Remark 3.1. From (3.2), it is not hard to see that both $b_1$ and $b_2$ will vanish if $\mu^-=\mu^+$ and $\beta^-=\beta^+$ . In this case, $\mathcal{C}_T$ reduces to an identity operator, and thus the proposed IFE functions are consistent with standard Nédélec functions. This feature is particularly useful for moving interface problems.

Proof. See Appendix A.1

Proof. See Appendix A.2.

Remark 3.2. Theorem 3.1 means that the unisolvence always holds regardless of interface location and the parameters $\beta$ , $\mu$ if the maximal angle is not greater than $\pi/2$ . Besides, if the maximal angle is close or equal to $\pi/2$ , the constants in Theorem 3.2 do not blow up. But for an obtuse triangle, it is possible to find some interface configuration and $\beta$ , $\mu$ to violate the unisolvence, and in this case, the constants in Theorem 3.2 may blow up.

Proof. The local results $\nabla\widetilde{S}_h(T)\subset \mathcal{IND}_h(T)$ are trivial due to the jump conditions. The global result is non-trivial due to the discontinuities on interface edges. For each interface edge $e\in\mathcal{E}_h$ , we need to show $\int_e[\nabla v_h\cdot\textbf{t}_e]ds=0$ , $v_h\in \widetilde{S}_{h}(\Omega)$ . Let $A_1$ and $A_2$ be the two nodes of e, let $\textbf{t}_e$ be oriented from $A_1$ to $A_2$ and let $T_1$ and $T_2$ be its neighbour elements. Then the continuity at the interface intersection point of e yields

(3.13) \begin{equation}\int_e \nabla v_h|_{T_1}\cdot \textbf{t}_e ds = \int_e \partial_{\textbf{t}_e} v_h|_{T_1} ds = v_h|_{T_1}(A_1) - v_h|_{T_1}(A_2).\end{equation}

A similar identity also holds for $T_2$ . Therefore, the continuity at mesh nodes yields the desired result.

Proof. The result is trivial on non-interface elements, and, on interface elements, it follows from the argument similar to (3.13) and the nodal continuity of $\tilde{I}_{h} v$ and $v\in\widetilde{H}^2_0(\Omega)$ .

Remark 3.3. By similar arguments to Theorem 3.3, the following commuting properties also hold

(3.14) \begin{equation}\mathbb{\Pi}_h \circ\nabla = \nabla \circ \mathbb{I}_h \,\,\, \text{on} \, \widetilde{S}_{h,0}(\Omega) \,\,\,\,\, \text{and} \,\,\,\,\, \mathbb{\Pi}^{-1}_h \circ\nabla = \nabla \circ \mathbb{I}^{-1}_h \,\,\, \text{on} \, S_{h,0}(\Omega).\end{equation}

Proof. Let us focus on an interface element T, and recall that $\boldsymbol{\psi}_i$ , $i=1,2,3$ , are the $\textbf{H}(\text{curl})$ IFE shape functions. By the identity in (3.6b), we let $\tau=\mu^{-1}\text{curl}\,\boldsymbol{\psi}_i$ , $i=1,2,3$ . For each $\textbf{z}_h\in \mathcal{ND}_{h,0}(\Omega)$ , we note that $\mu^{-1}\text{curl}\, \mathbb{\Pi}^{-1}_{h} \textbf{z}_h$ is a constant, and then the integration by parts yields

(3.15) \begin{equation}\begin{split}\mu^{-1} \text{curl}\, \mathbb{\Pi}^{-1}_{h} \textbf{z}_h = \sum_{i=1}^3 \int_{e_i} \textbf{z}_h\cdot\textbf{t}_i ds \mu^{-1} \text{curl}\, \boldsymbol{\psi}_i = \int_{\partial T} \textbf{z}_h\cdot\textbf{t} \tau ds = \int_T \text{curl}\,\textbf{z}_h dX \,\tau = |T| \tau \text{curl}\,\textbf{z}_h.\end{split}\end{equation}

The identity above shows that $\text{curl}\,\textbf{z}_h=0$ if and only if $ \text{curl}\, \mathbb{\Pi}^{-1}_{h} \textbf{z}_h = 0$ , and similar results also hold on non-interface elements. Thus, we have the desired result.

Remark 3.4. We note that $\text{Ker(curl)}\cap\mathcal{ND}_{h,0}(\Omega)$ and $\text{Ker(curl)}\cap\mathcal{IND}_{h,0}(\Omega)$ consist of piecewise constant vectors but functions of the latter one on interface elements are piecewise constant vectors on each subelement. If $\beta^-=\beta^+$ , then the piecewise constant vector on each interface element will reduce to a single constant vector. Therefore, by the self-preserving property of $\mathbb{\Pi}_h$ , there holds $\mathbb{\Pi}_h|_{\textrm{Ker(curl)}\cap\mathcal{IND}_{h,0}(\Omega)} = \mathcal{I}$ where $\mathcal{I}$ is the identity.

Proof. By Lemma 2, it is easy to see $\nabla \widetilde{S}_{h,0}(\Omega)\subset \text{Ker(curl)}\cap\mathcal{IND}_{h,0}(\Omega)$ . As for the reverse direction, for each $\textbf{z}_h\in\text{Ker(curl)}\cap\mathcal{IND}_{h,0}(\Omega)$ , Theorem 3.4 suggests ${\Pi}_h\textbf{z}_h\in\text{Ker(curl)}\cap\mathcal{ND}_{h,0}(\Omega)$ . Then, due to the exact sequence for $S_{h,0}(\Omega)$ and $\mathcal{ND}_{h,0}(\Omega)$ , there exists $s_h\in S_{h,0}(\Omega)$ such that $\nabla s_h = \mathbb{\Pi}_h\textbf{z}_h$ . Now we take $\tilde{s}_h=\mathbb{I}^{-1}_hs_h$ and use (3.14) to obtain $\nabla \tilde{s}_h = \nabla \mathbb{I}^{-1}_h s_h = \mathbb{\Pi}^{-1}_h \nabla s_h = \mathbb{\Pi}^{-1}_h \mathbb{\Pi}_h\textbf{z}_h = \textbf{z}_h$ which has finished the proof.

