Proof 1. *Proof of (3.42).* From the consistency properties (3.12), (3.13) and (3.16) of the full gradients and the definition (3.18) of $\underline{\mathbf{G}}_h^k$, it is readily inferred that $\underline{\mathbf{G}}_h^k(\underline{I}_{\text{grad},h}^k C) = \underline{0}$ for all $C \in \mathbb{R}$, hence $\underline{I}_{\text{grad},h}^k \mathbb{R} \subset \text{Ker } \underline{\mathbf{G}}_h^k$.

To prove converse inclusion Ker $\underline{\mathbf{G}}_h^k \subset \underline{I}_{\text{grad},h}^k \mathbb{R}$, let $q_h \in \underline{\mathcal{X}}_{\text{grad},h}^k$ be such that $\underline{\mathbf{G}}_h^k q_h = \underline{0}$. By the definitions (3.18) of $\underline{\mathbf{G}}_h^k$ and (3.8) of $\mathcal{G}_E^k$, this means that $q'_E = 0$ for all $E \in \mathcal{E}_h$, that is, $(q_{\mathcal{E}_h})|_E$ is constant over $E$. Since $\Omega$ has only one connected component, accounting for the single-valuedness of $q_{\mathcal{E}_h}$ at vertices, we thus infer the existence of $C \in \mathbb{R}$ such that $q_{\mathcal{E}_h} = C$. Let now $F \in \mathcal{F}_h$ and $\mathbf{w}_F \in \mathcal{R}^{c,k}(F)$. We have $\pi_{\mathcal{R},F}^{c,k}(\underline{\mathbf{G}}_F^k q_F) = \underline{0}$, and thus

$$ 
\begin{aligned}
0 &= \int_F \underline{\mathbf{G}}_F^k q_F \cdot \mathbf{w}_F \\
&= -\int_F q_F \operatorname{div}_F \mathbf{w}_F + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E q_{\mathcal{E}_F} (\mathbf{w}_F \cdot \mathbf{n}_{FE}) \\
&= \int_F (C - q_F) \operatorname{div}_F \mathbf{w}_F,
\end{aligned}
$$

where the second equality comes from the definition (3.9) of $\underline{\mathbf{G}}_F^k q_F$, and the conclusion is obtained accounting for the fact that $q_{\mathcal{E}_F} = C$ and integrating by parts. Since $\mathbf{w}_F$ is generic in $\mathcal{R}^{c,k}(F)$, recalling the isomorphism (2.9) this implies $\pi_{\mathcal{P},F}^{k-1}(q_F - C) = 0$, and thus $q_F = \pi_{\mathcal{P},F}^{k-1} C$. As, for all $F \in \mathcal{F}_h$, the previous results give $q_F = (q_F, q_{\mathcal{E}_F}) = \underline{I}_{\text{grad},F}^k C$, we also have $\gamma_F^{k+1} q_F = C$ by (3.14). Similarly, let $T \in \mathcal{T}_h$ and $\mathbf{w}_T \in \mathcal{R}^{c,k}(T)$. Writing the definition (3.11) of $\underline{\mathbf{G}}_T^k q_T$ for $\mathbf{w}_T \in \mathcal{R}^{c,k}(T)$, and accounting for $\pi_{\mathcal{R},T}^{c,k}(\underline{\mathbf{G}}_T^k q_T) = \underline{0}$ and $\gamma_F^{k+1} q_F = C$, it is inferred

$$ 
\begin{aligned}
0 &= \int_T \underline{\mathbf{G}}_T^k q_T \cdot \mathbf{w}_T \\
&= -\int_T q_T \operatorname{div} \mathbf{w}_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F C(\mathbf{w}_T \cdot \mathbf{n}_F) \\
&= \int_F (C - q_T) \operatorname{div} \mathbf{w}_T,
\end{aligned}
$$

which implies, invoking the isomorphism (2.9), $\pi_{\mathcal{P},T}^{k-1}(q_T - C) = 0$ since $\mathbf{w}_T$ is generic in $\mathcal{R}^{c,k}(T)$. Hence $q_T = \pi_{\mathcal{P},T}^{k-1} C$ for all $T \in \mathcal{T}_h$, which concludes the proof that $q_T = \underline{I}_{\text{grad},h}^k C$.

2. *Proof of (3.43).* The inclusion (3.43) follows from the local property:

$$ \operatorname{Im} \underline{\mathbf{G}}_T^k \subset \operatorname{Ker} \underline{\mathbf{C}}_T^k \quad \forall T \in \mathcal{T}_h, \qquad (3.46) $$

i.e., $\underline{\mathbf{C}}_T^k(\underline{\mathbf{G}}_T^k q_T) = \underline{0}$ for all $q_T \in \underline{\mathcal{X}}_{\text{grad},T}$. Let $T \in \mathcal{T}_h$. The relation (3.21) implies $\underline{C}_F^k(\underline{\mathbf{G}}_F^k q_F) = 0$ for all $F \in \mathcal{F}_T$. The fact that $\pi_{\mathcal{G},T}^{k-1}[\underline{\mathbf{C}}_T^k(\underline{\mathbf{G}}_T^k q_T)] = \underline{0}$ then follows
from (3.35). We next notice that it holds, for all $w_T \in \mathcal{G}^{c,k}(T)$,

$$ \int_T \mathbf{C}_T^k(\underline{\mathbf{G}}_T^k \underline{q}_T) \cdot \mathbf{w}_T = \int_T \pi_{\mathcal{R},T}^{k,A}(\mathbf{G}_T^k \underline{q}_T) \cdot \mathrm{curl} \, \mathbf{w}_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \mathbf{G}_F^k \underline{q}_F \cdot (\mathbf{w}_T \times \mathbf{n}_F) = 0, $$

where we have used the definition (3.28) of $\mathbf{C}_T^k$ and the property (3.26) of the tangential trace reconstruction in the first equality, the fact that $\mathrm{curl} \, w_T \in \mathcal{R}^{k-1}(T)$ to cancel the projector, and the link (3.17) between volume and face gradients to conclude. This shows that $\pi_{\mathcal{G},T}^{c,k}[\mathbf{C}_T^k(\underline{\mathbf{G}}_T^k \underline{q}_T)] = \mathbf{0}$ and concludes the proof of (3.46).

### 3. Proof of (3.44)
Immediate consequence of (3.34) after observing that $\underline{\mathbf{C}}_T^k$ and $D_h^k$ are the restrictions of $\underline{\mathbf{C}}_h^k$ and $D_h^k$ to $T$, respectively.

### 4. Proof of (3.45)
The inclusion $\mathrm{Im} \, D_h^k \subset \mathcal{P}^k(\mathcal{T}_h)$ is an obvious consequence of the definition (3.33) of the global divergence. To prove the converse inclusion, let $q_h \in \mathcal{P}^k(\mathcal{T}_h)$. Since the continuous divergence operator $\mathrm{div} : \mathbf{H}^1(\Omega) \to L^2(\Omega)$ is onto (see, e.g., [28, Lemma 8.3]), there exists $v \in \mathbf{H}^1(\Omega)$ such that $\mathrm{div} \, v = q_h$. Setting $\underline{v}_h := \underline{I}_{\mathrm{div},h}^k v \in \underline{\mathbf{X}}_{\mathrm{div},h}^k$, the commutation property (3.40) and the definition (3.33) of the global divergence yield $D_h^k \underline{v}_h = \pi_{\mathcal{P},h}^k(\mathrm{div} \, v) = \pi_{\mathcal{P},h}^k q_h = q_h$. This shows that $\mathcal{P}^k(\mathcal{T}_h) \subset \mathrm{Im} \, D_h^k$, thereby concluding the proof of (3.45). $\square$

**Remark 15 (Kernel of the full curl operator)** Combining the inclusion (3.21) with the relation (3.30) linking element and face curls, it is inferred that, for all $\underline{q}_T \in \underline{\mathbf{X}}_{\mathrm{grad},T}^k$, $\pi_{\mathcal{G},T}^k[\underline{\mathbf{C}}_T^k(\underline{\mathbf{G}}_T^k \underline{q}_T)] = \mathbf{0}$. On the other hand, (3.46) implies $\pi_{\mathcal{G},T}^{c,k}[\underline{\mathbf{C}}_T^k(\underline{\mathbf{G}}_T^k \underline{q}_T)] = \mathbf{0}$. Hence $\underline{\mathbf{C}}_T^k(\underline{\mathbf{G}}_T^k \underline{q}_T) = \mathbf{0}$ by (2.15) with $\underline{\mathbf{a}} = \underline{\mathbf{C}}_T^k(\underline{\mathbf{G}}_T^k \underline{q}_T)$ and $(S, S^c) = (\underline{\mathbf{G}}^k(T), \underline{\mathbf{G}}^{c,k}(T))$. This shows that $\mathrm{Im} \, \underline{\mathbf{G}}_T^k \subset \mathrm{Ker} \, \underline{\mathbf{C}}_T^k$.

The exactness properties of the DDR sequence, depending on the topology of the domain, are collected in the following theorem.

**Theorem 2 (Exactness)** Denoting by $(b_0, b_1, b_2, b_3)$ the Betti numbers of $\Omega$ (with $b_0 = 1$ since $\Omega$ is connected and $b_3 = 0$ since $\Omega \subset \mathbb{R}^3$), we have

$$ b_1 = 0 \implies \mathrm{Im} \, \underline{\mathbf{G}}_h^k = \mathrm{Ker} \, \underline{\mathbf{C}}_h^k, \qquad (3.47) $$

$$ b_2 = 0 \implies \mathrm{Im} \, \underline{\mathbf{C}}_h^k = \mathrm{Ker} \, D_h^k. \qquad (3.48) $$

**Remark 16 (Meaning of vanishing Betti numbers)** In broad terms, the condition $b_1 = 0$ means that $\Omega$ does not have any tunnel, while $b_2 = 0$ means that $\Omega$ does not enclose any void. A typical example of $\Omega$ that has $b_1 \neq 0$ is (the interior of) a torus, and an example of $\Omega$ with $b_2 \neq 0$ is a domain enclosed between two concentric spheres.
from (3.35). We next notice that it holds, for all $w_T \in \mathcal{G}^{c,k}(T)$,

$$ \int_T \mathbf{C}_T^k(\underline{\mathbf{G}}_T^k \underline{q}_T) \cdot \mathbf{w}_T = \int_T \pi_{\mathcal{R},T}^{k,A}(\mathbf{G}_T^k \underline{q}_T) \cdot \mathrm{curl} \mathbf{w}_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \mathbf{G}_F^k \underline{q}_F \cdot (\mathbf{w}_T \times \mathbf{n}_F) = 0, $$

where we have used the definition (3.28) of $\mathbf{C}_T^k$ and the property (3.26) of the tangential trace reconstruction in the first equality, the fact that $\mathrm{curl} \mathbf{w}_T \in \mathcal{R}^{k-1}(T)$ to cancel the projector, and the link (3.17) between volume and face gradients to conclude. This shows that $\pi_{\mathcal{G},T}^{c,k}[\mathbf{C}_T^k(\underline{\mathbf{G}}_T^k \underline{q}_T)] = \mathbf{0}$ and concludes the proof of (3.46).

3. *Proof of (3.44).* Immediate consequence of (3.34) after observing that $\underline{\mathbf{C}}_T^k$ and $D_h^k$ are the restrictions of $\underline{\mathbf{C}}_h^k$ and $D_h^k$ to $T$, respectively.

4. *Proof of (3.45).* The inclusion $\mathrm{Im} \, D_h^k \subset \mathcal{P}^k(\mathcal{T}_h)$ is an obvious consequence of the definition (3.33) of the global divergence. To prove the converse inclusion, let $q_h \in \mathcal{P}^k(\mathcal{T}_h)$. Since the continuous divergence operator $\mathrm{div} : \mathbf{H}^1(\Omega) \to L^2(\Omega)$ is onto (see, e.g., [28, Lemma 8.3]), there exists $v \in \mathbf{H}^1(\Omega)$ such that $\mathrm{div} v = q_h$. Setting $\underline{v}_h := \underline{I}_{\mathrm{div},h}^k v \in \underline{\mathbf{X}}_{\mathrm{div},h}^k$, the commutation property (3.40) and the definition (3.33) of the global divergence yield $D_h^k \underline{v}_h = \pi_{\mathcal{P},h}^k(\mathrm{div} v) = \pi_{\mathcal{P},h}^k q_h = q_h$. This shows that $\mathcal{P}^k(\mathcal{T}_h) \subset \mathrm{Im} \, D_h^k$, thereby concluding the proof of (3.45). $\square$

**Remark 15 (Kernel of the full curl operator)** Combining the inclusion (3.21) with the relation (3.30) linking element and face curls, it is inferred that, for all $\underline{q}_T \in \underline{\mathbf{X}}_{\mathrm{grad},T}^k$, $\pi_{\mathcal{G},T}^k[\underline{\mathbf{C}}_T^k(\underline{\mathbf{G}}_T^k \underline{q}_T)] = \mathbf{0}$. On the other hand, (3.46) implies $\pi_{\mathcal{G},T}^{c,k}[\underline{\mathbf{C}}_T^k(\underline{\mathbf{G}}_T^k \underline{q}_T)] = \mathbf{0}$. Hence $\underline{\mathbf{C}}_T^k(\underline{\mathbf{G}}_T^k \underline{q}_T) = \mathbf{0}$ by (2.15) with $\underline{\mathbf{a}} = \underline{\mathbf{C}}_T^k(\underline{\mathbf{G}}_T^k \underline{q}_T)$ and $(S, S^c) = (\underline{\mathbf{G}}^k(T), \underline{\mathbf{G}}^{c,k}(T))$. This shows that $\mathrm{Im} \, \underline{\mathbf{G}}_T^k \subset \mathrm{Ker} \, \underline{\mathbf{C}}_T^k$.

The exactness properties of the DDR sequence, depending on the topology of the domain, are collected in the following theorem.

**Theorem 2 (Exactness)** Denoting by $(b_0, b_1, b_2, b_3)$ the Betti numbers of $\Omega$ (with $b_0 = 1$ since $\Omega$ is connected and $b_3 = 0$ since $\Omega \subset \mathbb{R}^3$), we have

$$ b_1 = 0 \implies \mathrm{Im} \, \underline{\mathbf{G}}_h^k = \mathrm{Ker} \, \underline{\mathbf{C}}_h^k, \qquad (3.47) $$

$$ b_2 = 0 \implies \mathrm{Im} \, \underline{\mathbf{C}}_h^k = \mathrm{Ker} \, D_h^k. \qquad (3.48) $$

**Remark 16 (Meaning of vanishing Betti numbers)** In broad terms, the condition $b_1 = 0$ means that $\Omega$ does not have any tunnel, while $b_2 = 0$ means that $\Omega$ does not enclose any void. A typical example of $\Omega$ that has $b_1 \neq 0$ is (the interior of) a torus, and an example of $\Omega$ with $b_2 \neq 0$ is a domain enclosed between two concentric spheres.
Proof 1. *Proof of (3.47).* Recalling (3.43), we only have to show the inclusion

$$ \mathrm{Ker} \underline{\mathcal{C}}_h^k \subset \mathrm{Im} \underline{\mathcal{G}}_h^k, \qquad (3.49) $$

that is, for all $\underline{\mathbf{v}}_h \in \underline{\mathcal{X}}_h^k$ such that $\underline{\mathcal{C}}_h^k \underline{\mathbf{v}}_h = \underline{\mathbf{0}}$, there exists $\underline{q}_h \in \underline{\mathcal{X}}_{\mathrm{grad},h}^k$ such that $\underline{\mathbf{v}}_h = \underline{\mathcal{G}}_h^k \underline{q}_h$. In what follows, we show how to construct such a $\underline{q}_h$.

