Then, adding and subtracting
\(\textbf{u}_h\), it follows that
$$\begin{aligned} \Vert \nabla \times (\textbf{u}-\textbf{u}_h^{\textrm{QP}})\Vert ^2_{\mathcal {T}_h}&= (\nabla \times (\textbf{u}-\textbf{u}_h^{\textrm{QP}}), \nabla \times (\nabla \varphi + \textbf{v}_s))_{\mathcal {T}_h}\\&= (\nabla \times (\textbf{u}-\textbf{u}_h), \nabla \times \textbf{v}_s)_{\mathcal {T}_h}+ (\nabla \times (\textbf{u}_h-\textbf{u}_h^{\textrm{QP}}), \nabla \times \textbf{v}_s)_{\mathcal {T}_h}. \end{aligned}$$
For the second term, according to (
7d), we have that
where in the last step, we have used the facts that
\(\llbracket \widehat{\textbf{u}}_h^t\rrbracket =0\) on
\(\mathcal {E}_I\),
\(\llbracket \widehat{\textbf{u}}_h^t\rrbracket _{\textrm{QP}}=0\) on
\(\mathcal {E}_{\textrm{QP}}\) (Remark
1) and
\(\widehat{\textbf{u}}_h^t \times \textbf{n}=\textbf{g}\) on
\(\Gamma _0\). Thus, since
\(\nabla \times \textbf{v}_s=\nabla \times (\textbf{u}-\textbf{u}_h^{\textrm{QP}})\) and apply the Young inequality, as follows
$$\begin{aligned} \begin{aligned} \Vert \nabla \times (\textbf{u}-\textbf{u}_h^{\textrm{QP}})\Vert ^2_{\mathcal {T}_h}&\lesssim (\nabla \times (\textbf{u}-\textbf{u}_h), \nabla \times \textbf{v}_s)_{\mathcal {T}_h} +\Vert h^{-1/2} (\textbf{u}_h- \widehat{\textbf{u}}_h^t) \times \textbf{n}\Vert _{\partial \mathcal {T}_h}^2 . \end{aligned} \end{aligned}$$
(28)
Now, if we use the
\(\textbf{L}^2\)-projector over
\(\textbf{P}_0(\mathcal {T}_h)\),
\(\Pi _{\textbf{V}}^0\) (see [
35]), in the first term of (
28), apply the Green’s identity of
\(\textrm{H}(\textbf{curl};\mathcal {T}_h)\), use (
2a) and (
2b), it follows that
$$\begin{aligned} \begin{aligned} (\nabla \times (\textbf{u}-&\textbf{u}_h), \nabla \times \textbf{v}_s)_{\mathcal {T}_h} \\&= (\nabla \times (\textbf{u}-\textbf{u}_h), \nabla \times (\textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s))_{\mathcal {T}_h}\\&= (\nabla \times \nabla \times (\textbf{u}-\textbf{u}_h), \textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s)_{\mathcal {T}_h} \\&\qquad - \langle (\nabla \times (\textbf{u}-\textbf{u}_h))^t, (\textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s) \times \textbf{n} \rangle _{\partial \mathcal {T}_h}\\&= (\textbf{f}-\overline{\epsilon } \nabla p + \kappa ^2 \epsilon \textbf{u}-\nabla \times \nabla \times \textbf{u}_h,\textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s)_{\mathcal {T}_h} \\&\qquad - \langle (\textbf{v}-\nabla \times \textbf{u}_h)^t, (\textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s)\times \textbf{n} \rangle _{\partial \mathcal {T}_h}. \end{aligned} \end{aligned}$$
(29)
Now, by taking