Proof. Given each $\textbf{v}_h\in\mathcal{ND}_h(T_{\epsilon})$ , since $\textbf{v}_h$ is simply a polynomial, using (4.3), the trace inequality and the inverse inequality, we have

\begin{equation*}\begin{split}& h^{1/2}_T |\textbf{v}_h(X_m)\cdot\bar{\textbf{n}} | \leqslant \|\textbf{v}_h\cdot\bar{\textbf{n}} \|_{L^2\left(\Gamma^{T_{\epsilon}}_h\right)} \lesssim h^{-1/2}_T \| \textbf{v}_h \|_{L^2(T_{\epsilon})}, \\&\| \textbf{v}_h \cdot\bar{\textbf{t}} \|_{L^2\left(\Gamma^{T_{\epsilon}}_h\right)} \lesssim h_T^{-1/2}\| \textbf{v}_h \|_{L^2(T_{\epsilon})}, \,\,\, \| \text{curl} \, \textbf{v}_h \|_{L^2\left(\Gamma^{T_{\epsilon}}_h\right)} \lesssim h_T^{-1/2} \| \text{curl} \, \textbf{v}_h \|_{L^2(T_{\epsilon})}.\end{split}\end{equation*}

This yields $\vert\kern-1pt\vert\kern-1pt\vert\textbf{v}_h\vert\kern-1pt\vert\kern-1pt\vert_{T_{\epsilon}} \lesssim h^{-1/2}_T \| \textbf{v}_h \|_{\textbf{H}(\text{curl};\,T_{\epsilon})}$ . For the reverse direction, by the Taylor expansion, we have

(4.7) \begin{equation}\textbf{v}_h(X) = \textbf{v}_h(X_m) + \nabla \textbf{v}_h(X_m)(X-X_m) = \textbf{v}_h(X_m)\cdot\bar{\textbf{n}} + \textbf{v}_h(X_m)\cdot\bar{\textbf{t}} - \frac{1}{2}\text{curl}\,\textbf{v}_h \, \textbf{R}( X - X_m),\end{equation}

where $\textbf{R}=[0,1;\,{-}1,0]\in\mathbb{R}^{2\times2}$ , since $\text{curl}\,\textbf{v}_h$ is a constant. Then we have

(4.8) \begin{equation}\| \textbf{v}_h \|_{L^2(T_{\epsilon})} \lesssim h_T \vert{ \textbf{v}_h(X_m)\cdot\bar{\textbf{n}} } + h_T \vert{ \textbf{v}_h(X_m)\cdot\bar{\textbf{t}} } + h_T^2\vert{ \text{curl}\,\textbf{v}_h }.\end{equation}

We note that $\textbf{v}_h\cdot\bar{\textbf{t}}$ can be understood as a polynomial defined on $\Gamma^{T_{\epsilon}}_h$ , and $|\text{curl}\,\textbf{v}_h|$ is a constant, and thus they can be simply bounded by the standard trace inequality from $\Gamma^{T_{\epsilon}}_h$ to $T_{\epsilon}$ . Therefore, we have

(4.9) \begin{equation}\| \textbf{v}_h \|_{L^2(T_{\epsilon})} \lesssim h_T \vert{ \textbf{v}_h(X_m)\cdot\bar{\textbf{n}} } + h_T^{1/2} \| \textbf{v}_h \cdot\bar{\textbf{t}} \|_{L^2\left(\Gamma^{T_{\epsilon}}_h\right)} + h^{3/2}_T\| \text{curl}\,\textbf{v}_h \|_{L^2\left(\Gamma^{T_{\epsilon}}_h\right)} \lesssim h^{1/2}_T \vert\kern-1pt\vert\kern-1pt\vert\textbf{v}_h\vert\kern-1pt\vert\kern-1pt\vert_{T_{\epsilon}}.\end{equation}

In addition, there also holds

\begin{equation*}\| \text{curl} \, \textbf{v}_h \|_{L^2(T_{\epsilon})} = \left(|T|/|\Gamma^{T_{\epsilon}}_h|\right)^{1/2} \| \text{curl} \, \textbf{v}_h \|_{L^2\left(\Gamma^{T_{\epsilon}}_h\right)} \lesssim h^{1/2}_T \| \text{curl} \, \textbf{v}_h \|_{L^2\left(\Gamma^{T_{\epsilon}}_h\right)}\end{equation*}

because of the mesh regularity and (4.3), which has finished the proof.

Proof. Since the proof is basically the same as Lemma 3.3 in [Reference Guo and Lin35], we omit it here.

Proof. We need to estimate each term in the definition (4.6). First, by (4.3) and the trace inequality, we have

(4.12) \begin{equation}| \boldsymbol{\eta}_h(X_m)\cdot\bar{\textbf{n}} | = \bigg\vert{ \left(\Pi_{T_{\epsilon}}\textbf{u}^+_E - \frac{\beta^-}{\beta^+} \Pi_{T_{\epsilon}}\textbf{u}^-_E\right)(X_m)\cdot\bar{\textbf{n}} } \bigg\vert \lesssim h^{-1/2}_T \bigg\|{ \left(\Pi_{T_{\epsilon}}\textbf{u}^+_E - \frac{\beta^-}{\beta^+} \Pi_{T_{\epsilon}}\textbf{u}^-_E\right)\cdot\bar{\textbf{n}} }\bigg\|_{L^2\left(\Gamma^{T_{\epsilon}}_h\right)},\end{equation}

where we have also used (3.1c) and $\left(\Pi_{T_{\epsilon}}\textbf{u}^+_E - \frac{\beta^-}{\beta^+} \Pi_{T_{\epsilon}}\textbf{u}^-_E\right)\cdot\bar{\textbf{n}}|_{\Gamma^{T_{\epsilon}}_h}\in\mathbb{P}_1\left(\Gamma^{T_{\epsilon}}_h\right)$ . Since $\Pi_{T_{\epsilon}}\textbf{u}^+_E - \frac{\beta^-}{\beta^+} \Pi_{T_{\epsilon}}\textbf{u}^-_E $ is a polynomial, applying the trace inequality for polynomials [Reference Warburton and Hesthaven64], we induce from (4.12) that

\begin{equation*} \begin{split}& |\boldsymbol{\eta}_h(X_m)\cdot\bar{\textbf{n}}| \lesssim h^{-1}_T \bigg\|{ \left(\Pi_{T_{\epsilon}}\textbf{u}^+_E - \frac{\beta^-}{\beta^+} \Pi_{T_{\epsilon}}\textbf{u}^-_E \right)\cdot\bar{\textbf{n}} }\bigg\|_{L^2(T_{\epsilon})} \\ \leqslant & h^{-1}_T \left( \|{ \left(\Pi_{T_{\epsilon}}\textbf{u}^+_E - \textbf{u}^+_E \right)\cdot\bar{\textbf{n}} }\|_{L^2(T_{\epsilon})} + \bigg\|{ \left( \frac{\beta^-}{\beta^+} \Pi_{T_{\epsilon}}\textbf{u}^-_E - \frac{\beta^-}{\beta^+} \textbf{u}^-_E \right)\cdot\bar{\textbf{n}} }\bigg\|_{L^2(T_{\epsilon})} + \bigg\|{ \left( \frac{\beta^-}{\beta^+} \textbf{u}^-_E - \textbf{u}^+_E \right)\cdot\bar{\textbf{n}} }\bigg\|_{L^2(T_{\epsilon})} \right).\end{split}\end{equation*}

Applying the estimate (2.5) for $\Pi_{T_{\epsilon}}$ on $T_{\epsilon}$ and (4.10b) to the estimate above, we have