We start by constructing a function $q_{\mathcal{E}_h} \in \mathcal{P}_c^{k+1}(\mathcal{E}_h)$ such that $v_E = (q_{\mathcal{E}_h})'$ for all $E \in \mathcal{E}_h$. Let $V_0, V \in \mathcal{V}_h$ be two distinct mesh vertices of coordinates $\mathbf{x}_{V_0}$ and $\mathbf{x}_V$, respectively, and denote by $\mathcal{E}_P \subset \mathcal{E}_h$ a set of edges that form a connected path $P$ from $V_0$ to $V$ (such a path always exists since $\Omega$ is connected). By the fundamental theorem of calculus, there is a unique function $q_{\mathcal{E}_P} \in \mathcal{P}_c^{k+1}(\mathcal{E}_P)$ such that $q_{\mathcal{E}_P}(\mathbf{x}_{V_0}) = 0$ and $(q_{\mathcal{E}_P})'_{|_E} = v_E$ for all $E \in \mathcal{E}_P$, the derivative being taken in the direction of $E$ ($q_{\mathcal{E}_P}$ is obtained integrating, in the direction defined on each edge $E$ by $t_E$, the functions $(v_E)_{E \in \mathcal{E}_P}$). We want to show that the value $q_{\mathcal{E}_P}(\mathbf{x}_V)$ taken at $V$ is independent of the choice of the path $P$. To this end, denote by $\tilde{P}$ another path from $V_0$ to $V$ formed by the edges in $\mathcal{E}_{\tilde{P}}$, and denote by $-\tilde{P}$ the same path but with reversed orientation. We assume, for the moment, that $\mathcal{E}_P$ and $\mathcal{E}_{\tilde{P}}$ are disjoint. By similar considerations as before, there exists a unique $q_{\mathcal{E}_{\tilde{P}}} \in \mathcal{P}_c^{k+1}(\mathcal{E}_{\tilde{P}})$ such that $q_{\mathcal{E}_{\tilde{P}}}(\mathbf{x}_{V_0}) = 0$ and $(q_{\mathcal{E}_{\tilde{P}}})'_{|_E} = v_E$ for all $E \in \mathcal{E}_{\tilde{P}}$. Since $b_1 = 0$ (i.e., there is no “tunnel” crossing $\Omega$), the path $B := P - P'$ formed by the edges in $\mathcal{E}_B := \mathcal{E}_P \cup \mathcal{E}_{\tilde{P}}$ is a 1-boundary, i.e., there is a set of faces $\mathcal{F}_B \subset \mathcal{F}_h$ giving rise to a connected surface $S_B := \cup_{F \in \mathcal{F}_B} \overline{F}$ such that $B = \partial S_B$. We fix an orientation for $S_B$ and, for all $F \in \mathcal{F}_B$, we denote by $\omega_{BF} \in \{-1, 1\}$ the orientation of $F$ relative to $S_B$. For all $E \in \mathcal{E}_B$, there is a unique face $F \in \mathcal{F}_B$ such that $E \in \mathcal{E}_F$, and we let $\omega_{BE} := \omega_{BF}\omega_{FE}$ denote the orientation of $E$ relative to $S_B$. Since $C_F^k \underline{\mathbf{v}}_F = 0$ for all $F \in \mathcal{F}_B$, it holds

$$ 0 = \sum_{F \in \mathcal{F}_B} \omega_{BF} \int_F C_F^k \underline{\mathbf{v}}_F = - \sum_{F \in \mathcal{F}_B} \omega_{BF} \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E v_E = - \sum_{E \in \mathcal{E}_B} \omega_{BE} \int_E v_E, $$

where the second equality is obtained from (3.19) with $r_F$ identically equal to 1, while the conclusion follows observing that all the edges that are interior to $S_B$ appear exactly twice in the sum, with opposite signs. Thus, reasoning as in [31, Proposition 4.2], there exists $q_{\mathcal{E}_B} \in \mathcal{P}_c^{k+1}(\mathcal{E}_B)$ such that $(q_{\mathcal{E}_B})' = v_E$ for all $E \in \mathcal{E}_B$, which we can be uniquely identified by additionally prescribing that $q_{\mathcal{E}_B}(\mathbf{x}_{V_0}) = 0$. Under this condition, by uniqueness we infer $(q_{\mathcal{E}_B})_{|E} = (q_{\mathcal{E}_P})_{|E}$ for all $E \in \mathcal{E}_P$ and $(q_{\mathcal{E}_B})_{|E} = (q_{\mathcal{E}_{\tilde{P}}})_{|E}$ for all $E \in \mathcal{E}_{\tilde{P}}$. Since $q_{\mathcal{E}_B}$ is continuous at the vertices of $B$, this shows that $q_{\mathcal{E}_P}(\mathbf{x}_V) = q_{\mathcal{E}_{\tilde{P}}}(\mathbf{x}_V)$. This argument can be extended to paths $P$ and $\tilde{P}$ such that $\mathcal{E}_P \cap \mathcal{E}_{\tilde{P}} \neq \emptyset$, the only difference being that one should reason, in this case, on each connected component of the manifold $S_B$ (corresponding to a “loop” inside the path $B = P - \tilde{P}$).

Repeating this reasoning for each vertex $V \in \mathcal{V}_h$ and all possible paths connecting $V_0$ and $V$, we conclude that there exists a unique $q_{\mathcal{E}_h} \in \mathcal{P}_c^{k+1}(\mathcal{E}_h)$ such that $q_{\mathcal{E}_h}(\mathbf{x}_{V_0}) = 0$ and, recalling the notation (3.4),

$$ q'_E = G_E^k q_E = v_E \quad \forall E \in \mathcal{E}_h. \qquad (3.50) $$
Proof 1. *Proof of (3.47).* Recalling (3.43), we only have to show the inclusion

$$ \ker \underline{\mathcal{C}}_h^k \subset \operatorname{Im} \underline{\mathcal{G}}_h^k, \qquad (3.49) $$

that is, for all $\underline{\mathbf{v}}_h \in \underline{\mathcal{X}}_h^k$ such that $\underline{\mathcal{C}}_h^k \underline{\mathbf{v}}_h = \underline{\mathbf{0}}$, there exists $\underline{q}_h \in \underline{\mathcal{X}}_{\text{grad},h}^k$ such that $\underline{\mathbf{v}}_h = \underline{\mathcal{G}}_h^k \underline{q}_h$. In what follows, we show how to construct such a $\underline{q}_h$.

We start by constructing a function $q_{\mathcal{E}_h} \in \mathcal{P}_c^{k+1}(\mathcal{E}_h)$ such that $v_E = (q_{\mathcal{E}_h})'$ for all $E \in \mathcal{E}_h$. Let $V_0, V \in \mathcal{V}_h$ be two distinct mesh vertices of coordinates $\mathbf{x}_{V_0}$ and $\mathbf{x}_V$, respectively, and denote by $\mathcal{E}_P \subset \mathcal{E}_h$ a set of edges that form a connected path $P$ from $V_0$ to $V$ (such a path always exists since $\Omega$ is connected). By the fundamental theorem of calculus, there is a unique function $q_{\mathcal{E}_P} \in \mathcal{P}_c^{k+1}(\mathcal{E}_P)$ such that $q_{\mathcal{E}_P}(\mathbf{x}_{V_0}) = 0$ and $(q_{\mathcal{E}_P})'_{|_E} = v_E$ for all $E \in \mathcal{E}_P$, the derivative being taken in the direction of $E$ ($q_{\mathcal{E}_P}$ is obtained integrating, in the direction defined on each edge $E$ by $t_E$, the functions $(v_E)_{E \in \mathcal{E}_P}$). We want to show that the value $q_{\mathcal{E}_P}(\mathbf{x}_V)$ taken at $V$ is independent of the choice of the path $P$. To this end, denote by $\tilde{P}$ another path from $V_0$ to $V$ formed by the edges in $\mathcal{E}_{\tilde{P}}$, and denote by $-\tilde{P}$ the same path but with reversed orientation. We assume, for the moment, that $\mathcal{E}_P$ and $\mathcal{E}_{\tilde{P}}$ are disjoint. By similar considerations as before, there exists a unique $q_{\mathcal{E}_{\tilde{P}}} \in \mathcal{P}_c^{k+1}(\mathcal{E}_{\tilde{P}})$ such that $q_{\mathcal{E}_{\tilde{P}}}(\mathbf{x}_{V_0}) = 0$ and $(q_{\mathcal{E}_{\tilde{P}}})'_{|_E} = v_E$ for all $E \in \mathcal{E}_{\tilde{P}}$. Since $b_1 = 0$ (i.e., there is no “tunnel” crossing $\Omega$), the path $B := P - P'$ formed by the edges in $\mathcal{E}_B := \mathcal{E}_P \cup \mathcal{E}_{\tilde{P}}$ is a 1-boundary, i.e., there is a set of faces $\mathcal{F}_B \subset \mathcal{F}_h$ giving rise to a connected surface $S_B := \bigcup_{F \in \mathcal{F}_B} \overline{F}$ such that $B = \partial S_B$. We fix an orientation for $S_B$ and, for all $F \in \mathcal{F}_B$, we denote by $\omega_{BF} \in \{-1, 1\}$ the orientation of $F$ relative to $S_B$. For all $E \in \mathcal{E}_B$, there is a unique face $F \in \mathcal{F}_B$ such that $E \in \mathcal{E}_F$, and we let $\omega_{BE} := \omega_{BF}\omega_{FE}$ denote the orientation of $E$ relative to $S_B$. Since $C_F^k \underline{\mathbf{v}}_F = 0$ for all $F \in \mathcal{F}_B$, it holds

$$ 0 = \sum_{F \in \mathcal{F}_B} \omega_{BF} \int_F C_F^k \underline{\mathbf{v}}_F = - \sum_{F \in \mathcal{F}_B} \omega_{BF} \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E v_E = - \sum_{E \in \mathcal{E}_B} \omega_{BE} \int_E v_E, $$

where the second equality is obtained from (3.19) with $r_F$ identically equal to 1, while the conclusion follows observing that all the edges that are interior to $S_B$ appear exactly twice in the sum, with opposite signs. Thus, reasoning as in [31, Proposition 4.2], there exists $q_{\mathcal{E}_B} \in \mathcal{P}_c^{k+1}(\mathcal{E}_B)$ such that $(q_{\mathcal{E}_B})' = v_E$ for all $E \in \mathcal{E}_B$, which we can be uniquely identified by additionally prescribing that $q_{\mathcal{E}_B}(\mathbf{x}_{V_0}) = 0$. Under this condition, by uniqueness we infer $(q_{\mathcal{E}_B})_{|E} = (q_{\mathcal{E}_P})_{|E}$ for all $E \in \mathcal{E}_P$ and $(q_{\mathcal{E}_B})_{|E} = (q_{\mathcal{E}_{\tilde{P}}})_{|E}$ for all $E \in \mathcal{E}_{\tilde{P}}$. Since $q_{\mathcal{E}_B}$ is continuous at the vertices of $B$, this shows that $q_{\mathcal{E}_P}(\mathbf{x}_V) = q_{\mathcal{E}_{\tilde{P}}}(\mathbf{x}_V)$. This argument can be extended to paths $P$ and $\tilde{P}$ such that $\mathcal{E}_P \cap \mathcal{E}_{\tilde{P}} \neq \emptyset$, the only difference being that one should reason, in this case, on each connected component of the manifold $S_B$ (corresponding to a “loop” inside the path $B = P - \tilde{P}$).

Repeating this reasoning for each vertex $V \in \mathcal{V}_h$ and all possible paths connecting $V_0$ and $V$, we conclude that there exists a unique $q_{\mathcal{E}_h} \in \mathcal{P}_c^{k+1}(\mathcal{E}_h)$ such that $q_{\mathcal{E}_h}(\mathbf{x}_{V_0}) = 0$ and, recalling the notation (3.4),

$$ q'_E = G_E^k q_E = v_E \quad \forall E \in \mathcal{E}_h. \qquad (3.50) $$
Let now $F \in \mathcal{F}_h$. We look for a $q_F \in \mathcal{P}^{k-1}(F)$ such that $\underline{q}_F := (q_F, q_{\varepsilon_F}) \in \underline{\mathcal{X}}_{\text{grad},F}^k$ satisfies $\underline{v}_F = \underline{\mathbf{G}}_F^k \underline{q}_F$. Plugging $C_F^k \underline{v}_F = 0$ into (3.19), we infer, for all $r_F \in \mathcal{P}^k(F)$,

$$ 
\begin{align*}
\int_F \mathbf{v}_{\mathcal{R},F} \cdot \mathbf{rot}_F r_F &= \sum_{E \in \varepsilon_F} \omega_{FE} \int_E \mathbf{v}_E r_F \\
&= \sum_{E \in \varepsilon_F} \omega_{FE} \int_E G_E^k q_E r_F \\
&= \int_F \mathbf{G}_F^k \underline{q}_F \cdot \mathbf{rot}_F r_F,
\end{align*}
$$

where the second equality is a consequence of (3.50), while the conclusion follows from (3.27). This shows that $\mathbf{v}_{\mathcal{R},F} = \pi_{\mathcal{R},F}^{k-1}(\mathbf{G}_F^k \underline{q}_F)$ for all $q_F \in \mathcal{P}^{k-1}(F)$. Let us now enforce $\mathbf{v}_{\mathcal{R},F}^c = \pi_{\mathcal{R},F}^{c,k}(\mathbf{G}_F^k \underline{q}_F)$, that is, for all $w_F \in \mathcal{R}^{c,k}(F)$,

$$ 
\begin{align*}
\int_F \mathbf{v}_{\mathcal{R},F} \cdot \mathbf{w}_F &= \int_F \mathbf{G}_F^k \underline{q}_F \cdot \mathbf{w}_F \\
&= -\int_F q_F \operatorname{div}_F \mathbf{w}_F + \sum_{E \in \varepsilon_F} \omega_{FE} \int_E q_{\varepsilon_F} (\mathbf{w}_F \cdot \mathbf{n}_{FE}),
\end{align*}
$$

where we have used the definition (3.9) of $\mathbf{G}_F^k$ in the second equality. Recalling the isomorphism (2.9), the above condition defines the sought $q_F \in \mathcal{P}^{k-1}(F)$ uniquely.

Writing the definition (3.28) of $\mathbf{C}_T^k \underline{v}_T$ with $\mathbf{w}_T \in \mathcal{G}^{c,k}(T)$, using $\underline{v}_h \in \text{Ker } \mathcal{C}_h^k$ to see that $\pi_{\mathcal{G}_T, T}^{c,k}(\mathbf{C}_T^k \underline{v}_T) = \mathbf{0}$, invoking (3.26) to write $\gamma_{t,F}^k \underline{v}_F = \mathbf{G}_F^k \underline{q}_F$ and using the link (3.17) between element and face gradients with $z_T = \mathbf{w}_T$, we see that, for any $T \in \mathcal{T}_h$, $\mathbf{v}_{\mathcal{R},T} = \pi_{\mathcal{R},T}^{k-1}(\mathbf{G}_T^k \underline{q}_T)$ with $\underline{q}_T := (q_T, (q_F)_{F \in \mathcal{T}_T}, q_{\varepsilon_T})$ for all $q_T \in \mathcal{P}^{k-1}(T)$. Proceeding then as for $q_F$ above, we can select $q_T \in \mathcal{P}^{k-1}(T)$ to additionally have $\mathbf{v}_{\mathcal{R},T}^c = \pi_{\mathcal{R},T}^{c,k}(\mathbf{G}_T^k \underline{q}_T)$. This concludes the proof of (3.49).

2. *Proof of (3.48).* The proof can be obtained reasoning as in [29, Point 2b) of Theorem 3]. As a matter of fact, this argument is based on a local exactness property analogous to (3.34) together with a topological assembly of the mesh valid for domains that do not enclose voids ($b_2 = 0$), and it therefore does not depend on the specific choice of the complements in (2.5) and (2.3). □

## 4 Potential reconstructions and $L^2$-products on discrete spaces

The definitions of the element gradient $\mathbf{G}_T^k$ and curl $\mathbf{C}_T^k$ required us to introduce discrete scalar and tangential traces on the mesh faces. In this section, for each $T \in \mathcal{T}_h$ and $\bullet \in \{\text{grad}, \text{curl}, \text{div}\}$, we define discrete potential reconstructions inside $T$ acting on the discrete space $\underline{\mathcal{X}}_{\bullet, T}^k$. These potentials have consistency properties, and enable the design of discrete $L^2$-inner products on DDR spaces that are also consistent.
Let now $F \in \mathcal{F}_h$. We look for a $q_F \in \mathcal{P}^{k-1}(F)$ such that $\underline{q}_F := (q_F, q_{\mathcal{E}_F}) \in \underline{X}_{\text{grad},F}^k$ satisfies $\underline{v}_F = \underline{G}_F^k \underline{q}_F$. Plugging $C_F^k \underline{v}_F = 0$ into (3.19), we infer, for all $r_F \in \mathcal{P}^k(F)$,

$$ 
\begin{align*}
\int_F \mathbf{v}_{\mathcal{R},F} \cdot \mathbf{rot}_F r_F &= \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E \mathbf{v}_E r_F \\
&= \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E G_E^k q_E r_F \\
&= \int_F \mathbf{G}_F^k \underline{q}_F \cdot \mathbf{rot}_F r_F,
\end{align*}
$$

where the second equality is a consequence of (3.50), while the conclusion follows from (3.27). This shows that $\mathbf{v}_{\mathcal{R},F} = \pi_{\mathcal{R},F}^{k-1}(\mathbf{G}_F^k \underline{q}_F)$ for all $q_F \in \mathcal{P}^{k-1}(F)$.

Let us now enforce $\mathbf{v}_{\mathcal{R},F}^c = \pi_{\mathcal{R},F}^{c,k}(\mathbf{G}_F^k \underline{q}_F)$, that is, for all $w_F \in \mathcal{R}^{c,k}(F)$,

$$ 
\begin{align*}
\int_F \mathbf{v}_{\mathcal{R},F} \cdot \mathbf{w}_F &= \int_F \mathbf{G}_F^k \underline{q}_F \cdot \mathbf{w}_F \\
&= -\int_F q_F \operatorname{div}_F \mathbf{w}_F + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E q_{\mathcal{E}_F} (\mathbf{w}_F \cdot \mathbf{n}_{FE}),
\end{align*}
$$

where we have used the definition (3.9) of $\mathbf{G}_F^k$ in the second equality. Recalling the isomorphism (2.9), the above condition defines the sought $q_F \in \mathcal{P}^{k-1}(F)$ uniquely.

Writing the definition (3.28) of $\mathbf{C}_T^k \underline{v}_T$ with $\mathbf{w}_T \in \mathcal{G}^{c,k}(T)$, using $\underline{v}_h \in \text{Ker } \underline{\mathcal{C}}_h^k$ to see that $\pi_{\mathcal{G}_T, T}^{c,k}(\mathbf{C}_T^k \underline{v}_T) = \mathbf{0}$, invoking (3.26) to write $\gamma_{\text{t},F}^k \underline{v}_F = \mathbf{G}_F^k \underline{q}_F$ and using the link (3.17) between element and face gradients with $z_T = \mathbf{w}_T$, we see that, for any $T \in \mathcal{T}_h$, $\mathbf{v}_{\mathcal{R},T} = \pi_{\mathcal{R},T}^{k-1}(\mathbf{G}_T^k \underline{q}_T)$ with $\underline{q}_T := (q_T, (q_F)_{F \in \mathcal{T}_T}, q_{\mathcal{E}_T})$ for all $q_T \in \mathcal{P}^{k-1}(T)$. Proceeding then as for $q_F$ above, we can select $q_T \in \mathcal{P}^{k-1}(T)$ to additionally have $\mathbf{v}_{\mathcal{R},T}^c = \pi_{\mathcal{R},T}^{c,k}(\mathbf{G}_T^k \underline{q}_T)$. This concludes the proof of (3.49).