\(\textbf{z}:= \Pi _{\textbf{V}}^0 \textbf{v}_s\) in (
10) and applying the Green’s identity of
\(\textrm{H}(\textrm{div};\mathcal {T}_h)\) to the fourth term of the obtained equation, we have
$$\begin{aligned} 0= & {} (\textbf{v}-\textbf{v}_h, \nabla \times \Pi _{\textbf{V}}^0 \textbf{v}_s)_{\mathcal {T}_h} + \langle \textbf{v}^t-\widehat{\textbf{v}}_h^t, \Pi _{\textbf{V}}^0 \textbf{v}_s \times \textbf{n} \rangle _{\partial \mathcal {T}_h} - \kappa ^2 (\epsilon (\textbf{u}-\textbf{u}_h), \Pi _{\textbf{V}}^0 \textbf{v}_s)_{\mathcal {T}_h}\\{} & {} - (p-p_h, \nabla \cdot (\epsilon \Pi _{\textbf{V}}^0 \textbf{v}_s))_{\mathcal {T}_h} + \langle p-\widehat{p}_h, \epsilon \Pi _{\textbf{V}}^0 \textbf{v}_s\cdot \textbf{n} \rangle _{\partial \mathcal {T}_h}, \end{aligned}$$
from which,
$$\begin{aligned} 0=\langle \textbf{v}^t-\widehat{\textbf{v}}_h^{\,t}, \Pi _{\textbf{V}}^0 \textbf{v}_s \times \textbf{n} \rangle _{\partial \mathcal {T}_h} - \kappa ^2 (\epsilon (\textbf{u}-\textbf{u}_h), \Pi _{\textbf{V}}^0 \textbf{v}_s)_{\mathcal {T}_h} +\langle p-\widehat{p}_h,\epsilon \Pi _{\textbf{V}}^0 \textbf{v}_s\cdot \textbf{n} \rangle _{\partial \mathcal {T}_h},\nonumber \\ \end{aligned}$$
(30)
thanks to the fact that
\(\epsilon \) is a piecewise constant. Then, by using (
30), let us rewrite the second term on the right hand side of (
29), thus
$$\begin{aligned} \langle (\textbf{v}-\nabla \times \textbf{u}_h)^t&, (\textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s)\times \textbf{n} \rangle _{\partial \mathcal {T}_h}\\ =&\langle \textbf{v}^t, \textbf{v}_s\times \textbf{n}\rangle _{\partial \mathcal {T}_h} -\langle \textbf{v}^t, \Pi _{\textbf{V}}^0 \textbf{v}_s\times \textbf{n}\rangle _{\partial \mathcal {T}_h} -\langle (\nabla \times \textbf{u}_h)^t, (\textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s)\times \textbf{n}\rangle _{\partial \mathcal {T}_h}\\ =&\langle \textbf{v}^t, \textbf{v}_s \times \textbf{n}\rangle _{\partial \mathcal {T}_h} -\langle \widehat{\textbf{v}}_h^t, \Pi _{\textbf{V}}^0 \textbf{v}_s \times \textbf{n} \rangle _{\partial \mathcal {T}_h} - \kappa ^2 (\epsilon (\textbf{u}-\textbf{u}_h), \Pi _{\textbf{V}}^0 \textbf{v}_s)_{\mathcal {T}_h}\\&+\langle p-\widehat{p}_h, \epsilon \Pi _{\textbf{V}}^0 \textbf{v}_s\cdot \textbf{n}\rangle _{\partial \mathcal {T}_h} -\langle (\nabla \times \textbf{u}_h)^t, (\textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s)\times \textbf{n}\rangle _{\partial \mathcal {T}_h}. \end{aligned}$$
Let us note that the first term on the right hand side vanishes, since
\(\textbf{v}_s \in \textrm{H}_{\Gamma _0}(\textbf{curl};\Omega )\),
\(\textbf{u}\),
\(\textbf{u}_h^{\textrm{QP}}\) and
