By the $L_\eta^2(\Omega)$-orthogonality of the decomposition (52), there holds $\|\text{grad }\varphi\|_0 \le \eta_b^{-\frac{1}{2}} \|\eta^{\frac{1}{2}}\mathbf{v}_h\|_0$, $\|\text{curl }\psi\|_0 \le \eta_\#^{-\frac{1}{2}} \|\eta^{\frac{1}{2}}\mathbf{v}_h\|_0$, and $\|\text{gr\v{a}d }\theta\|_0 \le \eta_b^{-\frac{1}{2}} \|\eta^{\frac{1}{2}}\text{gr\v{a}d }\theta\|_0 \le \eta_b^{-\frac{1}{2}} \|\eta^{\frac{1}{2}}\mathbf{v}_h\|_0$. Plugging the two estimates (54) and (55) into (53), we eventually obtain (51). $\square$

**Remark 16** (Useful variant). Recall the definition (48) of the hybrid counterpart $\underline{\mathbf{X}}_h^\ell$ of the space $\mathbf{H}(\text{curl}; \Omega)$. Following Remark 6, assume that $\underline{\mathbf{v}}_h \in \underline{\mathbf{X}}_h^\ell$ is such that

$$ (\eta \mathbf{v}_h, \mathbf{z})_{\Omega} = 0 \quad \forall \mathbf{z} \in \text{grad} \left( H^1(\Omega) \cap L_0^2(\Omega) \right)^{\perp \eta} \oplus \mathbf{H}_n(\Omega). \quad (56) $$

Then, revisiting the proof of Theorem 15, one can show that

$$ \|\eta^{\frac{1}{2}} \mathbf{v}_h\|_0 \le \eta_{\#}^{\frac{1}{2}} |\underline{\mathbf{v}}_h|_{\text{curl},h}. \quad (57) $$

# 5 Discrete Maxwell compactness

Let $\ell, m, p \in \mathbb{N}$ be three given polynomial degrees.

## 5.1 Discrete differential operators

Recall the definitions (30) of the discrete space $\underline{\mathbf{X}}_h^\ell$, (34) of the local Hermitian forms $s_{\text{curl},T}$ and $s_{\text{div},T}$, and (35)–(36) of the hybrid semi-norms $|\cdot|_{\text{curl},h}$ and $|\cdot|_{\text{div},h}$.

We define the (global) rotational reconstruction operator $C_h^m: \underline{\mathbf{X}}_h^\ell \to \mathcal{P}^m(\mathcal{T}_h)$ through its local restrictions $C_T^m: \underline{\mathbf{X}}_T^\ell \to \mathcal{P}^m(T)$ to any mesh cell $T \in \mathcal{T}_h$: for all $\underline{\mathbf{v}}_T \in \underline{\mathbf{X}}_T^\ell$, $C_T^m(\underline{\mathbf{v}}_T) \in \mathcal{P}^m(T)$ is the unique solution to

$$ (C_T^m(\underline{\mathbf{v}}_T), \mathbf{q})_T = (\mathbf{v}_T, \text{curl } \mathbf{q})_T - \sum_{F \in \mathcal{T}_h} \varepsilon_{T,F} (\mathbf{v}_{F,\tau}, \mathbf{n}_F \times (\mathbf{q}|_F \times \mathbf{n}_F))_F \quad \forall \mathbf{q} \in \mathcal{P}^m(T), \quad (58) $$

where, as now standard, $\mathbf{n}_F \times (\mathbf{q}|_F \times \mathbf{n}_F)$ is identified to its two-dimensional (tangential) counterpart. Performing an integration by parts of the term $(\mathbf{v}_T, \text{curl } \mathbf{q})_T$, and using a discrete trace inequality along with Lemma 9, it is an easy matter to prove that

$$ \sum_{T \in \mathcal{T}_h} \left( \|C_T^m(\underline{\mathbf{v}}_T)\|_{0,T}^2 + s_{\text{curl},T}(\underline{\mathbf{v}}_T, \underline{\mathbf{v}}_T) \right) \le |\underline{\mathbf{v}}_h|^2_{\text{curl},h} \quad \forall \underline{\mathbf{v}}_h \in \underline{\mathbf{X}}_h^\ell. \quad (59) $$

If $m \ge \ell - 1$, and $\mathcal{Q}^\ell(F)$ (already satisfying (31)) also satisfies $\mathcal{P}^{\ell-1}(F) \subset \mathcal{Q}^\ell(F)$ for all $F \in \mathcal{F}_h$, one can also prove that $|\underline{\mathbf{v}}_h|^2_{\text{curl},h} \lesssim \sum_{T \in \mathcal{T}_h} (\|C_T^m(\underline{\mathbf{v}}_T)\|_{0,T}^2 + s_{\text{curl},T}(\underline{\mathbf{v}}_T, \underline{\mathbf{v}}_T))$.

In the same vein, we define the (global) divergence reconstruction operator $D_h^p: \underline{\mathbf{X}}_h^\ell \to \mathcal{P}^p(\mathcal{T}_h)$ through its local restrictions $D_T^p: \underline{\mathbf{X}}_T^\ell \to \mathcal{P}^p(T)$ to any mesh cell $T \in \mathcal{T}_h$: for all $\underline{\mathbf{v}}_T \in \underline{\mathbf{X}}_T^\ell$, $D_T^p(\underline{\mathbf{v}}_T) \in \mathcal{P}^p(T)$ is the unique solution to

$$ (D_T^p(\underline{\mathbf{v}}_T), \mathbf{q})_T = -(\eta_T \mathbf{v}_T, \text{grad } \mathbf{q})_T + \sum_{F \in \mathcal{T}_h} \varepsilon_{T,F} ([\eta \mathbf{v}]_{F,n}, q|_F)_F \quad \forall \mathbf{q} \in \mathcal{P}^p(T). \quad (60) $$

Performing an integration by parts of the term $(\eta_T \mathbf{v}_T, \text{grad } \mathbf{q})_T$, and using a discrete trace inequality, it is an easy matter to prove that

$$ \sum_{T \in \mathcal{T}_h} \left( \|D_T^p(\underline{\mathbf{v}}_T)\|_{0,T}^2 + s_{\text{div},T}(\underline{\mathbf{v}}_T, \underline{\mathbf{v}}_T) \right) \le |\underline{\mathbf{v}}_h|^2_{\text{div},h} \quad \forall \underline{\mathbf{v}}_h \in \underline{\mathbf{X}}_h^\ell. \quad (61) $$

If $p \ge \ell - 1$, one can also prove that $|\underline{\mathbf{v}}_h|^2_{\text{div},h} \lesssim \sum_{T \in \mathcal{T}_h} (\|D_T^p(\underline{\mathbf{v}}_T)\|_{0,T}^2 + s_{\text{div},T}(\underline{\mathbf{v}}_T, \underline{\mathbf{v}}_T))$.
## 5.2 Compactness in $\underline{\mathcal{X}}_{h,\tau}^\ell$

Recall the definitions (33) of the discrete space $\underline{\mathcal{X}}_{h,\tau}^\ell$, and (35)–(36) of the hybrid semi-norms $|\cdot|_{\text{curl},h}$ and $|\cdot|_{\text{div},h}$. Leveraging the first hybrid Weber inequality from Theorem 13, we infer that

$$ \|\underline{\nu}_h\|_{\underline{\mathcal{X}}_{\tau,h}}^2 := \eta_b^{-1} |\underline{\nu}_h|_{\text{div},h}^2 + \eta_\# |\underline{\nu}_h|_{\text{curl},h}^2 + \eta_b^{-1} \sum_{j=1}^{\beta_2} |(\gamma_{n,h}(\underline{\nu}_h), 1)_{\Gamma_j}|^2 \quad (62) $$

defines a norm on $\underline{\mathcal{X}}_{h,\tau}^\ell$, which can be proven uniformly (in $h$) equivalent to the $\|\cdot\|_{\mathcal{X},h}$-norm (37): there is $c_\tau > 0$, independent of $h$ and $\eta$, such that

$$ (c_\tau \kappa_\eta)^{-1} \|\underline{\nu}_h\|_{\underline{\mathcal{X}}_{\tau,h}}^2 \le \|\underline{\nu}_h\|_{\underline{\mathcal{X}}_{\tau,h}}^2 \le c_\tau \kappa_\eta \|\underline{\nu}_h\|_{\underline{\mathcal{X}}_{\tau,h}}^2 \quad \forall \underline{\nu}_h \in \underline{\mathcal{X}}_{h,\tau}^\ell. $$

**Theorem 17 (Maxwell compactness in $\underline{\mathcal{X}}_{h,\tau}^\ell$).** Let $(M_h)_{h \in \mathcal{H}}$ be a regular mesh family as defined in Section 3.1. Let $(\underline{\nu}_h)_{h \in \mathcal{H}}$ be a sequence of elements of $\underline{\mathcal{X}}_{h,\tau}^\ell$ for which there exists a real number $c_M > 0$ (independent of $h$) such that $\eta_\#^{-\frac{1}{2}} \|\underline{\nu}_h\|_{\underline{\mathcal{X}}_{\tau,h}} \le c_M$ for all $h \in \mathcal{H}$. Then, there exists an element $\nu \in \mathcal{X}_\tau(\Omega)$ such that, along a subsequence, as $h \to 0$,

(a) $\nu_h \to \nu$ strongly in $L^2(\Omega)$;

(b) $C_h^m(\underline{\nu}_h) \to \text{curl} \nu$ weakly in $L^2(\Omega)$;

(c) $D_h^p(\underline{\nu}_h) \to \text{div}(\eta\nu)$ weakly in $L^2(\Omega)$;

(d) $\gamma_{n,h}(\underline{\nu}_h) \to (\eta\nu)|_{\Gamma \cdot \mathbf{n}}$ weakly in $H^{-\frac{1}{2}}(\Gamma)$.