2. *Proof of* (3.48). The proof can be obtained reasoning as in [29, Point 2b) of Theorem 3]. As a matter of fact, this argument is based on a local exactness property analogous to (3.34) together with a topological assembly of the mesh valid for domains that do not enclose voids ($b_2 = 0$), and it therefore does not depend on the specific choice of the complements in (2.5) and (2.3). □

## 4 Potential reconstructions and $L^2$-products on discrete spaces

The definitions of the element gradient $\mathbf{G}_T^k$ and curl $\mathbf{C}_T^k$ required us to introduce discrete scalar and tangential traces on the mesh faces. In this section, for each $T \in \mathcal{T}_h$ and $\bullet \in \{\text{grad, curl, div}\}$, we define discrete potential reconstructions inside $T$ acting on the discrete space $\underline{X}_{\bullet, T}^k$. These potentials have consistency properties, and enable the design of discrete $L^2$-inner products on DDR spaces that are also consistent.
## 4.1 Scalar potential on $X_{\text{grad},T}^k$

The scalar potential reconstruction $P_{\text{grad},T}^{k+1} : X_{\text{grad},T}^k \to \mathcal{P}^{k+1}(T)$ is such that, for all $q_T \in X_{\text{grad},T}^k$,

$$ \int_T P_{\text{grad},T}^{k+1} q_T \, \text{div} \, v_T = - \int_T \mathbf{G}_T^k q_T \cdot v_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \gamma_F^{k+1} q_F (v_T \cdot n_F) \\ \forall v_T \in \mathcal{R}^{c,k+2}(T), \quad (4.1) $$

with $\gamma_F^{k+1}$ defined by (3.10). This relation defines $P_{\text{grad},T}^{k+1} q_T$ uniquely since $\text{div} : \mathcal{R}^{c,k+2}(T) \to \mathcal{P}^{k+1}(T)$ is an isomorphism by (2.9) with $l = k + 2$.

**Remark 17 (Validity of (4.1))** The definition (3.11) of $\mathbf{G}_T^k$ and the identity $\text{div} \, \text{curl} = 0$ show that both sides of (4.1) vanish when $v_T \in \mathcal{R}^k(T)$. Hence, (4.1) actually holds for any $v_T \in \mathcal{R}^k(T) \oplus \mathcal{R}^{c,k+2}(T) = \mathcal{P}^k(T) + \mathcal{R}^{c,k+2}(T)$, the second equality following from $\mathcal{R}^{c,k}(T) \subset \mathcal{R}^{c,k+2}(T)$ and the decomposition (2.6).

Using the polynomial consistency properties $\mathbf{G}_T^k(I_{\text{grad},T}^k q) = \text{grad} \, q$ and $\gamma_F^{k+1}(I_{\text{grad},F}^k q_F) = q_F$, valid for all $q \in \mathcal{P}^{k+1}(T)$ (see (3.16) and (3.14), respectively), the following polynomial consistency of $P_{\text{grad},T}^{k+1}$ is obtained:

$$ P_{\text{grad},T}^{k+1}(I_{\text{grad},T}^k q) = q \quad \forall q \in \mathcal{P}^{k+1}(T). \quad (4.2) $$

Moreover, applying (4.1) to $v_T \in \mathcal{R}^{c,k}(T)$ (see Remark 1), using the definition (3.11) of $\mathbf{G}_T^k$ with $w_T = v_T$, and recalling that $\text{div} : \mathcal{R}^{c,k}(T) \to \mathcal{P}^{k-1}(T)$ is onto, we obtain

$$ \pi_{\mathcal{P},T}^{k-1}(P_{\text{grad},T}^{k+1} q_T) = q_T \quad \forall q_T \in X_{\text{grad},T}^k. \quad (4.3) $$

## 4.2 Vector potential on $X_{\text{curl},T}^k$

Recalling the definition (2.17) of $\mathfrak{R}_{\mathcal{R},T}^k(\cdot, \cdot)$, the vector potential reconstruction $\mathbf{P}_{\text{curl},T}^k : X_{\text{curl},T}^k \to \mathcal{P}^k(T)$ is such that, for all $v_T \in X_{\text{curl},T}^k$,

$$ \mathbf{P}_{\text{curl},T}^k v_T := \mathfrak{R}_{\mathcal{R},T}^k (\mathbf{P}_{\text{curl},\mathcal{R},T}^k v_T, v_{\mathcal{R},T}^c), \quad (4.4) $$

where $\mathbf{P}_{\text{curl},\mathcal{R},T}^k v_T \in \mathcal{R}^k(T)$ is defined, using the fact that $\text{curl} : \mathbf{G}_{\mathcal{R},T}^{c,k+1}(T) \to \mathcal{R}^k(T)$ is an isomorphism (see (2.10)), by

$$ \int_T \mathbf{P}_{\text{curl},\mathcal{R},T}^k v_T \cdot \text{curl} \, w_T = \int_T \mathbf{C}_T^k v_T \cdot w_T - \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \gamma_{T,F}^k v_F \cdot (w_T \times n_F) \\ \forall w_T \in \mathbf{G}_{\mathcal{R},T}^{c,k+1}(T). \quad (4.5) $$
## 4.1 Scalar potential on $X_{\text{grad},T}^k$

The scalar potential reconstruction $P_{\text{grad},T}^{k+1} : X_{\text{grad},T}^k \to \mathcal{P}^{k+1}(T)$ is such that, for all $q_T \in X_{\text{grad},T}^k$,

$$ \int_T P_{\text{grad},T}^{k+1} q_T \, \text{div} \, v_T = - \int_T \mathbf{G}_T^k q_T \cdot v_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \gamma_F^{k+1} q_F (v_T \cdot n_F) \\ \forall v_T \in \mathcal{R}^{c,k+2}(T), \quad (4.1) $$

with $\gamma_F^{k+1}$ defined by (3.10). This relation defines $P_{\text{grad},T}^{k+1} q_T$ uniquely since $\text{div} : \mathcal{R}^{c,k+2}(T) \to \mathcal{P}^{k+1}(T)$ is an isomorphism by (2.9) with $l = k + 2$.

**Remark 17 (Validity of (4.1))** The definition (3.11) of $\mathbf{G}_T^k$ and the identity $\text{div} \, \text{curl} = 0$ show that both sides of (4.1) vanish when $v_T \in \mathcal{R}^k(T)$. Hence, (4.1) actually holds for any $v_T \in \mathcal{R}^k(T) \oplus \mathcal{R}^{c,k+2}(T) = \mathcal{P}^k(T) + \mathcal{R}^{c,k+2}(T)$, the second equality following from $\mathcal{R}^{c,k}(T) \subset \mathcal{R}^{c,k+2}(T)$ and the decomposition (2.6).

Using the polynomial consistency properties $\mathbf{G}_T^k(I_{\text{grad},T}^k q) = \text{grad} \, q$ and $\gamma_F^{k+1}(I_{\text{grad},F}^k q_F) = q_F$, valid for all $q \in \mathcal{P}^{k+1}(T)$ (see (3.16) and (3.14), respectively), the following polynomial consistency of $P_{\text{grad},T}^{k+1}$ is obtained:

$$ P_{\text{grad},T}^{k+1}(I_{\text{grad},T}^k q) = q \quad \forall q \in \mathcal{P}^{k+1}(T). \quad (4.2) $$

Moreover, applying (4.1) to $v_T \in \mathcal{R}^{c,k}(T)$ (see Remark 1), using the definition (3.11) of $\mathbf{G}_T^k$ with $w_T = v_T$, and recalling that $\text{div} : \mathcal{R}^{c,k}(T) \to \mathcal{P}^{k-1}(T)$ is onto, we obtain

$$ \pi_{\mathcal{P},T}^{k-1}(P_{\text{grad},T}^{k+1} q_T) = q_T \quad \forall q_T \in X_{\text{grad},T}^k. \quad (4.3) $$

## 4.2 Vector potential on $X_{\text{curl},T}^k$

Recalling the definition (2.17) of $\mathfrak{R}_{\mathcal{R},T}^k(\cdot, \cdot)$, the vector potential reconstruction $\mathbf{P}_{\text{curl},T}^k : X_{\text{curl},T}^k \to \mathcal{P}^k(T)$ is such that, for all $v_T \in X_{\text{curl},T}^k$,

$$ \mathbf{P}_{\text{curl},T}^k v_T := \mathfrak{R}_{\mathcal{R},T}^k (\mathbf{P}_{\text{curl},\mathcal{R},T}^k v_T, v_{\mathcal{R},T}^c), \quad (4.4) $$

where $\mathbf{P}_{\text{curl},\mathcal{R},T}^k v_T \in \mathcal{R}^k(T)$ is defined, using the fact that $\text{curl} : \mathbf{G}_{\mathcal{R},k+1}^k(T) \to \mathcal{R}^k(T)$ is an isomorphism (see (2.10)), by

$$ \int_T \mathbf{P}_{\text{curl},\mathcal{R},T}^k v_T \cdot \text{curl} \, w_T = \int_T \mathbf{C}_T^k v_T \cdot w_T - \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \gamma_{T,F}^k v_F \cdot (w_T \times n_F) \\ \forall w_T \in \mathbf{G}_{\mathcal{R},k+1}^c(T). \quad (4.5) $$
Remark 18 (Discrete integration by parts formula for $\mathbf{P}_{\text{curl},T}^k$) Formula (4.5) can be extended to test functions in the Nédélec space $\mathcal{N}^{k+1}(T)$ defined by (2.12). To check it, simply notice that both sides vanish whenever $\mathbf{w}_T \in \mathcal{G}^k(T)$ (use $\text{curl grad} = 0$ and the definition (3.28) of $\mathbf{C}_T^k$). Since $\pi_{\mathcal{R},T}^k(\mathbf{P}_{\text{curl},T}^k \underline{\mathbf{v}}_T) = \mathbf{P}_{\text{curl},\mathcal{R},T}^k \underline{\mathbf{v}}_T$ (see (4.4) and (2.14)), we infer that

$$ \int_T \mathbf{P}_{\text{curl},T}^k \underline{\mathbf{v}}_T \cdot \text{curl} \, z_T = \int_T \mathbf{C}_T^k \underline{\mathbf{v}}_T \cdot z_T - \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \gamma_{\text{t},F}^k \underline{\mathbf{v}}_F \cdot (z_T \times \mathbf{n}_F) \\ \forall z_T \in \mathcal{N}^{k+1}(T). \quad (4.6) $$

Apply (4.6) to $\underline{\mathbf{v}}_T = \underline{\mathbf{I}}_{\text{curl},T}^k \underline{\mathbf{v}}$ with $\underline{\mathbf{v}} \in \mathcal{P}^k(T)$, use the consistency properties $\gamma_{\text{t},F}^k(\underline{\mathbf{I}}_{\text{curl},F}^k \underline{\mathbf{v}}) = \pi_{\mathcal{P},F}^k \underline{\mathbf{v}}_{\text{t},F} = \underline{\mathbf{v}}_{\text{t},F}$ and $\mathbf{C}_T^k(\underline{\mathbf{I}}_{\text{curl},T}^k \underline{\mathbf{v}}) = \text{curl} \, \underline{\mathbf{v}}$ (see (3.25) and (3.29), respectively), and integrate by parts. Since $\text{curl} : \mathcal{N}^{k+1}(T) \to \mathcal{R}^k(T)$ is onto (due to the isomorphism property (2.10)), we obtain the relation $\pi_{\mathcal{R},T}^k[\mathbf{P}_{\text{curl},T}^k(\underline{\mathbf{I}}_{\text{curl},T}^k \underline{\mathbf{v}})] = \pi_{\mathcal{R},T}^k \underline{\mathbf{v}}$. The definition (4.4) and the property (2.14) of the recovery operator also yield $\pi_{\mathcal{R},T}^{c,k}[\mathbf{P}_{\text{curl},T}^k(\underline{\mathbf{I}}_{\text{curl},T}^k \underline{\mathbf{v}})] = \pi_{\mathcal{R},T}^{c,k} \underline{\mathbf{v}}$. As a consequence,

$$ \mathbf{P}_{\text{curl},T}^k(\underline{\mathbf{I}}_{\text{curl},T}^k \underline{\mathbf{v}}) = \underline{\mathbf{v}} \quad \forall \underline{\mathbf{v}} \in \mathcal{P}^k(T). \quad (4.7) $$

Using similar arguments as in the proof of Proposition 3, we also have

$$ \pi_{\mathcal{R},T}^{k-1}(\mathbf{P}_{\text{curl},T}^k \underline{\mathbf{v}}_T) = \underline{\mathbf{v}}_{\mathcal{R},T} \quad \text{and} \quad \pi_{\mathcal{R},T}^{c,k}(\mathbf{P}_{\text{curl},T}^k \underline{\mathbf{v}}_T) = \underline{\mathbf{v}}_{\mathcal{R},T}^c \quad \forall \underline{\mathbf{v}}_T \in \underline{\mathbf{X}}_{\text{curl},T}^k. \quad (4.8) $$

### 4.3 Vector potential on $\underline{\mathbf{X}}_{\text{div},T}^k$

Recalling the definition (2.17) of $\mathfrak{R}_{\mathcal{G},T}^k(\cdot, \cdot)$, the vector potential reconstruction $\mathbf{P}_{\text{div},T}^k : \underline{\mathbf{X}}_{\text{div},T}^k \to \mathcal{P}^k(T)$ is such that, for all $\underline{\mathbf{w}}_T \in \underline{\mathbf{X}}_{\text{div},T}^k$,

$$ \mathbf{P}_{\text{div},T}^k \underline{\mathbf{w}}_T = \mathfrak{R}_{\mathcal{G},T}^k (\mathbf{P}_{\text{div},\mathcal{G},T}^k \underline{\mathbf{w}}_T, \mathbf{w}_{\mathcal{G},T}^c), \quad (4.9) $$

where $\mathbf{P}_{\text{div},\mathcal{G},T}^k \underline{\mathbf{w}}_T \in \mathcal{G}^k(T)$ is defined by

$$ \int_T \mathbf{P}_{\text{div},\mathcal{G},T}^k \underline{\mathbf{w}}_T \cdot \text{grad} \, r_T = - \int_T D_T^k \underline{\mathbf{w}}_T \, r_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F w_F \, r_T \\ \forall r_T \in \mathcal{P}^{0,k+1}(T). \quad (4.10) $$

Remark 19 (Discrete integration by parts formula for $\mathbf{P}_{\text{div},T}^k$) Observing that $\mathbf{P}_{\text{div},\mathcal{G},T}^k = \pi_{\mathcal{G},T}^k \mathbf{P}_{\text{div},T}^k$ (use (2.14)) and that (4.10) holds for any $r_T \in \mathcal{P}^{k+1}(T)$ (as can be proved taking $r_T$ constant in $T$ and observing that both sides of this equation vanish due to the definition (3.32) of $D_T^k$), we infer

$$ \int_T \mathbf{P}_{\text{div},T}^k \underline{\mathbf{w}}_T \cdot \text{grad} \, r_T = - \int_T D_T^k \underline{\mathbf{w}}_T \, r_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F w_F \, r_T \\ \forall r_T \in \mathcal{P}^{k+1}(T). \quad (4.11) $$
Remark 18 (Discrete integration by parts formula for $P_{\text{curl},T}^k$) Formula (4.5) can be extended to test functions in the Nédélec space $\mathcal{N}^{k+1}(T)$ defined by (2.12). To check it, simply notice that both sides vanish whenever $w_T \in \mathcal{G}^k(T)$ (use $\text{curl grad} = 0$ and the definition (3.28) of $C_T^k$). Since $\pi_{\mathcal{R},T}^k(P_{\text{curl},T}^k \underline{v}_T) = P_{\text{curl},\mathcal{R},T}^k \underline{v}_T$ (see (4.4) and (2.14)), we infer that

$$ \int_T P_{\text{curl},T}^k \underline{v}_T \cdot \text{curl} z_T = \int_T C_T^k \underline{v}_T \cdot z_T - \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \gamma_{t,F}^k \underline{v}_F \cdot (z_T \times n_F) \\ \qquad \qquad \qquad \qquad \qquad \qquad \forall z_T \in \mathcal{N}^{k+1}(T). \quad (4.6) $$

Apply (4.6) to $\underline{v}_T = \underline{I}_{\text{curl},T}^k \underline{v}$ with $\underline{v} \in \mathcal{P}^k(T)$, use the consistency properties $\gamma_{t,F}^k(\underline{I}_{\text{curl},F}^k \underline{v}) = \pi_{\mathcal{P},F}^k \underline{v}_{t,F} = \underline{v}_{t,F}$ and $C_T^k(\underline{I}_{\text{curl},T}^k \underline{v}) = \text{curl} \underline{v}$ (see (3.25) and (3.29), respectively), and integrate by parts. Since $\text{curl} : \mathcal{N}^{k+1}(T) \to \mathcal{R}^k(T)$ is onto (due to the isomorphism property (2.10)), we obtain the relation $\pi_{\mathcal{R},T}^k[P_{\text{curl},T}^k(\underline{I}_{\text{curl},T}^k \underline{v})] = \pi_{\mathcal{R},T}^k \underline{v}$. The definition (4.4) and the property (2.14) of the recovery operator also yield $\pi_{\mathcal{R},T}^{c,k}[P_{\text{curl},T}^k(\underline{I}_{\text{curl},T}^k \underline{v})] = \pi_{\mathcal{R},T}^{c,k} \underline{v}$. As a consequence,

$$ P_{\text{curl},T}^k(\underline{I}_{\text{curl},T}^k \underline{v}) = \underline{v} \quad \forall \underline{v} \in \mathcal{P}^k(T). \quad (4.7) $$

