounded lipschitz domain. The space $H(\text{curl})$ is defined as the set of all $u\in L^2(\Omega)^3$ such that the distributional curl is an element of $L^2(\Omega)^3$. We then have a linear and bounded tangential trace operator $$\gamma_t:H(\text{curl}) \to H^{-\frac{1}{2}}(\partial \Omega)^3$$ (see Theorem 3.29). $L^2(\partial \Omega)^3$ can be regarded as a subset of $H^{-\frac{1}{2}}(\partial \Omega)^3$. If we have $\gamma_t(u)\in L^2(\partial \Omega)^3$ for $u\in H(\text{curl})$, then we will informally write $\gamma_t(u) = \nu \times u$.
Let $H_\text{imp}$ denote the space of all functions $u\in H(\text{curl})$ such that $\gamma_t(u)\in L^2(\partial \Omega)^3$ and $\nu\cdot (\nu \times u)=0$, which we equip with the norm $$\lvert\lvert u\rvert\rvert^2 := \lvert\lvert u\rvert\rvert_{L^2(\Omega)^3}^2 + \lvert\lvert\text{curl}\,u\rvert\rvert_{L^2(\Omega)^3}^2 + \lvert\lvert \nu \times u\rvert\rvert_{L^2(\partial \Omega)^3}^2.$$ It is well known that $C^\infty(\overline{\Omega})$ is dense in $H_\text{imp}$ (see Theorem 3.54). It is also well known that, if $\Omega$ is a Lipschitz polyhedron, $\{\mathcal{T}_h\}_{h>0}$ is a suitable family of triangulations of $\Omega$ and $$\textbf{ND}_h:= \{u_h \in H(\text{curl}) : u_h\rvert_T = a_T + b_T \times \cdot\text{ for some } a_T,b_T\in \mathbb{R}^3\,\forall T\in \mathcal{T}_h\}$$ is the Nédélec finite element space of the first family, that for all $u\in H(\text{curl})$, there exists a sequence $\{u_h\}_h\subset H(\text{curl})$ with $u_h\in \text{ND}_h$ for all $h>0$ and $$u_h\to u\quad \text{in } H(\text{curl})\quad \text{as } h\to 0.$$ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ My question: For $u\in H_{\text{imp}}$, does there exist a sequence as above with $$u_h\to u\quad \text{in } H_{\text{imp}}\quad \text{as } h\to 0?$$ If not, does there exist a sequence like above such that $$u_h\to u\quad \text{in } H(\text{curl})\quad \text{as } h\to 0$$ and $\nu \times u_h$ remains bounded? I have tried to argue with the interpolation operator $$r_h : H(\text{curl})\to \text{ND}_h$$ or the Hilbert projection, but wasn't successful.
If necessary, one can assume that $u\in H^1(\Omega)$ (or even more regularity).