Importantly, no relation is required between $m$, $p$, and $\ell$ for this result to hold true.

*Proof.* As in the continuous case (cf. Proposition 4), the proof proceeds in three steps.

**Step 1 (Weak convergence):** By assumption, $\|\underline{\nu}_h\|_{\mathcal{X}_{\tau,h}} \le c_M \eta_\#^{\frac{1}{2}}$ for all $h \in \mathcal{H}$. By definition (62) of the $\|\cdot\|_{\mathcal{X}_{\tau,h}}$-norm, by the boundedness results (59) and (61), and by the first hybrid Weber inequality (41), this implies that (i) $(\nu_h)_{h \in \mathcal{H}}$ and $(C_h^m(\underline{\nu}_h))_{h \in \mathcal{H}}$ are uniformly bounded in $L^2(\Omega)$, and (ii) $(D_h^p(\underline{\nu}_h))_{h \in \mathcal{H}}$ is uniformly bounded in $L^2(\Omega)$. By weak compactness, we then infer the existence of $\nu \in L^2(\Omega)$, $c \in L^2(\Omega)$, and $d \in L^2(\Omega)$ such that, along a subsequence (not relabelled), (i) $\nu_h \to \nu$ and $C_h^m(\underline{\nu}_h) \to c$ weakly in $L^2(\Omega)$, and (ii) $D_h^p(\underline{\nu}_h) \to d$ weakly in $L^2(\Omega)$.

Let us first prove that $\nu \in H_0(\text{curl}; \Omega)$ and that $c = \text{curl} \nu$. Let $z \in C^\infty(\bar{\Omega})$. There holds

$$ \begin{align} (C_h^m(\underline{\nu}_h), z)_\Omega &= \sum_{T \in \mathcal{T}_h} (C_T^m(\underline{\nu}_T), \pi_T^m(z|_T))_T \nonumber \\ &= \sum_{T \in \mathcal{T}_h} \left( (\text{curl} \nu_T, \pi_T^m(z|_T))_T \right. \nonumber \\ &\qquad \left. + \sum_{F \in \mathcal{F}_T} \epsilon_{T,F} (\nu_T|_F \times \mathbf{n}_F - \nu_{F,\tau}, \mathbf{n}_F \times (\pi_T^m(z|_T)|_F \times \mathbf{n}_F))_F \right) \nonumber \\ &= (\text{curl}_h \nu_h, z)_\Omega + \sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \epsilon_{T,F} (\nu_T|_F \times \mathbf{n}_F - \nu_{F,\tau}, \mathbf{n}_F \times (z|_F \times \mathbf{n}_F))_F + I_{\text{curl},h} \nonumber \\ &= (\nu_h, \text{curl} z)_\Omega + I_{\text{curl},h}, \tag{63} \end{align} $$
## 5.2 Compactness in $\underline{\mathcal{X}}_{h,\tau}^\ell$

Recall the definitions (33) of the discrete space $\underline{\mathcal{X}}_{h,\tau}^\ell$, and (35)–(36) of the hybrid semi-norms $|\cdot|_{\text{curl},h}$ and $|\cdot|_{\text{div},h}$. Leveraging the first hybrid Weber inequality from Theorem 13, we infer that

$$ \| \underline{\nu}_h \|^2_{\underline{\mathcal{X}}_{\tau,h}} := \eta_b^{-1} |\underline{\nu}_h|^2_{\text{div},h} + \eta_\# |\underline{\nu}_h|^2_{\text{curl},h} + \eta_b^{-1} \sum_{j=1}^{\beta_2} |(\gamma_{n,h}(\underline{\nu}_h), 1)_{\Gamma_j}|^2 \quad (62) $$

defines a norm on $\underline{\mathcal{X}}_{h,\tau}^\ell$, which can be proven uniformly (in $h$) equivalent to the $\|\cdot\|_{\mathcal{X},h}$-norm (37): there is $c_\tau > 0$, independent of $h$ and $\eta$, such that

$$ (c_\tau \kappa_\eta)^{-1} \|\underline{\nu}_h\|^2_{\mathcal{X},h} \le \|\underline{\nu}_h\|^2_{\mathcal{X},h} \le c_\tau \kappa_\eta \|\underline{\nu}_h\|^2_{\mathcal{X},h} \quad \forall \underline{\nu}_h \in \underline{\mathcal{X}}_{h,\tau}^\ell. $$

**Theorem 17 (Maxwell compactness in $\underline{\mathcal{X}}_{h,\tau}^\ell$).** Let $(M_h)_{h \in \mathcal{H}}$ be a regular mesh family as defined in Section 3.1. Let $(\underline{\nu}_h)_{h \in \mathcal{H}}$ be a sequence of elements of $\underline{\mathcal{X}}_{h,\tau}^\ell$ for which there exists a real number $c_M > 0$ (independent of $h$) such that $\eta_\#^{-\frac{1}{2}} \|\underline{\nu}_h\|_{\mathcal{X},h} \le c_M$ for all $h \in \mathcal{H}$. Then, there exists an element $\nu \in \mathcal{X}_\tau(\Omega)$ such that, along a subsequence, as $h \to 0$,

(a) $\nu_h \to \nu$ strongly in $L^2(\Omega)$;

(b) $C_h^m(\underline{\nu}_h) \to \text{curl} \nu$ weakly in $L^2(\Omega)$;

(c) $D_h^p(\underline{\nu}_h) \to \text{div}(\eta\nu)$ weakly in $L^2(\Omega)$;

(d) $\gamma_{n,h}(\underline{\nu}_h) \to (\eta\nu)|_{\Gamma \cdot \mathbf{n}}$ weakly in $H^{-\frac{1}{2}}(\Gamma)$.

Importantly, no relation is required between $m$, $p$, and $\ell$ for this result to hold true.

*Proof.* As in the continuous case (cf. Proposition 4), the proof proceeds in three steps.

**Step 1 (Weak convergence):** By assumption, $\|\underline{\nu}_h\|_{\mathcal{X}_\tau,h} \le c_M \eta_\#^{\frac{1}{2}}$ for all $h \in \mathcal{H}$. By definition (62) of the $\|\cdot\|_{\mathcal{X}_\tau,h}$-norm, by the boundedness results (59) and (61), and by the first hybrid Weber inequality (41), this implies that (i) $(\nu_h)_{h \in \mathcal{H}}$ and $(C_h^m(\underline{\nu}_h))_{h \in \mathcal{H}}$ are uniformly bounded in $L^2(\Omega)$, and (ii) $(D_h^p(\underline{\nu}_h))_{h \in \mathcal{H}}$ is uniformly bounded in $L^2(\Omega)$. By weak compactness, we then infer the existence of $\nu \in L^2(\Omega)$, $c \in L^2(\Omega)$, and $d \in L^2(\Omega)$ such that, along a subsequence (not relabelled), (i) $\nu_h \to \nu$ and $C_h^m(\underline{\nu}_h) \to c$ weakly in $L^2(\Omega)$, and (ii) $D_h^p(\underline{\nu}_h) \to d$ weakly in $L^2(\Omega)$.