(4.13) \begin{equation}|\boldsymbol{\eta}_h(X_m)\cdot\bar{\textbf{n}}| \lesssim \left( \| \textbf{u}^+_E \|_{\textbf{H}^1(\text{curl};\,T_{\epsilon})} + \| \textbf{u}^-_E \|_{\textbf{H}^1(\text{curl};\,T_{\epsilon})} \right).\end{equation}

By similar arguments, using (4.10a) and (4.10c), we have the following estimates

(4.14) \begin{equation}\begin{split}\| \boldsymbol{\eta}_h \cdot\bar{\textbf{t}} \|_{L^2\left(\Gamma^{T_{\epsilon}}_h\right)} + \| \text{curl}\,\boldsymbol{\eta}_h \|_{L^2\left(\Gamma^{T_{\epsilon}}_h\right)} \lesssim h^{1/2}_T \left( \| \textbf{u}^+_E \|_{\textbf{H}^1(\text{curl};\,T_{\epsilon})} + \| \textbf{u}^-_E \|_{\textbf{H}^1(\text{curl};\,T_{\epsilon})} \right).\end{split}\end{equation}

Then the desired result follows from (4.13) to (4.14) together with the assumption that $T_{\epsilon}\subseteq\omega_T$ .

Proof. By the definition in (4.4), the estimate of $J^-_{T} \textbf{u} - \textbf{u}^-_E= \Pi_{T_{\epsilon}} \textbf{u}^-_E - \textbf{u}^-_E $ in $T^-_{\epsilon}$ directly follows from applying (2.5) to $\Pi_{T_{\epsilon}}$ which is simply the right-hand side of the diagram in Figure 8. So we only need to estimate the error on $T^+_{\epsilon}$ . By (4.5), we first have the following error decomposition

(4.16) \begin{equation}\begin{split}\| J^+_{T} \textbf{u} - \textbf{u}^+_E \|_{\textbf{H}(\text{curl};\,T_{\epsilon})}\leqslant \| \boldsymbol{\eta}_h \|_{\textbf{H}(\text{curl};\,T_{\epsilon})} +\| \Pi_{T_{\epsilon}} \textbf{u}^+_E - \textbf{u}^+_E \|_{\textbf{H}(\text{curl};\,T_{\epsilon})}.\end{split}\end{equation}

Again the second term in the right-hand side of (4.16) follows from applying (2.5) to $\Pi_{T_{\epsilon}}$ . For the first term, using the norm equivalence in Lemaa 3 together with the estimate in Lemaa 5, we have

(4.17) \begin{equation}\begin{split}\| \boldsymbol{\eta}_h \|_{\textbf{H}(\text{curl};\,T_{\epsilon})} \lesssim h^{1/2}_T \vert\kern-1pt\vert\kern-1pt\vert\boldsymbol{\eta}_h\vert\kern-1pt\vert\kern-1pt\vert_{T_{\epsilon}} \lesssim h_T \left( \| \textbf{u}_E^+ \|_{\textbf{H}^1(\text{curl};\,\omega_T)} + \| \textbf{u}_E^- \|_{\textbf{H}^1(\text{curl};\,\omega_T)} \right),\end{split}\end{equation}

which has finished the proof by the assumption that $T_{\epsilon}\subseteq\omega_T$ .

Proof. Given an interface element T with the edges $e_i$ , $i=1,2,3$ , the triangular inequality yields

(4.20) \begin{equation}\| \widetilde{\Pi}_{T} \textbf{u} - \textbf{u} \|_{L^2(T)} \leqslant \| \widetilde{\Pi}_{T} \textbf{u} - J_{T}\textbf{u} \|_{L^2(T)} + \| J_{T}\textbf{u} - \textbf{u} \|_{L^2(T)}.\end{equation}

The estimate of the second term in (4.20) directly follows from Theorem 4.1. So we only need to estimate the first term. Note that $\widetilde{\Pi}_{T} \textbf{u} - J_{T}\textbf{u} = \widetilde{\Pi}_{T}\left( \textbf{u} - J_{T}\textbf{u} \right)$ . Using the IFE shape functions, we can write

(4.21) \begin{equation}\widetilde{\Pi}_{T}(\textbf{u} - J_{T}\textbf{u}) = \sum_{i=1}^3 \int_{e_i} (\textbf{u} - J_{T}\textbf{u})\cdot\textbf{t}_i ds \boldsymbol{\psi}_i = \sum_{i=1}^3 \int_{e_i} \textbf{w}\cdot\textbf{t}_i ds \boldsymbol{\psi}_i .\end{equation}

By Hölder’s inequality and the boundedness (3.6a), we have

(4.22) \begin{equation}\begin{split}\| \widetilde{\Pi}_{T} (\textbf{u} - J_{T}\textbf{u}) \|_{L^2(T)} & \leqslant \sum_{i=1}^3 \bigg\vert{ \int_{e_i} \textbf{w}\cdot\textbf{t}_i ds } \bigg\vert\| \boldsymbol{\psi}_i \|_{L^2(T)} \lesssim h^{1/2}_T \sum_{i=1}^3 \left( \| \textbf{w}^-\cdot\textbf{t} \|_{L^2(e_i)} + \| \textbf{w}^+\cdot\textbf{t} \|_{L^2(e_i)} \right).\end{split}\end{equation}

Then, by the scaling arguments for Nédélec elements in [Reference Ciarlet and Zou25, Lemma 3.2], we have

(4.23) \begin{equation}\begin{split}\| \textbf{w}^- \cdot\textbf{t} \|_{L^2(e_i)} & = \|{ (\Pi_{T_{\epsilon}} \textbf{u}^-_E - \textbf{u}^-_E)\cdot\textbf{t} }\|_{L^2(e_i)} \lesssim h_T^{1/2} \left( \| \textbf{u}_E^+ \|_{\textbf{H}^1(\text{curl};\,T_{\epsilon})} + \| \textbf{u}_E^- \|_{\textbf{H}^1(\text{curl};\,T_{\epsilon})} \right).\end{split}\end{equation}

In addition, the triangular inequality yields

(4.24) \begin{equation}\begin{split}\| \textbf{w}^+\cdot\textbf{t} \|_{L^2(e_i)} \leqslant \|{ \boldsymbol{\eta}_h\cdot\textbf{t} }\|_{L^2(e_i)}+ \|{ ( \Pi_{T_{\epsilon}} \textbf{u}^+_E - \textbf{u}^+_E )\cdot\textbf{t} }\|_{L^2(e_i)}.\end{split}\end{equation}

Since $\boldsymbol{\eta}_h$ is a polynomial, by the trace inequality for polynomials [Reference Warburton and Hesthaven64], the estimate in Lemma 5 and the norm equivalence in Lemma 3, we have

(4.25) \begin{equation}\begin{split}\|{ \boldsymbol{\eta}_h\cdot\textbf{t} }\|_{L^2(e_i)} & \lesssim h^{-1/2}_T \|{ \boldsymbol{\eta}_h }\|_{L^2(T)} \lesssim \vert\kern-1pt\vert\kern-1pt\vert\boldsymbol{\eta}_h\vert\kern-1pt\vert\kern-1pt\vert_{T_{\epsilon}} \lesssim h_T^{1/2} \left( \| \textbf{u}_E^+ \|_{\textbf{H}^1(\text{curl};\,T_{\epsilon})} + \| \textbf{u}_E^- \|_{\textbf{H}^1(\text{curl};\,T_{\epsilon})} \right).\end{split}\end{equation}

The estimate for $ \|{ (\Pi_{T_{\epsilon}} \textbf{u}^+_E - \textbf{u}^+_E)\cdot\textbf{t} }\|_{L^2(e_i)}$ in (4.24) is the same as (4.23). Putting these two estimates into (4.24) and substituting (4.24) with (4.23) into (4.22), we have the estimate of $\| \widetilde{\Pi}_{T}(\textbf{u} - J_T \textbf{u}) \|_{L^2(T)}$ . Then the proof is finished by using $T_{\epsilon}\subseteq\omega_T$ .

