+ 2\sqrt{2} M) \inf_{({\boldsymbol{v}}, \varphi)\in \mathbf{X}_h\times V_h} {\lVert({\boldsymbol{h}}- {\boldsymbol{v}}, \psi -\varphi, 0)\rVert}_*.$$

The well posedness of Problem  follows immediately from Proposition \[discElip\].

Moreover we deduce from Proposition \[discElip\] and the consistency of the scheme that $$\begin{array}{l}
\frac{1}{2} {\lVert({\boldsymbol{h}}_{h}- {\boldsymbol{v}}, \psi_h - \varphi, k_h-m)\rVert}^2 \\[.1cm]
\qquad \leq \text{Re} \left[ (1 - \imath) A_h(({\boldsymbol{h}}_{h}- {\boldsymbol{v}}, \psi_h - \varphi, k_h-m),({\boldsymbol{h}}_{C,h}- {\boldsymbol{v}}, \psi_h - \varphi, k_h-m)) \right]  \\[.1cm]
\qquad = \text{Re} \left[ (1 - \imath) A_h(({\boldsymbol{h}}- {\boldsymbol{v}}, \psi - \varphi, k-m),({\boldsymbol{h}}- {\boldsymbol{v}}, \psi - \varphi, k-m)) \right]
\end{array}$$ for all $({\boldsymbol{v}}, \varphi,m)\in \mathbf{X}_h\times V_h\times {\mathbb C}$. Then from Proposition \[boundedness\] we have $${\lVert({\boldsymbol{h}}_{h}- {\boldsymbol{v}}, \psi_h - \varphi, k_h-m)\rVert} \leq 2 \sqrt{2} M {\lVert({\boldsymbol{h}}- {\boldsymbol{v}}, \psi - \varphi,k-m)\rVert}_\ast.$$ The result follows now from the triangle inequality.

Asymptotic error estimates {#sec5}
==========================

We denote by ${\boldsymbol{\Pi}}_{h,m}^{\text{curl}}$ the $m$-order $\mathbf{H}({\mathbf{curl}}, \Omega_C)$-conforming Nédélec interpolation operator of the second kind, see for example [@NED86] or [@Monk Section 8.2]. It is well known that ${\boldsymbol{\Pi}}_{h,m}^{\text{curl}}$ is bounded on $\mathbf{H}({\mathbf{curl}}, \Omega_C)\cap \mathbf{H}^s({\mathbf{curl}}, {\mathcal{T}}_h^{\Omega_C})$ for $s>1/2$, where $$\mathbf{H}^s({\mathbf{curl}}, {\mathcal{T}}_h^{\Omega_C}) := {\left\{{\boldsymbol{v}}\in \mathbf{H}^s({\mathcal{T}}_h^{\Omega_C});\quad {\mathbf{curl}}_h {\boldsymbol{v}}\in \mathbf{H}^{s}({\mathcal{T}}_h^{\Omega_C})\right\}}.$$ Moreover, there exists a constant $C_1>0$ independent of $h$ such that (cf. [@AVbook]) $$\label{errorInterp1}
{\lVert{\boldsymbol{u}}- {\boldsymbol{\Pi}}_{h,m}^{\text{curl}} {\boldsymbol{u}}\rVert}_{0, \Omega_C} + {\lVert{\mathbf{curl}}({\boldsymbol{u}}- {\boldsymbol{\Pi}}_{h,m}^{\text{curl}}{\boldsymbol{u}})\rVert}_{0, \Omega_C} \leq C_1 h^{\min(s, m)} \big(  