Let us first prove that $\nu \in H_0(\text{curl}; \Omega)$ and that $c = \text{curl} \nu$. Let $z \in C^\infty(\bar{\Omega})$. There holds

$$ \begin{align} (C_h^m(\underline{\nu}_h), z)_\Omega &= \sum_{T \in \mathcal{T}_h} (C_T^m(\underline{\nu}_T), \pi_T^m(z|_T))_T \nonumber \\ &= \sum_{T \in \mathcal{T}_h} \left( (\text{curl} \nu_T, \pi_T^m(z|_T))_T \right. \nonumber \\ &\qquad \left. + \sum_{F \in \mathcal{F}_T} \epsilon_{T,F} (\nu_T|_F \times \mathbf{n}_F - \nu_{F,\tau}, \mathbf{n}_F \times (\pi_T^m(z|_T)|_F \times \mathbf{n}_F))_F \right) \nonumber \\ &= (\text{curl}_h \nu_h, z)_\Omega + \sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \epsilon_{T,F} (\nu_T|_F \times \mathbf{n}_F - \nu_{F,\tau}, \mathbf{n}_F \times (z|_F \times \mathbf{n}_F))_F + I_{\text{curl},h} \nonumber \\ &= (\nu_h, \text{curl} z)_\Omega + I_{\text{curl},h}, \tag{63} \end{align} $$
with $\mathcal{I}_{\text{curl},h}$ given by

$$ \mathcal{I}_{\text{curl},h} := ( \text{curl}_h v_h, \pi_h^m(z) - z )_{\Omega} \\ + \sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \varepsilon_{T,F} (v_T |_F \times n_F - v_{F,\tau}, n_F \times ((\pi_T^m(z|_T) - z)|_F \times n_F))_F, $$

where we have (i) used that $C_T^m(\underline{v}_T) \in \mathcal{P}^m(T)$ for all $T \in \mathcal{T}_h$ in the first line, (ii) applied the definition (58) of $C_T^m(\underline{v}_T)$ along with an integration by parts of the term $(v_T, \text{curl} \pi_T^m(z|_T))_T$ in the second line, (iii) added/subtracted $(\text{curl}_h v_h, z)_{\Omega}$ and $\sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \varepsilon_{T,F} (v_T|_F \times n_F - v_{F,\tau}, n_F \times (z|_F \times n_F))_F$ in the third line, and (iv) performed cell-by-cell integration by parts of the term $(\text{curl}_h v_h, z)_{\Omega}$ and used that $\sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \varepsilon_{T,F} (v_{F,\tau}, n_F \times (z|_F \times n_F))_F = 0$ (since $z$ is single-valued at interfaces and $v_{F,\tau} = 0$ for all $F \in \mathcal{F}_h^\partial$ for $\underline{v}_h \in \underline{X}_{h,\tau}^\ell$) in the fourth line. For the term $\mathcal{I}_{\text{curl},h}$ in the right-hand side of (63), standard approximation properties for $\pi_h^m$ on mesh cells (cf. e.g. [22, Thm. 1.45]) on the one side, and repeated applications of the triangle and Cauchy-Schwarz inequalities along with the estimate (38) and approximation properties for $\pi_h^m$ on mesh faces on the other side, yield

$$ |\mathcal{I}_{\text{curl},h}| \lesssim h^{m+1} |\underline{v}_h|_{\text{curl},h} |z|_{m+1}. $$

Starting from (63), and using that $v_h \to v$ and $C_h^m(\underline{v}_h) \to c$ weakly in $L^2(\Omega)$, combined with the fact that $\mathcal{I}_{\text{curl},h} \to 0$ as $h \to 0$ (since $|\underline{v}_h|_{\text{curl},h} \le c_M$ for all $h \in \mathcal{H}$ and $m+1 \ge 1$), we infer that

$$ (c, z)_{\Omega} = (v, \text{curl} z)_{\Omega} \quad \forall z \in C^{\infty}(\bar{\Omega}). $$

Since the latter relation is, in particular, valid for all $z \in C_0^\infty(\Omega)$, we infer that $v \in H(\text{curl}; \Omega)$ and that $c = \text{curl} v$. Then, integrating by parts the left-hand side of the relation, we get that $\langle v|_{\Gamma} \times n, n \times (z|_{\Gamma} \times n) \rangle_{\Gamma} = 0$ for all $z \in C^\infty(\bar{\Omega})$, which implies that $v \in H_0(\text{curl}; \Omega)$.

Let us now prove that $v \in H(\text{div}_\eta; \Omega)$ and that $d = \text{div}(\eta v)$, along with the weak convergence result $\gamma_{n,h}(\underline{v}_h) \to (\eta v)|_{\Gamma} \cdot n$ in $H^{-\frac{1}{2}}(\Gamma)$. Let $z \in C^\infty(\bar{\Omega})$. Adapting the arguments used in (63) in the case of the rotational operator to the case of the divergence operator, we infer that

$$ (D_h^p(\underline{v}_h), z)_{\Omega} = -(\eta v_h, \text{grad } z)_{\Omega} + (\gamma_{n,h}(\underline{v}_h), z|_{\Gamma})_{\Gamma} + \mathcal{I}_{\text{div},h}, \quad (64) $$

where $\mathcal{I}_{\text{div},h}$ is given by

$$ \mathcal{I}_{\text{div},h} := ( \text{div}_h(\eta v_h), \pi_h^p(z) - z )_{\Omega} + \sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \varepsilon_{T,F} ([\eta v]_{F,n} - \eta_T v_T |_F \cdot n_F, (\pi_T^p(z|_T) - z)|_F)_F. $$

Adapting the arguments from the rotational case, it is an easy matter to prove that

$$ |\mathcal{I}_{\text{div},h}| \lesssim h^{p+1} |\underline{v}_h|_{\text{div},h} |z|_{p+1}. $$

Let us first choose $z \in C_0^\infty(\Omega)$. Remark that the boundary term in (64) vanishes. Starting from (64), and using that $v_h \to v$ weakly in $L^2(\Omega)$ and $D_h^p(\underline{v}_h) \to d$ weakly in $L^2(\Omega)$, combined with the fact that $\mathcal{I}_{\text{div},h} \to 0$ as $h \to 0$ (since $|\underline{v}_h|_{\text{div},h} \le c_M \eta_\#^\ell$ for all $h \in \mathcal{H}$ and $p+1 \ge 1$), we infer that

$$ (d, z)_{\Omega} = -(\eta v, \text{grad } z)_{\Omega} \quad \forall z \in C_0^{\infty}(\Omega). $$

The latter relation directly implies that $v \in H(\text{div}_\eta; \Omega)$ and that $d = \text{div}(\eta v)$. Now, passing to the limit $h \to 0$ in (64) for a generic $z \in C^\infty(\bar{\Omega})$, we infer that

$$ \lim_{h \to 0} (\gamma_{n,h}(\underline{v}_h), z|_{\Gamma})_{\Gamma} = (\text{div}(\eta v), z)_{\Omega} + (\eta v, \text{grad } z)_{\Omega} = \langle (\eta v)|_{\Gamma} \cdot n, z|_{\Gamma} \rangle_{\Gamma}, $$
with $\mathcal{I}_{\text{curl},h}$ given by

$$ \mathcal{I}_{\text{curl},h} := ( \mathbf{curl}_h v_h, \pi_h^m(z) - z )_{\Omega} \\ + \sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \varepsilon_{T,F} (v_T |_F \times n_F - v_{F,\tau}, n_F \times ((\pi_T^m(z|_T) - z)|_F \times n_F))_F, $$

where we have (i) used that $C_T^m(\underline{v}_T) \in \mathcal{P}^m(T)$ for all $T \in \mathcal{T}_h$ in the first line, (ii) applied the definition (58) of $C_T^m(\underline{v}_T)$ along with an integration by parts of the term $(v_T, \text{curl} \pi_T^m(z|_T))_T$ in the second line, (iii) added/subtracted $(\text{curl}_h v_h, z)_{\Omega}$ and $\sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \varepsilon_{T,F} (v_T|_F \times n_F - v_{F,\tau}, n_F \times (z|_F \times n_F))_F$ in the third line, and (iv) performed cell-by-cell integration by parts of the term $(\text{curl}_h v_h, z)_{\Omega}$ and used that $\sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \varepsilon_{T,F} (v_{F,\tau}, n_F \times (z|_F \times n_F))_F = 0$ (since $z$ is single-valued at interfaces and $v_{F,\tau} = 0$ for all $F \in \mathcal{F}_h^\partial$ for $\underline{v}_h \in \underline{X}_{h,\tau}^\ell$) in the fourth line. For the term $\mathcal{I}_{\text{curl},h}$ in the right-hand side of (63), standard approximation properties for $\pi_h^m$ on mesh cells (cf. e.g. [22, Thm. 1.45]) on the one side, and repeated applications of the triangle and Cauchy-Schwarz inequalities along with the estimate (38) and approximation properties for $\pi_h^m$ on mesh faces on the other side, yield

$$ |\mathcal{I}_{\text{curl},h}| \lesssim h^{m+1} |\underline{v}_h|_{\text{curl},h} |z|_{m+1}. $$

Starting from (63), and using that $v_h \to v$ and $C_h^m(\underline{v}_h) \to c$ weakly in $L^2(\Omega)$, combined with the fact that $\mathcal{I}_{\text{curl},h} \to 0$ as $h \to 0$ (since $|\underline{v}_h|_{\text{curl},h} \le c_M$ for all $h \in \mathcal{H}$ and $m+1 \ge 1$), we infer that

$$ (c, z)_{\Omega} = (v, \text{curl} z)_{\Omega} \quad \forall z \in C^{\infty}(\bar{\Omega}). $$

Since the latter relation is, in particular, valid for all $z \in C_0^\infty(\Omega)$, we infer that $v \in H(\text{curl}; \Omega)$ and that $c = \text{curl} v$. Then, integrating by parts the left-hand side of the relation, we get that $\langle v|_{\Gamma} \times n, n \times (z|_{\Gamma} \times n) \rangle_{\Gamma} = 0$ for all $z \in C^\infty(\bar{\Omega})$, which implies that $v \in H_0(\text{curl}; \Omega)$.

