Discontinuous Galerkin Approximations for Computing Electromagnetic Bloch Modes in Photonic Crystals

Abstract

We analyze discontinuous Galerkin finite element discretizations of the Maxwell equations with periodic coefficients. These equations are used to model the behavior of light in photonic crystals, which are materials containing a spatially periodic variation of the refractive index commensurate with the wavelength of light. Depending on the geometry, material properties and lattice structure these materials exhibit a photonic band gap in which light of certain frequencies is completely prohibited inside the photonic crystal. By Bloch/Floquet theory, this problem is equivalent to a modified Maxwell eigenvalue problem with periodic boundary conditions, which is discretized with a mixed discontinuous Galerkin (DG) formulation using modified Nédélec basis functions. We also investigate an alternative primal DG interior penalty formulation and compare this method with the mixed DG formulation. To guarantee the non-pollution of the numerical spectrum, we prove a discrete compactness property for the corresponding DG space. The convergence rate of the numerical eigenvalues is twice the minimum of the order of the polynomial basis functions and the regularity of the solution of the Maxwell equations. We present both 2D and 3D numerical examples to verify the convergence rate of the mixed DG method and demonstrate its application to computing the band structure of photonic crystals.

Appendices

Appendix

Appendix A: Continuity and Semi-Ellipticity

Lemma 10.1

For all $\varvec{v}\in {\varvec{V}}(h)$ and $q \in Q(h)$,

$$\begin{aligned} \begin{aligned} \Vert \epsilon ^{-\frac{1}{2}} \mathcal {L}({\varvec{v}}) \Vert _{0 ,\varOmega }&\le C \Vert {{e}}^{-\frac{1}{2}}{{h}}^{-\frac{1}{2}}[[ \varvec{v} ]] _T \Vert _{0 ,{\mathcal {F}_h }},\\ \Vert \mathcal {M}({q}) \Vert _{0 ,\varOmega }&\le C \Vert {{h}}^{-\frac{1}{2}}{[[ q ]] _N} \Vert _{0 ,{\mathcal {F}_h }}. \end{aligned} \end{aligned}$$

with a constant $C>0$ that is independent of the mesh size and the coefficient $\epsilon$.

Proof

$$\begin{aligned} \begin{aligned} \Vert \epsilon ^{-\frac{1}{2}} \mathcal {L}({\varvec{v}}) \Vert _{0 ,\varOmega }&=\sup _{\varvec{w}\in {\varvec{V}}^{\varvec{\alpha }}_h}\frac{\int _ {\varOmega }\epsilon ^{-\frac{1}{2}} \mathcal {L}({\varvec{v}}) \cdot \overline{ \varvec{w}}d\varvec{x}}{\Vert \varvec{w} \Vert _{0 ,\varOmega }}\\&=\sup _{\varvec{w}\in {\varvec{V}}^{\varvec{\alpha }}_h}\frac{\int _ {{\mathcal {F}_h }}\epsilon ^{-\frac{1}{2}}[[ \varvec{v} ]] _T\cdot \overline{ \{\{ \varvec{w} \}\}}ds}{\Vert \varvec{w} \Vert _{0 ,\varOmega }}\\&\le \sup _{\varvec{w}\in {\varvec{V}}^{\varvec{\alpha }}_h}\frac{\Vert {{e}}^{-\frac{1}{2}}{{h}}^{-\frac{1}{2}}[[ \varvec{v} ]] _T \Vert _{0 ,{\mathcal {F}_h }} \Vert {{h}}^\frac{1}{2}\{\{ \varvec{w} \}\} \Vert _{0 ,{\mathcal {F}_h }}}{\Vert \varvec{w} \Vert _{0 ,\varOmega }}\\&\le C \Vert {{e}}^{-\frac{1}{2}}{{h}}^{-\frac{1}{2}}[[ \varvec{v} ]] _T \Vert _{0 ,{\mathcal {F}_h }}. \end{aligned} \end{aligned}$$

Here we use the inverse inequality Lemma 11 in [41]:

$$\begin{aligned} \begin{aligned} \Vert {{h}}^\frac{1}{2}\{\{ \varvec{w} \}\} \Vert _{0 ,{\mathcal {F}_h }}^2 \le C \sum _{K\in {\mathcal {T}_h }} h_K \Vert \varvec{w} \Vert _{0,\partial K} ^2 \le C \sum _{K\in {\mathcal {T}_h }} \Vert \varvec{w} \Vert _{0,K}^2= C \Vert \varvec{w} \Vert _{0 ,\varOmega } ^2. \end{aligned} \end{aligned}$$

The proof of the other estimate is similar. $\square$

Theorem 10.1

There exist constants $a_1>0$ and $a_2>0$, independent of the mesh size and the coefficient $\epsilon$, such that

$$\begin{aligned} \begin{aligned} \left| A_h(\varvec{u},{\varvec{\lambda }};\varvec{v},{\varvec{\eta }}) \right|&\le a_1 \Vert ( \varvec{u},{\varvec{\lambda }}) \Vert _{{\varvec{U}}(h)}\Vert ( \varvec{v},{\varvec{\eta }}) \Vert _{{\varvec{U}}(h)}\quad \;\;\; \forall \; (\varvec{u},{\varvec{\lambda }}), (\varvec{v},{\varvec{\eta }}) \in {\varvec{U}}(h),\\ \left| B_h(\varvec{v},{\varvec{\eta }};q) \right|&\le a_2 \Vert ( \varvec{v},{\varvec{\eta }}) \Vert _{{\varvec{U}}(h)}\Vert q \Vert _{Q(h)} \quad \quad \quad \;\,\, \forall \; (\varvec{v},{\varvec{\eta }}) \in {\varvec{U}}(h), \; \forall \; q \in Q(h). \end{aligned} \end{aligned}$$

Proof

$$\begin{aligned} \begin{aligned} \left| A_h(\varvec{u},{\varvec{\lambda }};\varvec{v},{\varvec{\eta }}) \right| \le&\, \Vert \epsilon ^{-\frac{1}{2}}\nabla _{{\varvec{\alpha }},h}\times \varvec{u} \Vert _{0 ,\varOmega }\Vert \epsilon ^{-\frac{1}{2}}\nabla _{{\varvec{\alpha }},h}\times \varvec{v} \Vert _{0 ,\varOmega }\\&+C(\Vert {{e}}^{-\frac{1}{2}}{{h}}^{-\frac{1}{2}}[[ \varvec{u} ]] _T \Vert _{0 ,{\mathcal {F}_h }} \Vert \epsilon ^{-\frac{1}{2}}\nabla _{{\varvec{\alpha }},h}\times \varvec{v} \Vert _{0 ,\varOmega }\\&+\Vert {{e}}^{-\frac{1}{2}}{{h}}^{-\frac{1}{2}}[[ \varvec{v} ]] _T \Vert _{0 ,\varOmega } \Vert \epsilon ^{-\frac{1}{2}}\nabla _{{\varvec{\alpha }},h}\times \varvec{u} \Vert _{0 ,\varOmega } )\\&+ {\mathfrak {a}}\Vert {{e}}^{-\frac{1}{2}}{{h}}^{-\frac{1}{2}}[[ \varvec{u} ]] _T \Vert _{0 ,{\mathcal {F}_h }} \Vert {{e}}^{-\frac{1}{2}}{{h}}^{-\frac{1}{2}}[[ \varvec{v} ]] _T \Vert _{0 ,{\mathcal {F}_h }}\\&+ {\mathfrak {b}}\Vert {{h}}^\frac{1}{2}{[[ \varvec{u} ]] _N} \Vert _{0 ,{\mathcal {F}_h }} \Vert {{h}}^\frac{1}{2}{[[ \varvec{v} ]] _N} \Vert _{0 ,{\mathcal {F}_h }}\\&+{\mathfrak {c}}\Vert {{h}}^{-\frac{1}{2}}{\varvec{\lambda }} \Vert _{0 ,{\mathcal {F}_h }} \Vert {{h}}^{-\frac{1}{2}}{\varvec{\eta }} \Vert _{0 ,{\mathcal {F}_h }}\\ \le&\, a_1 \Vert ( \varvec{u},{\varvec{\lambda }}) \Vert _{{\varvec{U}}(h)}\Vert ( \varvec{v},{\varvec{\eta }}) \Vert _{{\varvec{U}}(h)}. \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \left| B_h(\varvec{v},{\varvec{\eta }};q) \right| \le&\, \Vert \varvec{v} \Vert _{0 ,\varOmega }\Vert \nabla _{{\varvec{\alpha }},h}q \Vert _{0 ,\varOmega } + C \Vert \varvec{v} \Vert _{0 ,\varOmega } \Vert {{h}}^{-\frac{1}{2}}{[[ q ]] _N} \Vert _{0 ,{\mathcal {F}_h }} \\&+ {\mathfrak {c}}\Vert {{h}}^{-\frac{1}{2}}{\varvec{\eta }} \Vert _{0 ,{\mathcal {F}_h }} \Vert {{h}}^{-\frac{1}{2}}{[[ q ]] _N} \Vert _{0 ,{\mathcal {F}_h }}\\ \le&\, a_2 \Vert ( \varvec{v},{\varvec{\eta }}) \Vert _{{\varvec{U}}(h)}\Vert q \Vert _{Q(h)}. \end{aligned} \end{aligned}$$