\(\varphi \) satisfies quasi-periodic conditions. In fact, by using the definitions of
\(\textbf{v}\) [cf. (
2a)] and
\(\textbf{v}_s\) [cf. (
26)], we have that
$$\begin{aligned} \langle \textbf{v}^t, \textbf{v}_s\times \textbf{n}\rangle _{\partial \mathcal {T}_h} =&\langle \textbf{v}^t, \textbf{v}_s \times \textbf{n}\rangle _{\partial \mathcal {T}_h \setminus \Gamma } +\langle \textbf{v}^t, \textbf{v}_s \times \textbf{n}\rangle _{\Gamma _0} +\langle \textbf{v}^t, \textbf{v}_s \times \textbf{n}\rangle _{\Gamma _{\textrm{QP}}}\\ =&\langle \textbf{v}^t, \textbf{v}_s \times \textbf{n}\rangle _{\Gamma _{\textrm{QP}}} =\langle (\nabla \times \textbf{u})^t, \textbf{v}_s \times \textbf{n}\rangle _{\Gamma _{\textrm{QP}}}\\ =&\langle (\nabla \times \textbf{u})^t, (\textbf{u}-\textbf{u}_h^{\textrm{QP}})\times \textbf{n}\rangle _{\Gamma _{\textrm{QP}}} -\langle (\nabla \times \textbf{u})^t, \nabla \varphi \times \textbf{n}\rangle _{\Gamma _{\textrm{QP}}} \\=&\langle (\nabla \times \textbf{u})^t, (\textbf{u}-\textbf{u}_h^{\textrm{QP}})\times \textbf{n}\rangle _{\Gamma _1\cup \Gamma _2} + \langle (\nabla \times \textbf{u})^t, (\textbf{u}-\textbf{u}_h^{\textrm{QP}})\times \textbf{n}\rangle _{\Gamma _3\cup \Gamma _4} \\ =&\langle (\nabla \times \textbf{u})^t, (\textbf{u}-\textbf{u}_h^{\textrm{QP}}) \times \textbf{n}\rangle _{\Gamma _1}-|e ^{i \alpha L}| \langle (\nabla \times \textbf{u})^t, (\textbf{u}-\textbf{u}_h^{\textrm{QP}}) \times \textbf{n}\rangle _{\Gamma _1}\\ {}&+ \langle (\nabla \times \textbf{u})^t, (\textbf{u}-\textbf{u}_h^{\textrm{QP}}) \times \textbf{n}\rangle _{\Gamma _3}-| e ^{i \beta L}| \langle (\nabla \times \textbf{u})^t, (\textbf{u}-\textbf{u}_h^{\textrm{QP}}) \times \textbf{n}\rangle _{\Gamma _3}=0. \end{aligned}$$
In addition, if we add
\(0=\langle \widehat{\textbf{v}}_h^t, \textbf{v}_s \times \textbf{n} \rangle _{\partial \mathcal {T}_h}\) in the second term, it is obtained that
$$\begin{aligned} \langle (\textbf{v}-\nabla \times \textbf{u}_h)^t, (\textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s)\times \textbf{n}\rangle _{\partial \mathcal {T}_h} = -\langle \widehat{\textbf{v}}_h^t, (\Pi _{\textbf{V}}^0 \textbf{v}_s -\textbf{v}_s) \times \textbf{n} \rangle _{\partial \mathcal {T}_h} \\ - \kappa ^2 (\epsilon (\textbf{u}-\textbf{u}_h), \Pi _{\textbf{V}}^0 \textbf{v}_s)_{\mathcal {T}_h} +\langle p-\widehat{p}_h, \epsilon \Pi _{\textbf{V}}^0 \textbf{v}_s\cdot \textbf{n} \rangle _{\partial \mathcal {T}_h} -\langle (\nabla \times \textbf{u}_h)^t, (\textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s)\times \textbf{n}\rangle _{\partial \mathcal {T}_h}. \end{aligned}$$
Then, after replacing the above expression in (