Let us now prove that $v \in H(\text{div}_\eta; \Omega)$ and that $d = \text{div}(\eta v)$, along with the weak convergence result $\gamma_{n,h}(\underline{v}_h) \to (\eta v)|_{\Gamma} \cdot n$ in $H^{-\frac{1}{2}}(\Gamma)$. Let $z \in C^\infty(\bar{\Omega})$. Adapting the arguments used in (63) in the case of the rotational operator to the case of the divergence operator, we infer that

$$ (D_h^p(\underline{v}_h), z)_{\Omega} = -(\eta v_h, \text{grad } z)_{\Omega} + (\gamma_{n,h}(\underline{v}_h), z|_{\Gamma})_{\Gamma} + \mathcal{I}_{\text{div},h}, \quad (64) $$

where $\mathcal{I}_{\text{div},h}$ is given by

$$ \mathcal{I}_{\text{div},h} := ( \text{div}_h(\eta v_h), \pi_h^p(z) - z )_{\Omega} + \sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \varepsilon_{T,F} ( [\eta v]_{F,n} - \eta_T v_T |_F \cdot n_F, (\pi_T^p(z|_T) - z)|_F )_F. $$

Adapting the arguments from the rotational case, it is an easy matter to prove that

$$ |\mathcal{I}_{\text{div},h}| \lesssim h^{p+1} |\underline{v}_h|_{\text{div},h} |z|_{p+1}. $$

Let us first choose $z \in C_0^\infty(\Omega)$. Remark that the boundary term in (64) vanishes. Starting from (64), and using that $v_h \to v$ weakly in $L^2(\Omega)$ and $D_h^p(\underline{v}_h) \to d$ weakly in $L^2(\Omega)$, combined with the fact that $\mathcal{I}_{\text{div},h} \to 0$ as $h \to 0$ (since $|\underline{v}_h|_{\text{div},h} \le c_M \eta_\#^\ell$ for all $h \in \mathcal{H}$ and $p+1 \ge 1$), we infer that

$$ (d, z)_{\Omega} = -(\eta v, \text{grad } z)_{\Omega} \quad \forall z \in C_0^{\infty}(\Omega). $$

The latter relation directly implies that $v \in H(\text{div}_\eta; \Omega)$ and that $d = \text{div}(\eta v)$. Now, passing to the limit $h \to 0$ in (64) for a generic $z \in C^\infty(\bar{\Omega})$, we infer that

$$ \lim_{h \to 0} (\gamma_{n,h}(\underline{v}_h), z|_{\Gamma})_{\Gamma} = (\text{div}(\eta v), z)_{\Omega} + (\eta v, \text{grad } z)_{\Omega} = \langle (\eta v)|_{\Gamma} \cdot n, z|_{\Gamma} \rangle_{\Gamma}, $$
which implies that $\gamma_{n,h}(\underline{v}_h) \to (\eta\underline{v})_{|\Gamma \cdot n}$ weakly in $H^{-\frac{1}{2}}(\Gamma)$.

**Step 2 (Characterization of the limit):** This step is identical to Step 2 from the continuous case (cf. the proof of Proposition 4), up to the replacement of $m \in \mathbb{N}$ by $h \in \mathcal{H}$ (and $m \to \infty$ by $h \to 0$) and of $v_m \in X_\tau(\Omega)$ by $v_h \in \mathcal{P}^\ell(\mathcal{T}_h)$. For any $h \in \mathcal{H}$, by (7), there exist $\varphi_h \in H_0^1(\Omega)$, $\psi_h \in H^1(\Omega) \cap L_0^2(\Omega)$, and $w_h \in H_\tau(\Omega)$, such that

$$v_h = \text{grad } \varphi_h + \frac{\eta_{\#}}{\eta} \text{curl } \psi_h + w_h, \qquad (65)$$

with $\|\psi_h\|_1 \le C_{\Omega,1} \|\text{curl } \psi_h\|_0$ by (8). Proceeding as in the continuous case, one can infer the existence of $w \in H_\tau(\Omega)$ such that, up to extraction (not relabelled), $w_h \to w$ strongly in $L^2(\Omega)$, and of $\varphi \in H_0^1(\Omega)$ and $\psi \in H^1(\Omega) \cap L_0^2(\Omega)$ such that, up to extractions (not relabelled), $\text{grad } \varphi_h \to \text{grad } \varphi$ and $\text{curl } \psi_h \to \text{curl } \psi$ weakly in $L^2(\Omega)$. By linearity, we have thus proven that, along a subsequence (not relabelled), $v_h \to \text{grad } \varphi + \frac{\eta_{\#}}{\eta} \text{curl } \psi + w$ weakly in $L^2(\Omega)$. By uniqueness of the weak limit, $v = \text{grad } \varphi + \frac{\eta_{\#}}{\eta} \text{curl } \psi + w$.

**Step 3 (Strong convergence):** Here again, the arguments are inspired by what is done in Step 3 from the continuous case, but with the important difference that here $v_h \in \mathcal{P}^\ell(\mathcal{T}_h) \not\subset X_\tau(\Omega)$. By Rellich's compactness theorem, we actually have that, along the same subsequence (not relabelled) as in Step 2 above, $\varphi_h \to \varphi$ strongly in $L^2(\Omega)$ and $\psi_h \to \psi$ strongly in $L^2(\Omega)$. We now want to prove the strong convergences of $(\text{grad } \varphi_h)_{h \in \mathcal{H}}$ and $(\text{curl } \psi_h)_{h \in \mathcal{H}}$ in $L^2(\Omega)$ (we remind the reader that we already proved strong convergence for $(w_h)_{h \in \mathcal{H}}$).

Recalling the expression of $v \in X_\tau(\Omega)$ derived in Step 2, since the decomposition (65) is $L_\eta^2(\Omega)$-orthogonal, there holds

$$ \begin{aligned} \|\eta^{\frac{1}{2}}(\text{grad } \varphi_h - \text{grad } \varphi)\|_0^2 &= (\eta(v_h - v), \text{grad}(\varphi_h - \varphi))_{\Omega} \\ &= \sum_{T \in \mathcal{T}_h} \left( - (\text{div}(\eta_T(v_T - v)), \varphi_h - \varphi)_T \right. \\ & \qquad \left. + \sum_{F \in \mathcal{F}_T} (\eta_T v_T |_F \cdot n_{T,F}, (\varphi_h - \varphi)|_F)_F \right) \\ &= - (\text{div}_h(\eta(v_h - v)), \varphi_h - \varphi)_{\Omega} \\ & \qquad + \sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \varepsilon_{T,F} (\eta_T v_T |_F \cdot n_F - [\eta v]_{F,n}, (\varphi_h - \varphi)|_F)_F, \end{aligned} $$

where, to pass from the first to the second line, we have performed cell-by-cell integration by parts and used that $\sum_{T \in \mathcal{T}_h} \langle (\eta v)|_{\partial T} \cdot n|_{\partial T}, (\varphi_h - \varphi)|_{\partial T} \rangle_{\partial T} = 0$ (as a consequence of the fact that $v \in H(\text{div}_\eta; \Omega)$ and that $\varphi_h, \varphi \in H_0^1(\Omega)$) and, to pass from the second to the third line, we have again used the fact that $(\varphi_h - \varphi)$ is single-valued at interfaces and vanishes on the boundary of the domain to subtract the zero contribution $\sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \varepsilon_{T,F} ([\eta v]_{F,n}, (\varphi_h - \varphi)|_F)_F$. By Cauchy-Schwarz inequality, combined with a continuous trace inequality (along with mesh regularity), and the fact that $|\underline{v}_h|_{\text{div},h} \le c_M \eta_{\#}$ for all $h \in \mathcal{H}$ and $\|\text{div}(\eta v)\|_0 \le \liminf_{h \to 0} |\underline{v}_h|_{\text{div},h}$ (by weak convergence of $(D_h^P(\underline{v}_h))_{h \in \mathcal{H}}$ to $\text{div}(\eta v)$ in $L^2(\Omega)$ and the uniform bound (61)), we infer that

$$ \|\eta^{\frac{1}{2}}(\text{grad } \varphi_h - \text{grad } \varphi)\|_0^2 \lesssim \eta_{\#} (\|\varphi_h - \varphi\|_0 + h \|\text{grad } \varphi_h - \text{grad } \varphi\|_0). \qquad (66) $$

Since $\|\text{grad } \varphi_h\|_0 \le \eta_b^{-\frac{1}{2}} \|\eta^{\frac{1}{2}}v_h\|_0$ (by $L_\eta^2(\Omega)$-orthogonality of the decomposition (65)), $\|\eta^{\frac{1}{2}}v_h\|_0 \le c_{W,1} \sqrt{3\kappa_\eta^{\frac{1}{2}}} c_M \eta_{\#}^{\frac{1}{2}}$ (by (41), (62)), and $\|\underline{v}_h\|_{X_\tau,h} \le c_M \eta_{\#}^{\frac{1}{2}}$, and $\|\text{grad } \varphi\|_0 \le \liminf_{h \to 0} \|\text{grad } \varphi_h\|_0$
which implies that $\gamma_{n,h}(\underline{v}_h) \to (\eta\underline{v})_{|\Gamma \cdot n}$ weakly in $H^{-\frac{1}{2}}(\Gamma)$.