Using similar arguments as in the proof of Proposition 3, we also have

$$ \pi_{\mathcal{R},T}^{k-1}(P_{\text{curl},T}^k \underline{v}_T) = \underline{v}_{\mathcal{R},T} \quad \text{and} \quad \pi_{\mathcal{R},T}^{c,k}(P_{\text{curl},T}^k \underline{v}_T) = \underline{v}_{\mathcal{R},T}^c \quad \forall \underline{v}_T \in \underline{X}_{\text{curl},T}^k. \quad (4.8) $$

### 4.3 Vector potential on $\underline{X}_{\text{div},T}^k$

Recalling the definition (2.17) of $\mathfrak{R}_{\mathcal{G},T}^k(\cdot, \cdot)$, the vector potential reconstruction $P_{\text{div},T}^k : \underline{X}_{\text{div},T}^k \to \mathcal{P}^k(T)$ is such that, for all $w_T \in \underline{X}_{\text{div},T}^k$,

$$ P_{\text{div},T}^k \underline{w}_T = \mathfrak{R}_{\mathcal{G},T}^k (P_{\text{div},\mathcal{G},T}^k \underline{w}_T, w_{\mathcal{G},T}^c), \quad (4.9) $$

where $P_{\text{div},\mathcal{G},T}^k \underline{w}_T \in \mathcal{G}^k(T)$ is defined by

$$ \int_T P_{\text{div},\mathcal{G},T}^k \underline{w}_T \cdot \text{grad} r_T = - \int_T D_T^k \underline{w}_T r_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F w_F r_T \\ \qquad \qquad \qquad \qquad \qquad \qquad \forall r_T \in \mathcal{P}^{0,k+1}(T). \quad (4.10) $$

Remark 19 (Discrete integration by parts formula for $P_{\text{div},T}^k$) Observing that $P_{\text{div},\mathcal{G},T}^k = \pi_{\mathcal{G},T}^k P_{\text{div},T}^k$ (use (2.14)) and that (4.10) holds for any $r_T \in \mathcal{P}^{k+1}(T)$ (as can be proved taking $r_T$ constant in $T$ and observing that both sides of this equation vanish due to the definition (3.32) of $D_T^k$), we infer

$$ \int_T P_{\text{div},T}^k \underline{w}_T \cdot \text{grad} r_T = - \int_T D_T^k \underline{w}_T r_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F w_F r_T \\ \qquad \qquad \qquad \qquad \qquad \qquad \forall r_T \in \mathcal{P}^{k+1}(T). \quad (4.11) $$
Writing this relation for $\underline{w}_T = \underline{I}_{\text{div},T}^k w$ with $w \in \mathcal{R}\mathcal{T}^{k+1}(T)$, observing that $D_T^k(\underline{I}_{\text{div},T}^k w) = \pi_{\mathcal{P},T}^k(\text{div } w) = \text{div } w$ by (3.40) and $\pi_{\mathcal{P},F}^k(w|_F \cdot n_F) = w|_F \cdot n_F$ for all $F \in \mathcal{F}_T$ by (A.4), and integrating by parts the right-hand side of the resulting expression, we infer $\pi_{\mathcal{G},T}^k[\underline{P}_{\text{div},T}^k(\underline{I}_{\text{div},T}^k w)] = \pi_{\mathcal{G},T}^k w$; since $\underline{\pi}_{\mathcal{G},T}^{c,k}[\underline{P}_{\text{div},T}^k(\underline{I}_{\text{div},T}^k w)] = \underline{\pi}_{\mathcal{G},T}^{c,k} w$ by definition of $\underline{P}_{\text{div},T}^k$, $\underline{I}_{\text{div},T}^k$ and (2.14), we deduce that

$$ \underline{P}_{\text{div},T}^k(\underline{I}_{\text{div},T}^k w) = \underline{\pi}_{\mathcal{P},T}^k w \quad \forall w \in \mathcal{R}\mathcal{T}^{k+1}(T). \quad (4.12) $$

Moreover, using similar arguments as in Proposition 3 we get

$$ \underline{\pi}_{\mathcal{G},T}^{k-1}(\underline{P}_{\text{div},T}^k \underline{w}_T) = \underline{w}_{\mathcal{G},T} \quad \text{and} \quad \underline{\pi}_{\mathcal{G},T}^{c,k}(\underline{P}_{\text{div},T}^k \underline{w}_T) = \underline{w}_{\mathcal{G},T}^c \quad \forall \underline{w}_T \in \underline{\mathcal{X}}_{\text{div},T}^k. \quad (4.13) $$

## 4.4 Discrete L²-products

We now define discrete L²-inner products on the DDR spaces. These products are all constructed in a similar way: by assembling local contributions composed of a consistent term based on the potential reconstruction and a stabilisation term that provides a control over the polynomial components on the lower dimensional geometrical objects. Specifically, each L²-product $(\cdot, \cdot)_{\text{grad},h} : \underline{X}_{\text{grad},h}^k \times \underline{X}_{\text{grad},h}^k \to \mathbb{R}$, $(\cdot, \cdot)_{\text{curl},h} : \underline{X}_{\text{curl},h}^k \times \underline{X}_{\text{curl},h}^k \to \mathbb{R}$, and $(\cdot, \cdot)_{\text{div},h} : \underline{X}_{\text{div},h}^k \times \underline{X}_{\text{div},h}^k \to \mathbb{R}$ is the sum over $T \in \mathcal{T}_h$ of its local counterpart defined by:

$$ (\underline{r}_T, \underline{q}_T)_{\text{grad},T} := \int_T \underline{P}_{\text{grad},T}^{k+1} \underline{r}_T \underline{P}_{\text{grad},T}^{k+1} \underline{q}_T + \text{s}_{\text{grad},T}(\underline{r}_T, \underline{q}_T) \quad \forall (\underline{r}_T, \underline{q}_T) \in \underline{X}_{\text{grad},T}^k \times \underline{X}_{\text{grad},T}^k, \quad (4.14) $$

$$ (\underline{w}_T, \underline{v}_T)_{\text{curl},T} := \int_T \underline{P}_{\text{curl},T}^k \underline{w}_T \cdot \underline{P}_{\text{curl},T}^k \underline{v}_T + \text{s}_{\text{curl},T}(\underline{w}_T, \underline{v}_T) \quad \forall (\underline{w}_T, \underline{v}_T) \in \underline{X}_{\text{curl},T}^k \times \underline{X}_{\text{curl},T}^k, \quad (4.15) $$

$$ (\underline{w}_T, \underline{v}_T)_{\text{div},T} := \int_T \underline{P}_{\text{div},T}^k \underline{w}_T \cdot \underline{P}_{\text{div},T}^k \underline{v}_T + \text{s}_{\text{div},T}(\underline{w}_T, \underline{v}_T) \quad \forall (\underline{w}_T, \underline{v}_T) \in \underline{X}_{\text{div},T}^k \times \underline{X}_{\text{div},T}^k, \quad (4.16) $$

with symmetric, positive semidefinite stabilisation bilinear forms $s_{\bullet,T}$, $\bullet \in \{\text{grad}, \text{curl}, \text{div}\}$ defined as follows:

$$ s_{\text{grad},T}(\underline{r}_T, \underline{q}_T) := \sum_{F \in \mathcal{F}_T} h_F \int_F (P_{\text{grad},T}^{k+1} \underline{r}_T - \gamma_F^{k+1} \underline{r}_F)(P_{\text{grad},T}^{k+1} \underline{q}_T - \gamma_F^{k+1} \underline{q}_F) \\ + \sum_{E \in \mathcal{E}_T} h_E^2 \int_E (P_{\text{grad},T}^{k+1} \underline{r}_T - r_E)(P_{\text{grad},T}^{k+1} \underline{q}_T - q_E), \quad (4.17) $$
Writing this relation for $\underline{w}_T = \underline{I}_{\text{div},T}^k w$ with $w \in \mathcal{R}\mathcal{T}^{k+1}(T)$, observing that $D_T^k(\underline{I}_{\text{div},T}^k w) = \pi_{\mathcal{P},T}^k(\text{div } w) = \text{div } w$ by (3.40) and $\pi_{\mathcal{P},F}^k(w_{|F} \cdot n_F) = w_{|F} \cdot n_F$ for all $F \in \mathcal{F}_T$ by (A.4), and integrating by parts the right-hand side of the resulting expression, we infer $\pi_{\mathcal{G},T}^k[P_{\text{div},T}^k(\underline{I}_{\text{div},T}^k w)] = \pi_{\mathcal{G},T}^k w$; since $\underline{\pi}_{\mathcal{G},T}^{c,k}[P_{\text{div},T}^k(\underline{I}_{\text{div},T}^k w)] = \underline{\pi}_{\mathcal{G},T}^{c,k} w$ by definition of $P_{\text{div},T}^k$, $\underline{I}_{\text{div},T}^k$ and (2.14), we deduce that

$$ \mathbf{P}_{\text{div},T}^{k}(\underline{\mathbf{I}}_{\text{div},T}^{k} \mathbf{w}) = \underline{\pi}_{\mathcal{P},T}^{k} \mathbf{w} \quad \forall \mathbf{w} \in \mathcal{R}\mathcal{T}^{k+1}(T). \quad (4.12) $$

Moreover, using similar arguments as in Proposition 3 we get

$$ \underline{\pi}_{\mathcal{G},T}^{k-1}(\mathbf{P}_{\text{div},T}^k \underline{\mathbf{w}}_T) = \underline{\mathbf{w}}_{\mathcal{G},T} \quad \text{and} \quad \underline{\pi}_{\mathcal{G},T}^{c,k}(\mathbf{P}_{\text{div},T}^k \underline{\mathbf{w}}_T) = \underline{\mathbf{w}}_{\mathcal{G},T}^c \quad \forall \underline{\mathbf{w}}_T \in \underline{\mathbf{X}}_{\text{div},T}^k. \quad (4.13) $$

## 4.4 Discrete L²-products

We now define discrete L²-inner products on the DDR spaces. These products are all constructed in a similar way: by assembling local contributions composed of a consistent term based on the potential reconstruction and a stabilisation term that provides a control over the polynomial components on the lower dimensional geometrical objects. Specifically, each L²-product $(\cdot, \cdot)_{\text{grad},h} : \underline{\mathbf{X}}_{\text{grad},h}^k \times \underline{\mathbf{X}}_{\text{grad},h}^k \to \mathbb{R}$, $(\cdot, \cdot)_{\text{curl},h} : \underline{\mathbf{X}}_{\text{curl},h}^k \times \underline{\mathbf{X}}_{\text{curl},h}^k \to \mathbb{R}$, and $(\cdot, \cdot)_{\text{div},h} : \underline{\mathbf{X}}_{\text{div},h}^k \times \underline{\mathbf{X}}_{\text{div},h}^k \to \mathbb{R}$ is the sum over $T \in \mathcal{T}_h$ of its local counterpart defined by:

$$ (\underline{r}_T, \underline{q}_T)_{\text{grad},T} := \int_T P_{\text{grad},T}^{k+1} \underline{r}_T P_{\text{grad},T}^{k+1} \underline{q}_T + s_{\text{grad},T}(\underline{r}_T, \underline{q}_T) \\ \qquad \forall (\underline{r}_T, \underline{q}_T) \in \underline{\mathbf{X}}_{\text{grad},T}^k \times \underline{\mathbf{X}}_{\text{grad},T}^k, \quad (4.14) $$

$$ (\underline{w}_T, \underline{v}_T)_{\text{curl},T} := \int_T \mathbf{P}_{\text{curl},T}^k \underline{w}_T \cdot \mathbf{P}_{\text{curl},T}^k \underline{v}_T + s_{\text{curl},T}(\underline{w}_T, \underline{v}_T) \\ \qquad \forall (\underline{w}_T, \underline{v}_T) \in \underline{\mathbf{X}}_{\text{curl},T}^k \times \underline{\mathbf{X}}_{\text{curl},T}^k, \quad (4.15) $$

$$ (\underline{w}_T, \underline{v}_T)_{\text{div},T} := \int_T \mathbf{P}_{\text{div},T}^k \underline{w}_T \cdot \mathbf{P}_{\text{div},T}^k \underline{v}_T + s_{\text{div},T}(\underline{w}_T, \underline{v}_T) \\ \qquad \forall (\underline{w}_T, \underline{v}_T) \in \underline{\mathbf{X}}_{\text{div},T}^k \times \underline{\mathbf{X}}_{\text{div},T}^k, \quad (4.16) $$

with symmetric, positive semidefinite stabilisation bilinear forms $s_{\bullet,T}$, $\bullet \in \{\text{grad}, \text{curl}, \text{div}\}$ defined as follows:

$$ s_{\text{grad},T}(\underline{r}_T, \underline{q}_T) := \sum_{F \in \mathcal{F}_T} h_F \int_F (P_{\text{grad},T}^{k+1} \underline{r}_T - \gamma_F^{k+1} \underline{r}_F)(P_{\text{grad},T}^{k+1} \underline{q}_T - \gamma_F^{k+1} \underline{q}_F) \\ + \sum_{E \in \mathcal{E}_T} h_E^2 \int_E (P_{\text{grad},T}^{k+1} \underline{r}_T - r_E)(P_{\text{grad},T}^{k+1} \underline{q}_T - q_E), \quad (4.17) $$
$$ s_{\text{curl},T}(\underline{\mathbf{w}}_T, \underline{\mathbf{v}}_T) := \sum_{F \in \mathcal{F}_T} h_F \int_F \left( (\mathbf{P}_{\text{curl},T}^k \underline{\mathbf{w}}_T)_{t,F} - \gamma_{t,F}^k \underline{\mathbf{w}}_F \right) \cdot \left( (\mathbf{P}_{\text{curl},T}^k \underline{\mathbf{v}}_T)_{t,F} - \gamma_{t,F}^k \underline{\mathbf{v}}_F \right) \\ + \sum_{E \in \mathcal{E}_T} h_E^2 \int_E \left( \mathbf{P}_{\text{curl},T}^k \underline{\mathbf{w}}_T \cdot \mathbf{t}_E - w_E \right) \left( \mathbf{P}_{\text{curl},T}^k \underline{\mathbf{v}}_T \cdot \mathbf{t}_E - v_E \right), \quad (4.18) $$

where we recall that the index $t, F$ denotes the tangential trace on $F$, and

$$ s_{\text{div},T}(\underline{\mathbf{w}}_T, \underline{\mathbf{v}}_T) := \sum_{F \in \mathcal{F}_T} h_F \int_F (\mathbf{P}_{\text{div},T}^k \underline{\mathbf{w}}_T \cdot \mathbf{n}_F - w_F)(\mathbf{P}_{\text{div},T}^k \underline{\mathbf{v}}_T \cdot \mathbf{n}_F - v_F). \quad (4.19) $$

These local stabilisation bilinear forms $s_{\bullet,T}$ are polynomially consistent, i.e., they vanish whenever one of their arguments is the interpolate of a polynomial of total degree $\le k+1$ for $\bullet = \text{grad}$, or $\le k$ for $\bullet \in \{\text{curl}, \text{div}\}$. The consistency properties on interpolates of smooth functions of the potential reconstructions and stabilisation forms proved in Section 6 below make these discrete $L^2$-products natural candidates for use in the discretisation of PDEs in weak formulation; see the application of DDR to the magnetostatic problem in Section 7.

For $\bullet \in \{\text{grad}, \text{curl}, \text{div}\}$, we denote by $\|\cdot\|_{\bullet,T}$ the norm on $X_{\bullet,T}^k$ induced by the corresponding local discrete $L^2$-product $(\cdot, \cdot)_{\bullet,T}$, and by $\|\cdot\|_{\bullet,h}$ the norm on $X_{\bullet,h}^k$ corresponding to the global discrete $L^2$-product $(\cdot, \cdot)_{\bullet,h}$.