$\square$

Lemma 10.2

For ${\varvec{\alpha }}\in K \text { with } {\varvec{\alpha }}\not = \varvec{{0}}$, given that ${\mathfrak {a}}>0$ is large enough, ${\mathfrak {b}}>0$ and ${\mathfrak {c}}>0$, there exists a $C>0$ independent of h, such that

$$\begin{aligned} \begin{aligned} A_h(\varvec{u},{\varvec{\lambda }};\varvec{u},{\varvec{\lambda }}) \ge C |( \varvec{u},{\varvec{\lambda }}) |_{{\varvec{U}}(h)}^2 \quad \quad \forall (\varvec{u},{\varvec{\lambda }}) \in {\varvec{U}}^{\varvec{\alpha }}_h. \end{aligned} \end{aligned}$$

(10.1)

Proof

$$\begin{aligned} \begin{aligned} A_h(\varvec{u},{\varvec{\lambda }};\varvec{u},{\varvec{\lambda }}) \ge&\, \Vert \epsilon ^{-\frac{1}{2}}\nabla _{{\varvec{\alpha }},h}\times \varvec{u} \Vert _{0 ,\varOmega }^2 -2C \Vert {{e}}^{-\frac{1}{2}}{{h}}^{-\frac{1}{2}}[[ \varvec{u} ]] _T \Vert _{0 ,{\mathcal {F}_h }} \Vert \epsilon ^{-\frac{1}{2}}\nabla _{{\varvec{\alpha }},h}\times \varvec{u} \Vert _{0 ,\varOmega }\\&+ {\mathfrak {a}}\Vert {{e}}^{-\frac{1}{2}}{{h}}^{-\frac{1}{2}}[[ \varvec{u} ]] _T \Vert _{0 ,{\mathcal {F}_h }}^2 +{\mathfrak {b}}\Vert {{h}}^\frac{1}{2}{[[ \varvec{u} ]] _N} \Vert _{0 ,{\mathcal {F}_h }}^2 +{\mathfrak {c}}\Vert {{h}}^{-\frac{1}{2}}{\varvec{\lambda }} \Vert _{0 ,{\mathcal {F}_h }} ^2\\ {=}&\, \frac{1}{2}\Vert \epsilon ^{-\frac{1}{2}}\nabla _{{\varvec{\alpha }},h}{\times } \varvec{u} \Vert _{0 ,\varOmega }^2 {+}\frac{1}{2} \left( \Vert \epsilon ^{-\frac{1}{2}}\nabla _{{\varvec{\alpha }},h}\times \varvec{u} \Vert _{0 ,\varOmega } {-}2C \Vert {{e}}^{-\frac{1}{2}}{{h}}^{-\frac{1}{2}}[[ \varvec{u} ]] _T \Vert _{0 ,{\mathcal {F}_h }} \right) ^2\\&+({\mathfrak {a}}-2C^2)\Vert {{e}}^{-\frac{1}{2}}{{h}}^{-\frac{1}{2}}[[ \varvec{u} ]] _T \Vert _{0 ,{\mathcal {F}_h }}^2 +{\mathfrak {b}}\Vert {{h}}^\frac{1}{2}{[[ \varvec{u} ]] _N} \Vert _{0 ,{\mathcal {F}_h }}^2 +{\mathfrak {c}}\Vert {{h}}^{-\frac{1}{2}}{\varvec{\lambda }} \Vert _{0 ,{\mathcal {F}_h }} ^2\\ \ge&\, C |( \varvec{u},{\varvec{\lambda }}) |_{{\varvec{U}}(h)}^2. \end{aligned} \end{aligned}$$

where we used the estimate in Lemma 10.1. $\square$

Appendix B: Inf-Sup Condition

For the proof of Lemma 5.5, we first need the following auxiliary result.

Lemma 10.3

Given N real numbers $\{\alpha _1,\dots ,\alpha _N\}$ let $\beta =\frac{1}{N}\sum _{j=1}^{N}\alpha _j$. Then,

$$\begin{aligned} \begin{aligned} \sum _{j=1}^{N} \left| \alpha _j-\beta \right| ^2\le C \sum _{j=1}^{N-1} \left| \alpha _{j+1}-\alpha _j \right| ^2, \end{aligned} \end{aligned}$$

(10.2)

where $C>0$ depends only on N.

Proof

For any j, the Cauchy-Schwarz inequality gives

$$\begin{aligned} \begin{aligned} \left| \alpha _j-\beta \right| ^2 = \frac{1}{N^2} \left| \sum _{i=1}^{N}(\alpha _j-\alpha _i) \right| ^2 \le \frac{N-1}{N^2} \sum _{i=1}^{N}\left| \alpha _j-\alpha _i \right| ^2. \end{aligned} \end{aligned}$$

Summing over j, we obtain

$$\begin{aligned} \begin{aligned} \sum _{j=1}^{N} \left| \alpha _j-\beta \right| ^2&\le \frac{2(N-1)^2}{N^2}\sum _{j=1}^{N} \sum _{i=1}^{j-1}\sum _{k=i}^{j-1} \left| \alpha _{k+1}-\alpha _k \right| ^2\\&\le \frac{2(N-1)^2}{N^2}\sum _{j=1}^{N} \sum _{i=1}^{N-1}\sum _{k=1}^{N-1} \left| \alpha _{k+1}-\alpha _k \right| ^2 \\&\le \frac{2(N-1)^4}{N^2}\sum _{j=1}^{N-1}\left| \alpha _{j+1}-\alpha _j \right| ^2 \\&= C(N)\sum _{j=1}^{N-1}\left| \alpha _{j+1}-\alpha _j \right| ^2. \end{aligned} \end{aligned}$$

$\square$

Proof of Lemma 5.5

Given $q_h\in Q^{\varvec{\alpha }}_h$, we construct a function $\chi \in Q^{{\varvec{\alpha }},c}_h$ as follows: At every node of the mesh ${\mathcal {T}_h }$ corresponding to a Lagrangian type degree of freedom for $Q^{{\varvec{\alpha }},c}_h$, the value of $\chi$ is set to the average of the values of $q_h$ at that node.