Proof. Similar to (4.20), we have

(4.28) \begin{equation}\| \text{curl}(\widetilde{\Pi}_{T} \textbf{u} - \textbf{u}) \|_{L^2(T)} \leqslant \| \text{curl}( \widetilde{\Pi}_{T} \textbf{u} - J_{T}\textbf{u} ) \|_{L^2(T)} + \| \text{curl}(J_{T}\textbf{u} - \textbf{u}) \|_{L^2(T)}.\end{equation}

The second term in (4.28) directly follows from Theorem 4.1. For the first term, we also consider the piecewise-defined function $\textbf{w}$ in (4.18). By the identity in (3.6b), we let $\tau=\mu^{-1}\text{curl}\,\boldsymbol{\psi}_i$ , $i=1,2,3$ . Then the similar derivation to (3.15), i.e., the integration by parts on $T^{\pm}_h$ , leads to

(4.29) \begin{equation}\begin{split}\frac{1}{\mu} \text{curl}\, \widetilde{\Pi}_{T}(\textbf{u} - J_{T} \textbf{u})= \int_{\partial T} (\textbf{u} - J_{T} \textbf{u})\cdot\textbf{t}_i ds \, \tau= \int_{\partial T} \textbf{w}\cdot\textbf{t}_i ds \, \tau = \int_{\Gamma^T_h} [\textbf{w}\cdot\bar{\textbf{t}}]_{\Gamma^T_h} ds \,\tau + \int_T \text{curl}\,\textbf{w} dX \,\tau,\end{split}\end{equation}

where $\bar{\textbf{t}}$ denotes the unit tangential vector to $\Gamma^T_h$ in the clockwise orientation of $T^-_h$ . Then, applying the integration by parts on the subregion $\widetilde{T}$ , using the jump conditions (3.1a), (1.1b), Hölder’s inequality and the first one in (2.1), we have

(4.30) \begin{equation}\begin{split}\bigg\vert{ \int_{\Gamma^T_h} [\textbf{w}\cdot\bar{\textbf{t}}]_{\Gamma^T_h} ds }\bigg\vert = \bigg\vert{ \int_{\widetilde{T}} \text{curl}(\textbf{u}^-_E - \textbf{u}^+_E ) dX }\bigg\vert \lesssim h^{3/2}_T \left( \| \text{curl}\, \textbf{u}^-_E \|_{L^2(\widetilde{T})} + \| \text{curl}\, \textbf{u}^+_E \|_{L^2(\widetilde{T})} \right).\end{split}\end{equation}

Also, by Hölder’s inequality and Theorem 4.1, we have

(4.31) \begin{equation}\bigg\vert{ \int_T \text{curl}\,\textbf{w} dX }\bigg\vert \leqslant \| \text{curl}\,\textbf{w} \|_{L^2(T)} |T|^{1/2} \lesssim h^2_T \left( \| \textbf{u}_E^+ \|_{\textbf{H}^1(\text{curl};\,\omega_T)} + \| \textbf{u}_E^- \|_{\textbf{H}^1(\text{curl};\,\omega_T)} \right).\end{equation}

Moreover, by (4.30) we have $\|\tau\|_{L^2(T)} \lesssim h^{-1}_T$ . Now putting it together with (4.30) and (4.31) into (4.29), we have the estimate of $\| \mu^{-1} \text{curl}\,\widetilde{\Pi}_{T} (\textbf{u} - J_{T} \textbf{u}) \|_{L^2(T)}$ which yields the desired result with (4.28).

Proof. By the first estimate in (2.1), there exists a $\delta$ -strip $S_{\delta}$ such that $\cup_{T\in\mathcal{T}^i_h}\widetilde{T}\subseteq S_{\delta}$ with $\delta \lesssim h^2$ . Then, the desired result follows from Lemma 6, Theorems 4.2, 4.3 and the standard estimates (2.5) on non-interface elements with the finite overlapping property of patches and the bounds of extensions by Theorem 2.1.

Proof. For simplicity, we only prove the first one in (5.5a). Let $D=[x_{D,1},x_{D,2}]^t$ in the explicit formula in (3.2). Note that $\text{curl}\,\textbf{v}$ is a constant. Then if $|A_1D|\ge\frac{1}{2}|A_1A_2|$ and $|A_1E|\ge\frac{1}{2}|A_1A_3|$ , the formula of $b_1$ in (3.2) leads to

(5.6) \begin{equation}|b_1| \lesssim | \text{curl}\,\textbf{v}_h | \lesssim h^{-1}_T \| \text{curl}\,\textbf{v}_h \|_{L^2(T)} \lesssim h^{-2}_T \| \textbf{v}_h \|_{L^2(T)} \lesssim h^{-2}_T \| \textbf{v}_h \|_{L^2\left(T^-_h\right)},\end{equation}

where in the last two inequalities above we have used the inverse inequality and norm equivalence (5.4a). In addition, we note that there exists a constant C such that $|\triangle A_1DE|/|DE|\ge Ch$ where C is independent of interface location. So by the trace inequalities for polynomials [Reference Warburton and Hesthaven64] we have $|\textbf{v}_h(X_m)\cdot\bar{\textbf{n}}|\lesssim h^{-1}_T\|\textbf{v}_h \|_{L^2(\triangle A_1DE)} \simeq h^{-1}_T \| \textbf{v}_h \|_{L^2\left(T^-_h\right)}$ . Putting this estimate and (5.6) into the formula of $b_2$ in (3.2), we obtain

(5.7) \begin{equation}|b_2| \lesssim h^{-1}_T \| \textbf{v}_h \|_{L^2\left(T^-_h\right)} + h_T |b_1| \lesssim h^{-1}_T \| \textbf{v}_h \|_{L^2\left(T^-_h\right)} .\end{equation}

Now using the expression of $\mathcal{C}_T(\textbf{v}_h)$ in (3.2), we have

(5.8) \begin{equation} \| \mathcal{C}_T(\textbf{v}_h) \|_{L^2\left(T^-_h\right)} \lesssim \| \textbf{v}_h \|_{L^2\left(T^-_h\right)} + h^2_T |b_1| + h_T |b_2| \lesssim \| \textbf{v}_h \|_{L^2\left(T^-_h\right)},\end{equation}

which has finished the proof of the first one in (5.5a). The argument for other estimates is similar.