**Step 2 (Characterization of the limit):** This step is identical to Step 2 from the continuous case (cf. the proof of Proposition 4), up to the replacement of $m \in \mathbb{N}$ by $h \in \mathcal{H}$ (and $m \to \infty$ by $h \to 0$) and of $v_m \in X_\tau(\Omega)$ by $v_h \in \mathcal{P}^\ell(\mathcal{T}_h)$. For any $h \in \mathcal{H}$, by (7), there exist $\varphi_h \in H_0^1(\Omega)$, $\psi_h \in H^1(\Omega) \cap L_0^2(\Omega)$, and $w_h \in H_\tau(\Omega)$, such that

$$v_h = \text{grad } \varphi_h + \frac{\eta_{\#}}{\eta} \text{curl } \psi_h + w_h, \qquad (65)$$

with $\|\psi_h\|_1 \le C_{\Omega,1} \|\text{curl } \psi_h\|_0$ by (8). Proceeding as in the continuous case, one can infer the existence of $w \in H_\tau(\Omega)$ such that, up to extraction (not relabelled), $w_h \to w$ strongly in $L^2(\Omega)$, and of $\varphi \in H_0^1(\Omega)$ and $\psi \in H^1(\Omega) \cap L_0^2(\Omega)$ such that, up to extractions (not relabelled), $\text{grad } \varphi_h \to \text{grad } \varphi$ and $\text{curl } \psi_h \to \text{curl } \psi$ weakly in $L^2(\Omega)$. By linearity, we have thus proven that, along a subsequence (not relabelled), $v_h \to \text{grad } \varphi + \frac{\eta_{\#}}{\eta} \text{curl } \psi + w$ weakly in $L^2(\Omega)$. By uniqueness of the weak limit, $v = \text{grad } \varphi + \frac{\eta_{\#}}{\eta} \text{curl } \psi + w$.

**Step 3 (Strong convergence):** Here again, the arguments are inspired by what is done in Step 3 from the continuous case, but with the important difference that here $v_h \in \mathcal{P}^\ell(\mathcal{T}_h) \not\subset X_\tau(\Omega)$. By Rellich's compactness theorem, we actually have that, along the same subsequence (not relabelled) as in Step 2 above, $\varphi_h \to \varphi$ strongly in $L^2(\Omega)$ and $\psi_h \to \psi$ strongly in $L^2(\Omega)$. We now want to prove the strong convergences of $(\text{grad } \varphi_h)_{h \in \mathcal{H}}$ and $(\text{curl } \psi_h)_{h \in \mathcal{H}}$ in $L^2(\Omega)$ (we remind the reader that we already proved strong convergence for $(w_h)_{h \in \mathcal{H}}$).

Recalling the expression of $v \in X_\tau(\Omega)$ derived in Step 2, since the decomposition (65) is $L_\eta^2(\Omega)$-orthogonal, there holds

$$ \begin{aligned} \|\eta^{\frac{1}{2}}(\text{grad } \varphi_h - \text{grad } \varphi)\|_0^2 &= (\eta(v_h - v), \text{grad}(\varphi_h - \varphi))_{\Omega} \\ &= \sum_{T \in \mathcal{T}_h} \left( - (\text{div}(\eta_T(v_T - v)), \varphi_h - \varphi)_T \right. \\ & \qquad \left. + \sum_{F \in \mathcal{F}_T} (\eta_T v_T |_F \cdot n_{T,F}, (\varphi_h - \varphi)|_F)_F \right) \\ &= - (\text{div}_h(\eta(v_h - v)), \varphi_h - \varphi)_{\Omega} \\ & \qquad + \sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \varepsilon_{T,F} (\eta_T v_T |_F \cdot n_F - [\eta v]_{F,n}, (\varphi_h - \varphi)|_F)_F, \end{aligned} $$

where, to pass from the first to the second line, we have performed cell-by-cell integration by parts and used that $\sum_{T \in \mathcal{T}_h} \langle (\eta v)|_{\partial T} \cdot n|_{\partial T}, (\varphi_h - \varphi)|_{\partial T} \rangle_{\partial T} = 0$ (as a consequence of the fact that $v \in H(\text{div}_\eta; \Omega)$ and that $\varphi_h, \varphi \in H_0^1(\Omega)$) and, to pass from the second to the third line, we have again used the fact that $(\varphi_h - \varphi)$ is single-valued at interfaces and vanishes on the boundary of the domain to subtract the zero contribution $\sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \varepsilon_{T,F} ([\eta v]_{F,n}, (\varphi_h - \varphi)|_F)_F$. By Cauchy-Schwarz inequality, combined with a continuous trace inequality (along with mesh regularity), and the fact that $|\underline{v}_h|_{\text{div},h} \le c_M \eta_{\#}$ for all $h \in \mathcal{H}$ and $\|\text{div}(\eta v)\|_0 \le \liminf_{h \to 0} |\underline{v}_h|_{\text{div},h}$ (by weak convergence of $(D_h^P(\underline{v}_h))_{h \in \mathcal{H}}$ to $\text{div}(\eta v)$ in $L^2(\Omega)$ and the uniform bound (61)), we infer that

$$\|\eta^{\frac{1}{2}}(\text{grad } \varphi_h - \text{grad } \varphi)\|_0^2 \lesssim \eta_{\#} (\|\varphi_h - \varphi\|_0 + h \|\text{grad } \varphi_h - \text{grad } \varphi\|_0). \qquad (66)$$

Since $\|\text{grad } \varphi_h\|_0 \le \eta_b^{-\frac{1}{2}} \|\eta^{\frac{1}{2}}v_h\|_0$ (by $L_\eta^2(\Omega)$-orthogonality of the decomposition (65)), $\|\eta^{\frac{1}{2}}v_h\|_0 \le c_{W,1} \sqrt{3} \kappa_\eta^{\frac{1}{2}} c_M \eta_{\#}^{\frac{1}{2}}$ (by (41), (62)), and $\|\underline{v}_h\|_{X_\tau,h} \le c_M \eta_{\#}^{\frac{1}{2}}$, and $\|\text{grad } \varphi\|_0 \le \liminf_{h \to 0} \|\text{grad } \varphi_h\|_0$
(since grad $\varphi_h \to$ grad $\varphi$ weakly in $L^2(\Omega)$), we infer by the triangle inequality that

$$ \| \text{grad } \varphi_h - \text{grad } \varphi \|_0 \le 2c_{W,1}\sqrt{3\kappa_{\eta}c_M}. $$

Combining this uniform boundedness result to the fact that $\varphi_h \to \varphi$ strongly in $L^2(\Omega)$ to pass to the limit $h \to 0$ in (66), we conclude that $\text{grad } \varphi_h \to \text{grad } \varphi$ strongly in $L^2(\Omega)$.

Adapting the latter arguments, there also holds

$$ \begin{aligned} \eta_{\#} \|\eta^{-\frac{1}{2}}(\text{curl } \psi_h - \text{curl } \psi)\|_{0}^{2} &= (v_h - v, \text{curl}(\psi_h - \psi))_{\Omega} \\ &= (\text{curl}_h(v_h - v), \psi_h - \psi)_{\Omega} \\ &\quad + \sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \varepsilon_{T,F} (v_{T|F} \times n_F - v_{F,\tau}, n_F \times ((\psi_h - \psi)_{F} \times n_F))_{F}, \end{aligned} $$

where we have used that $\underline{v}_h \in \underline{\mathcal{X}}_{h,\tau}^{\ell}$, $v \in \mathbf{H}_0(\text{curl}; \Omega)$, and $\psi_h, \psi \in \mathbf{H}^1(\Omega)$. Additionally invoking (38), this implies that

$$ \eta_{\#} \|\eta^{-\frac{1}{2}}(\text{curl } \psi_h - \text{curl } \psi)\|_{0}^{2} \lesssim (\|\psi_h - \psi\|_{0} + h|\psi_h - \psi|_{1}), $$

which eventually yields, since $|\psi_h - \psi|_1 \le 2C_{\Omega,1}c_{W,1}\sqrt{3\kappa_{\eta}c_M}$ (here, we additionally used the estimate $|\psi_h|_1 \le C_{\Omega,1} \|\text{curl } \psi_h\|_0$ and the fact that $\psi_h \to \psi$ weakly in $\mathbf{H}^1(\Omega)$), and $\psi_h \to \psi$ strongly in $L^2(\Omega)$, that $\text{curl } \psi_h \to \text{curl } \psi$ strongly in $L^2(\Omega)$.