## 4.5 Component $L^2$-norms, bounds, and equivalence properties

The analysis of the stability and consistency properties of the DDR sequence is facilitated by the introduction of $L^2$-like norms naturally associated with the choices of polynomial components in the DDR spaces. Specifically we set, for all $q_h \in X_{\text{grad},h}^k$,

$$ \begin{align} \|q_h\|_{\text{grad},h} &:= \left( \sum_{T \in \mathcal{T}_h} \|q_T\|_{\text{grad},T}^2 \right)^{\frac{1}{2}} & \text{with} \nonumber \\ & \forall T \in \mathcal{T}_h, \forall q_T \in X_{\text{grad},T}^k, \nonumber \\ \|q_T\|_{\text{grad},T} &:= \left( \|q_T\|_{L^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F \|q_F\|_{\text{grad},F}^2 \right)^{\frac{1}{2}} & \quad (4.20) \\ \forall F \in \mathcal{F}_h, \forall q_F \in X_{\text{grad},F}^k, \nonumber \\ \|q_F\|_{\text{grad},F} &:= \left( \|q_F\|_{L^2(F)}^2 + \sum_{E \in \mathcal{E}_F} h_E \|q_E\|_{L^2(E)}^2 \right)^{\frac{1}{2}}. & \nonumber \end{align} $$
$$
\begin{equation}
\begin{split}
\mathrm{s}_{\mathrm{curl},T}(\underline{\mathbf{w}}_T, \underline{\mathbf{v}}_T) := & \sum_{F \in \mathcal{F}_T} h_F \int_F \left( (\mathbf{P}_{\mathrm{curl},T}^k \underline{\mathbf{w}}_T)_{t,F} - \gamma_{t,F}^k \underline{\mathbf{w}}_F \right) \cdot \left( (\mathbf{P}_{\mathrm{curl},T}^k \underline{\mathbf{v}}_T)_{t,F} - \gamma_{t,F}^k \underline{\mathbf{v}}_F \right) \right. \\
& \left. + \sum_{E \in \mathcal{E}_T} h_E^2 \int_E (\mathbf{P}_{\mathrm{curl},T}^k \underline{\mathbf{w}}_T \cdot \mathbf{t}_E - w_E)(\mathbf{P}_{\mathrm{curl},T}^k \underline{\mathbf{v}}_T \cdot \mathbf{t}_E - v_E),
\end{split}
\tag{4.18}
\end{equation}
$$

where we recall that the index t, F denotes the tangential trace on F, and

$$
\mathrm{s}_{\mathrm{div},T}(\underline{\mathbf{w}}_T, \underline{\mathbf{v}}_T) := \sum_{F \in \mathcal{F}_T} h_F \int_F (\mathbf{P}_{\mathrm{div},T}^k \underline{\mathbf{w}}_T \cdot \mathbf{n}_F - w_F)(\mathbf{P}_{\mathrm{div},T}^k \underline{\mathbf{v}}_T \cdot \mathbf{n}_F - v_F). \quad (4.19)
$$

These local stabilisation bilinear forms $s_{\bullet,T}$ are polynomially consistent, i.e., they vanish whenever one of their arguments is the interpolate of a polynomial of total degree $\le k+1$ for $\bullet = \text{grad}$, or $\le k$ for $\bullet \in \{\text{curl}, \text{div}\}$. The consistency properties on interpolates of smooth functions of the potential reconstructions and stabilisation forms proved in Section 6 below make these discrete $L^2$-products natural candidates for use in the discretisation of PDEs in weak formulation; see the application of DDR to the magnetostatic problem in Section 7.

For $\bullet \in \{\text{grad}, \text{curl}, \text{div}\}$, we denote by $\|\cdot\|_{\bullet,T}$ the norm on $X_{\bullet,T}^k$ induced by the corresponding local discrete $L^2$-product $(\cdot, \cdot)_{\bullet,T}$, and by $\|\cdot\|_{\bullet,h}$ the norm on $X_{\bullet,h}^k$ corresponding to the global discrete $L^2$-product $(\cdot, \cdot)_{\bullet,h}$.

4.5 Component L²-norms, bounds, and equivalence properties

The analysis of the stability and consistency properties of the DDR sequence
is facilitated by the introduction of L²-like norms naturally associated with
the choices of polynomial components in the DDR spaces. Specifically we set,
for all $\underline{q}_h \in \underline{X}_{\text{grad},h}^k$,

$$
\|\underline{q}_h\|_{\text{grad},h} := \left( \sum_{T \in \mathcal{T}_h} \|\underline{q}_T\|_{\text{grad},T}^2 \right)^{\frac{1}{2}} \quad \text{with}
$$

$$
\begin{equation}
\begin{gathered}
\forall T \in \mathcal{T}_h, \forall \underline{q}_T \in \underline{X}_{\text{grad},T}^k, \\
\|\underline{q}_T\|_{\text{grad},T} := \left( \|\underline{q}_T\|_{L^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F \|\underline{q}_F\|_{\text{grad},F}^2 \right)^{\frac{1}{2}} \quad (4.20)
\end{gathered}
\end{equation}
$$

$$
\forall F \in \mathcal{F}_h, \forall \underline{q}_F \in \underline{X}_{\text{grad},F}^k,
$$
$$
\|\underline{q}_F\|_{\text{grad},F} := \left( \|\underline{q}_F\|_{L^2(F)}^2 + \sum_{E \in \mathcal{E}_F} h_E \|\underline{q}_E\|_{L^2(E)}^2 \right)^{\frac{1}{2}}.
$$
Similarly, for all $\underline{v}_h \in \underline{X}_{\text{curl},h}^k$,

$$ \|\underline{v}_h\|_{\text{curl},h} := \left( \sum_{T \in \mathcal{T}_h} \|\underline{v}_T\|_{\text{curl},T}^2 \right)^{\frac{1}{2}} \quad \text{with} $$

$$ \forall T \in \mathcal{T}_h, \forall \underline{v}_T \in \underline{X}_{\text{curl},T}^k, $$

$$ \|\underline{v}_T\|_{\text{curl},T} := \left( \|\underline{v}_{\mathcal{R},T}\|_{\mathcal{L}^2(T)}^2 + \|\underline{v}_{\mathcal{R},T}^c\|_{\mathcal{L}^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F \|\underline{v}_F\|_{\text{curl},F}^2 \right)^{\frac{1}{2}} \quad (4.21) $$

$$ \forall F \in \mathcal{F}_h, \forall \underline{v}_F \in \underline{X}_{\text{curl},F}^k, $$

$$ \|\underline{v}_F\|_{\text{curl},F} := \left( \|\underline{v}_{\mathcal{R},F}\|_{\mathcal{L}^2(F)}^2 + \|\underline{v}_{\mathcal{R},F}^c\|_{\mathcal{L}^2(F)}^2 + \sum_{E \in \mathcal{E}_F} h_E \|\underline{v}_E\|_{\mathcal{L}^2(E)}^2 \right)^{\frac{1}{2}}. $$

Finally, for all $\underline{w}_h \in \underline{X}_{\text{div},h}^k$,

$$ \|\underline{w}_h\|_{\text{div},h} := \left( \sum_{T \in \mathcal{T}_h} \|\underline{w}_T\|_{\text{div},T}^2 \right)^{\frac{1}{2}} \quad \text{with} $$

$$ \forall T \in \mathcal{T}_h, \forall \underline{w}_T \in \underline{X}_{\text{div},T}^k, $$

$$ \|\underline{w}_T\|_{\text{div},T} := \left( \|\underline{w}_{\mathcal{G},T}\|_{\mathcal{L}^2(T)}^2 + \|\underline{w}_{\mathcal{G},T}^c\|_{\mathcal{L}^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F \|\underline{w}_F\|_{\mathcal{L}^2(F)}^2 \right)^{\frac{1}{2}}. $$

**Remark 20 (Alternative stabilisations)** Each of the component norms $|||\cdot|||_{\bullet,T}$, for $\bullet \in \{\text{grad}, \text{curl}, \text{div}\}$, is a Euclidean norm on the corresponding local space $\underline{X}_{\bullet,T}^k$. It is therefore associated to an inner product $[\cdot, \cdot]_{\bullet,T}$, which can be used to design an alternative stabilisation to $s_{\bullet,T}$ by setting $\tilde{s}_{\bullet,T}(\underline{v}_T, \underline{w}_T) := [\underline{v}_T - \underline{I}_{\bullet,T} \mathbf{P}_{\bullet,T} \underline{v}_T, \underline{w}_T - \underline{I}_{\bullet,T} \mathbf{P}_{\bullet,T} \underline{w}_T]_{\bullet,T}$, where $\underline{I}_{\bullet,T}$ and $\mathbf{P}_{\bullet,T}$ are respectively the interpolator and potential reconstruction on $\underline{X}_{\bullet,T}^k$. This alternative stabilisation is the one chosen for $\underline{X}_{\text{curl},T}^k$ in [29]. We also note that $s_{\text{div},T} = \tilde{s}_{\text{div},T}$.

The next proposition follows from (2.19) and Lemma 9 in Appendix A, in a similar way as in the proof of [29, Proposition 13].

**Proposition 6 (Boundedness of local potentials)** It holds, for all $T \in \mathcal{T}_h$ and all $F \in \mathcal{F}_T$,

$$ \left\{ \begin{array}{l} \|\gamma_F^{k+1} \underline{q}_F\|_{\mathcal{L}^2(F)} \lesssim \|\underline{q}_F\|_{\text{grad},F} \\ \|P_{\text{grad},T}^{k+1} \underline{q}_T\|_{\mathcal{L}^2(T)} \lesssim \|\underline{q}_T\|_{\text{grad},T} \end{array} \right. \quad \forall \underline{q}_T \in \underline{X}_{\text{grad},T}^k, \quad (4.22) $$

$$ \left\{ \begin{array}{l} \|\gamma_{t,F}^k \underline{v}_F\|_{\mathcal{L}^2(F)} \lesssim \|\underline{v}_F\|_{\text{curl},F} \\ \|\mathbf{P}_{\text{curl},T}^k \underline{v}_T\|_{\mathcal{L}^2(T)} \lesssim \|\underline{v}_T\|_{\text{curl},T} \end{array} \right. \quad \forall \underline{v}_T \in \underline{X}_{\text{curl},T}^k, \quad (4.23) $$

$$ \|\mathbf{P}_{\text{div},T}^k \underline{w}_T\|_{\mathcal{L}^2(T)} \lesssim \|\underline{w}_T\|_{\text{div},T} \quad \forall \underline{w}_T \in \underline{X}_{\text{div},T}^k. \quad (4.24) $$
Similarly, for all $\underline{v}_h \in \underline{X}_{\text{curl},h}^k$,

$$ \|\underline{v}_h\|_{\text{curl},h} := \left( \sum_{T \in \mathcal{T}_h} \|\underline{v}_T\|_{\text{curl},T}^2 \right)^{\frac{1}{2}} \quad \text{with} $$

$$ \forall T \in \mathcal{T}_h, \forall \underline{v}_T \in \underline{X}_{\text{curl},T}^k, $$

$$ \|\underline{v}_T\|_{\text{curl},T} := \left( \|\underline{v}_{\mathcal{R},T}\|_{\mathcal{L}^2(T)}^2 + \|\underline{v}_{\mathcal{R},T}^c\|_{\mathcal{L}^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F \|\underline{v}_F\|_{\text{curl},F}^2 \right)^{\frac{1}{2}} \quad (4.21) $$

$$ \forall F \in \mathcal{F}_h, \forall \underline{v}_F \in \underline{X}_{\text{curl},F}^k, $$

$$ \|\underline{v}_F\|_{\text{curl},F} := \left( \|\underline{v}_{\mathcal{R},F}\|_{\mathcal{L}^2(F)}^2 + \|\underline{v}_{\mathcal{R},F}^c\|_{\mathcal{L}^2(F)}^2 + \sum_{E \in \mathcal{E}_F} h_E \|\underline{v}_E\|_{\mathcal{L}^2(E)}^2 \right)^{\frac{1}{2}}. $$

Finally, for all $\underline{w}_h \in \underline{X}_{\text{div},h}^k$,

$$ \|\underline{w}_h\|_{\text{div},h} := \left( \sum_{T \in \mathcal{T}_h} \|\underline{w}_T\|_{\text{div},T}^2 \right)^{\frac{1}{2}} \quad \text{with} $$

$$ \forall T \in \mathcal{T}_h, \forall \underline{w}_T \in \underline{X}_{\text{div},T}^k, $$

$$ \|\underline{w}_T\|_{\text{div},T} := \left( \|\underline{w}_{\mathcal{G},T}\|_{\mathcal{L}^2(T)}^2 + \|\underline{w}_{\mathcal{G},T}^c\|_{\mathcal{L}^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F \|\underline{w}_F\|_{\mathcal{L}^2(F)}^2 \right)^{\frac{1}{2}}. $$

**Remark 20 (Alternative stabilisations)** Each of the component norms $|||\cdot|||_{\bullet,T}$, for $\bullet \in \{\text{grad}, \text{curl}, \text{div}\}$, is a Euclidean norm on the corresponding local space $\underline{X}_{\bullet,T}^k$. It is therefore associated to an inner product $[\cdot, \cdot]_{\bullet,T}$, which can be used to design an alternative stabilisation to $s_{\bullet,T}$ by setting $\tilde{s}_{\bullet,T}(\underline{v}_T, \underline{w}_T) := [\underline{v}_T - \underline{I}_{\bullet,T} \mathbf{P}_{\bullet,T} \underline{v}_T, \underline{w}_T - \underline{I}_{\bullet,T} \mathbf{P}_{\bullet,T} \underline{w}_T]_{\bullet,T}$, where $\underline{I}_{\bullet,T}$ and $\mathbf{P}_{\bullet,T}$ are respectively the interpolator and potential reconstruction on $\underline{X}_{\bullet,T}^k$. This alternative stabilisation is the one chosen for $\underline{X}_{\text{curl},T}^k$ in [29]. We also note that $s_{\text{div},T} = \tilde{s}_{\text{div},T}$.

The next proposition follows from (2.19) and Lemma 9 in Appendix A, in a similar way as in the proof of [29, Proposition 13].

**Proposition 6 (Boundedness of local potentials)** It holds, for all $T \in \mathcal{T}_h$ and all $F \in \mathcal{F}_T$,

$$ \left\{ \begin{array}{l} \|\gamma_F^{k+1} \underline{q}_F\|_{\mathcal{L}^2(F)} \lesssim \|\underline{q}_F\|_{\text{grad},F} \\ \|P_{\text{grad},T}^{k+1} \underline{q}_T\|_{\mathcal{L}^2(T)} \lesssim \|\underline{q}_T\|_{\text{grad},T} \end{array} \right. \quad \forall \underline{q}_T \in \underline{X}_{\text{grad},T}^k, \quad (4.22) $$

$$ \left\{ \begin{array}{l} \|\gamma_{t,F}^k \underline{v}_F\|_{\mathcal{L}^2(F)} \lesssim \|\underline{v}_F\|_{\text{curl},F} \\ \|\mathbf{P}_{\text{curl},T}^k \underline{v}_T\|_{\mathcal{L}^2(T)} \lesssim \|\underline{v}_T\|_{\text{curl},T} \end{array} \right. \quad \forall \underline{v}_T \in \underline{X}_{\text{curl},T}^k, \quad (4.23) $$

$$ \|\mathbf{P}_{\text{div},T}^k \underline{w}_T\|_{\mathcal{L}^2(T)} \lesssim \|\underline{w}_T\|_{\text{div},T} \quad \forall \underline{w}_T \in \underline{X}_{\text{div},T}^k. \quad (4.24) $$
We next establish the equivalence of the norms corresponding to the discrete L²-products and the component norms.

**Lemma 5 (Equivalence of norms)** Let $\bullet \in \{\text{grad}, \text{curl}, \text{div}\}$. We have, for any $T \in \mathcal{T}_h$,

$$ \|\underline{z}_T\|_{\bullet,T} \approx \|\underline{z}_T\|_{\bullet,T} \quad \forall \underline{z}_T \in \underline{X}_{\bullet,T}^k. \qquad (4.25) $$

*Proof* We only prove the result for $\bullet = \text{div}$, the other cases being similar (see also [29, Proposition 14] for $\bullet = \text{curl}$ in the case of orthogonal complements, instead of the Koszul complements (2.3), (2.5)). Let $T \in \mathcal{T}_h$ and $\underline{w}_T \in \underline{X}_{\text{div},T}^k$. By definition (4.16) of the L²-product on $\underline{X}_{\text{div},T}^k$, we have

$$ \begin{aligned} \|\underline{w}_T\|_{\text{div},T}^2 &= \|\underline{P}_{\text{div},T}^k \underline{w}_T\|_{\text{L}^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F \|\underline{P}_{\text{div},T}^k \underline{w}_T \cdot \underline{n}_F - w_F\|_{\text{L}^2(F)}^2 \\ &\lesssim \|\underline{P}_{\text{div},T}^k \underline{w}_T\|_{\text{L}^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F \|w_F\|_{\text{L}^2(F)}^2 \lesssim \|\underline{w}_T\|_{\text{div},T}^2, \end{aligned} $$

where the first inequality follows from a triangle inequality together with the discrete trace inequality $h_F \|\underline{P}_{\text{div},T}^k \underline{w}_T\|_{\text{L}^2(F)}^2 \lesssim \|\underline{P}_{\text{div},T}^k \underline{w}_T\|_{\text{L}^2(T)}^2$ (see [28, Lemma 1.32]), while the conclusion is a consequence of (4.24) together with the definition of $\|\underline{w}_T\|_{\text{div},T}$. This proves $\ge$ in (4.25).