Proof. Without loss of generality, we only consider the case that $|A_1D|\ge\frac{1}{2}|A_1A_2|$ and $|A_1E|\ge\frac{1}{2}|A_1A_3|$ as shown by Case 1 in Figure 9. On the interface edge $A_1A_2$ , the standard trace inequality [Reference Warburton and Hesthaven64] yields

(5.10) \begin{equation}\| \textbf{z}_h \|_{L^2(A_1D)} \lesssim h^{-1/2}_T \| \textbf{z}_h \|_{L^2(\triangle A_1DE)} \,\,\, \text{and} \,\,\, \| \textbf{z}_h \|_{L^2(DA_2)} \lesssim h^{-1/2}_T \| \textbf{z}_h \|_{L^2(\triangle DA_2E)},\end{equation}

since the distance from the point E to $A_1A_2$ is bounded below regardless of interface location, which yields the estimate on this edge. The argument for the interface edge $A_1A_3$ is similar. For the non-interface edge $A_2A_3$ , by the standard trace inequality and (5.5a), we obtain

(5.11) \begin{equation}\begin{split}\| \textbf{z}_h^+ \|_{L^2(A_2A_3)} & \lesssim h^{-1/2}_T \| \textbf{z}^+_h \|_{L^2(T)} = h^{-1/2}_T ( \| \mathcal{C}^{-1}_T(\textbf{z}^-_h) \|_{L^2\left(T^-_h\right)} + \| \textbf{z}^+_h \|_{L^2(T^+_h)} ) \\& \lesssim h^{-1/2}_T ( \| \textbf{z}^-_h \|_{L^2\left(T^-_h\right)} + \| \textbf{z}^+_h \|_{L^2(T^+_h)} ) ,\end{split}\end{equation}

which yields the desired result. The argument for the Case 2 in Figure 9 is similar and relies on (5.5b).

Proof. These inequalities for non-interface elements are trivial, and we only consider the interface elements here. For (5.12a), using the argument similar to (4.22) with (3.6a), we obtain

(5.13) \begin{equation}\|{\mathbb{\Pi}_{h}\textbf{z}_h}\|_{L^2(T)} = \bigg\|{\sum_{i=1}^3 \int_{e_i} \textbf{z}_h\cdot\textbf{t}_i ds \boldsymbol{\psi}_i}\bigg\|_{L^2(T)}\lesssim \sum_{i=1}^3 h^{1/2}_T \| \textbf{z}_h \|_{L^2(e_i)}.\end{equation}

Applying the trace inequality in Theorem 5.2 to (5.13) yields the right inequality of (5.12a). The left one can be obtained by applying this argument to $\mathbb{\Pi}^{-1}_{h}$ . In addition, (5.12b) is a direct consequence of the identity in (3.15).

Proof. The proof is lengthy, and we decompose it into several steps.

Step 1. We first focus on $\textbf{u}_h\in\mathcal{ND}_{h,0}(\Omega)$ . We need to construct a function $\textbf{v}_h\in\mathcal{ND}_{h,0}(\Omega^{\Gamma,+}_h\cup\Omega^i_h)\cap\text{Ker(curl)}$ such that its trace on $\gamma$ matches $\textbf{u}_h$ except one edge denoted by $e^{\ast}$ of the polyline $\gamma$ in (5.16). Note that $\textbf{v}_h$ should have vanishing trace on $\partial( \Omega^{\Gamma,+}_h\cup\Omega^i_h)$ , i.e., the two polyline near $\gamma$ .

Let us denote the edges on $\gamma$ by $e_1$ , $e_2$ , …, $e_N$ with the clockwise orientation with the nodes $X_1$ , $X_2$ ,…, $X_N$ . Without loss of generality, we assume $e^{\ast}=e_N$ , as shown in the right plot in Figure 10. We consider a linear finite element function $v_h$ such that $v_h(X_1)=0$ , $v_h(X_{n+1})= \sum_{j=1}^{n} \int_{e_j} \textbf{u}_h\cdot\textbf{t} ds$ , $n=1,2,...,N-1$ , and it vanishes at all other nodes. Then, $v_h\in S_{h,0}(\Omega^{\Gamma,+}_h\cup\Omega^i_h)$ , and $\textbf{v}_h=\nabla v_h\in \mathcal{ND}_{h,0}(\Omega^{\Gamma,+}_h\cup\Omega^i_h)$ . Thus, we clearly have $\int_{e_n}\textbf{v}_h\cdot\textbf{t} ds = \int_{e_n}\textbf{u}_h\cdot\textbf{t} ds$ , $n=1,2,...,N-1$ . Note that $\int_{\gamma}\textbf{v}_h\cdot\textbf{t} ds =0$ but $\textbf{u}_h$ may not have this property, so $\int_{e^{\ast}} \textbf{v}_h\cdot\textbf{t} ds$ may not equal $\int_{e^{\ast}} \textbf{u}_h\cdot\textbf{t} ds$ .

Step 2. Let us assume that the closed polyline $\gamma$ partitions the whole domain $\Omega$ into $\Omega_{h,+}$ and $\Omega_{h,-}$ where we have $\Omega_{h,+}\subseteq\Omega^+$ and $\Omega^-\subseteq\Omega_{h,-}$ . Since the triangulation is regular, $\gamma$ is a Lipschitz curve, and thus both $\Omega_{h,\pm}$ have Lipschitz boundary. Denote $\textbf{k}_h = \textbf{u}_h - \textbf{v}_h$ and $\textbf{k}^{\pm}_h = (\textbf{u}_h - \textbf{v}_h)|_{\Omega_{h,\pm}}$ . By the discussion in Step 1, $\textbf{k}^{+}_h$ has the zero trace on $\partial\Omega$ and $\gamma\backslash e^{\ast}$ , and $\textbf{k}^{-}_h$ simply has the zero trace on $\gamma\backslash e^{\ast}$ . Then, we apply the discrete regular decomposition of Theorem 11 in [Reference Hiptmair and Pechstein43] (also see the related discussion in [Reference Hiptmair40, Reference Hiptmair and Pechstein42]) to $\textbf{k}^{\pm}_h$ on $\Omega_{h,\pm}$ which gives

\begin{equation*}\textbf{k}^{\pm}_h = \textbf{R}^{\ast}_{\pm} \textbf{z}^{\pm}_h + \textbf{r}^{\pm}_h + \textbf{h}^{\pm}_h, \,\,\,\, \text{on} \, \Omega_{h,\pm},\end{equation*}