For each $K\in {\mathcal {T}_h }$, let $\mathcal {N}_K=\{ \varvec{x}^{(j)}_K,j=1,\ldots ,m \}$ be the Lagrange nodes (points) of K and $\{ \phi ^{(j)}_K,j=1,\ldots ,m \}$ the corresponding (local) basis functions satisfying $\phi ^{(j)}_K(\varvec{x}^{(i)}_K)=\delta _{ij}$. Set $\mathcal {N}=\cup _{K\in {\mathcal {T}_h }} \mathcal {N}_K$. We view $\mathcal {N}$ as the union of two classes:

$$\begin{aligned} \begin{aligned} \mathcal {N}_i&=\{ \nu \in \mathcal {N}:\nu \text { is interior to an element}\},\\ \mathcal {N}_f&=\{\nu \in \mathcal {N}: \nu \in \partial K, \text { for some } K \in {\mathcal {T}_h }\},\\ \end{aligned} \end{aligned}$$

We note that $\mathcal {N}_f$ can be divided into two sets $\mathcal {N}_f^i$ and $\mathcal {N}_f^b$: $\mathcal {N}_f^i$ is the set of nodes on interior faces, while $\mathcal {N}_f^b$ is the set of nodes on the boundary of a face $f \subset \partial \varOmega$. As $\varOmega$ and ${\mathcal {T}_h }$ are both periodic, for every $\nu ^1\in \mathcal {N}_f^b$, there exist a unique $\nu ^2$ also in $\mathcal {N}_f^b$ being the corresponding periodic point of $\nu ^1$. From the definition $Q^{{\varvec{\alpha }},c}_h=Q^{\varvec{\alpha }}_h\cap {H^1_{\mathrm {per}}(\varOmega ) }$, $Q^{{\varvec{\alpha }},c}_h$ is a periodic conforming finite element space. To satisfy the periodicity of $Q^{{\varvec{\alpha }},c}_h$, we can let $\nu ^1$ and $\nu ^2$ share the same degree of freedom. Then we regard $\nu ^1$ and $\nu ^2$ as the ’same’ node in our computational domain. Furthermore, the nodes in $\mathcal {N}_f^b$ can therefore be considered as nodes in $\mathcal {N}_f^i$. In the following discussion, we consider the nodes in $\mathcal {N}_f^b$ and $\mathcal {N}_f^i$ therefore in the same way.

For each $\nu \in \mathcal {N}$, let $\omega _\nu =\{K \in {\mathcal {T}_h }| \nu \in K \}$ and denote its cardinality by $\left| \omega _\nu \right|$. If $\nu \in \mathcal {N}_i$, then $\left| \omega _\nu \right| =1$, and if $\nu \in \mathcal {N}_f$, $\left| \omega _\nu \right| \ge 1$. Then the basis function $\phi ^{(\nu )}$ in $Q^{{\varvec{\alpha }},c}_h$ at the node $\nu \in \mathcal {N}$ can be constructed as

$$\begin{aligned} \begin{aligned} \text {supp }\phi ^{(\nu )}= \bigcup _{K\in \omega _\nu }, \quad \phi ^{(\nu )}|_K=\phi ^{(j)}_K,\quad \varvec{x}^{(j)}_K=\nu . \end{aligned} \end{aligned}$$

Now, given $q_h\in Q^{\varvec{\alpha }}_h$, written as $q_h=\sum _{K\in {\mathcal {T}_h }} \sum _{j=1}^{m}\alpha ^{(j)}_K\phi ^{(j)}_K$, we define the function $\chi \in Q^{{\varvec{\alpha }},c}_h$ by

$$\begin{aligned} \begin{aligned} \chi&=\sum _{\nu \in \mathcal {N}} \beta ^{(\nu )}\phi ^{(\nu )}, \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} \beta ^{(\nu )}&=\frac{1}{\left| \omega _\nu \right| } \sum _{\varvec{x}^{(j)}_K=\nu } \alpha ^{(j)}_K\qquad \text { for } \nu \in \mathcal {N}. \end{aligned} \end{aligned}$$

Now set $\beta ^{(j)}_K=\beta ^{(\nu )}$ whenever $\varvec{x}^{(j)}_K=\nu$. A simple scaling argument shows that $\Vert \nabla _{\varvec{\alpha }}\phi ^{(j)}_K \Vert ^2_K \le ch^{d-2}_K$. Hence

$$\begin{aligned} \begin{aligned} \sum _{K\in {\mathcal {T}_h }} \Vert \nabla _{\varvec{\alpha }}(q_h-\chi ) \Vert _{0,K}^2&\le C\,m \sum _{K\in {\mathcal {T}_h }} h_K^{d-2} \sum _{j=1}^{m} \left| \alpha ^{(j)}_K-\beta ^{(j)}_K \right| ^2\\&\le C \sum _{\nu \in \mathcal {N}} h_\nu ^{d-2} \sum _{\varvec{x}^{(j)}_K=\nu ,\;\varvec{x}^{(j)}_K\in \mathcal {N}_K,\;K\in {\mathcal {T}_h }} \left| \alpha ^{(j)}_K-\beta ^{(\nu )} \right| ^2\\&= C \sum _{\nu \in \mathcal {N}_f} h_\nu ^{d-2} \sum _{\varvec{x}^{(j)}_K=\nu ,\;\varvec{x}^{(j)}_K\in \mathcal {N}_K,\;K\in {\mathcal {T}_h }} \left| \alpha ^{(j)}_K-\beta ^{(\nu )} \right| ^2,\\ \end{aligned} \end{aligned}$$

(10.3)

where in the last step, we remove the nodes in $\mathcal {N}_i$ as they have no contribution by the definition of $\beta ^{(\nu )}$.

We now temporarily focus on the case $d = 2$. For $\nu \in \mathcal {N}_f$ we enumerate the elements of $\omega _\nu$ as $\{ K_{1}, \dots , K_{\left| \omega _\nu \right| } \}$ so that any consecutive pair $K_{i}$, $K_{i+1}$ in that list shares an edge. Then from Lemma 10.3, with some constant C depending only on $\left| \omega _\nu \right|$, we have

$$\begin{aligned} \begin{aligned} \sum _{\varvec{x}^{(j)}_K=\nu } \left| \alpha ^{(j)}_K-\beta ^{(\nu )} \right| ^2 \le C \sum _{i=1}^{\left| \omega _\nu \right| -1} \left| \alpha _{K_{i}}^{j_{i}}-\alpha _{K_{i+1}}^{j_{i+1}} \right| ^2. \end{aligned} \end{aligned}$$

(10.4)

For $d=3$, it may not be possible to enumerate $\omega _\nu$ in such a way. However, by allowing some repetitions of its elements, we can write $\omega _\nu =\{ K_{l_1}, \ldots ,K_{l_{n(\nu )}} \}$ for some $n(\nu )$, so that in this case also $K_{l_i}$ and $K_{l_{i+1}}$ share a face or an edge. Having done so, by applying Lemma 10.3 to the list obtained by removing all repetitions of elements of $\omega _\nu$, we obtain

$$\begin{aligned} \begin{aligned} \sum _{\varvec{x}^{(j)}_K=\nu } \left| \alpha ^{(j)}_K-\beta ^{(\nu )} \right| ^2 \le C \sum _{i=1}^{n(\nu )-1} \left| \alpha _{K_{i}}^{j_{i}}-\alpha _{K_{i+1}}^{j_{i+1}} \right| ^2. \end{aligned} \end{aligned}$$

(10.5)

Using (10.4) for $d=2$, or (10.5) if $d=3$, from (10.3) we have

$$\begin{aligned} \begin{aligned} \sum _{K\in {\mathcal {T}_h }} \Vert \nabla _{\varvec{\alpha }}(q_h-\chi ) \Vert _{0,K}^2 \le&C \sum _{f\in {\mathcal {F}_h }}\sum _{\nu \in f} h_\nu ^{d-2}\left| \alpha _{K^+}^{j_{\nu }^+}-\alpha _{K^-}^{j_{\nu }^-} \right| ^2,\\ \end{aligned} \end{aligned}$$

(10.6)

with $\varvec{x}^{j^{+}_{\nu }}_{K_+}=\varvec{x}^{j^{-}_{\nu }}_{K_-}=\nu$. Note that $\alpha ^{j^{+}_{\nu }}_{K_+}-\alpha ^{j^{-}_{\nu }}_{K_-}$ is the jump in the values of $q_h$ at $\nu$ across f. Also, since the mesh ${\mathcal {T}_h }$ is locally quasi-uniform, it follows that

$$\begin{aligned} \begin{aligned} \sum _{\nu \in f} h_\nu ^{d-2} \left| \alpha ^{j^{+}_{\nu }}_{K_+}-\alpha ^{j^{-}_{\nu }}_{K_-} \right| ^2&\le C h_f^{d-2}\Vert {[[ q_h ]] _N} \Vert _{L^{\infty }(f)}^2\\&\le C h_f^{-1}\Vert {[[ q_h ]] _N} \Vert _{0,f}^2, \end{aligned} \end{aligned}$$