To prove the converse inequality, we start from (4.13) to write

$$ \begin{aligned} \|\underline{w}_T\|_{\text{div},T}^2 &= \|\underline{\pi}_{\mathcal{G},T}^{k-1}(\underline{P}_{\text{div},T}^k \underline{w}_T)\|_{\text{L}^2(T)}^2 + \|\underline{\pi}_{\mathcal{G},T}^{c,k}(\underline{P}_{\text{div},T}^k \underline{w}_T)\|_{\text{L}^2(T)}^2 \\ &\quad + \sum_{F \in \mathcal{F}_T} h_F \|w_F\|_{\text{L}^2(F)}^2 \\ &\lesssim \|\underline{P}_{\text{div},T}^k \underline{w}_T\|_{\text{L}^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F \|w_F - \underline{P}_{\text{div},T}^k \underline{w}_T \cdot \underline{n}_F\|_{\text{L}^2(T)}^2 \\ &\quad + \sum_{F \in \mathcal{F}_T} h_F \|\underline{P}_{\text{div},T}^k \underline{w}_T \cdot \underline{n}_F\|_{\text{L}^2(T)}^2 \\ &\lesssim \|\underline{w}_T\|_{\text{div},T}^2, \end{aligned} $$

where the first inequality follows from the L²-boundedness of the orthogonal projectors $\underline{\pi}_{\mathcal{G},T}^{k-1}$ and $\underline{\pi}_{\mathcal{G},T}^{c,k}$ together with a triangle inequality, and the conclusion is obtained invoking the same discrete trace inequality as before together with the definition of $\|\underline{w}_T\|_{\text{div},T}$. This proves $\le$ in (4.25). $\square$

**Lemma 6 (Boundedness of local interpolators)** It holds, for all $T \in \mathcal{T}_h$,

$$ \|\underline{I}_{\text{grad},T}^k q\|_{\text{grad},T} \lesssim \|q\|_{\text{L}^2(T)} + h_T |q|_{\text{H}^1(T)} + h_T^2 |q|_{\text{H}^2(T)} \quad \forall q \in \text{H}^2(T), \quad (4.26) $$

$$ \|\underline{I}_{\text{curl},T}^k \nu\|_{\text{curl},T} \lesssim \|\nu\|_{\text{L}^2(T)} + h_T |\nu|_{\text{H}^1(T)} + h_T^2 |\nu|_{\text{H}^2(T)} \quad \forall \nu \in \text{H}^2(T), \quad (4.27) $$

$$ \|\underline{I}_{\text{div},T}^k \mathbf{w}\|_{\text{div},T} \lesssim \|\mathbf{w}\|_{\text{L}^2(T)} + h_T |\mathbf{w}|_{\text{H}^1(T)} \quad \forall \mathbf{w} \in \text{H}^1(T). \quad (4.28) $$
We next establish the equivalence of the norms corresponding to the discrete L²-products and the component norms.

**Lemma 5 (Equivalence of norms)** Let $\bullet \in \{\text{grad}, \text{curl}, \text{div}\}$. We have, for any $T \in \mathcal{T}_h$,

$$ \|\underline{z}_T\|_{\bullet,T} \approx \|\underline{z}_T\|_{\bullet,T} \quad \forall \underline{z}_T \in \underline{X}_{\bullet,T}^k. \qquad (4.25) $$

*Proof* We only prove the result for $\bullet = \text{div}$, the other cases being similar (see also [29, Proposition 14] for $\bullet = \text{curl}$ in the case of orthogonal complements, instead of the Koszul complements (2.3), (2.5)). Let $T \in \mathcal{T}_h$ and $\underline{w}_T \in \underline{X}_{\text{div},T}^k$. By definition (4.16) of the L²-product on $\underline{X}_{\text{div},T}^k$, we have

$$ \begin{aligned} \|\underline{w}_T\|_{\text{div},T}^2 &= \|\underline{P}_{\text{div},T}^k \underline{w}_T\|_{\text{L}^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F \|\underline{P}_{\text{div},T}^k \underline{w}_T \cdot \underline{n}_F - w_F\|_{\text{L}^2(F)}^2 \\ &\lesssim \|\underline{P}_{\text{div},T}^k \underline{w}_T\|_{\text{L}^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F \|w_F\|_{\text{L}^2(F)}^2 \lesssim \|\underline{w}_T\|_{\text{div},T}^2, \end{aligned} $$

where the first inequality follows from a triangle inequality together with the discrete trace inequality $h_F \|\underline{P}_{\text{div},T}^k \underline{w}_T\|_{\text{L}^2(F)}^2 \lesssim \|\underline{P}_{\text{div},T}^k \underline{w}_T\|_{\text{L}^2(T)}^2$ (see [28, Lemma 1.32]), while the conclusion is a consequence of (4.24) together with the definition of $\|\underline{w}_T\|_{\text{div},T}$. This proves $\ge$ in (4.25).

To prove the converse inequality, we start from (4.13) to write

$$ \begin{aligned} \|\underline{w}_T\|_{\text{div},T}^2 &= \|\underline{\pi}_{\underline{G},T}^{k-1}(\underline{P}_{\text{div},T}^k \underline{w}_T)\|_{\text{L}^2(T)}^2 + \|\underline{\pi}_{\underline{G},T}^{c,k}(\underline{P}_{\text{div},T}^k \underline{w}_T)\|_{\text{L}^2(T)}^2 \\ &\quad + \sum_{F \in \mathcal{F}_T} h_F \|w_F\|_{\text{L}^2(F)}^2 \\ &\lesssim \|\underline{P}_{\text{div},T}^k \underline{w}_T\|_{\text{L}^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F \|w_F - \underline{P}_{\text{div},T}^k \underline{w}_T \cdot \underline{n}_F\|_{\text{L}^2(T)}^2 \\ &\quad + \sum_{F \in \mathcal{F}_T} h_F \|\underline{P}_{\text{div},T}^k \underline{w}_T \cdot \underline{n}_F\|_{\text{L}^2(T)}^2 \\ &\lesssim \|\underline{w}_T\|_{\text{div},T}^2, \end{aligned} $$

where the first inequality follows from the L²-boundedness of the orthogonal projectors $\underline{\pi}_{\underline{G},T}^{k-1}$ and $\underline{\pi}_{\underline{G},T}^{c,k}$ together with a triangle inequality, and the conclusion is obtained invoking the same discrete trace inequality as before together with the definition of $\|\underline{w}_T\|_{\text{div},T}$. This proves $\le$ in (4.25). $\square$

**Lemma 6 (Boundedness of local interpolators)** It holds, for all $T \in \mathcal{T}_h$,

$$ \|\underline{I}_{\text{grad},T}^k q\|_{\text{grad},T} \le \|q\|_{\text{L}^2(T)} + h_T |q|_{\text{H}^1(T)} + h_T^2 |q|_{\text{H}^2(T)} \quad \forall q \in \text{H}^2(T), \quad (4.26) $$

$$ \|\underline{I}_{\text{curl},T}^k \nu\|_{\text{curl},T} \le \|\nu\|_{\text{L}^2(T)} + h_T |\nu|_{\text{H}^1(T)} + h_T^2 |\nu|_{\text{H}^2(T)} \quad \forall \nu \in \text{H}^2(T), \quad (4.27) $$

$$ \|\underline{I}_{\text{div},T}^k \mathbf{w}\|_{\text{div},T} \le \|\mathbf{w}\|_{\text{L}^2(T)} + h_T |\mathbf{w}|_{\text{H}^1(T)} \quad \forall \mathbf{w} \in \text{H}^1(T). \quad (4.28) $$
**Remark 21 (Boundedness in other norms)** The boundedness of $I_{-\text{grad},T}^k$ and $I_{\text{curl},T}^k$ could easily be stated using norms in larger spaces (typically, $C^0(\bar{T})$ for $I_{-\text{grad},T}^k$, and the same spaces on which usual Nédélec interpolators are defined for $I_{\text{curl},T}^k$ – see [10, Section 2.5.3]). However, the role of Lemma 6 is to enable primal consistency estimates (Theorem 6); since these estimates require higher regularity on the solutions, the bounds (4.26) and (4.27) stated in non-minimal norms are sufficient to our purpose.

**Proof (Lemma 6)** The definition (3.5) of $I_{-\text{grad},T}^k$ shows that $\|I_{\text{grad},T}^k q\|_{\text{grad},T} \lesssim |T|^{1/2} \max_T |q|$. By [28, Eq. (5.110)], it holds

$$ \max_T |q| \lesssim |T|^{-\frac{1}{2}} \sum_{r=0}^{2} h_T^r |q|_{H^r(T)}, $$

which concludes the proof of (4.26). The estimate (4.27) is obtained the same way. As for (4.28), by the continuous trace inequality of [28, Lemma 1.31], we have

$$ \|\pi_{\mathcal{P},F}^k (\mathbf{w} \cdot \mathbf{n}_F)\|_{L^2(F)} \le \|\mathbf{w}\|_{L^2(F)} \lesssim h_F^{-\frac{1}{2}} \|\mathbf{w}\|_{L^2(T)} + h_F^{\frac{1}{2}} |\mathbf{w}|_{H^1(T)}. $$

Using this bound in the definition (3.7) of $I_{\text{div},T}^k$ yields (4.28). □

## 4.6 Links between discrete vector potentials and vector calculus operators

In the next proposition, we show that the element gradient and curl can be recovered applying the suitable potential reconstruction to the corresponding discrete vector calculus operator, in a similar way as in (3.26) for the tangential face reconstruction and face gradients.

**Proposition 7 (Link between discrete vector potentials and vector calculus operators)** For all $T \in \mathcal{T}_h$, it holds

$$ \mathbf{P}_{\text{curl},T}^k (\underline{\mathbf{G}}_T^k \underline{q}_T) = \mathbf{G}_T^k \underline{q}_T \quad \forall \underline{q}_T \in \underline{\mathcal{X}}_{\text{grad},T}^k, \qquad (4.29) $$

$$ \mathbf{P}_{\text{div},T}^k (\underline{\mathbf{C}}_T^k \underline{\mathbf{v}}_T) = \mathbf{C}_T^k \underline{\mathbf{v}}_T \quad \forall \underline{\mathbf{v}}_T \in \underline{\mathcal{X}}_{\text{curl},T}^k. \qquad (4.30) $$

**Proof 1.** *Proof of (4.29).* By the second projection property in (4.8), we have $\pi_{\mathcal{R},T}^{c,k}[\mathbf{P}_{\text{curl},T}^k(\underline{\mathbf{G}}_T^k \underline{q}_T)] = \pi_{\mathcal{R},T}^{c,k}(\mathbf{G}_T^k \underline{q}_T)$. To infer the conclusion, it then suffices to prove that

$$ \pi_{\mathcal{R},T}^k [\mathbf{P}_{\text{curl},T}^k (\underline{\mathbf{G}}_T^k \underline{q}_T)] = \pi_{\mathcal{R},T}^k (\mathbf{G}_T^k \underline{q}_T) \qquad (4.31) $$

and invoke (2.15). To prove (4.31), we take $z_T \in \mathcal{N}^{k+1}(T)$ and apply (4.6) with $\underline{\mathbf{v}}_T = \underline{\mathbf{G}}_T^k \underline{q}_T$. Using the inclusion $\text{Im} \underline{\mathbf{G}}_T^k \subset \text{Ker} \mathbf{C}_T^k$ (see Remark 15) and
**Remark 21 (Boundedness in other norms)** The boundedness of $I_{-\text{grad},T}^k$ and $I_{\text{curl},T}^k$ could easily be stated using norms in larger spaces (typically, $C^0(\bar{T})$ for $I_{-\text{grad},T}^k$, and the same spaces on which usual Nédélec interpolators are defined for $I_{\text{curl},T}^k$ – see [10, Section 2.5.3]). However, the role of Lemma 6 is to enable primal consistency estimates (Theorem 6); since these estimates require higher regularity on the solutions, the bounds (4.26) and (4.27) stated in non-minimal norms are sufficient to our purpose.

**Proof (Lemma 6)** The definition (3.5) of $I_{-\text{grad},T}^k$ shows that $\|\| I_{\text{grad},T}^k q\|\|_{\text{grad},T} \lesssim |T|^{1/2} \max_T |q|$. By [28, Eq. (5.110)], it holds

$$ \max_T |q| \lesssim |T|^{-\frac{1}{2}} \sum_{r=0}^{2} h_T^r |q|_{H^r(T)}, $$

which concludes the proof of (4.26). The estimate (4.27) is obtained the same way. As for (4.28), by the continuous trace inequality of [28, Lemma 1.31], we have

$$ \| \pi_{\mathcal{P},F}^k (\mathbf{w} \cdot \mathbf{n}_F) \|_{L^2(F)} \le \| \mathbf{w} \|_{L^2(F)} \lesssim h_F^{-\frac{1}{2}} \| \mathbf{w} \|_{L^2(T)} + h_F^{\frac{1}{2}} |\mathbf{w}|_{H^1(T)}. $$

Using this bound in the definition (3.7) of $I_{\text{div},T}^k$ yields (4.28). □

## 4.6 Links between discrete vector potentials and vector calculus operators

In the next proposition, we show that the element gradient and curl can be recovered applying the suitable potential reconstruction to the corresponding discrete vector calculus operator, in a similar way as in (3.26) for the tangential face reconstruction and face gradients.

**Proposition 7 (Link between discrete vector potentials and vector calculus operators)** For all $T \in \mathcal{T}_h$, it holds

$$ \mathbf{P}_{\text{curl},T}^k (\underline{\mathbf{G}}_T^k \underline{q}_T) = \mathbf{G}_T^k \underline{q}_T \quad \forall \underline{q}_T \in \underline{\mathcal{X}}_{\text{grad},T}^k, \qquad (4.29) $$

$$ \mathbf{P}_{\text{div},T}^k (\underline{\mathbf{C}}_T^k \underline{\mathbf{v}}_T) = \mathbf{C}_T^k \underline{\mathbf{v}}_T \quad \forall \underline{\mathbf{v}}_T \in \underline{\mathcal{X}}_{\text{curl},T}^k. \qquad (4.30) $$

**Proof 1.** *Proof of (4.29).* By the second projection property in (4.8), we have $\pi_{\mathcal{R},T}^{c,k}[\mathbf{P}_{\text{curl},T}^k(\underline{\mathbf{G}}_T^k \underline{q}_T)] = \pi_{\mathcal{R},T}^{c,k}(\mathbf{G}_T^k \underline{q}_T)$. To infer the conclusion, it then suffices to prove that

$$ \pi_{\mathcal{R},T}^k [\mathbf{P}_{\text{curl},T}^k (\underline{\mathbf{G}}_T^k \underline{q}_T)] = \pi_{\mathcal{R},T}^k (\mathbf{G}_T^k \underline{q}_T) \qquad (4.31) $$

and invoke (2.15). To prove (4.31), we take $z_T \in \mathcal{N}^{k+1}(T)$ and apply (4.6) with $\underline{\mathbf{v}}_T = \underline{\mathbf{G}}_T^k \underline{q}_T$. Using the inclusion $\text{Im} \underline{\mathbf{G}}_T^k \subset \text{Ker} \mathbf{C}_T^k$ (see Remark 15) and
the relation $\gamma_{\mathcal{T},F}^k(\underline{\mathbf{G}}_F^k \underline{\mathbf{q}}_F) = \mathbf{G}_F^k \underline{\mathbf{q}}_F$ valid for all $F \in \mathcal{F}_T$ (see Proposition 3), we obtain

$$ \int_T \mathbf{P}_{\text{curl},T}^k (\underline{\mathbf{G}}_T^k \underline{\mathbf{q}}_T) \cdot \text{curl} \, \mathbf{z}_T = - \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \mathbf{G}_F^k \underline{\mathbf{q}}_F \cdot (\mathbf{z}_T \times \mathbf{n}_F) \\ = \int_T \mathbf{G}_T^k \underline{\mathbf{q}}_T \cdot \text{curl} \, \mathbf{z}_T, $$

where the conclusion follows from the link between element and face gradients established in Proposition 1. By the isomorphism (2.10) with $\ell = k + 1$ and since $\mathbf{G}^{c,k+1}(T) \subset \mathcal{N}^{k+1}(T)$, this establishes (4.31) and concludes the proof of (4.29).