where the regular components $\textbf{z}^{\pm}_h\in \left[S_{h}(\Omega_{h,\pm})\right]^2$ , the curl-free components $\textbf{h}^{\pm}_h\in \mathcal{ND}_{h}(\Omega_{h,\pm})\cap\text{Ker(curl)}$ , the reminders $\textbf{r}^{\pm}_h\in\mathcal{ND}_{h}(\Omega_{h,\pm})$ and $\textbf{R}^{\ast}_{\pm}\,:\,S_{h}(\Omega_{h,\pm})\rightarrow \mathcal{ND}_{h}(\Omega_{h,\pm})$ are special local projection operators preserving zero boundary conditions. Moreover, these components inherit the zero traces of $\textbf{k}^{\pm}_h$ , namely, $\textbf{z}^{+}_h$ , $\textbf{r}^{+}_h$ and $\textbf{h}^{+}_h$ also have the corresponding zero traces on $\partial\Omega\cup(\gamma\backslash e^{\ast})$ and the “ $-$ ” components have the zero traces on $\gamma\backslash e^{\ast}$ . Furthermore, since $e^{\ast}$ is a single edge, the continuity suggests that $\textbf{z}^{\pm}_h$ and $\textbf{h}^{\pm}_h$ must have the zero traces on the whole $\gamma$ . Then we have $\textbf{r}^+_h\cdot \textbf{t}=\textbf{k}_h\cdot\textbf{t} = \textbf{r}^-_h\cdot \textbf{t}$ on every edge of $\gamma$ . So all these three decomposed components must be tangentially continuous on $\gamma$ . Therefore, we can put these components together to define $\textbf{z}_h = \textbf{z}^{\pm}_h$ in $\Omega_{h,\pm}$ belonging to $\left[S_{h,0}(\Omega)\right]^2$ , $\textbf{h}_h = \textbf{h}^{\pm}_h$ in $\Omega_{h,\pm}$ belonging to $\mathcal{ND}_{h,0}(\Omega)\cap\text{Ker(curl)}$ and $\textbf{r}_h = \textbf{r}^{\pm}_h$ in $\Omega_{h,\pm}$ belonging to $\mathcal{ND}_{h,0}(\Omega)$ which satisfy

(5.19) \begin{equation}\textbf{u}_h - \textbf{v}_h = \textbf{k}_h = \textbf{R}^{\ast}\textbf{z}_h + \textbf{r}_h + \textbf{h}_h\end{equation}

with $\textbf{R}^{\ast} = \textbf{R}^{\ast}_{\pm}$ in $\Omega_{h,\pm}$ . By this definition, using the estimates of the components $\textbf{z}^{\pm}_h$ , $\textbf{h}^{\pm}_h$ and $\textbf{r}^{\pm}_h$ in in [Reference Hiptmair and Pechstein43, Theorem 11], and noticing that $\textbf{v}_h$ is curl-free, we have

(5.20a) \begin{align} \| \nabla \textbf{z}_h \|_{L^2(\Omega)} \lesssim \| \text{curl} \, \textbf{u}_h \|_{L^2(\Omega)}, \,\,\| \textbf{r}_h \|_{L^2(\Omega)} \lesssim h \| \text{curl}\, \textbf{u}_h \|_{L^2(\Omega)} \,\,\, \text{and} \,\,\, \| \textbf{h}_h \|_{L^2(\Omega)} \lesssim \| \text{curl}\, \textbf{u}_h \|_{L^2(\Omega)},\end{align}

and for each patch $\omega_T$ of an element T, there holds

(5.20b) \begin{align}&\| \textbf{R}^{\ast} \textbf{z}_h \|_{L^2(T)} \lesssim \| \textbf{z}_h \|_{L^2(\omega_T)} + h_T \| \text{curl}\, \textbf{z}_h \|_{L^2(\omega_T)}.\end{align}

Step 3. We estimate $\mathbf{ R}^{\ast}\textbf{z}_h$ on $\Omega^{\Gamma}_h$ , i.e., the subdomain given in (5.15). By (5.20b), we obtain

(5.21) \begin{equation}\begin{split}\| \textbf{R}^{\ast} \textbf{z}_h \|^2_{L^2\left(\Omega^{\Gamma}_h\right)} &= \sum_{T\in\mathcal{T}^{\Gamma}_h} \| \textbf{R}^{\ast} \textbf{z}_h \|^2_{L^2(T)}\lesssim \sum_{T\in\mathcal{T}^{\Gamma}_h} \left( \| \textbf{z}_h \|_{L^2(\omega_T)} + h_T \| \text{curl}\,\textbf{z}_h \|_{L^2(\omega_T)} \right)^2.\end{split}\end{equation}

For each patch $\omega_T$ , $T\in\mathcal{T}^{\Gamma}_h$ , without loss of generality we assume $T\notin{\mathcal{T}}^i_h$ , i.e., it is not an interface element, and thus there is at least one interface element denoted by T’ such that T’ and T share at least one node denoted by A as shown in the right plot in Figure 10. Since T’ has one node on $\gamma$ and $\textbf{z}_h$ must vanish at this node, then the estimate of $\textbf{z}_h$ on T’ is straightforward through the Poincaré-type inequality:

(5.22) \begin{equation}\| \textbf{z}_h \|_{L^2(T^{\prime})} \lesssim h_{T^{\prime}} \| \nabla \textbf{z}_h \|_{L^2(T^{\prime})}.\end{equation}

Then, we also have the estimate for $|\textbf{z}_h(A)|$ through the trace inequality and (5.22)

(5.23) \begin{equation}| \textbf{z}_h(A) | \lesssim h^{-1}_{T^{\prime}}\| \textbf{z}_h \|_{L^2(T^{\prime})} \lesssim \| \nabla \textbf{z}_h \|_{L^2(T^{\prime})}.\end{equation}

In addition, on T we can write $\textbf{z}_h$ as $\textbf{z}_h(X)=\textbf{z}_h(A) + \nabla\textbf{z}_h(X-A)$ where $\nabla\textbf{z}_h$ is a 2-by-2 constant matrix. Thus (5.23) together with the continuity at A leads to

(5.24) \begin{equation}\| \textbf{z}_h \|_{L^2(T)} \lesssim h_T (|\textbf{z}_h(A)| + \| \nabla \textbf{z}_h \|_{L^2(T)} ) \lesssim h_T \| \nabla \textbf{z}_h \|_{L^2(T^{\prime}\cup T)} \lesssim h \| \nabla\textbf{z}_h \|_{L^2(\omega_T)} .\end{equation}

Furthermore, we note that any element T” in $\omega_T$ must share at least one node with T denoted by B as shown in the right plot in Figure 10 for an example. Then the similar argument to (5.23) and (5.24) gives the estimate for $\| \textbf{z}_h \|_{L^2(T^{\prime\prime})}$ . The results above give the estimate of $\textbf{z}_h$ on $\omega_T$ . Applying it to (5.21) and using the finite overlapping property together with the first estimate in (5.20a), we obtain

(5.25) \begin{equation}\begin{split}\| \textbf{R}^{\ast} \textbf{z}_h \|_{L^2\left(\Omega^{\Gamma}_h\right)} \lesssim \sum_{T\in\mathcal{T}^{\Gamma}_h} h\left( \| \nabla \textbf{z}_h \|_{L^2(\omega_T)} + \| \text{curl}\,\textbf{z}_h \|_{L^2(\omega_T)} \right) \lesssim h \| \text{curl}\,\textbf{z}_h \|_{L^2(\Omega)}.\end{split}\end{equation}

Therefore, by (5.25) and the second estimate in (5.20a), setting $\textbf{u}^{\ast}_h = \mathbf{ R}^{\ast}_h\textbf{z}_h + \textbf{r}_h$ and ${\mathop{\textbf{u}}\limits^{\tiny\circ}}_h=\textbf{v}_h + \textbf{h}_h$ fulfills the decomposition (5.17) and (5.18) for the case $\textbf{u}_h\in\mathcal{ND}_{h,0}(\Omega)$ .