(10.7)

where the constant C depends on the number of nodes in f . The required result now follows from (10.6)–(10.7). $\square$

Proof of Theorem 5.2

Fix $0 \not = q \in Q^{\varvec{\alpha }}_h$, and use the $Q^{\varvec{\alpha }}_h$-decomposition as $q = q_0 + q_1$ with $q_0\in Q^{{\varvec{\alpha }},c}_h$ and $q_1\in Q_h^{{\varvec{\alpha }},\perp }$. Choose $\varvec{v}_0= - \nabla _{\varvec{\alpha }}q_0 \in {\varvec{V}}^{\varvec{\alpha }}_h\cap {{\varvec{H}}_{\mathrm {per}}(\mathrm {curl}^0_{\varvec{\alpha }};\varOmega ) }$, then we have

$$\begin{aligned} \begin{aligned} B_h(\varvec{v}_0, \varvec{{0}}; q_0)=\Vert \nabla _{\varvec{\alpha }}q_0\Vert ^2_{0,\varOmega }=\Vert q_0\Vert ^2_{Q(h)}. \end{aligned} \end{aligned}$$

(10.8)

$$\begin{aligned} \begin{aligned} \Vert ( \varvec{v}_0,\varvec{{0}}) \Vert _{{\varvec{U}}(h)}^2&=\Vert {{h}}^\frac{1}{2}{[[ \varvec{v}_0 ]] _N} \Vert _{0,{\mathcal {F}_h }}^2 +\Vert \varvec{v}_0 \Vert ^2_{0, \varOmega }\\&\le C \sum _{K\in T_h} h_K \Vert \nabla _{\varvec{\alpha }}q_0\Vert ^2_{0,\partial K} +\Vert \nabla _{\varvec{\alpha }}q_0 \Vert _{0 ,\varOmega }^2\\&\le C\Vert \nabla _{\varvec{\alpha }}q_0 \Vert _{0 ,\varOmega }^2 = C \Vert q_0 \Vert _{Q(h)} ^2, \end{aligned} \end{aligned}$$

(10.9)

where we use $\Vert \nabla _{\varvec{\alpha }}\phi \Vert _{0, \partial K}\le C h_K^{- \frac{1}{2}}\Vert \nabla _{\varvec{\alpha }}\phi \Vert _{0,K}$, for any $\phi = {e^{i{\varvec{\alpha }}\cdot \varvec{x}}}\tilde{\phi }$ and $\tilde{\phi }\in \mathcal {S}_l(K)$ with $C>0$, which we obtain from the trace inequality $\Vert \nabla \tilde{\phi }\Vert _{0, \partial K}\le C h_K^{- \frac{1}{2}}\Vert \nabla \tilde{\phi }\Vert _{0,K}$, with $C>0$. Let ${\varvec{\nu }}_1= - {[[ q_1 ]] _N}$. Using Lemma 5.6, we obtain

$$\begin{aligned} \begin{aligned} B_h(\varvec{{0}},{\varvec{\nu }}_1; q_1)&={\mathfrak {c}}\int _ {{\mathcal {F}_h }}{{h}}^{-1}{[[ q_1 ]] _N}^2 ds \ge {\mathfrak {c}}C_1^2 \Vert q_1\Vert _{Q(h)}^2,\\ \Vert ( \varvec{{0}},{\varvec{\nu }}_1) \Vert _{{\varvec{U}}(h)}&\le \Vert q_1 \Vert _{Q(h)}. \end{aligned} \end{aligned}$$

(10.10)

Let $(\varvec{v}, {\varvec{\nu }})=(\varvec{v}_0,\varvec{{0}})+\delta (\varvec{{0}}, {\varvec{\nu }}_1)$ with $\delta >0$. Since $q_0\in Q^{{\varvec{\alpha }},c}_h$, ${[[ q_0 ]] _N} =\varvec{{0}}$ on ${\mathcal {F}_h }$ and $B_h(\varvec{{0}},{\varvec{\nu }}_1;q_0)=c \int _ {{\mathcal {F}_h }}{{h}}^{-1}{[[ q_0 ]] _N}\cdot \bar{{\varvec{\nu }}}_1 ds =0$, we have

$$\begin{aligned} \begin{aligned} B_h(\varvec{v}, {\varvec{\nu }};q)&= B_h(\varvec{v}_0, \varvec{{0}}; q_0)+ B_h(\varvec{v}_0,\varvec{{0}}; q_1)+\delta B_h(\varvec{{0}}, {\varvec{\nu }}_1; q_1)\\&\ge \Vert q_0 \Vert ^2_{Q(h)} \quad + \delta {\mathfrak {c}}C_1^2 \Vert q_1 \Vert ^2_{Q(h)} - | B_h (\varvec{v}_0,0,q_1)|. \end{aligned} \end{aligned}$$

Using Theorem 10.1 and (10.9), we obtain

$$\begin{aligned} \begin{aligned} | B_h( \varvec{v}_0, \varvec{{0}};q_1)|&\le C \Vert ( \varvec{v}_0,\varvec{{0}}) \Vert _{{\varvec{U}}(h)}\Vert q_1 \Vert _{Q(h)}\\&\le C \zeta \Vert q_0 \Vert _{Q(h)}^2 + \frac{C}{\zeta } \Vert q_1 \Vert _{Q(h)}^2,\\ \end{aligned} \end{aligned}$$

with any $\zeta >0$. Choosing suitable $\delta$ and $\zeta$, we have

$$\begin{aligned} \begin{aligned} B_h(\varvec{v},{\varvec{\nu }}; q ) \ge (1-C \zeta ) \Vert q_0 \Vert _{Q(h)}^2 +(\delta {\mathfrak {c}}C_1^2-\frac{C}{\zeta })\Vert q_1 \Vert _{Q(h)}^2 \ge k_1 \Vert q \Vert _{Q(h)}^2, \end{aligned} \end{aligned}$$

(10.11)

with $k_1>0$. From (10.9) and (10.10), we have

$$\begin{aligned} \begin{aligned} \Vert ( \varvec{v},{\varvec{\nu }}) \Vert _{{\varvec{U}}(h)}=\Vert ( \varvec{v}_0,\varvec{{0}}) \Vert _{{\varvec{U}}(h)}+\delta \Vert ( \varvec{{0}},{\varvec{\nu }}_1) \Vert _{{\varvec{U}}(h)} \le k_2 \Vert q \Vert _{Q(h)}. \end{aligned} \end{aligned}$$

Then the result follows with $k=k_1/k_2$. $\square$

Appendix C: Ellipticity on the Kernel

Lemma 10.4

$$\begin{aligned} \nabla _{\varvec{\alpha }}\times {{\varvec{H}}_{\mathrm {per}}(\mathrm {curl};\varOmega ) }= \nabla _{\varvec{\alpha }}\times {{\varvec{H}}^1_{\mathrm {per}}(\varOmega ) }. \end{aligned}$$

Proof

Let $\varvec{v}\in {{\varvec{H}}_{\mathrm {per}}(\mathrm {curl};\varOmega ) }$. By [16, Theorem 3.1], there exists $\varvec{w}\in {{\varvec{H}}^1_{\mathrm {per}}(\varOmega ) }$ and $\phi \in {H^1_{\mathrm {per}}(\varOmega ) }$ such that

$$\begin{aligned}\nabla _{\varvec{\alpha }}\times \varvec{v}= \nabla _{\varvec{\alpha }}\times \varvec{w}+ \nabla _{\varvec{\alpha }}\phi , \quad \nabla _{\varvec{\alpha }}\cdot \varvec{w}=0.\end{aligned}$$