Thus, along a subsequence (not relabelled), $v_h \to v$ strongly in $L^2(\Omega)$, yielding the conclusion. □

## 5.3 Compactness in $\underline{\mathcal{X}}_{h,n}^{\ell}$

Recall the definitions (33) of the discrete space $\underline{\mathcal{X}}_{h,n}^{\ell}$, and (35)–(36) of the hybrid semi-norms $|\cdot|_{\text{curl},h}$ and $|\cdot|_{\text{div},h}$. Leveraging the second hybrid Weber inequality from Theorem 15, we infer that

$$ \|\underline{v}_h\|_{\mathbf{X}_{n,h}}^2 := \eta_b^{-1} |\underline{v}_h|_{\text{div},h}^2 + \eta_{\#} |\underline{v}_h|_{\text{curl},h}^2 + \eta_b^{-1} \sum_{i=1}^{\beta_1} |(\gamma_{n,h}(\underline{v}_h), 1)_{\Sigma_i}|^2 \quad (67) $$

defines a norm on $\underline{\mathcal{X}}_{h,n}^{\ell}$, which can be proven uniformly (in $h$) equivalent to the $\|\cdot\|_{\mathbf{X},h}$-norm (37): there is $c_n > 0$, independent of $h$ and $\eta$, such that

$$ (c_n \kappa_{\eta})^{-1} \|\underline{v}_h\|_{\mathbf{X},h}^2 \le \|\underline{v}_h\|_{\mathbf{X},h}^2 \le c_n \kappa_{\eta} \|\underline{v}_h\|_{\mathbf{X},h}^2 \quad \forall \underline{v}_h \in \underline{\mathcal{X}}_{h,n}^{\ell}. $$

**Theorem 18** (Maxwell compactness in $\underline{\mathcal{X}}_{h,n}^{\ell}$). Let $(M_h)_{h \in \mathcal{H}}$ be a regular mesh family as defined in Section 3.1. Let $(\underline{v}_h)_{h \in \mathcal{H}}$ be a sequence of elements of $\underline{\mathcal{X}}_{h,n}^{\ell}$ for which there exists a real number $c_M > 0$ (independent of $h$) such that $\eta_{\#}^{-\frac{1}{2}} \|\underline{v}_h\|_{\mathbf{X}_{n,h}} \le c_M$ for all $h \in \mathcal{H}$. Then, there exists an element $v \in \mathbf{X}_n(\Omega)$ such that, along a subsequence, as $h \to 0$,

(a) $v_h \to v$ strongly in $L^2(\Omega)$;

(b) $C_h^m(\underline{v}_h) \to \text{curl } v$ weakly in $L^2(\Omega)$;

(c) $D_h^p(\underline{v}_h) \to \text{div}(\eta v)$ weakly in $L^2(\Omega)$;

(d) $\gamma_{\tau,h}(\underline{v}_h) \to v|_{\Gamma} \times n$ weakly in $\mathbf{H}^{-\frac{1}{2}}(\Gamma)$.

Importantly, no relation is required between $m$, $p$, and $\ell$ for this result to hold true.

*Proof.* The proof proceeds in three steps, as in the continuous case (cf. Proposition 7). For the sake of brevity, since, up to inverting the roles of the rotational- and divergence-related parts in the proof, the three steps are essentially identical to those from Theorem 17, we do not detail them. □
(since grad $\varphi_h \to$ grad $\varphi$ weakly in $L^2(\Omega)$), we infer by the triangle inequality that

$$ \| \mathrm{grad} \varphi_h - \mathrm{grad} \varphi \|_0 \le 2c_{W,1} \sqrt{3\kappa_\eta c_M}. $$

Combining this uniform boundedness result to the fact that $\varphi_h \to \varphi$ strongly in $L^2(\Omega)$ to pass to the limit $h \to 0$ in (66), we conclude that $\mathrm{grad} \varphi_h \to \mathrm{grad} \varphi$ strongly in $L^2(\Omega)$.

Adapting the latter arguments, there also holds

$$ \begin{aligned} \eta_{\#} \|\eta^{-\frac{1}{2}}(\mathrm{curl} \psi_h - \mathrm{curl} \psi)\|_{0}^{2} &= (v_h - v, \mathrm{curl}(\psi_h - \psi))_{\Omega} \\ &= (\mathrm{curl}_h(v_h - v), \psi_h - \psi)_{\Omega} \\ &\quad + \sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \varepsilon_{T,F} (v_{T|F} \times n_F - v_{F,\tau}, n_F \times ((\psi_h - \psi)_{F} \times n_F))_{F}, \end{aligned} $$

where we have used that $\underline{v}_h \in \underline{X}_{h,\tau}^{\ell}$, $v \in H_0(\mathrm{curl}; \Omega)$, and $\psi_h, \psi \in H^1(\Omega)$. Additionally invoking (38), this implies that

$$ \eta_{\#} \|\eta^{-\frac{1}{2}}(\mathrm{curl} \psi_h - \mathrm{curl} \psi)\|_{0}^{2} \lesssim (\|\psi_h - \psi\|_{0} + h|\psi_h - \psi|_{1}), $$

which eventually yields, since $|\psi_h - \psi|_1 \le 2C_{\Omega,1}c_{W,1}\sqrt{3\kappa_\eta^{\frac{1}{2}}}c_M$ (here, we additionally used the estimate $|\psi_h|_1 \le C_{\Omega,1} \|\mathrm{curl} \psi_h\|_0$ and the fact that $\psi_h \to \psi$ weakly in $H^1(\Omega)$), and $\psi_h \to \psi$ strongly in $L^2(\Omega)$, that $\mathrm{curl} \psi_h \to \mathrm{curl} \psi$ strongly in $L^2(\Omega)$.

Thus, along a subsequence (not relabelled), $v_h \to v$ strongly in $L^2(\Omega)$, yielding the conclusion. □

## 5.3 Compactness in $\underline{X}_{h,n}^\ell$

Recall the definitions (33) of the discrete space $\underline{X}_{h,n}^\ell$, and (35)–(36) of the hybrid semi-norms $|\cdot|_{\mathrm{curl},h}$ and $|\cdot|_{\mathrm{div},h}$. Leveraging the second hybrid Weber inequality from Theorem 15, we infer that

$$ \| \underline{v}_h \|_{\underline{X}_{n,h}}^2 := \eta_b^{-1} |\underline{v}_h|_{\mathrm{div},h}^2 + \eta_{\#} |\underline{v}_h|_{\mathrm{curl},h}^2 + \eta_b^{-1} \sum_{i=1}^{\beta_1} |(\gamma_{n,h}(\underline{v}_h), 1)_{\Sigma_i}|^2 \quad (67) $$

defines a norm on $\underline{X}_{h,n}^\ell$, which can be proven uniformly (in $h$) equivalent to the $\|\cdot\|_{X,h}$-norm (37): there is $c_n > 0$, independent of $h$ and $\eta$, such that

$$ (c_n \kappa_\eta)^{-1} \|\underline{v}_h\|_{X,h}^2 \le \|\underline{v}_h\|_{X,n,h}^2 \le c_n \kappa_\eta \|\underline{v}_h\|_{X,h}^2 \quad \forall \underline{v}_h \in \underline{X}_{h,n}^\ell. $$

**Theorem 18** (Maxwell compactness in $\underline{X}_{h,n}^\ell$). Let $(M_h)_{h \in \mathcal{H}}$ be a regular mesh family as defined in Section 3.1. Let $(\underline{v}_h)_{h \in \mathcal{H}}$ be a sequence of elements of $\underline{X}_{h,n}^\ell$ for which there exists a real number $c_M > 0$ (independent of $h$) such that $\eta_{\#}^{-\frac{1}{2}} \|\underline{v}_h\|_{X,n,h} \le c_M$ for all $h \in \mathcal{H}$. Then, there exists an element $v \in X_n(\Omega)$ such that, along a subsequence, as $h \to 0$,

(a) $v_h \to v$ strongly in $L^2(\Omega)$;

(b) $C_h^m(\underline{v}_h) \to \mathrm{curl} v$ weakly in $L^2(\Omega)$;

(c) $D_h^p(\underline{v}_h) \to \mathrm{div}(\eta v)$ weakly in $L^2(\Omega)$;

(d) $\gamma_{\tau,h}(\underline{v}_h) \to v|_{\Gamma} \times n$ weakly in $H^{-\frac{1}{2}}(\Gamma)$.

Importantly, no relation is required between $m$, $p$, and $\ell$ for this result to hold true.

*Proof.* The proof proceeds in three steps, as in the continuous case (cf. Proposition 7). For the sake of brevity, since, up to inverting the roles of the rotational- and divergence-related parts in the proof, the three steps are essentially identical to those from Theorem 17, we do not detail them. □
# Acknowledgments

The work of the authors is supported by the action "Préservation de l'emploi de R&D" from the "Plan de Relance" recovery plan of the French State. Support from the LabEx CEMPI (ANR-11-LABX-0007) is also acknowledged.