Step 4. Finally, if $\textbf{u}_h\in\mathcal{IND}_{h,0}(\Omega)$ , we have $\mathbb{\Pi}_h\textbf{u}_h\in\mathcal{ND}_{h,0}(\Omega)$ , and then the previous analysis shows the existence of functions $\textbf{w}^{\ast}_h \in\mathcal{ND}_{h,0}(\Omega)$ and ${\mathop{\textbf{w}}\limits^{\tiny\circ}}_h\in\mathcal{ND}_{h,0}(\Omega)\cap\text{Ker(curl)}$ such that $\mathbb{\Pi}_h\textbf{u}_h= \textbf{w}^{\ast}_h + {\mathop{\textbf{w}}\limits^{\tiny\circ}}_h$ . So we obtain $\textbf{u}_h = \textbf{u}^{\ast}_h + {\mathop{\textbf{u}}\limits^{\tiny\circ}}_h$ with $\textbf{u}^{\ast}_h = \mathbb{\Pi}^{-1}_h\textbf{w}^{\ast}_h$ and ${\mathop{\textbf{u}}\limits^{\tiny\circ}}_h = \mathbb{\Pi}^{-1}_h{\mathop{\textbf{w}}\limits^{\tiny\circ}}_h$ . Here, $\textbf{u}^{\ast}_h$ satisfies (5.18) thanks to Theorem 5.3 and ${\mathop{\textbf{u}}\limits^{\tiny\circ}}_h\in\mathcal{IND}_{h,0}(\Omega)$ thanks to the curl-free-preserving property of the isomorphism in Theorem 3.4.

Proof. It is trivial on non-interface elements. On an interface element, we note that $\mu^{-1}\text{curl}\,\textbf{z}_h$ is a constant, and then the integration by part yields

(5.27) \begin{equation}\begin{split} \mu^{-1}\text{curl}\,\textbf{z}_h \int_{\partial T} \mathbb{\Pi}_{h}\textbf{z}_h\cdot\textbf{t} \, ds = \mu^{-1}\text{curl}\, \textbf{z}_h \int_{\partial T} \textbf{z}_h\cdot\textbf{t} \, ds = \int_T \mu^{-1} \text{curl}\,\textbf{z}_h\,\text{curl}\,\textbf{z}_h dX,\end{split}\end{equation}

where we have also used the continuous tangential jump condition (3.1a).

Proof. The argument is technical based on direct calculation, so we put it in Appendix B.

Proof. First of all, Theorem 5.4 directly yields

(5.29) \begin{equation}(\mu^{-1} \,\text{curl}\,\textbf{u}_h, \text{curl}\,\mathbb{\Pi}_h\textbf{u}_h)_{L^2(\Omega)} = (\mu^{-1} \,\text{curl}\,\textbf{u}_h, \text{curl}\,\textbf{u}_h)_{L^2(\Omega)} \ge 1/\min\{\mu^-,\mu^+\} \| \text{curl}\,\textbf{u}_h \|^2_{L^2(\Omega)}.\end{equation}

Since $\mathbb{\Pi}_h\textbf{u}_h=\textbf{u}_h$ on $\Omega\backslash\Omega^{\Gamma}_h$ , we certainly have

(5.30) \begin{equation}(\beta\, \textbf{u}_h, \mathbb{\Pi}_h\textbf{u}_h)_{L^2\left(\Omega\backslash\Omega^{\Gamma}_h\right)} \ge \min\{\beta^-,\beta^+\} \| \textbf{u}_h \|^2_{L^2\left(\Omega\backslash\Omega^{\Gamma}_h\right)}.\end{equation}

As for $\Omega^{\Gamma}_h$ , applying the decomposition in Lemma 9, the estimate in Lemma 10 and the stability in Theorem 5.3 together with the arithmetic inequality, for h sufficiently small we have

(5.31) \begin{equation}\begin{split}& (\beta\,(\textbf{u}^{\ast}_h+{\mathop{\textbf{u}}\limits^{\tiny\circ}}_h), \mathbb{\Pi}_h(\textbf{u}^{\ast}_h+{\mathop{\textbf{u}}\limits^{\tiny\circ}}_h))_{L^2\left(\Omega^{\Gamma}_h\right)} \\& \quad = (\beta \textbf{u}^{\ast}_h, \textbf{u}^{\ast}_h)_{L^2\left(\Omega^{\Gamma}_h\right)} + (\beta\,{\mathop{\textbf{u}}\limits^{\tiny\circ}}_h, \mathbb{\Pi}_h{\mathop{\textbf{u}}\limits^{\tiny\circ}}_h)_{L^2\left(\Omega^{\Gamma}_h\right)} \\& \qquad + (\beta \textbf{u}^{\ast}_h, \mathbb{\Pi}_h\textbf{u}^{\ast}_h - \textbf{u}^{\ast}_h)_{L^2\left(\Omega^{\Gamma}_h\right)} + (\beta\,\textbf{u}^{\ast}_h, \mathbb{\Pi}_h{\mathop{\textbf{u}}\limits^{\tiny\circ}}_h)_{L^2\left(\Omega^{\Gamma}_h\right)} + (\beta\,{\mathop{\textbf{u}}\limits^{\tiny\circ}}_h, \mathbb{\Pi}_h\textbf{u}^{\ast}_h)_{L^2\left(\Omega^{\Gamma}_h\right)} \\& \quad \ge \min\{\beta^-,\beta^+\} \| \textbf{u}^{\ast}_h \|^2_{L^2\left(\Omega^{\Gamma}_h\right)} + C_4 \| {\mathop{\textbf{u}}\limits^{\tiny\circ}}_h \|^2_{L^2\left(\Omega^{\Gamma}_h\right)} \\ & \qquad - \max\{\beta^-,\beta^+\}\left( 2C^2_3C_2 h^2 \| \text{curl}\, \textbf{u}_h \|^2_{L^2(\Omega)} + C_2C_3 h \| \text{curl}\, \textbf{u}_h \|^2_{L^2(\Omega)} + C_2C_3 h\| {\mathop{\textbf{u}}\limits^{\tiny\circ}}_h \|^2_{L^2\left(\Omega^{\Gamma}_h\right)}\right) \\ & \quad \ge \min\{\beta^-,\beta^+\} \| \textbf{u}^{\ast}_h \|^2_{L^2\left(\Omega^{\Gamma}_h\right)} + (C_4 - hC_2C_3\max\{\beta^-,\beta^+\} ) \| {\mathop{\textbf{u}}\limits^{\tiny\circ}}_h \|^2_{L^2\left(\Omega^{\Gamma}_h\right)} \\ &\qquad - \max\{\beta^-,\beta^+\}C_2C_3 (2C_3h+1) h \| \text{curl}\, \textbf{u}_h \|^2_{L^2(\Omega)} \\ & \quad \ge \min\{\beta^-,\beta^+,C_4\}/2 \| \textbf{u}_h \|^2_{L^2\left(\Omega^{\Gamma}_h\right)} - \max\{\beta^-,\beta^+\}C_2C_3 (2C_3h+1) h \| \text{curl}\, \textbf{u}_h \|^2_{L^2(\Omega)} ,\end{split}\end{equation}