By Lemma 3.1, since $\nabla _{\varvec{\alpha }}\cdot \nabla _{\varvec{\alpha }}\times \varvec{v}=0$, we obtain $\phi =0$. Therefore

$$\begin{aligned} \nabla _{\varvec{\alpha }}\times \varvec{v}= \nabla _{\varvec{\alpha }}\times \varvec{w}\in \nabla _{\varvec{\alpha }}\times {{\varvec{H}}^1_{\mathrm {per}}(\varOmega ) }. \end{aligned}$$

implying $\nabla _{\varvec{\alpha }}\times {{\varvec{H}}_{\mathrm {per}}(\mathrm {curl};\varOmega ) }\subset \nabla _{\varvec{\alpha }}\times {{\varvec{H}}^1_{\mathrm {per}}(\varOmega ) }$. The other inclusion is obvious as ${{\varvec{H}}^1_{\mathrm {per}}(\varOmega ) }\subset {{\varvec{H}}_{\mathrm {per}}(\mathrm {curl};\varOmega ) }$. $\square$

Proof of Lemma 5.7

Lemma 10.4 implies that $\nabla _{\varvec{\alpha }}\times$ maps ${{\varvec{H}}^1_{\mathrm {per}}(\varOmega ) }$ onto $\nabla _{\varvec{\alpha }}\times {{\varvec{H}}_{\mathrm {per}}(\mathrm {curl};\varOmega ) }$. Let K denote the orthogonal complement of the kernel of $\nabla _{\varvec{\alpha }}\times$ in ${{\varvec{H}}^1_{\mathrm {per}}(\varOmega ) }$. Then, the restriction $\nabla _{\varvec{\alpha }}\times |_{K}$ of $\nabla _{\varvec{\alpha }}\times$ to K also maps ${{\varvec{H}}^1_{\mathrm {per}}(\varOmega ) }$ onto $\nabla _{\varvec{\alpha }}\times {{\varvec{H}}_{\mathrm {per}}(\mathrm {curl};\varOmega ) }$. In addition to being onto, $\nabla _{\varvec{\alpha }}\times |_{K}$ is continuous, one-to-one and has a continuous inverse due to [16, Theorem 3.1]. The operator $R=(\nabla _{\varvec{\alpha }}\times |_{K})^{-1}\nabla _{\varvec{\alpha }}\times$ satisfies the conclusion of the lemma. $\square$

Lemma 10.5

For $\varvec{u}\in {\varvec{L}}^2(\varOmega )$, we have the following estimate for the auxiliary problem (10.12):

$$\begin{aligned} \begin{aligned} \Vert \epsilon ^{-1}\nabla _{\varvec{\alpha }}\times \varvec{z} \Vert _{0 ,\varOmega }+\Vert \varvec{z} \Vert _{0 ,\varOmega } +\Vert \nabla _{\varvec{\alpha }}\times \epsilon ^{-1}\nabla _{\varvec{\alpha }}\times \varvec{z} \Vert _{0 ,\varOmega }\\ +\Vert \nabla _{\varvec{\alpha }}\psi \Vert _{0 ,\varOmega } + \Vert \psi \Vert _{0 ,\varOmega }\le C_m \Vert \varvec{u} \Vert _{0 ,\varOmega }. \end{aligned} \end{aligned}$$

Proof

Taking the periodic boundary conditions into consideration and integrating by parts, we have

$$\begin{aligned} \begin{aligned} \Vert \varvec{u} \Vert _{0 ,\varOmega } ^2&=(\varvec{u},\varvec{u})\\&=\,(\nabla _{\varvec{\alpha }}\times \epsilon ^{-1}\nabla _{\varvec{\alpha }}\times \varvec{z}-\nabla _{\varvec{\alpha }}\psi ,\nabla _{\varvec{\alpha }}\times \epsilon ^{-1}\nabla _{\varvec{\alpha }}\times \varvec{z}-\nabla _{\varvec{\alpha }}\psi )\\&=\,(\nabla _{\varvec{\alpha }}\times \epsilon ^{-1}\nabla _{\varvec{\alpha }}\times \varvec{z},\nabla _{\varvec{\alpha }}\times \epsilon ^{-1}\nabla _{\varvec{\alpha }}\times \varvec{z})+(\nabla _{\varvec{\alpha }}\psi ,\nabla _{\varvec{\alpha }}\psi )\\&-2Re(\nabla _{\varvec{\alpha }}\times \epsilon ^{-1}\nabla _{\varvec{\alpha }}\times \varvec{z},\nabla _{\varvec{\alpha }}\psi )\\&=\,(\nabla _{\varvec{\alpha }}\times \epsilon ^{-1}\nabla _{\varvec{\alpha }}\times \varvec{z},\nabla _{\varvec{\alpha }}\times \epsilon ^{-1}\nabla _{\varvec{\alpha }}\times \varvec{z})+(\nabla _{\varvec{\alpha }}\psi ,\nabla _{\varvec{\alpha }}\psi )\\&-2Re( \epsilon ^{-1}\nabla _{\varvec{\alpha }}\times \varvec{z},\nabla _{\varvec{\alpha }}\times (\nabla _{\varvec{\alpha }}\psi ))\\&=\,\Vert \nabla _{\varvec{\alpha }}\times \epsilon ^{-1}\nabla _{\varvec{\alpha }}\times \varvec{z} \Vert _{0 ,\varOmega }^2+\Vert \nabla _{\varvec{\alpha }}\psi \Vert _{0 ,\varOmega }^2,\\ \end{aligned} \end{aligned}$$

where $\nabla _{\varvec{\alpha }}\times (\nabla _{\varvec{\alpha }}\psi )=\varvec{{0}}$. Combining with the estimate given in Theorem 3.3 gives the result. $\square$

Proof of Theorem 5.3

From the seminorm ellipticity in Lemma 10.2 , it is sufficient to show that there exist $C>0$, such that

$$\begin{aligned} \Vert \varvec{u} \Vert _{0 ,\varOmega } \le C |( \varvec{u},{\varvec{\nu }}) |_{{\varvec{U}}(h)}\quad \forall (\varvec{u},{\varvec{\nu }}) \in {\mathrm {Ker}(B_h)}.\end{aligned}$$

Now fix $(\varvec{u}, {\varvec{\nu }}) \in {\mathrm {Ker}(B_h)}$, and let $(\varvec{z}, \psi )\in {\varvec{V}}\times Q$ satisfying

$$\begin{aligned} \begin{aligned} \nabla _{\varvec{\alpha }}\times \epsilon ^{-1}\nabla _{\varvec{\alpha }}\times \varvec{z}-\nabla _{\varvec{\alpha }}\psi&=\varvec{u},\\ \nabla _{\varvec{\alpha }}\cdot \varvec{z}&=0, \end{aligned} \end{aligned}$$

(10.12)

with periodic boundary conditions. Thereby,

$$\begin{aligned} \begin{aligned} \Vert \epsilon ^{-1}\nabla _{\varvec{\alpha }}\times \varvec{z} \Vert _{0 ,\varOmega }+\Vert \varvec{z} \Vert _{0 ,\varOmega } +\Vert \nabla _{\varvec{\alpha }}\times \epsilon ^{-1}\nabla _{\varvec{\alpha }}\times \varvec{z} \Vert _{0 ,\varOmega }\\ +\,\Vert \nabla _{\varvec{\alpha }}\psi \Vert _{0 ,\varOmega } + \Vert \psi \Vert _{0 ,\varOmega } \le C_m \Vert \varvec{u} \Vert _{0 ,\varOmega }, \end{aligned} \end{aligned}$$

(10.13)

where the detailed derivation of (10.13) is given in Lemma 10.5. Set $\varvec{w}= \epsilon ^{-1} \nabla _{\varvec{\alpha }}\times \varvec{z}$, clearly $\varvec{w}\in {{\varvec{H}}_{\mathrm {per}}(\mathrm {curl};\varOmega ) }$. Then, from Theorem 5.7 and the inequality (10.13) there exists $\varvec{w}_0 \in {{\varvec{H}}^1_{\mathrm {per}}(\varOmega ) }$ such that

$$\begin{aligned} \begin{aligned} \nabla _{\varvec{\alpha }}\times \varvec{w}_0 =&\nabla _{\varvec{\alpha }}\times \varvec{w},\\ \Vert \varvec{w}_0 \Vert _{1,\varOmega } \le C \Vert \varvec{w}\Vert _{{{\varvec{H}}_{\mathrm {per}}(\mathrm {curl};\varOmega ) }} \le&\, C_m \Vert \varvec{u}\Vert _{0, \varOmega } . \end{aligned} \end{aligned}$$