## A Continuity modulus of $\mathbf{div}^{-1}$ on star-shaped cells

**Lemma 19** (Continuity modulus of $\mathbf{div}^{-1}$). Let $\ell \in \mathbb{N}$. Let $T \in \mathcal{T}_h$, and assume that $T$ is star-shaped with respect to $\mathbf{x}_T$. Then, for all $d \in \mathcal{P}^{\ell-1}(T)$,

$$ \| \mathbf{div}^{-1} d \|_{0,T} \le \frac{2}{3} h_T \| d \|_{0,T}. \quad (68) $$

*Proof.* The mapping $\mathrm{div} : \mathcal{R}^{c,\ell}(T) \to \mathcal{P}^{\ell-1}(T)$ being an isomorphism, let $r \in \mathcal{P}^{\ell-1}(T)$ be the unique solution to

$$ d = \mathrm{div}(r(\mathbf{x} - \mathbf{x}_T)). $$

Using a simple vector calculus identity to expand the right-hand side, there holds

$$ d = 3r + \mathrm{grad} \, r \cdot (\mathbf{x} - \mathbf{x}_T). $$

Now, multiplying both sides by $\bar{r}$, integrating over $T$, and taking the real part, we get

$$ \begin{aligned} \Re((d, r)_T) &= 3\|r\|_{0,T}^2 + \frac{1}{2}(\mathrm{grad}(|r|^2), (\mathbf{x} - \mathbf{x}_T))_T \\ &= 3\|r\|_{0,T}^2 - \frac{3}{2}\|r\|_{0,T}^2 + \frac{1}{2}(|r|^2, (\mathbf{x} - \mathbf{x}_T) \cdot \mathbf{n})_{\partial T})_{\partial T} \\ &\ge \frac{3}{2}\|r\|_{0,T}^2, \end{aligned} $$

where we have used an integration by parts formula to pass to the second line, and the fact that $T$ is star-shaped with respect to $\mathbf{x}_T$ to conclude. Finally, applying the Cauchy-Schwarz inequality to $(d, r)_T$, we infer

$$ \| \mathbf{div}^{-1} d \|_{0,T} = \| r(\mathbf{x} - \mathbf{x}_T) \|_{0,T} \le h_T \| r \|_{0,T} \le \frac{2}{3} h_T \| d \|_{0,T}, $$

which proves (68). □

# References

[1] C. Amrouche, C. Bernardi, M. Dauge, and V. Girault. Vector potentials in three-dimensional non-smooth domains. *Math. Methods Appl. Sci.*, 21(9):823–864, 1998.

[2] D. N. Arnold. *Finite element exterior calculus*, volume 93 of *CBMS-NSF Regional Conference Series in Applied Mathematics*. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2018.

[3] D. N. Arnold, R. S. Falk, and R. Winther. Finite element exterior calculus, homological techniques, and applications. *Acta Numer.*, 15:1–155, 2006.

[4] F. Assous, P. Ciarlet Jr., and S. Labrunie. *Mathematical foundations of computational electromagnetism*, volume 198 of *Applied Mathematical Sciences*. Springer, Cham, 2018.

[5] B. Ayuso de Dios, K. Lipnikov, and G. Manzini. The nonconforming virtual element method. *ESAIM Math. Model. Numer. Anal.*, 50(3):879–904, 2016.
# Acknowledgments

The work of the authors is supported by the action "Préservation de l'emploi de R&D" from the "Plan de Relance" recovery plan of the French State. Support from the LabEx CEMPI (ANR-11-LABX-0007) is also acknowledged.

## A Continuity modulus of $\mathbf{div}^{-1}$ on star-shaped cells

**Lemma 19** (Continuity modulus of $\mathbf{div}^{-1}$). Let $\ell \in \mathbb{N}$. Let $T \in \mathcal{T}_h$, and assume that $T$ is star-shaped with respect to $\mathbf{x}_T$. Then, for all $d \in \mathcal{P}^{\ell-1}(T)$,

$$ \|\mathbf{div}^{-1} d\|_{0,T} \le \frac{2}{3} h_T \|d\|_{0,T}. \qquad (68) $$

*Proof.* The mapping $\mathrm{div} : \mathcal{R}^{c,\ell}(T) \to \mathcal{P}^{\ell-1}(T)$ being an isomorphism, let $r \in \mathcal{P}^{\ell-1}(T)$ be the unique solution to

$$ d = \mathrm{div}(r(\mathbf{x} - \mathbf{x}_T)). $$

Using a simple vector calculus identity to expand the right-hand side, there holds

$$ d = 3r + \mathrm{grad} \, r(\mathbf{x} - \mathbf{x}_T). $$

Now, multiplying both sides by $\bar{r}$, integrating over $T$, and taking the real part, we get

$$ \begin{aligned} \Re((d, r)_T) &= 3\|r\|_{0,T}^2 + \frac{1}{2}(\mathrm{grad}(|r|^2), (\mathbf{x} - \mathbf{x}_T))_T \\ &= 3\|r\|_{0,T}^2 - \frac{3}{2}\|r\|_{0,T}^2 + \frac{1}{2}(|r|^2, (\mathbf{x} - \mathbf{x}_T) \cdot \mathbf{n})_{\partial T})_{\partial T} \\ &\ge \frac{3}{2}\|r\|_{0,T}^2, \end{aligned} $$

where we have used an integration by parts formula to pass to the second line, and the fact that $T$ is star-shaped with respect to $\mathbf{x}_T$ to conclude. Finally, applying the Cauchy-Schwarz inequality to $(d, r)_T$, we infer

$$ \|\mathbf{div}^{-1} d\|_{0,T} = \|r(\mathbf{x} - \mathbf{x}_T)\|_{0,T} \le h_T \|r\|_{0,T} \le \frac{2}{3} h_T \|d\|_{0,T}, $$

which proves (68). □

# References

[1] C. Amrouche, C. Bernardi, M. Dauge, and V. Girault. Vector potentials in three-dimensional non-smooth domains. *Math. Methods Appl. Sci.*, 21(9):823–864, 1998.

[2] D. N. Arnold. *Finite element exterior calculus*, volume 93 of *CBMS-NSF Regional Conference Series in Applied Mathematics*. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2018.

[3] D. N. Arnold, R. S. Falk, and R. Winther. Finite element exterior calculus, homological techniques, and applications. *Acta Numer.*, 15:1–155, 2006.

[4] F. Assous, P. Ciarlet Jr., and S. Labrunie. *Mathematical foundations of computational electromagnetism*, volume 198 of *Applied Mathematical Sciences*. Springer, Cham, 2018.

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[13] G. Chen, J. Cui, and L. Xu. Analysis of a Hybridizable Discontinuous Galerkin method for the Maxwell operator. *ESAIM Math. Model. Numer. Anal.*, 53(1):301–324, 2019.

[14] H. Chen, W. Qiu, K. Shi, and M. Solano. A superconvergent HDG method for the Maxwell equations. *J. Sci. Comput.*, 70(3):1010–1029, 2017.

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[19] M. Costabel. A remark on the regularity of solutions of Maxwell’s equations on Lipschitz domains. *Math. Methods Appl. Sci.*, 12(4):365–368, 1990.

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[23] D. A. Di Pietro and J. Droniou. An arbitrary-order Discrete De Rham complex on polyhedral meshes: Exactness, Poincaré inequalities, and consistency. *Found. Comput. Math.*, 23(1):85–164, 2023.

[24] D. A. Di Pietro and A. Ern. A hybrid high-order locking-free method for linear elasticity on general meshes. *Comput. Methods Appl. Mech. Engrg.*, 283:1–21, 2015.

[25] D. A. Di Pietro, A. Ern, and S. Lemaire. An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. *Comput. Methods Appl. Math.*, 14(4):461–472, 2014.

[26] S. Du and F.-J. Sayas. A unified error analysis of Hybridizable Discontinuous Galerkin methods for the static Maxwell equations. *SIAM J. Numer. Anal.*, 58(2):1367–1391, 2020.
[6] S. Bauer, D. Pauly, and M. Schomburg. The Maxwell compactness property in bounded weak Lipschitz domains with mixed boundary conditions. *SIAM J. Math. Anal.*, 48(4):2912–2943, 2016.

[7] L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and A. Russo. Basic principles of virtual element methods. *Math. Models Methods Appl. Sci.*, 23(1):199–214, 2013.

[8] F. Bonaldi, D. A. Di Pietro, J. Droniou, and K. Hu. An exterior calculus framework for polytopal methods, 2023. Submitted, preprint arXiv:2303.11093.

[9] A. Bonito, J.-L. Guermond, and F. Luddens. Regularity of the Maxwell equations in heterogeneous media and Lipschitz domains. *J. Math. Anal. Appl.*, 408(2):498–512, 2013.

[10] A. Bossavit. *Computational electromagnetism – Variational formulations, complementarity, edge elements*. Electromagnetism. Academic Press, Inc., San Diego, CA, 1998.