2. *Proof of (4.30).* The second projection property in (4.13) ensures that

$$ \pi_{\mathcal{G},T}^{c,k}[\mathbf{P}_{\text{div},T}^k(\underline{\mathbf{C}}_T^k \underline{\mathbf{v}}_T)] = \pi_{\mathcal{G},T}^{c,k}(\underline{\mathbf{C}}_T^k \underline{\mathbf{v}}_T). $$

As before, it therefore remains to analyse the projections on $\mathbf{G}^k(T)$. Apply (4.11) to $\underline{\mathbf{w}}_T = \underline{\mathbf{C}}_T^k \underline{\mathbf{v}}_T$ and a generic $r_T \in \mathcal{P}^{k+1}(T)$, and use the inclusion $\text{Im} \underline{\mathbf{C}}_T^k \subset \text{Ker} \, D_T^k$ (see Proposition 5) to get

$$ \int_T \mathbf{P}_{\text{div},T}^k (\underline{\mathbf{C}}_T^k \underline{\mathbf{v}}_T) \cdot \text{grad} \, r_T = \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F C_F^k \underline{\mathbf{v}}_F \cdot r_T = \int_T \mathbf{C}_T^k \underline{\mathbf{v}}_T \cdot \text{grad} \, r_T, $$

where the conclusion is obtained applying the link between element and face curls of Proposition 4. This yields $\pi_{\mathcal{G},T}^k[\mathbf{P}_{\text{div},T}^k(\underline{\mathbf{C}}_T^k \underline{\mathbf{v}}_T)] = \pi_{\mathcal{G},T}^k(\underline{\mathbf{C}}_T^k \underline{\mathbf{v}}_T)$, proving (4.30). $\square$

**Corollary 1 (Bounds on discrete gradients and curl)** For all $F \in \mathcal{F}_h$, it holds

$$ \| \mathbf{G}_F^k \underline{\mathbf{q}}_F \|_{\mathcal{L}^2(F)}^2 + \sum_{E \in \mathcal{E}_F} h_E \| G_E^k q_E \|_{\mathcal{L}^2(E)}^2 \lesssim \| \underline{\mathbf{G}}_F^k \underline{\mathbf{q}}_F \|_{\text{curl},F} \quad \forall \underline{\mathbf{q}}_F \in \underline{\mathbf{X}}_{\text{grad},F}^k. \quad (4.32) $$

For all $T \in \mathcal{T}_h$, it holds

$$ \| \mathbf{G}_T^k \underline{\mathbf{q}}_T \|_{\mathcal{L}^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F \| \mathbf{G}_F^k \underline{\mathbf{q}}_F \|_{\mathcal{L}^2(F)}^2 + \sum_{E \in \mathcal{E}_T} h_E^2 \| G_E^k q_E \|_{\mathcal{L}^2(E)}^2 \lesssim \| \underline{\mathbf{G}}_T^k \underline{\mathbf{q}}_T \|_{\text{curl},T} \\ \forall \underline{\mathbf{q}}_T \in \underline{\mathbf{X}}_{\text{grad},T}^k, \quad (4.33) $$

$$ \| \mathbf{C}_T^k \underline{\mathbf{v}}_T \|_{\mathcal{L}^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F \| \mathbf{C}_F^k \underline{\mathbf{v}}_F \|_{\mathcal{L}^2(F)}^2 \lesssim \| \underline{\mathbf{C}}_T^k \underline{\mathbf{v}}_T \|_{\text{div},T} \quad \forall \underline{\mathbf{v}}_T \in \underline{\mathbf{X}}_{\text{curl},T}^k. \quad (4.34) $$

*Proof* The definitions of $\|\cdot\|_{\text{curl},F}$, $\|\cdot\|_{\text{curl},T}$, $\underline{\mathbf{G}}_F^k$ and $\underline{\mathbf{G}}_T^k$ show that the edge gradient contributions in the left-hand sides of (4.32) and (4.33) are bounded by the corresponding right-hand sides. To bound the face and element gradient contributions in the left-hand sides of (4.32) and (4.33), simply apply (4.23) to $\underline{\mathbf{v}}_T = \underline{\mathbf{G}}_T^k \underline{\mathbf{q}}_T$ and use (3.26) along with (4.29). The estimate (4.34) is established in a similar way, using (4.30). $\square$
the relation $\gamma_{\mathcal{T},F}^k(\underline{G}_F^k \underline{q}_F) = \underline{G}_F^k \underline{q}_F$ valid for all $F \in \mathcal{F}_T$ (see Proposition 3), we obtain

$$ 
\begin{aligned}
\int_T \mathbf{P}_{\mathrm{curl},T}^k (\underline{\underline{G}}_T^k \underline{q}_T) \cdot \mathrm{curl} \, z_T &= - \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \underline{\underline{G}}_F^k \underline{q}_F \cdot (z_T \times \mathbf{n}_F) \\
&= \int_T \underline{\underline{G}}_T^k \underline{q}_T \cdot \mathrm{curl} \, z_T,
\end{aligned}
$$

where the conclusion follows from the link between element and face gradients established in Proposition 1. By the isomorphism (2.10) with $\ell = k + 1$ and since $\underline{\underline{G}}^{c,k+1}(T) \subset \mathcal{N}^{k+1}(T)$, this establishes (4.31) and concludes the proof of (4.29).

2. *Proof of (4.30).* The second projection property in (4.13) ensures that

$$ \pi_{\underline{\underline{G}},T}^{c,k}[\mathbf{P}_{\mathrm{div},T}^k(\underline{\underline{C}}_T^k \underline{\underline{v}}_T)] = \pi_{\underline{\underline{G}},T}^{c,k}(\underline{\underline{C}}_T^k \underline{\underline{v}}_T). $$

As before, it therefore remains to analyse the projections on $\underline{\underline{G}}^k(T)$. Apply (4.11) to $\underline{\underline{w}}_T = \underline{\underline{C}}_T^k \underline{\underline{v}}_T$ and a generic $r_T \in \mathcal{P}^{k+1}(T)$, and use the inclusion $\mathrm{Im} \, \underline{\underline{C}}_T^k \subset \mathrm{Ker} \, D_T^k$ (see Proposition 5) to get

$$ \int_T \mathbf{P}_{\mathrm{div},T}^k (\underline{\underline{C}}_T^k \underline{\underline{v}}_T) \cdot \mathrm{grad} \, r_T = \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F C_F^k \underline{\underline{v}}_F \cdot r_T = \int_T \underline{\underline{C}}_T^k \underline{\underline{v}}_T \cdot \mathrm{grad} \, r_T, $$

where the conclusion is obtained applying the link between element and face curls of Proposition 4. This yields $\pi_{\underline{\underline{G}},T}^k [\mathbf{P}_{\mathrm{div},T}^k(\underline{\underline{C}}_T^k \underline{\underline{v}}_T)] = \pi_{\underline{\underline{G}},T}^k(\underline{\underline{C}}_T^k \underline{\underline{v}}_T)$, proving (4.30). $\square$

**Corollary 1 (Bounds on discrete gradients and curl)** For all $F \in \mathcal{F}_h$, it holds

$$ \| \underline{\underline{G}}_F^k \underline{\underline{q}}_F \|_{\mathcal{L}^2(F)}^2 + \sum_{E \in \mathcal{E}_F} h_E \| G_E^k q_E \|_{\mathcal{L}^2(E)}^2 \lesssim \| \underline{\underline{G}}_F^k \underline{\underline{q}}_F \|_{\mathrm{curl},F} \quad \forall \underline{\underline{q}}_F \in \underline{\underline{X}}_{\mathrm{grad},F}^k. \quad (4.32) $$

For all $T \in \mathcal{T}_h$, it holds

$$ \begin{aligned} \| \underline{\underline{G}}_T^k \underline{\underline{q}}_T \|_{\mathcal{L}^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F \| \underline{\underline{G}}_F^k \underline{\underline{q}}_F \|_{\mathcal{L}^2(F)}^2 + \sum_{E \in \mathcal{E}_T} h_E^2 \| G_E^k q_E \|_{\mathcal{L}^2(E)}^2 &\lesssim \| \underline{\underline{G}}_T^k \underline{\underline{q}}_T \|_{\mathrm{curl},T} \\ \forall \underline{\underline{q}}_T \in \underline{\underline{X}}_{\mathrm{grad},T}^k, \end{aligned} \quad (4.33) $$

$$ \| \underline{\underline{C}}_T^k \underline{\underline{v}}_T \|_{\mathcal{L}^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F \| \underline{\underline{C}}_F^k \underline{\underline{v}}_F \|_{\mathcal{L}^2(F)}^2 \lesssim \| \underline{\underline{C}}_T^k \underline{\underline{v}}_T \|_{\mathrm{div},T} \quad \forall \underline{\underline{v}}_T \in \underline{\underline{X}}_{\mathrm{curl},T}^k. \quad (4.34) $$

*Proof* The definitions of $\|\cdot\|_{\mathrm{curl},F}$, $\|\cdot\|_{\mathrm{curl},T}$, $\underline{\underline{G}}_F^k$ and $\underline{\underline{G}}_T^k$ show that the edge gradient contributions in the left-hand sides of (4.32) and (4.33) are bounded by the corresponding right-hand sides. To bound the face and element gradient contributions in the left-hand sides of (4.32) and (4.33), simply apply (4.23) to $\underline{\underline{v}}_T = \underline{\underline{G}}_T^k \underline{\underline{q}}_T$ and use (3.26) along with (4.29). The estimate (4.34) is established in a similar way, using (4.30). $\square$
## 5 Poincaré inequalities

In this section we state and prove Poincaré-type inequalities for the operators in the DDR sequence. Notice that we consider here the complex without boundary conditions, but one could alternatively consider the complex with (homogeneous) boundary conditions, for which similar inequalities are expected to hold. The details are left for a future work.

### 5.1 Discrete Poincaré inequalities

**Theorem 3 (Poincaré inequality for the gradient)** Let $q_h \in \underline{X}_{\text{grad},h}^k$ be such that

$$ \sum_{T \in \mathcal{T}_h} \int_T p_{\text{grad},T}^{k+1} \underline{q}_T = 0. \qquad (5.1) $$

Then, there exists a real number $C > 0$ independent of $h$ and $\underline{q}_h$, and depending only on $\Omega$, $k$, and the mesh regularity parameter, such that

$$ \| \underline{q}_h \|_{\text{grad},h} \le C \| \underline{G}_h^k \underline{q}_h \|_{\text{curl},h}. \qquad (5.2) $$

*Proof* See Section 5.2.

**Remark 22 (Condition (5.1))** For $k \ge 1$, owing to (4.3) the condition (5.1) is equivalent to

$$ \sum_{T \in \mathcal{T}_h} \int_T q_T = 0. $$

For $k = 0$, the absence of element components means that translating (5.1) in terms of the components of $\underline{q}_h$ is less straightforward. Assuming that, for all $Y \in \mathcal{T}_h \cup \mathcal{F}_h$, $Y$ is star-shaped with respect to $\mathbf{x}_Y := \frac{1}{|Y|} \int_Y \mathbf{x}$, and that this point is selected in the definition of the complements in Section 2.4, condition (5.1) becomes

$$ \sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \sum_{E \in \mathcal{E}_F} |P_{TFE}| \pi_{\mathcal{P},E}^0 q_E = \frac{1}{2} \sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \sum_{E \in \mathcal{E}_F} \sum_{V \in \mathcal{V}_E} |P_{TFE}| q_{\mathcal{E}_F}(\mathbf{x}_V) = 0, $$

where, for any mesh element $T \in \mathcal{T}_h$, face $F \in \mathcal{F}_T$, and edge $E \in \mathcal{E}_F$ of vertices $V_1$ and $V_2$, $P_{TFE}$ is the tetrahedron of vertices $\mathbf{x}_T$, $\mathbf{x}_F$, $\mathbf{x}_{V_1}$, and $\mathbf{x}_{V_2}$. This corresponds to the construction on the dual barycentric mesh of [12, Section 4.1].

We also notice, in passing, that condition (5.1) is not needed when considering the subspace of $\underline{X}_{\text{grad},h}^k$ with homogeneous boundary conditions.

For the sake of completeness, we state in what follows Poincaré inequalities for the curl and the divergence that are easy consequences of the results of [29].
## 5 Poincaré inequalities

In this section we state and prove Poincaré-type inequalities for the opera-
tors in the DDR sequence. Notice that we consider here the complex with-
out boundary conditions, but one could alternatively consider the complex
with (homogeneous) boundary conditions, for which similar inequalities are
expected to hold. The details are left for a future work.

### 5.1 Discrete Poincaré inequalities

**Theorem 3 (Poincaré inequality for the gradient)** Let $q_h \in \underline{X}_{\text{grad},h}^k$ be such that

$$ \sum_{T \in \mathcal{T}_h} \int_T p_{\text{grad},T}^{k+1} \underline{q}_T = 0. \qquad (5.1) $$

Then, there exists a real number $C > 0$ independent of $h$ and $\underline{q}_h$, and depending only on $\Omega$, $k$, and the mesh regularity parameter, such that

$$ \| \underline{q}_h \|_{\text{grad},h} \le C \| \underline{G}_h^k \underline{q}_h \|_{\text{curl},h}. \qquad (5.2) $$

*Proof* See Section 5.2.

**Remark 22 (Condition (5.1))** For $k \ge 1$, owing to (4.3) the condition (5.1) is equivalent to

$$ \sum_{T \in \mathcal{T}_h} \int_T q_T = 0. $$

For $k=0$, the absence of element components means that translating (5.1) in terms of the components of $\underline{q}_h$ is less straightforward. Assuming that, for all $Y \in \mathcal{T}_h \cup \mathcal{F}_h$, $Y$ is star-shaped with respect to $\mathbf{x}_Y := \frac{1}{|Y|} \int_Y \mathbf{x}$, and that this point is selected in the definition of the complements in Section 2.4, condition (5.1) becomes

$$ \sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \sum_{E \in \mathcal{E}_F} |P_{TFE}| \pi_{\mathcal{P},E}^0 q_E = \frac{1}{2} \sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \sum_{E \in \mathcal{E}_F} \sum_{V \in \mathcal{V}_E} |P_{TFE}| q_{\mathcal{E}_F}(\mathbf{x}_V) = 0, $$

where, for any mesh element $T \in \mathcal{T}_h$, face $F \in \mathcal{F}_T$, and edge $E \in \mathcal{E}_F$ of vertices $V_1$ and $V_2$, $P_{TFE}$ is the tetrahedron of vertices $\mathbf{x}_T$, $\mathbf{x}_F$, $\mathbf{x}_{V_1}$, and $\mathbf{x}_{V_2}$. This corresponds to the construction on the dual barycentric mesh of [12, Section 4.1].

We also notice, in passing, that condition (5.1) is not needed when considering the subspace of $\underline{X}_{\text{grad},h}^k$ with homogeneous boundary conditions.

For the sake of completeness, we state in what follows Poincaré inequalities for the curl and the divergence that are easy consequences of the results of [29].
**Theorem 4 (Poincaré inequality for the curl)** Denote by $(b_0, b_1, b_2, b_3)$ the Betti numbers of $\Omega$ (with $b_0 = 1$ and $b_3 = 0$) and assume $b_2 = 0$. Let (Ker $\underline{C}_h^k$)⊥ be the orthogonal of Ker $\underline{C}_h^k$ in $\underline{X}_{\text{curl},h}$ for an inner product whose norm is, uniformly in $h$, equivalent to $|||\cdot|||_{\text{curl},h}$. Then, $\underline{C}_h^k : (\text{Ker } \underline{C}_h^k)^\perp \to \text{Ker } D_h^k$ is an isomorphism. Further assuming that $b_1 = 0$, there exists $C > 0$ independent of $h$, and depending only on $\Omega$, $k$ and the mesh regularity parameter, such that

$$ \|\underline{v}_h\|_{\text{curl},h} \le C \|\underline{C}_h^k \underline{v}_h\|_{\text{div},h} \quad \forall \underline{v}_h \in (\text{Ker } \underline{C}_h^k)^\perp. \quad (5.3) $$

*Proof* The isomorphism property is a consequence of (3.48). In order to prove the Poincaré inequality (5.3), combine [29, Theorem 20] with [29, Proposition 16] (which requires the additional assumption $b_1 = 0$) and the norm equivalence (4.25). □

**Theorem 5 (Poincaré inequality for the divergence)** Let $(\text{Ker } D_h^k)^\perp$ be the orthogonal of $\text{Ker } D_h^k$ in $\underline{X}_{\text{div},h}$ for an inner product whose norm is, uniformly in $h$, equivalent to $|||\cdot|||_{\text{div},h}$. Then, $D_h^k : (\text{Ker } D_h^k)^\perp \to \mathcal{P}^k(\mathcal{T}_h)$ is an isomorphism and there exists $C > 0$ independent of $h$, and depending only on $\Omega$, $k$ and the mesh regularity parameter, such that

$$ \|\underline{w}_h\|_{\text{div},h} \le C \|D_h^k \underline{w}_h\|_{L^2(\Omega)} \quad \forall \underline{w}_h \in (\text{Ker } D_h^k)^\perp. \quad (5.4) $$

*Proof* The isomorphism property is a consequence of (3.45). The Poincaré inequality (5.4) follows from [29, Theorem 18] accounting for the norm equivalence (4.25). □

## 5.2 Proof of the discrete Poincaré inequality for the gradient

We first prove a preliminary result, which will also be useful to establish adjoint consistency properties for the discrete gradient operator in Section 6.4.

**Lemma 7 (Estimates on local $H^1$-seminorms of potentials)** For all $F \in \mathcal{F}_h$ and all $\underline{q}_F \in \underline{X}_{\text{grad},F}^k$, it holds

$$ \| \text{grad} \, \gamma_F^{k+1} \underline{q}_F \|_{L^2(F)}^2 + \sum_{E \in \mathcal{E}_F} h_E^{-1} \| \gamma_F^{k+1} \underline{q}_F - q_E \|_{L^2(E)}^2 \lesssim \| \underline{G}_F^k \underline{q}_F \|_{\text{curl},F}^2. \quad (5.5) $$

For all $T \in \mathcal{T}_h$ and all $\underline{q}_T \in \underline{X}_{\text{grad},T}^k$, it holds

$$ \| \text{grad} \, P_{\text{grad},T}^{k+1} \underline{q}_T \|_{L^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F^{-1} \| P_{\text{grad},T}^{k+1} \underline{q}_T - \gamma_F^{k+1} \underline{q}_F \|_{L^2(F)}^2 \lesssim \| \underline{G}_T^k \underline{q}_T \|_{\text{curl},T}^2. \quad (5.6) $$
**Theorem 4 (Poincaré inequality for the curl)** Denote by $(b_0, b_1, b_2, b_3)$ the Betti numbers of $\Omega$ (with $b_0 = 1$ and $b_3 = 0$) and assume $b_2 = 0$. Let (Ker $\underline{C}_h^k$)⊥ be the orthogonal of Ker $\underline{C}_h^k$ in $\underline{X}_{\text{curl},h}$ for an inner product whose norm is, uniformly in $h$, equivalent to $\|\cdot\|_{\text{curl},h}$. Then, $\underline{C}_h^k : (\text{Ker } \underline{C}_h^k)^\perp \to \text{Ker } D_h^k$ is an isomorphism. Further assuming that $b_1 = 0$, there exists $C > 0$ independent of $h$, and depending only on $\Omega$, $k$ and the mesh regularity parameter, such that

$$ \|\underline{v}_h\|_{\text{curl},h} \le C \|\underline{C}_h^k \underline{v}_h\|_{\text{div},h} \quad \forall \underline{v}_h \in (\text{Ker } \underline{C}_h^k)^\perp. \quad (5.3) $$

*Proof* The isomorphism property is a consequence of (3.48). In order to prove the Poincaré inequality (5.3), combine [29, Theorem 20] with [29, Proposition 16] (which requires the additional assumption $b_1 = 0$) and the norm equivalence (4.25). □

**Theorem 5 (Poincaré inequality for the divergence)** Let $(\text{Ker } D_h^k)^\perp$ be the orthogonal of $\text{Ker } D_h^k$ in $\underline{X}_{\text{div},h}$ for an inner product whose norm is, uniformly in $h$, equivalent to $\|\cdot\|_{\text{div},h}$. Then, $D_h^k : (\text{Ker } D_h^k)^\perp \to \mathcal{P}^k(\mathcal{T}_h)$ is an isomorphism and there exists $C > 0$ independent of $h$, and depending only on $\Omega$, $k$ and the mesh regularity parameter, such that

$$ \|\underline{w}_h\|_{\text{div},h} \le C \|D_h^k \underline{w}_h\|_{L^2(\Omega)} \quad \forall \underline{w}_h \in (\text{Ker } D_h^k)^\perp. \quad (5.4) $$

*Proof* The isomorphism property is a consequence of (3.45). The Poincaré inequality (5.4) follows from [29, Theorem 18] accounting for the norm equivalence (4.25). □

## 5.2 Proof of the discrete Poincaré inequality for the gradient

We first prove a preliminary result, which will also be useful to establish adjoint consistency properties for the discrete gradient operator in Section 6.4.