29), add
\(0=\langle p-\widehat{p}_h, \epsilon \textbf{v}_s\cdot \textbf{n}\rangle _{\partial \mathcal {T}_h}\) and by adding and subtracting
\(\kappa ^2 \epsilon \textbf{u}_h\) and
\(\overline{\epsilon } \nabla p_h\) in the first term, we can form the residual associated to (
2b), as follows
$$\begin{aligned} (\nabla \times (\textbf{u}-\textbf{u}_h), \nabla \times \textbf{v}_s)_{\mathcal {T}_h}= & {} (\textbf{f}-\overline{\epsilon } \nabla p_h + \kappa ^2 \epsilon \textbf{u}_h-\nabla \times \nabla \times \textbf{u}_h, \textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s)_{\mathcal {T}_h}\\{} & {} -\kappa ^2 (\epsilon (\textbf{u}-\textbf{u}_h), \textbf{v}_s)_{\mathcal {T}_h} \\{} & {} -\langle \widehat{\textbf{v}}_h^t-(\nabla \times \textbf{u}_h)^t,(\textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s)\times \textbf{n}\rangle _{\partial \mathcal {T}_h}\\{} & {} +\langle p-\widehat{p}_h, \epsilon (\textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s)\cdot \textbf{n}\rangle _{\partial \mathcal {T}_h}\\{} & {} -(\nabla (p-p_h),\epsilon (\textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s)) _{\mathcal {T}_h}, \end{aligned}$$
using Green’s identity and recalling that
\(\epsilon (\textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s)\) is divergence free on each element, it is obtained that
$$\begin{aligned} (\nabla \times ({\textbf {u}}-{\textbf {u}}_h), \nabla \times {\textbf {v}}_s)_{\mathcal {T}_h}{} & {} = ({\textbf {f}}-\overline{\epsilon } \nabla p_h + \kappa ^2 \epsilon {\textbf {u}}_h +\nabla \times \nabla \times {\textbf {u}}_h, {\textbf {v}}_s-\Pi _{{\textbf {V}}}^0 {\textbf {v}}_s)_{\mathcal {T}_h}\\ {}{} & {} \quad - \kappa ^2 (\epsilon ({\textbf {u}}-{\textbf {u}}_h),{\textbf {v}}_s)_{\mathcal {T}_h} \\ {}{} & {} \quad -\langle \widehat{{\textbf {v}}}_h^t-(\nabla \times {\textbf {u}}_h)^t,({\textbf {v}}_s-\Pi _{{\textbf {V}}}^0 {\textbf {v}}_s)\times {\textbf {n}}\rangle _{\partial \mathcal {T}_h}\\{} & {} \quad +\langle p_h-\widehat{p}_h, \epsilon ({\textbf {v}}_s-\Pi _{{\textbf {V}}}^0 {\textbf {v}}_s)\cdot {\textbf {n}}\rangle _{\partial \mathcal {T}_h} \end{aligned}$$
Now, if we use (
26) to rewrite
\(\textbf{v}_s\) in the second term, adding and subtracting
\(\textbf{v}_h^t\) in the third term and taking account the numerical flux (
4h), it holds