(10.14)

Multiplying the first equation of (10.12) by $\varvec{u}$ and integrating by parts, we obtain

$$\begin{aligned} \begin{aligned} \Vert \varvec{u}\Vert _{0, \varOmega }^2=&\int _ {\varOmega }\varvec{w}_0 \cdot \overline{ \nabla _{{\varvec{\alpha }},h}\times \varvec{u}} d\varvec{x}- \int _ {{\mathcal {F}_h }}\varvec{w}_0 \cdot {\overline{[[ \varvec{u} ]]}_T}d s + \int _ {\varOmega }\psi \overline{\nabla _{{\varvec{\alpha }},h}\cdot \varvec{u}} d\varvec{x}\\&- \int _ {{\mathcal {F}_h }}\{\{ \psi \}\} { \overline{[[ \varvec{u} ]]} }ds. \end{aligned} \end{aligned}$$

Since $(\varvec{u},{\varvec{\nu }}) \in {\mathrm {Ker}(B_h)}$, we choose $\psi _h$ as the $L^2$-projection of $\psi$ in $Q^{\varvec{\alpha }}_h$, then we have $B_h(\varvec{u}, {\varvec{\nu }};\psi _h)=0$. Using the fact that $\psi \in Q$ in the auxiliary problem (10.12) and $[[ \psi ]]_N=0$ on ${\mathcal {F}_h }$,

$$\begin{aligned} \begin{aligned} \Vert \varvec{u}\Vert _{0, \varOmega }^2=&\int _ {\varOmega }\varvec{w}_0 \cdot \overline{ \nabla _{{\varvec{\alpha }},h}\times \varvec{u}} d\varvec{x}- \int _ {{\mathcal {F}_h }}\varvec{w}_0 \cdot {\overline{[[ \varvec{u} ]]}_T} d s + \int _ {\varOmega }(\psi -\psi _h) \overline{\nabla _{{\varvec{\alpha }},h}\cdot \varvec{u}} d\varvec{x}\\&- \int _ {{\mathcal {F}_h }}\{\{ \psi -\psi _h \}\} {\overline{[[ \varvec{u} ]]}} ds - \int _ {{\mathcal {F}_h }}{\mathfrak {c}}{{h}}^{-1} {\varvec{\nu }}\cdot { \overline{[[ \psi -\psi _h ]]}} ds. \end{aligned} \end{aligned}$$

Using (10.14), we have

$$\begin{aligned} \begin{aligned} \left| \int _ {\varOmega }\varvec{w}_0 \cdot \overline{ \nabla _{{\varvec{\alpha }},h}\times \varvec{u}}d\varvec{x}\right| \le \Vert \varvec{w}_0 \Vert _{1,\varOmega }\Vert \nabla _{{\varvec{\alpha }},h}\times \varvec{u}\Vert _{0, \varOmega } \le C_m \Vert \varvec{u}\Vert _{0, \varOmega } | \varvec{u}|_{V(h)}. \end{aligned} \end{aligned}$$

Using trace inequalities and (10.14), we have

$$\begin{aligned} \begin{aligned} \left| \int _ {{\mathcal {F}_h }}\varvec{w}_0 \cdot { \overline{[[ \varvec{u} ]]}_T} d s \right|&\le C \left( \sum _{K \in T_h} h_K \epsilon _K\Vert \varvec{w}_0\Vert _{0, \partial K}^2\right) ^\frac{1}{2}\Vert {{e}}^{-\frac{1}{2}}{{h}}^{-\frac{1}{2}}[[ \varvec{u} ]]_T \Vert _{0,{\mathcal {F}_h }}\\&\le C \Vert \varvec{w}_0\Vert _{1, \varOmega }\Vert {{e}}^{-\frac{1}{2}}{{h}}^{-\frac{1}{2}}[[ \varvec{u} ]]_T \Vert _{0,{\mathcal {F}_h }} \le C \Vert \varvec{u}\Vert _{0, \varOmega } | \varvec{u}|_{V(h)}. \end{aligned} \end{aligned}$$

Since $\psi _h$ is the $L^2$-projection of $\psi$, the third term is zero. Using (10.13), we obtain the following estimate for the last two terms:

$$\begin{aligned}\nonumber \begin{aligned} \left| \int _ {{\mathcal {F}_h }}\{\{ \psi -\psi _h \}\} {\overline{[[ \varvec{u} ]]} }ds \right|&\le C \left( \sum _{K \in T_h} h_k^{-1} \Vert \psi -\psi _h \Vert _{0, \partial K}^2\right) ^\frac{1}{2}\left( \sum _{K \in T_h} \Vert {{h}}^\frac{1}{2}{[[ \varvec{u} ]] _N} \Vert ^2_{0,\partial K} \right) ^\frac{1}{2}\\&\le C \left( \sum _{K \in T_h} h_k^{-1} \Vert \psi -\psi _h \Vert _{0, \partial K}^2\right) ^\frac{1}{2}| \varvec{u}|_{V(h)}\\&\le C \left( \sum _{K \in T_h} \Vert \nabla _{{\varvec{\alpha }},h}\psi \Vert _{0, K}^2 + \Vert \psi \Vert _{0, K}^2 \right) ^\frac{1}{2}| \varvec{u}|_{V(h)}\\&\le C \Vert \varvec{u}\Vert _{0, \varOmega } | \varvec{u}|_{V(h)}. \end{aligned}\\ \begin{aligned} \left| \int _ {{\mathcal {F}_h }}{{h}}^{-1} {\varvec{\nu }}\cdot { \overline{[[ \psi -\psi _h ]]}} ds \right|&\le C \left( \sum _{K \in T_h} h_K^{-1} \Vert \psi -\psi _h \Vert _{0, \partial K}^2\right) ^\frac{1}{2}\left( \int _ {{\mathcal {F}_h }}{{h}}^{-1} |{\varvec{\nu }}|^2 ds \right) ^\frac{1}{2}\\&\le C \Vert \varvec{u}\Vert _{0, \varOmega } \Vert {\varvec{\nu }} \Vert _{M^{\varvec{\alpha }}_h}. \end{aligned} \end{aligned}$$

From the results above, we have $\Vert \varvec{u}\Vert _{0, \varOmega } \le C |( \varvec{u},{\varvec{\nu }}) |_{{\varvec{U}}(h)}$. $\square$

Appendix D: The Convergence of the Operator

Proof of Theorem 5.4

Let $(\varvec{u},p)$ be the analytical solution of (3.7), and $(\varvec{u}_h, {\varvec{\lambda }}_h, p_h)$ be the numerical solution of (5.4). By the triangle inequality and the definition of $\Vert ( \cdot ,\cdot ) \Vert _{{\varvec{U}}(h)}$, we have

$$\begin{aligned} \begin{aligned} \Vert ( \varvec{u}-\varvec{u}_h,{\varvec{\lambda }}_h) \Vert _{{\varvec{U}}(h)} \le \Vert ( \varvec{u}-\varvec{v},{\varvec{\eta }}) \Vert _{{\varvec{U}}(h)}+\Vert ( \varvec{v}-\varvec{u}_h,{\varvec{\eta }}-{\varvec{\lambda }}_h) \Vert _{{\varvec{U}}(h)}, \end{aligned} \end{aligned}$$