[11] D. Braess and J. Schöberl. Equilibrated residual error estimator for edge elements. *Math. Comp.*, 77(262):651–672, 2008.

[12] F. Chave, D. A. Di Pietro, and S. Lemaire. A discrete Weber inequality on three-dimensional hybrid spaces with application to the HHO approximation of magnetostatics. *Math. Models Methods Appl. Sci.*, 32(1):175–207, 2022.

[13] G. Chen, J. Cui, and L. Xu. Analysis of a Hybridizable Discontinuous Galerkin method for the Maxwell operator. *ESAIM Math. Model. Numer. Anal.*, 53(1):301–324, 2019.

[14] H. Chen, W. Qiu, K. Shi, and M. Solano. A superconvergent HDG method for the Maxwell equations. *J. Sci. Comput.*, 70(3):1010–1029, 2017.

[15] S. H. Christiansen and M. W. Licht. Poincaré–Friedrichs inequalities of complexes of discrete distribu-
tional differential forms. *BIT Numer. Math.*, 60(2):345–371, 2020.

[16] M. Cicuttin, A. Ern, and N. Pignet. *Hybrid high-order methods – A primer with applications to solid mechanics*. SpringerBriefs in Mathematics. Springer, Cham, 2021.

[17] B. Cockburn, D. A. Di Pietro, and A. Ern. Bridging the Hybrid High-Order and Hybridizable Discontin-
uous Galerkin methods. *ESAIM Math. Model. Numer. Anal.*, 50(3):635–650, 2016.

[18] B. Cockburn, J. Gopalakrishnan, and R. Lazarov. Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. *SIAM J. Numer. Anal.*, 47(2):1319–1365, 2009.

[19] M. Costabel. A remark on the regularity of solutions of Maxwell’s equations on Lipschitz domains. *Math. Methods Appl. Sci.*, 12(4):365–368, 1990.

[20] M. Costabel, M. Dauge, and S. Nicaise. Singularities of Maxwell interface problems. *M2AN Math. Model. Numer. Anal.*, 33(3):627–649, 1999.

[21] R. Dautray and J.-L. Lions. *Mathematical analysis and numerical methods for science and technology – Vol. 3: Spectral theory and applications*. Springer-Verlag, Berlin, 1990. With the collaboration of Michel Artola and Michel Cessenat. Translated from the French by John C. Amson.

[22] D. A. Di Pietro and J. Droniou. *The Hybrid High-Order method for polytopal meshes – Design, analysis, and applications*, volume 19 of *Modeling, Simulation and Applications*. Springer, Cham, 2020.

[23] D. A. Di Pietro and J. Droniou. An arbitrary-order Discrete De Rham complex on polyhedral meshes: Exactness, Poincaré inequalities, and consistency. *Found. Comput. Math.*, 23(1):85–164, 2023.

[24] D. A. Di Pietro and A. Ern. A hybrid high-order locking-free method for linear elasticity on general meshes. *Comput. Methods Appl. Mech. Engrg.*, 283:1–21, 2015.

[25] D. A. Di Pietro, A. Ern, and S. Lemaire. An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. *Comput. Methods Appl. Math.*, 14(4):461–472, 2014.

[26] S. Du and F.-J. Sayas. A unified error analysis of Hybridizable Discontinuous Galerkin methods for the static Maxwell equations. *SIAM J. Numer. Anal.*, 58(2):1367–1391, 2020.
[27] A. Ern and J.-L. Guermond. Quasi-optimal nonconforming approximation of elliptic PDEs with contrasted coefficients and $H^{1+r}$, $r > 0$, regularity. *Found. Comput. Math.*, 22(5):1273–1308, 2022.

[28] P. Fernandes and G. Gilardi. Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. *Math. Models Methods Appl. Sci.*, 7(7):957–991, 1997.

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[32] F. Jochmann. Regularity of weak solutions of Maxwell’s equations with mixed boundary conditions. *Math. Methods Appl. Sci.*, 22(14):1255–1274, 1999.

[33] C. Lehrenfeld and J. Schöberl. High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows. *Comput. Methods Appl. Mech. Engrg.*, 307:339–361, 2016.

[34] S. Lemaire. Bridging the Hybrid High-Order and Virtual Element methods. *IMA J. Numer. Anal.*, 41(1):549–593, 2021.

[35] M. W. Licht. Complexes of discrete distributional differential forms and their homology theory. *Found. Comput. Math.*, 17(4):1085–1122, 2017.

[36] P. Monk. *Finite element methods for Maxwell's equations*. Numerical Mathematics and Scientific Computation. Oxford University Press, New York, 2003.

[37] L. Mu, J. Wang, X. Ye, and S. Zhang. A weak Galerkin finite element method for the Maxwell equations. *J. Sci. Comput.*, 65(1):363–386, 2015.

[38] J.-C. Nédélec. Mixed finite elements in $\mathbb{R}^3$. *Numer. Math.*, 35(3):315–341, 1980.

[39] J.-C. Nédélec. A new family of mixed finite elements in $\mathbb{R}^3$. *Numer. Math.*, 50(1):57–81, 1986.

[40] N. C. Nguyen, J. Peraire, and B. Cockburn. Hybridizable Discontinuous Galerkin methods for the time-harmonic Maxwell's equations. *J. Comput. Phys.*, 230(19):7151–7175, 2011.

[41] J. Wang and X. Ye. A weak Galerkin finite element method for second-order elliptic problems. *J. Comput. Appl. Math.*, 241:103–115, 2013.

[42] C. Weber. A local compactness theorem for Maxwell's equations. *Math. Methods Appl. Sci.*, 2(1):12–25, 1980.

[43] N. Weck. Maxwell's boundary value problem on Riemannian manifolds with nonsmooth boundaries. *J. Math. Anal. Appl.*, 46:410–437, 1974.

[44] H. Whitney. *Geometric integration theory*. Princeton University Press, Princeton, N. J., 1957.
[27] A. Ern and J.-L. Guermond. Quasi-optimal nonconforming approximation of elliptic PDEs with contrasted coefficients and $H^{1+r}$, $r > 0$, regularity. *Found. Comput. Math.*, 22(5):1273–1308, 2022.

[28] P. Fernandes and G. Gilardi. Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. *Math. Models Methods Appl. Sci.*, 7(7):957–991, 1997.

[29] V. Girault and P.-A. Raviart. *Finite element methods for Navier–Stokes equations – Theory and algorithms, volume 5 of Springer Series in Computational Mathematics*. Springer-Verlag, Berlin, 1986.

[30] P. W. Gross and P. R. Kotiuga. *Electromagnetic theory and computation: a topological approach*, volume 48 of *Mathematical Sciences Research Institute Publications*. Cambridge University Press, Cambridge, 2004.

[31] F. Jochmann. A compactness result for vector fields with divergence and curl in $L^q(\Omega)$ involving mixed boundary conditions. *Appl. Anal.*, 66(1-2):189–203, 1997.

[32] F. Jochmann. Regularity of weak solutions of Maxwell’s equations with mixed boundary conditions. *Math. Methods Appl. Sci.*, 22(14):1255–1274, 1999.

[33] C. Lehrenfeld and J. Schöberl. High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows. *Comput. Methods Appl. Mech. Engrg.*, 307:339–361, 2016.

[34] S. Lemaire. Bridging the Hybrid High-Order and Virtual Element methods. *IMA J. Numer. Anal.*, 41(1):549–593, 2021.

[35] M. W. Licht. Complexes of discrete distributional differential forms and their homology theory. *Found. Comput. Math.*, 17(4):1085–1122, 2017.

[36] P. Monk. *Finite element methods for Maxwell's equations*. Numerical Mathematics and Scientific Computation. Oxford University Press, New York, 2003.

[37] L. Mu, J. Wang, X. Ye, and S. Zhang. A weak Galerkin finite element method for the Maxwell equations. *J. Sci. Comput.*, 65(1):363–386, 2015.

[38] J.-C. Nédélec. Mixed finite elements in $\mathbb{R}^3$. *Numer. Math.*, 35(3):315–341, 1980.

[39] J.-C. Nédélec. A new family of mixed finite elements in $\mathbb{R}^3$. *Numer. Math.*, 50(1):57–81, 1986.

[40] N. C. Nguyen, J. Peraire, and B. Cockburn. Hybridizable Discontinuous Galerkin methods for the time-harmonic Maxwell's equations. *J. Comput. Phys.*, 230(19):7151–7175, 2011.

[41] J. Wang and X. Ye. A weak Galerkin finite element method for second-order elliptic problems. *J. Comput. Appl. Math.*, 241:103–115, 2013.

[42] C. Weber. A local compactness theorem for Maxwell's equations. *Math. Methods Appl. Sci.*, 2(1):12–25, 1980.

[43] N. Weck. Maxwell's boundary value problem on Riemannian manifolds with nonsmooth boundaries. *J. Math. Anal. Appl.*, 46:410–437, 1974.

[44] H. Whitney. *Geometric integration theory*. Princeton University Press, Princeton, N. J., 1957.