$$\begin{aligned} (\nabla \times (\textbf{u}-\textbf{u}_h), \nabla \times \textbf{v}_s)_{\mathcal {T}_h}&= (\textbf{f}-\overline{\epsilon } \nabla p_h + \kappa ^2 \epsilon \textbf{u}_h-\nabla \times \nabla \times \textbf{u}_h, \textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s)_{\mathcal {T}_h} \\&\quad -\kappa ^2 (\epsilon (\textbf{u}-\textbf{u}_h), \textbf{u}-\textbf{u}_h^{\textrm{QP}})_{\mathcal {T}_h} \\&\quad + \kappa ^2 (\epsilon (\textbf{u}-\textbf{u}_h), \nabla \varphi )_{\mathcal {T}_h} +\langle p_h-\widehat{p}_h, \epsilon (\textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s)\cdot \textbf{n}\rangle _{\partial \mathcal {T}_h} \\&\quad +\langle \tau (\widehat{\textbf{u}}_h^t-\textbf{u}_h^t) ,\textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s\rangle _{\partial \mathcal {T}_h} \\&\quad +\langle \textbf{v}_h^t-(\nabla \times \textbf{u}_h)^t,(\textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s)\times \textbf{n}\rangle _{\partial \mathcal {T}_h} . \end{aligned}$$
Since
\(- \kappa ^2 (\epsilon (\textbf{u}-\textbf{u}_h), \textbf{u}-\textbf{u}_h^{\textrm{QP}})_{\mathcal {T}_h} = - \kappa ^2 \Vert \epsilon ^{1/2} (\textbf{u}-\textbf{u}_h)\Vert _{\mathcal {T}_h}^2 - \kappa ^2 (\epsilon (\textbf{u}-\textbf{u}_h), \textbf{u}_h-\textbf{u}_h^{\textrm{QP}})_{\mathcal {T}_h}\), from the above equality we obtain that
$$\begin{aligned} \begin{aligned}&\kappa ^2 \Vert \epsilon ^{1/2} (\textbf{u}-\textbf{u}_h)\Vert _{\mathcal {T}_h}^2+(\nabla \times (\textbf{u}-\textbf{u}_h), \nabla \times \textbf{v}_s)_{\mathcal {T}_h} \\&\quad = (\textbf{f}-\overline{\epsilon } \nabla p_h + \kappa ^2 \epsilon \textbf{u}_h-\nabla \times \nabla \times \textbf{u}_h, \textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s)_{\mathcal {T}_h}\\&\qquad -\kappa ^2 (\epsilon (\textbf{u}-\textbf{u}_h), \textbf{u}_h-\textbf{u}_h^{\textrm{QP}})_{\mathcal {T}_h} \\&\qquad + \kappa ^2 (\epsilon (\textbf{u}-\textbf{u}_h), \nabla \varphi )_{\mathcal {T}_h} +\langle p_h-\widehat{p}_h,\epsilon (\textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s)\cdot \textbf{n} \rangle _{\partial \mathcal {T}_h}\\&\qquad +\langle \tau (\widehat{\textbf{u}}_h^t-\textbf{u}_h^t) ,\textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s\rangle _{\partial \mathcal {T}_h} +\langle \textbf{v}_h^t-(\nabla \times \textbf{u}_h)^t,(\textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s)\times \textbf{n}\rangle _{\partial \mathcal {T}_h}. \end{aligned} \end{aligned}$$
(31)
In what follows, we bound each term on the right hand side of (
31), by applying the Cauchy-Schwarz inequality, the definitions of the error indicators (
6a), (
6e), the relation (
4g), the approximation properties of
\(\Pi _{\textbf{V}}^0\) and the inverse inequality, we have
\(\star \):