(10.15)

for any $(\varvec{v},{\varvec{\eta }}) \in {\varvec{U}}^{\varvec{\alpha }}_h$. First, we take $(\varvec{v},{\varvec{\eta }}) \in {\mathrm {Ker}(B_h)}$. Since $(\varvec{v}-\varvec{u}_h,{\varvec{\eta }}-{\varvec{\lambda }}_h) \in {\mathrm {Ker}(B_h)}$, employing the ellipticity property of Theorem 5.3 and the definition of $R^1_h$, we have

$$\begin{aligned} \begin{aligned} b\Vert ( \varvec{v}-\varvec{u}_h,{\varvec{\eta }}-{\varvec{\lambda }}_h) \Vert _{{\varvec{U}}(h)}^2 \le&\, A_h(\varvec{v}-\varvec{u}_h,{\varvec{\eta }}-{\varvec{\lambda }}_h;\varvec{v}-\varvec{u}_h,{\varvec{\eta }}-{\varvec{\lambda }}_h) \\ =\,&A_h(\varvec{v}-\varvec{u},{\varvec{\eta }};\varvec{v}-\varvec{u}_h,{\varvec{\eta }}-{\varvec{\lambda }}_h) \\&\quad +A_h(\varvec{u}-\varvec{u}_h,-{\varvec{\lambda }}_h;\varvec{v}-\varvec{u}_h,{\varvec{\eta }}-{\varvec{\lambda }}_h) \\ =\,&A_h(\varvec{v}-\varvec{u},{\varvec{\eta }};\varvec{v}-\varvec{u}_h,{\varvec{\eta }}-{\varvec{\lambda }}_h) \\&- B_h(\varvec{v}-\varvec{u}_h,{\varvec{\eta }}-{\varvec{\lambda }}_h;p-p_h) \\&\quad +R^1_h(\varvec{u}-\varvec{u}_h,p-p_h;\varvec{v}-\varvec{u}_h,{\varvec{\eta }}-{\varvec{\lambda }}_h)\\ =\,&A_h(\varvec{v}-\varvec{u},{\varvec{\eta }};\varvec{v}-\varvec{u}_h,{\varvec{\eta }}-{\varvec{\lambda }}_h) \\&- B_h(\varvec{v}-\varvec{u}_h,{\varvec{\eta }}-{\varvec{\lambda }}_h;p-q) +R^1_h(\varvec{u},p;\varvec{v}-\varvec{u}_h,{\varvec{\eta }}-{\varvec{\lambda }}_h)\\ \le&\, a_1 \Vert ( \varvec{v}-\varvec{u}_h,{\varvec{\eta }}-{\varvec{\lambda }}_h) \Vert _{{\varvec{U}}(h)}\Vert ( \varvec{v}-\varvec{u},{\varvec{\eta }}) \Vert _{{\varvec{U}}(h)} \\&{+}\, a_2 \Vert ( \varvec{v}{-}\varvec{u}_h,{\varvec{\eta }}{-}{\varvec{\lambda }}_h) \Vert _{{\varvec{U}}(h)} \Vert p{-}q \Vert _{Q(h)}{+}R^1_h(\varvec{u},p;\varvec{v}{-}\varvec{u}_h,{\varvec{\eta }}{-}{\varvec{\lambda }}_h), \end{aligned} \end{aligned}$$

(10.16)

for any $q\in Q^{\varvec{\alpha }}_h$. Combining (10.15) and (10.16), we have

$$\begin{aligned} \begin{aligned} \Vert ( \varvec{u}-\varvec{u}_h,{\varvec{\lambda }}_h) \Vert _{{\varvec{U}}(h)} \le&\left( {1+\frac{a_1}{b}}\right) \inf _{(\varvec{v},{\varvec{\eta }}) \in {\mathrm {Ker}(B_h)}} \Vert ( \varvec{u}-\varvec{v},{\varvec{\eta }}) \Vert _{{\varvec{U}}(h)}\\&+ \frac{a_2}{b} \inf _{q\in Q^{\varvec{\alpha }}_h}\Vert p-q \Vert _{Q(h)}+\frac{1}{b} \mathcal {R}^1_h(\varvec{u},p) . \end{aligned} \end{aligned}$$

(10.17)

Next, we prove that

$$\begin{aligned} \begin{aligned} \inf _{(\varvec{v},{\varvec{\eta }}) \in {\mathrm {Ker}(B_h)}} \Vert ( \varvec{u}-\varvec{v},{\varvec{\eta }}) \Vert _{{\varvec{U}}(h)} \le \left( {1+\frac{a_2}{k}}\right) \inf _{(\varvec{v},{\varvec{\eta }}) \in {\varvec{U}}^{\varvec{\alpha }}_h} \Vert ( \varvec{u}-\varvec{v},{\varvec{\eta }}) \Vert _{{\varvec{U}}(h)} +\frac{1}{k} \mathcal {R}^2_h(\varvec{u}) . \end{aligned} \end{aligned}$$

(10.18)

Let $(\varvec{v},{\varvec{\eta }}) \in {\varvec{U}}^{\varvec{\alpha }}_h$, and consider the following problem: find $(\varvec{w},\nu ) \in {\varvec{U}}(h)$ such that

$$\begin{aligned} \begin{aligned} B_h(\varvec{w},\nu ;q) =B_h(\varvec{u}-\varvec{v},-{\varvec{\eta }};q) - R^2_h(\varvec{u};q) \qquad \forall q\in Q^{\varvec{\alpha }}_h. \end{aligned} \end{aligned}$$

(10.19)

Problem (10.19) admits a solution in ${\varvec{U}}(h)$ that is unique up to elements in ${\mathrm {Ker}(B_h)}$. The discrete inf-sup condition of Theorem 5.2 guarantees the existence of a solution $(\varvec{w},\nu ) \in {\varvec{U}}(h)$ satisfying

$$\begin{aligned} \begin{aligned} \Vert ( \varvec{w},\nu ) \Vert _{{\varvec{U}}(h)}&\le \frac{1}{k} \left( {\sup _{q\in Q^{\varvec{\alpha }}_h} \frac{B_h(\varvec{u}-\varvec{v},-{\varvec{\eta }};q ) }{\Vert q \Vert _{Q(h)}} + \sup _{q\in Q^{\varvec{\alpha }}_h}\frac{R^2_h(\varvec{u};q)}{\Vert q \Vert _{Q(h)}} }\right) \\&\le \frac{a_2}{k}\Vert ( \varvec{u}-\varvec{v},{\varvec{\eta }}) \Vert _{{\varvec{U}}(h)} +\frac{1}{k}\mathcal {R}^2_h(\varvec{u}) ,\\ \end{aligned} \end{aligned}$$

(10.20)

where we have used the continuity of $B_h(\cdot ,\cdot ;\cdot )$, the definition of the norm $\Vert ( \cdot ,\cdot ) \Vert _{{\varvec{U}}(h)}$, and the definition of $\mathcal {R}^2_h(\cdot )$. From (10.19), $B_h(\varvec{w}+\varvec{v},\nu +{\varvec{\eta }};q) =0$, for any $q\in Q^{\varvec{\alpha }}_h$, so that $(\varvec{w}+\varvec{v},\nu +{\varvec{\eta }}) \in {\mathrm {Ker}(B_h)}$. Therefore, since

$$\begin{aligned} \Vert ( \varvec{u}-(\varvec{v}+\varvec{w}),{\varvec{\eta }}+ \nu ) \Vert _{{\varvec{U}}(h)} \le \Vert ( \varvec{u}-\varvec{v},{\varvec{\eta }}) \Vert _{{\varvec{U}}(h)} +\Vert ( \varvec{w},\nu ) \Vert _{{\varvec{U}}(h)}, \end{aligned}$$

for any $(\varvec{v},{\varvec{\eta }})\in {\varvec{U}}(h)$, taking into account (10.20), we obtain (10.18). This, together with (10.17), yields

$$\begin{aligned} \begin{aligned} \Vert ( \varvec{u}-\varvec{u}_h,{\varvec{\lambda }}_h) \Vert _{{\varvec{U}}(h)} \le&\, C \Big ( \inf _{(\varvec{v},{\varvec{\eta }}) \in {\varvec{U}}^{\varvec{\alpha }}_h} \Vert ( \varvec{u}-\varvec{v},{\varvec{\eta }}) \Vert _{{\varvec{U}}(h)}\\&+\inf _{q \in Q^{\varvec{\alpha }}_h} \Vert p-q \Vert _{Q(h)} + \mathcal {R}^1_h(\varvec{u},p) + \mathcal {R}^2_h(\varvec{u}) \Big ) , \end{aligned} \end{aligned}$$

where the constant C depends on $a_1$, $a_2$ and $k_1$. Choosing ${\varvec{\eta }}=\varvec{{0}}$ gives the error bound for $(\varvec{u}-\varvec{u}_h,{\varvec{\lambda }}_h)$.