where $C_2$ , $C_3$ and $C_4$ inherit from Theorem 5.3, Lemmas 9 and 10, respectively. Let $C_5= \max\{\beta^-,\beta^+\}$ $C_2C_3 (2C_3h+1)$ . Noticing that ${\mathop{\textbf{u}}\limits^{\tiny\circ}}_h$ is curl-free, we finally obtain from Theorem 5.4 and (5.29)–(5.31) that

(5.32) \begin{equation}\begin{split}a_h(\textbf{u}_h,\mathbb{\Pi}_h\textbf{u}_h) & = (\mu\,\text{curl}\,\textbf{u}_h, \text{curl}\,\mathbb{\Pi}_h\textbf{u}_h)_{L^2(\Omega)} + (\beta\,\textbf{u}_h, \mathbb{\Pi}_h \textbf{u}_h)_{L^2\left(\Omega\backslash\Omega^{\Gamma}_h\right)} + (\beta\,\textbf{u}_h, \mathbb{\Pi}_h \textbf{u}_h)_{L^2\left(\Omega^{\Gamma}_h\right)} \\& \ge ( 1/\min\{\mu^-,\mu^+\}- C_5 h) \| \text{curl}\, \textbf{u}_h \|^2_{L^2(\Omega)} \\& \quad + \min\{\beta^-,\beta^+,C_4\}/2 \| \textbf{u}_h \|^2_{L^2\left(\Omega^{\Gamma}_h\right)} + \min\{\beta^-,\beta^+\} \| \textbf{u}_h \|^2_{L^2\left(\Omega\backslash\Omega^{\Gamma}_h\right)} \ge C_s \| \textbf{u}_h \|^2_{\textbf{H}(\text{curl};\,\Omega)},\end{split}\end{equation}

which yields the desired result for h sufficiently small.

Remark 5.1. (Comments on the analysis of the inf-sup stability.) If $\beta^-=\beta^+$ , Remark 3.4 indicates the curl-free subspaces of FE and IFE spaces are identical. So we have a Poincaré-type inequality:

(5.33) \begin{equation}\| \textbf{u}_h - \mathbb{\Pi}_h\textbf{u}_h \|_{L^2(T)} \lesssim h_T \| \textrm{curl}\, \textbf{u}_h \|_{L^2(T)}, \,\,\, \textbf{u}_h\in\mathcal{IND}_h(T).\end{equation}

Then the analysis of the inf-sup stability is straightforward by using (5.33) to estimate the term $(\beta\textbf{u}_h,\mathbb{\Pi}_h\textbf{u}_h)_{L^2(\Omega)}$ . However, the estimate (5.33) is only true for continuous conductivity. In fact, $\mathcal{IND}_h(T)$ cannot recover the local spaces of constant vectors when the conductivity is discontinuous.

By inspecting the proof of Theorem 5.5, one critical ingredient is the inf-sup stability on the curl-free subspace, i.e., Lemma 10. In this lemma, the Cartesian-mesh assumption is for relatively easy calculation, and we expect that some other bounds can be also obtained for various-shaped elements through a similar derivation, but we note that the critical upper bound value should depend on the mesh geometry. In fact, from (B.4) and the related calculation, the constant $C_4$ in Lemma 10 can be specified as $C_4 = C^{\prime}_4(10.65- \max\{\beta^-/\beta^+,\beta^+/\beta^-\})$ where $C^{\prime}_4$ is independent of the mesh size, interface location and parameters. So $C_4$ may be close to 0 if the contrast of $\beta$ gets close to this critical upper bound, which causes the loss of coercivity. Indeed, numerical results suggest that $(\beta\textbf{u}_h,\mathbb{\Pi}_h\textbf{u}_h)_{L^2(\Omega)}$ fails to satisfy the inf-sup condition when the contrast is beyond this bound.

Nevertheless, in numerical experiments, we have not observed any instability issue for large contrast of conductivity. So for analysis, $\mathbb{\Pi}_h$ may not be a suitable operator to generate the test function, and some more appropriate test functions are demanded. Due to the exact sequence in Theorem 3.5, this is highly related to the inf-sup stability for the PG-IFE method for the $H^1$ -elliptic interface problems [Reference Hou, Song, Wang and Zhao44] which remains open for years. Lemma 10 gives the estimate under some conditions.

Furthermore, we note that the condition of h being sufficiently small seems to be rather essential not just for analysis, as the matrix $\mathbf{ A}+\mathbf{ A}^t$ may have negative eigenvalues on coarse meshes for certain interface shape where $\mathbf{ A}$ is the resulting matrix given by (5.34). But again, this does not mean the original inf-sup condition needs the requirement of fine meshes, as a more appropriate test function may be chosen, see the discussion above.

Proof. Based on (5.1) and (5.3), we use (5.35b) and (5.36b) to obtain

(5.37) \begin{equation} \inf_{\textbf{u}_h\in\mathcal{IND}_{h,0}(\Omega)} \sup_{\textbf{v}_h\in\mathcal{ND}_{h,0}(\Omega)} \frac{a(\textbf{u}_h,\textbf{v}_h)}{ \| \textbf{u}_h \|_{L^2(\Omega)} \| \textbf{v}_h \|_{L^2(\Omega)}} \ge C_s m^2_2 C^{-2}_2 c^2_2, a(\textbf{u}_h,\textbf{v}_h) \leqslant C_1 M^2_2 c^{-2}_2C^2_2 h^{-2} \| \textbf{u}_h \|_{L^2(\Omega)} \| \textbf{v}_h \|_{L^2(\Omega)}. \end{equation}

Then, applying Theorem 3.1 in [Reference Ern and Guermond30] together with (5.35a) and (5.36a), we arrive at

(5.38) \begin{equation}\kappa_2({\textbf{A}}) \leqslant \left( C_1M^2_1 M^2_2 C^5_2 \right) \left(C_s m^2_1 m^2_2 c^5_2 \right)^{-1} h^{-2}.\end{equation}