$$\begin{aligned} (\textbf{f}-\overline{\epsilon } \nabla p_h + \kappa ^2 \epsilon \textbf{u}_h -\nabla \times \nabla \times \textbf{u}_h, \textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s)_{\mathcal {T}_h}&\le \sum _{K\in \mathcal {T}_h}h_K^{-1}\eta _{K,1} \Vert \textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s\Vert _K\\&\lesssim \sum _{K \in \mathcal {T}_h} h_K^{-1}\ \eta _{K,1}\ h_K^{\ell }\Vert \textbf{v}_s\Vert _{\ell ,K}\\&\le h^{\ell -1}\Vert \textbf{v}_s\Vert _{\ell ,\Omega } \sum _{K \in \mathcal {T}_h}\ \eta _{K,1}, \end{aligned}$$
\(\star \):
$$\begin{aligned} \langle \tau (\widehat{\textbf{u}}_h^t-\textbf{u}_h^t) ,\textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s\rangle _{\partial \mathcal {T}_h} \lesssim h^{\ell }\Vert \textbf{v}_s\Vert _{\ell ,\Omega }\ \Vert {h^{-1/2}}\tau (\widehat{\textbf{u}}_h^t-\textbf{u}_h^t)\Vert _{\partial \mathcal {T}_h} \end{aligned}$$
\(\star \):
$$\begin{aligned} \langle \textbf{v}_h^t-(\nabla \times \textbf{u}_h)^t,(\textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s)\times \textbf{n}\rangle _{\partial \mathcal {T}_h}&=\langle \textbf{n}\times (\textbf{v}_h-\nabla \times \textbf{u}_h),\textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s\rangle _{\partial \mathcal {T}_h} \\&\le \sum _{K\in \mathcal {T}_h}\Vert \textbf{n}\times (\textbf{v}_h-\nabla \times \textbf{u}_h)\Vert _{\partial K} \Vert \textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s\Vert _{\partial K}\\&\le \sum _{K\in \mathcal {T}_h}\Vert \textbf{n}\times (\textbf{v}_h-\nabla \times \textbf{u}_h)\Vert _{\partial K}\ h_K^{-1/2} \Vert \textbf{v}_s-\Pi _{\textbf{V}}^0 \textbf{v}_s\Vert _{K}\\&\lesssim \sum _{K\in \mathcal {T}_h}\left( \ h_K^{\ell -1/2} \Vert \textbf{v}_s\Vert _{\ell ,K}\ \sum _{F\in \partial K}\Vert \textbf{n}\times (\textbf{v}_h-\nabla \times \textbf{u}_h)\Vert _{F} \right) \\&\le \sum _{K\in \mathcal {T}_h}\left( \ h_K^{\ell -1/2} \Vert \textbf{v}_s\Vert _{\ell ,K}\ \sum _{F\in \partial K} h_F^{-1/2}\eta _{F,3} \right) \\&\lesssim h^{\ell -1}\Vert \textbf{v}_s\Vert _{\ell ,\Omega }\ \sum _{K\in \mathcal {T}_h}\sum _{F\in \partial K}\eta _{F,3} \end{aligned}$$
\(\star \):
$$\begin{aligned} (\epsilon (\textbf{u}-\textbf{u}_h), \textbf{u}_h-\textbf{u}_h^{\textrm{QP}})_{\mathcal {T}_h}&\le \Vert \epsilon (\textbf{u}-\textbf{u}_h)\Vert _{\mathcal {T}_h} \Vert \textbf{u}_h-\textbf{u}_h^{\textrm{QP}}\Vert _{\mathcal {T}_h}\\&\lesssim h^{1/2} \Vert \epsilon (\textbf{u}-\textbf{u}_h)\Vert _{\mathcal {T}_h} \Vert (\textbf{u}_h- \widehat{\textbf{u}}_h^t) \times \textbf{n}\Vert _{\partial \mathcal {T}_h}, \end{aligned}$$
\(\star \):
$$\begin{aligned} \langle \epsilon (\Pi _{\textbf{V}}^0 \textbf{v}_s \cdot \textbf{n} -\textbf{v}_s \cdot \textbf{n}),p_h - \widehat{p}_h \rangle _{\partial \mathcal {T}_h} \lesssim h^{\ell -1/2}\Vert \textbf{v}_s\Vert _{\ell ,\Omega }\ \Vert p_h-\widehat{p}_h\Vert _{\partial \mathcal {T}_h}. \end{aligned}$$