We now turn to the bound for $p-p_h$. Again by the triangle inequality, we have

$$\begin{aligned} \begin{aligned} \Vert p-p_h \Vert _{Q(h)} \le \Vert p-q \Vert _{Q(h)} +\Vert q-p_h \Vert _{Q(h)}, \end{aligned} \end{aligned}$$

(10.21)

for any $q \in Q^{\varvec{\alpha }}_h$. Since

$$\begin{aligned} \begin{aligned} A_h(\varvec{u}-\varvec{u}_h,-{\varvec{\lambda }}_h;\varvec{v},{\varvec{\eta }}) +B_h(\varvec{v},{\varvec{\eta }};p-q) +B_h(\varvec{v},{\varvec{\eta }};q-p_h) =R^1_h(\varvec{u},p;\varvec{v},{\varvec{\eta }}), \end{aligned} \end{aligned}$$

for any $(\varvec{v},{\varvec{\eta }}) \in {\varvec{U}}(h)$, the discrete inf-sup condition of $B_h(\cdot ,\cdot ;\cdot )$ gives

$$\begin{aligned} \begin{aligned} \Vert q-p_h \Vert _{Q(h)}&\le \frac{1}{k} \sup _{(\varvec{{0}},\varvec{{0}}) \not =(\varvec{v},{\varvec{\eta }})\in {\varvec{U}}^{\varvec{\alpha }}_h} \frac{B_h(\varvec{v},{\varvec{\eta }};q-p_h) }{\Vert ( \varvec{v},{\varvec{\eta }}) \Vert _{{\varvec{U}}(h)}}\\&=\frac{1}{k}\sup _{(\varvec{{0}},\varvec{{0}}) \not =(\varvec{v},{\varvec{\eta }})\in {\varvec{U}}^{\varvec{\alpha }}_h} \frac{-A_h(\varvec{u}{-}\varvec{u}_h,{-}{\varvec{\lambda }}_h;\varvec{v},{\varvec{\eta }}) {-}B_h(\varvec{v},{\varvec{\eta }};p-q) +R^1_h(\varvec{u},p;\varvec{v},{\varvec{\eta }})}{\Vert ( \varvec{v},{\varvec{\eta }}) \Vert _{{\varvec{U}}(h)}}\\&\le \frac{a_1}{k} \Vert ( \varvec{u}{-}\varvec{u}_h,{\varvec{\lambda }}_h) \Vert _{{\varvec{U}}(h)} +\frac{a_2}{k} \Vert p-q \Vert _{Q(h)} + \frac{1}{k}\mathcal {R}^1_h(\varvec{u},p) . \end{aligned} \end{aligned}$$

This, together with (10.21), gives a bound for $p-p_h$. $\square$

Proof of Lemma 5.8

Let $(\varvec{u},p)$ be the analytical solution of (3.7). Let ${\Pi _{{\varvec{V}}^{\varvec{\alpha }}_h}}$ be the $L^2$-projection onto ${\varvec{V}}^{\varvec{\alpha }}_h$. For $(\varvec{v},{\varvec{\eta }}) \in {\varvec{U}}^{\varvec{\alpha }}_h$,

$$\begin{aligned} \begin{aligned} \left| R^1_h(\varvec{u},p;\varvec{v},{\varvec{\eta }}) \right|&=\left| A_h(\varvec{u},\varvec{{0}};\varvec{v},{\varvec{\eta }}) +B_h(\varvec{v},{\varvec{\eta }};p) -a_h(\varvec{u},\varvec{v}) -b_h(\varvec{v},p) \right| \\&= \left| -\int _ {\varOmega }\overline{ \mathcal {L}({\varvec{v}}) } \cdot (\epsilon ^{-1}\nabla _{\varvec{\alpha }}\times \varvec{u}) d\varvec{x}+ \int _ {{\mathcal {F}_h }}\overline{[[ \varvec{v} ]]}_T\cdot (\epsilon ^{-1}\nabla _{\varvec{\alpha }}\times \varvec{u})d\varvec{x} \right| \\&= \left| -\int _ {\varOmega }\overline{ \mathcal {L}({\varvec{v}}) } \cdot {\Pi _{{\varvec{V}}^{\varvec{\alpha }}_h}}(\epsilon ^{-1}\nabla _{\varvec{\alpha }}\times \varvec{u}) d\varvec{x}+ \int _ {{\mathcal {F}_h }}\overline{[[ \varvec{v} ]]}_T\cdot (\epsilon ^{-1}\nabla _{\varvec{\alpha }}\times \varvec{u})ds \right| \\&= \left| \int _ {{\mathcal {F}_h }}\{\{ \epsilon ^{-1}\nabla _{\varvec{\alpha }}\times \varvec{u}-{\Pi _{{\varvec{V}}^{\varvec{\alpha }}_h}}(\epsilon ^{-1}\nabla _{\varvec{\alpha }}\times \varvec{u}) \}\} \cdot \overline{[[ \varvec{v} ]]}_Tds \right| \\&\le C \left( {\sum _{K \in {\mathcal {T}_h }} h_K \Vert \epsilon ^{-1}\nabla _{\varvec{\alpha }}\times \varvec{u}-{\Pi _{{\varvec{V}}^{\varvec{\alpha }}_h}}(\epsilon ^{-1}\nabla _{\varvec{\alpha }}\times \varvec{u}) \Vert _{0 ,\partial K}^2}\right) ^\frac{1}{2}\Vert ( \varvec{v},{\varvec{\eta }}) \Vert _{{\varvec{U}}(h)}\\&\le C h^{\min \{s, k+1\}} \Vert \epsilon ^{-1}\nabla _{\varvec{\alpha }}\times \varvec{u} \Vert _{s, \varOmega } \Vert ( \varvec{v},{\varvec{\eta }}) \Vert _{{\varvec{U}}(h)}. \end{aligned} \end{aligned}$$

Similarly, for $q \in Q^{\varvec{\alpha }}_h$,

$$\begin{aligned} \begin{aligned} \left| R^2_h(\varvec{u};q) \right|&=\left| \overline{B_h(\varvec{u},\varvec{{0}};q) } - \overline{b_h(\varvec{u},q) } + c_h(p,q) \right| \\&\le \left| \int _ {\varOmega }\overline{\varvec{u}}\cdot { \mathcal {M}({q}) } d\varvec{x}- \int _ {{\mathcal {F}_h }}\overline{\{\{ \varvec{u} \}\}} \cdot {[[ q ]] _N} ds \right| \\&= \left| \int _ {\varOmega }\overline{{\Pi _{{\varvec{V}}^{\varvec{\alpha }}_h}}\varvec{u}} \cdot { \mathcal {M}({q}) } d\varvec{x}- \int _ {{\mathcal {F}_h }}\overline{ \{\{ \varvec{u} \}\}} \cdot {[[ q ]] _N} ds \right| \\&\le \left| \int _ {{\mathcal {F}_h }}\overline{\{\{ \varvec{u}-{\Pi _{{\varvec{V}}^{\varvec{\alpha }}_h}}\varvec{u} \}\}} \cdot {[[ q ]] _N} ds \right| \\&\le C \left( {\sum _{K\in {\mathcal {T}_h }}h_K \Vert \varvec{u}-{\Pi _{{\varvec{V}}^{\varvec{\alpha }}_h}}\varvec{u} \Vert ^2_{0 ,\partial K} }\right) ^\frac{1}{2}\Vert q \Vert _{Q(h)} \\&\le C h^{\min \{s, k+1\}}\Vert \varvec{u} \Vert _{s, \varOmega } \Vert q \Vert _{Q(h)}. \end{aligned} \end{aligned}$$

$\square$