\(\star \):
$$\begin{aligned} (\epsilon (\textbf{u}-\textbf{u}_h), \nabla \varphi )_{\mathcal {T}_h}&\le \Vert \epsilon (\textbf{u}-\textbf{u}_h)\Vert _{\mathcal {T}_h} \Vert \nabla \varphi \Vert _{\mathcal {T}_h}\\&\lesssim \Vert \epsilon (\textbf{u}-\textbf{u}_h)\Vert _{\mathcal {T}_h} \left( h^{1/2} \Vert \tau _n (p_h-\widehat{p}_h)\Vert _{\partial \mathcal {T}_h} +\Vert h^{1/2} (\textbf{u}_h- \widehat{\textbf{u}}_h^t) \times \textbf{n}\Vert _{\partial \mathcal {T}_h}\right) , \end{aligned}$$
Where in the last inequality, we have used (
23) for
\(\psi \) in place of
\(\varphi \). Using Young’s inequality in the above equations and replacing in (
31), we get
$$\begin{aligned} \kappa ^2 \Vert \epsilon (\textbf{u}-\textbf{u}_h)\Vert _{\mathcal {T}_h}^2&+(\nabla \times (\textbf{u}-\textbf{u}_h),\nabla \times \textbf{v}_s)_{\mathcal {T}_h} \lesssim \ \delta ^2\Vert \epsilon (\textbf{u}-\textbf{u}_h)\Vert _{\mathcal {T}_h}^2 +\left( h^{\ell -1} \sum _{K \in \mathcal {T}_h}\ \eta _{K,1}\right) ^2\\&\quad +\left( h^{\ell }\Vert {h^{-1/2}}\tau (\widehat{\textbf{u}}_h^t-\textbf{u}_h^t)\Vert _{\partial \mathcal {T}_h}\right) ^2 +\left( h^{\ell -1} \sum _{K\in \mathcal {T}_h}\sum _{F\in \partial K}\eta _{F,3}\right) ^2\\&\quad +\left( h^{1/2} \Vert \tau _n (p_h-\widehat{p}_h)\Vert _{\partial \mathcal {T}_h}\right) ^2\\&\quad +\left( \Vert h^{1/2} (\textbf{u}_h- \widehat{\textbf{u}}_h^t) \times \textbf{n}\Vert _{\partial \mathcal {T}_h}\right) ^2 +\left( h^{\ell -1/2}\Vert p_h-\widehat{p}_h\Vert _{\partial \mathcal {T}_h}\right) ^2 +\Vert \textbf{v}_s\Vert _{\ell ,\Omega }^2 . \end{aligned}$$
According with the continuous embedding, we rewrite
\(\textbf{v}_s\) by using (
26) and (
23), we have
$$\begin{aligned} \Vert \textbf{v}_s\Vert ^2_{\ell ,\Omega }&\lesssim \Vert \textbf{v}_s\Vert ^2_{\Omega } +\Vert \nabla \times \textbf{v}_s\Vert ^2_{\Omega } =\Vert \textbf{u}-\textbf{u}_h^{\textrm{QP}}-\nabla \varphi \Vert ^2_{\mathcal {T}_h} +\Vert \nabla \times (\textbf{u}-\textbf{u}_h^{\textrm{QP}})\Vert ^2_{\mathcal {T}_h}\\&\lesssim \Vert \nabla \varphi \Vert ^2_{\Omega } +\Vert \textbf{u}-\textbf{u}_h^{\textrm{QP}}\Vert ^2_{\mathcal {T}_h} +\Vert \nabla \times (\textbf{u}-\textbf{u}_h^{\textrm{QP}})\Vert ^2_{\mathcal {T}_h} \\&\lesssim \left( h^{1/2} \Vert \tau _n (p_h-\widehat{p}_h)\Vert _{\partial \mathcal {T}_h} \right) ^2 +\left( \Vert h^{1/2} (\textbf{u}_h- \widehat{\textbf{u}}_h^t) \times \textbf{n}\Vert _{\partial \mathcal {T}_h}\right) ^2\\&\quad +\Vert \epsilon (\textbf{u}-\textbf{u}_h)\Vert ^2_{\mathcal {T}_h} +\Vert \epsilon (\textbf{u}_h-\textbf{u}_h^{\textrm{QP}})\Vert ^2_{\mathcal {T}_h} +\Vert \nabla \times (\textbf{u}-\textbf{u}_h^{\textrm{QP}})\Vert ^2_{\mathcal {T}_h}\\&\lesssim \left( h^{1/2} \Vert \tau _n (p_h-\widehat{p}_h)\Vert _{\partial \mathcal {T}_h} \right) ^2 +\left( \Vert h^{1/2} (\textbf{u}_h- \widehat{\textbf{u}}_h^t) \times \textbf{n}\Vert _{\partial \mathcal {T}_h}\right) ^2 +\Vert \epsilon (\textbf{u}-\textbf{u}_h)\Vert ^2_{\mathcal {T}_h}\\&\quad +\Vert \nabla \times (\textbf{u}-\textbf{u}_h^{\textrm{QP}})\Vert ^2_{\mathcal {T}_h} \end{aligned}$$
thus
$$\begin{aligned} \Vert \epsilon (\textbf{u}-\textbf{u}_h)\Vert _{\mathcal {T}_h}^2&+(\nabla \times (\textbf{u}-\textbf{u}_h),\nabla \times \textbf{v}_s)_{\mathcal {T}_h} \lesssim \ \hat{\delta }^2\Vert \epsilon (\textbf{u}-\textbf{u}_h)\Vert _{\mathcal {T}_h}^2\\&\quad +\left( h^{\ell -1} \sum _{K \in \mathcal {T}_h}\ \eta _{K,1}\right) ^2+\left( h^{\ell }\Vert {h^{-1/2}}\tau (\widehat{\textbf{u}}_h^t-\textbf{u}_h^t)\Vert _{\partial \mathcal {T}_h}\right) ^2\\&\quad +\left( h^{\ell -1} \sum _{K\in \mathcal {T}_h}\sum _{F\in \partial K}\eta _{F,3}\right) ^2+\left( h^{1/2} \Vert \tau _n (p_h-\widehat{p}_h)\Vert _{\partial \mathcal {T}_h}\right) ^2\\&\quad +\left( \Vert h^{1/2} (\textbf{u}_h- \widehat{\textbf{u}}_h^t) \times \textbf{n}\Vert _{\partial \mathcal {T}_h}\right) ^2\\&\quad +\left( h^{\ell -1/2}\Vert p_h-\widehat{p}_h\Vert _{\partial \mathcal {T}_h}\right) ^2+\Vert \nabla \times (\textbf{u}-\textbf{u}_h^{\textrm{QP}})\Vert ^2_{\mathcal {T}_h}. \end{aligned}$$
Finally, replacing in (
28), we conclude
$$\begin{aligned} \Vert \nabla \times (\textbf{u}-\textbf{u}_h^{\textrm{QP}})\Vert ^2_{\mathcal {T}_h} \lesssim&\ \Vert h^{-1/2} (\textbf{u}_h- \widehat{\textbf{u}}_h^t) \times \textbf{n}\Vert _{\partial \mathcal {T}_h}^2 +\left( h^{\ell -1} \sum _{K \in \mathcal {T}_h}\ \eta _{K,1}\right) ^2\\&\quad +\left( h^{\ell }\Vert {h^{-1/2}}\tau (\widehat{\textbf{u}}_h^t-\textbf{u}_h^t)\Vert _{\partial \mathcal {T}_h}\right) ^2\\&\quad +\left( h^{\ell -1} \sum _{K\in \mathcal {T}_h}\sum _{F\in \partial K}\eta _{F,3}\right) ^2 +\left( h^{1/2} \Vert \tau _n (p_h-\widehat{p}_h)\Vert _{\partial \mathcal {T}_h}\right) ^2\\&\quad +\left( \Vert h^{1/2} (\textbf{u}_h- \widehat{\textbf{u}}_h^t) \times \textbf{n}\Vert _{\partial \mathcal {T}_h}\right) ^2 +\left( h^{\ell -1/2}\Vert p_h-\widehat{p}_h\Vert _{\partial \mathcal {T}_h}\right) ^2\\&\quad +\hat{\delta }^2\Vert \epsilon (\textbf{u}-\textbf{u}_h)\Vert _{\mathcal {T}_h}^2. \end{aligned}$$
Then, choosing
\(\hat{\delta }\) small enough and using the fact that
\(0<h<1\), (
24) is deduced from the last inequality.
\(\square \)