**Lemma 7 (Estimates on local $H^1$-seminorms of potentials)** For all $F \in \mathcal{F}_h$ and all $\underline{q}_F \in \underline{X}_{\text{grad},F}^k$, it holds

$$ \| \text{grad} \gamma_F^{k+1} \underline{q}_F \|_{L^2(F)}^2 + \sum_{E \in \mathcal{E}_F} h_E^{-1} \| \gamma_F^{k+1} \underline{q}_F - q_E \|_{L^2(E)}^2 \lesssim \| \underline{G}_F^k \underline{q}_F \|_{\text{curl},F}^2. \quad (5.5) $$

For all $T \in \mathcal{T}_h$ and all $\underline{q}_T \in \underline{X}_{\text{grad},T}^k$, it holds

$$ \| \text{grad} P_{\text{grad},T}^{k+1} \underline{q}_T \|_{L^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F^{-1} \| P_{\text{grad},T}^{k+1} \underline{q}_T - \gamma_F^{k+1} \underline{q}_F \|_{L^2(F)}^2 \lesssim \| \underline{G}_T^k \underline{q}_T \|_{\text{curl},T}^2. \quad (5.6) $$
Proof

1. *Proof of (5.5).* Let $\underline{q}_F \in \underline{X}_{\text{grad},F}^k$ and define $A_{q,\partial F} \in \mathbb{R}$ as the average of $q_{\mathcal{E}_F}$ over $\partial F$. Introducing $A_{q,\partial F} = \gamma_F^{k+1}(\underline{I}_{\text{grad},F}^k A_{q,\partial F})$ (see (3.14)), using $h_E \approx h_F$ and card($\mathcal{E}_F) \le 1$, and invoking a discrete trace inequality on $\gamma_F^{k+1}(\underline{q}_F - A_{q,\partial F})$, we have

$$ \sum_{E \in \mathcal{E}_F} h_E^{-1} \| \gamma_F^{k+1} \underline{q}_F - q_E \|_{L^2(E)}^2 \le \sum_{E \in \mathcal{E}_F} h_E^{-1} \| q_E - A_{q,\partial F} \|_{L^2(E)}^2 + h_F^{-2} \| \gamma_F^{k+1} (\underline{q}_F - A_{q,\partial F}) \|_{L^2(F)}^2. \quad (5.7) $$

Since $q_{\mathcal{E}_F}$ is continuous, recalling that $q_E = (q_{\mathcal{E}_h})|_E$ for all $E \in \mathcal{E}_F$ and using a Poincaré–Wirtinger inequality along $\partial F$ followed by the definition (4.21) of $\|\cdot\|_{\text{curl},F}$ yields

$$ \sum_{E \in \mathcal{E}_F} h_E^{-1} \|q_E - A_{q,\partial F}\|_{L^2(E)}^2 \le h_F \sum_{E \in \mathcal{E}_F} \|G_E^k q_E\|_{L^2(E)}^2 \le \|\underline{G}_F^k \underline{q}_F\|_{\text{curl},F}^2. \quad (5.8) $$

We now turn to the second term in (5.7). Using the isomorphism property (2.9), we select $v_F \in \mathcal{R}^{c,k+2}(F)$ such that $\text{div}_F v_F = \gamma_F^{k+1}(\underline{q}_F - \underline{I}_{\text{grad},F}^k A_{q,\partial F})$. By Lemma 9 in Appendix A, we have

$$ \|v_F\|_{L^2(F)} \le h_F \|\gamma_F^{k+1}(\underline{q}_F - \underline{I}_{\text{grad},F}^k A_{q,\partial F})\|_{L^2(F)}. $$

The discrete trace inequality of [28, Lemma 1.32] and the consistency property (3.13) of $\mathbf{G}_F^k$ then yield

$$ \|v_F\|_{L^2(F)} + \left( \sum_{E \in \mathcal{E}_F} h_E \|v_F\|_{L^2(E)}^2 \right)^{\frac{1}{2}} \le h_F \|\gamma_F^{k+1}(\underline{q}_F - \underline{I}_{\text{grad},F}^k A_{q,\partial F})\|_{L^2(F)}, \\ \mathbf{G}_F^k(\underline{q}_F - \underline{I}_{\text{grad},F}^k A_{q,\partial F}) = \mathbf{G}_F^k \underline{q}_F. $$

Hence, applying the definition (3.10) of $\gamma_F^{k+1}$ to $\underline{q}_F - \underline{I}_{\text{grad},F}^k A_{q,\partial F} \in \underline{X}_{\text{grad},F}^k$, taking $v_F$ above as a test function, and using Cauchy–Schwarz inequalities, we obtain

$$ \|\gamma_F^{k+1}(\underline{q}_F - \underline{I}_{\text{grad},F}^k A_{q,\partial F})\|_{L^2(F)}^2 \\ \le h_F \|\mathbf{G}_F^k \underline{q}_F\|_{L^2(F)} \|\gamma_F^{k+1}(\underline{q}_F - \underline{I}_{\text{grad},F}^k A_{q,\partial F})\|_{L^2(F)} \\ + \left( \sum_{E \in \mathcal{E}_F} h_E^{-1} \|q_E - A_{q,\partial F}\|_{L^2(E)}^2 \right)^{\frac{1}{2}} h_F \|\gamma_F^{k+1}(\underline{q}_F - \underline{I}_{\text{grad},F}^k A_{q,\partial F})\|_{L^2(F)}. $$

Simplifying and recalling (4.32) and (5.8), we infer $\|\gamma_F^{k+1}(\underline{q}_F - A_{q,\partial F})\|_{L^2(F)} \le h_F \|\underline{G}_F^k \underline{q}_F\|_{\text{curl},F}$ which, plugged together with (5.8) into (5.7), gives the following estimate on the second term in the left-hand side of (5.5):

$$ \sum_{E \in \mathcal{E}_F} h_E^{-1} \| \gamma_F^{k+1} \underline{q}_F - q_E \|_{L^2(E)}^2 \le \| \underline{G}_F^k \underline{q}_F \|_{\text{curl},F}^2. \quad (5.9) $$
Proof

1. *Proof of (5.5).* Let $q_F \in \underline{X}_{\text{grad},F}^k$ and define $A_{q,\partial F} \in \mathbb{R}$ as the average of $q_{\mathcal{E}_F}$ over $\partial F$. Introducing $A_{q,\partial F} = \gamma_F^{k+1}(\underline{I}_{\text{grad},F}^k A_{q,\partial F})$ (see (3.14)), using $h_E \approx h_F$ and card($\mathcal{E}_F) \le 1$, and invoking a discrete trace inequality on $\gamma_F^{k+1}(\underline{q}_F - A_{q,\partial F})$, we have

$$ \sum_{E \in \mathcal{E}_F} h_E^{-1} \| \gamma_F^{k+1} \underline{q}_F - q_E \|_{L^2(E)}^2 \le \sum_{E \in \mathcal{E}_F} h_E^{-1} \| q_E - A_{q,\partial F} \|_{L^2(E)}^2 + h_F^{-2} \| \gamma_F^{k+1} (\underline{q}_F - A_{q,\partial F}) \|_{L^2(F)}^2. \quad (5.7) $$

Since $q_{\mathcal{E}_F}$ is continuous, recalling that $q_E = (q_{\mathcal{E}_h})|_E$ for all $E \in \mathcal{E}_F$ and using a Poincaré–Wirtinger inequality along $\partial F$ followed by the definition (4.21) of $\|\cdot\|_{\text{curl},F}$ yields

$$ \sum_{E \in \mathcal{E}_F} h_E^{-1} \|q_E - A_{q,\partial F}\|_{L^2(E)}^2 \le h_F \sum_{E \in \mathcal{E}_F} \|G_E^k q_E\|_{L^2(E)}^2 \le \|\underline{G}_F^k \underline{q}_F\|_{\text{curl},F}^2. \quad (5.8) $$

We now turn to the second term in (5.7). Using the isomorphism property (2.9), we select $v_F \in \mathcal{R}^{c,k+2}(F)$ such that $\text{div}_F v_F = \gamma_F^{k+1}(\underline{q}_F - \underline{I}_{\text{grad},F}^k A_{q,\partial F})$. By Lemma 9 in Appendix A, we have

$$ \|v_F\|_{L^2(F)} \le h_F \|\gamma_F^{k+1}(\underline{q}_F - \underline{I}_{\text{grad},F}^k A_{q,\partial F})\|_{L^2(F)}. $$

The discrete trace inequality of [28, Lemma 1.32] and the consistency property (3.13) of $\mathbf{G}_F^k$ then yield

$$ \|v_F\|_{L^2(F)} + \left( \sum_{E \in \mathcal{E}_F} h_E \|v_F\|_{L^2(E)}^2 \right)^{\frac{1}{2}} \le h_F \|\gamma_F^{k+1}(\underline{q}_F - \underline{I}_{\text{grad},F}^k A_{q,\partial F})\|_{L^2(F)}, \\ \mathbf{G}_F^k(\underline{q}_F - \underline{I}_{\text{grad},F}^k A_{q,\partial F}) = \mathbf{G}_F^k \underline{q}_F. $$

Hence, applying the definition (3.10) of $\gamma_F^{k+1}$ to $\underline{q}_F - \underline{I}_{\text{grad},F}^k A_{q,\partial F} \in \underline{X}_{\text{grad},F}^k$, taking $v_F$ above as a test function, and using Cauchy–Schwarz inequalities, we obtain

$$ \|\gamma_F^{k+1}(\underline{q}_F - \underline{I}_{\text{grad},F}^k A_{q,\partial F})\|_{L^2(F)}^2 \\ \le h_F \|\mathbf{G}_F^k \underline{q}_F\|_{L^2(F)} \|\gamma_F^{k+1}(\underline{q}_F - \underline{I}_{\text{grad},F}^k A_{q,\partial F})\|_{L^2(F)} \\ + \left( \sum_{E \in \mathcal{E}_F} h_E^{-1} \|q_E - A_{q,\partial F}\|_{L^2(E)}^2 \right)^{\frac{1}{2}} h_F \|\gamma_F^{k+1}(\underline{q}_F - \underline{I}_{\text{grad},F}^k A_{q,\partial F})\|_{L^2(F)}. $$

Simplifying and recalling (4.32) and (5.8), we infer $\|\gamma_F^{k+1}(\underline{q}_F - A_{q,\partial F})\|_{L^2(F)} \le h_F \|\underline{G}_F^k \underline{q}_F\|_{\text{curl},F}$ which, plugged together with (5.8) into (5.7), gives the following estimate on the second term in the left-hand side of (5.5):

$$ \sum_{E \in \mathcal{E}_F} h_E^{-1} \| \gamma_F^{k+1} \underline{q}_F - q_E \|_{L^2(E)}^2 \le \| \underline{G}_F^k \underline{q}_F \|_{\text{curl},F}^2. \quad (5.9) $$
Integrating by parts the definition (3.10) of $\gamma_F^{k+1}$ applied to a generic $v_F \in \mathcal{P}^k(F)$ (see Remark 8), we have

$$ \int_F \mathbf{grad}_F \gamma_F^{k+1} \underline{q}_F \cdot \mathbf{v}_F = \int_F \mathbf{G}_F^k \underline{q}_F \cdot \mathbf{v}_F + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E (\gamma_F^{k+1} \underline{q}_F - q_E)(\mathbf{v}_F \cdot \mathbf{n}_{FE}). $$

Making $v_F = \mathbf{grad}_F \gamma_F^{k+1} \underline{q}_F$, using Cauchy-Schwarz inequalities, (4.32), a discrete trace inequality, and (5.9) then yields the bound on the first term in the left-hand side of (5.5).

2. *Proof of (5.6).* The ideas are similar to those used to prove (5.5), but first we need to establish a Poincaré-Wirtinger inequality for face potentials (which is not straightforward given their discontinuity). Let

$$ A_{q, \partial T} := \frac{1}{|\partial T|} \sum_{F \in \mathcal{F}_T} |F| A_{q,F} \quad \text{with} \quad A_{q,F} := \frac{1}{|F|} \int_F \gamma_F^{k+1} \underline{q}_F $$

denote the average over $\partial T$ of the piecewise polynomial function defined by $(\gamma_F^{k+1} \underline{q}_F)_{F \in \mathcal{F}_T}$. We write, using triangle inequalities,

$$ \begin{aligned} \sum_{F \in \mathcal{F}_T} h_F^{-1} \| P_{\mathrm{grad},T} \underline{q}_T - \gamma_F^{k+1} \underline{q}_F \|_{\mathrm{L}^2(F)}^2 & \\ \le & \sum_{F \in \mathcal{F}_T} h_F^{-1} \| \gamma_F^{k+1} \underline{q}_F - A_{q,F} \|_{\mathrm{L}^2(F)}^2 \\ & + \sum_{F \in \mathcal{F}_T} h_F^{-1} \| A_{q,F} - A_{q,\partial T} \|_{\mathrm{L}^2(F)}^2 \\ & + \sum_{F \in \mathcal{F}_T} h_F^{-1} \| P_{\mathrm{grad},T} \underline{q}_T - A_{q,\partial T} \|_{\mathrm{L}^2(F)}^2 =: \mathfrak{I}_1 + \mathfrak{I}_2 + \mathfrak{I}_3. \end{aligned} \quad (5.10) $$

The first term is estimated using a Poincaré-Wirtinger inequality on $\gamma_F^{k+1} \underline{q}_F$ and invoking (5.5) together with the definition (4.21) of $\|\cdot\|_{\mathrm{curl},T}$ to get

$$ \begin{aligned} \mathfrak{I}_1 & \le \sum_{F \in \mathcal{F}_T} h_F^{-1} \left( h_F \| \mathbf{grad}_F \gamma_F^{k+1} \underline{q}_F \|_{\mathrm{L}^2(F)} \right)^2 \\ & \le \sum_{F \in \mathcal{F}_T} h_F \| \underline{G}_F^k \underline{q}_F \|_{\mathrm{curl},F}^2 \\ & \le \| \underline{G}_T^k \underline{q}_T \|_{\mathrm{curl},T}^2. \end{aligned} \quad (5.11) $$

Let us turn to the second term in (5.10). Since $A_{q,\partial T}$ is a weighted average of all $(A_{q,F})_{F \in \mathcal{F}_T}$, the bound

$$ \mathfrak{I}_2 \le \| \underline{G}_T^k \underline{q}_T \|_{\mathrm{curl},T}^2 \quad (5.12) $$
Integrating by parts the definition (3.10) of $\gamma_F^{k+1}$ applied to a generic $v_F \in \mathcal{P}^k(F)$ (see Remark 8), we have

$$ \int_F \mathbf{grad}_F \gamma_F^{k+1} \underline{q}_F \cdot v_F = \int_F \mathbf{G}_F^k \underline{q}_F \cdot v_F + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E (\gamma_F^{k+1} \underline{q}_F - q_E)(v_F \cdot n_{FE}). $$

Making $v_F = \mathbf{grad}_F \gamma_F^{k+1} \underline{q}_F$, using Cauchy-Schwarz inequalities, (4.32), a discrete trace inequality, and (5.9) then yields the bound on the first term in the left-hand side of (5.5).

2. *Proof of (5.6).* The ideas are similar to those used to prove (5.5), but first we need to establish a Poincaré-Wirtinger inequality for face potentials (which is not straightforward given their discontinuity). Let

$$ A_{q, \partial T} := \frac{1}{|\partial T|} \sum_{F \in \mathcal{F}_T} |F| A_{q,F} \quad \text{with} \quad A_{q,F} := \frac{1}{|F|} \int_F \gamma_F^{k+1} \underline{q}_F $$

denote the average over $\partial T$ of the piecewise polynomial function defined by $(\gamma_F^{k+1} \underline{q}_F)_{F \in \mathcal{F}_T}$. We write, using triangle inequalities,

$$ \begin{aligned} \sum_{F \in \mathcal{F}_T} h_F^{-1} \| P_{\mathrm{grad},T}^{k+1} \underline{q}_T - \gamma_F^{k+1} \underline{q}_F \|_{\mathrm{L}^2(F)}^2 & \\ \le & \sum_{F \in \mathcal{F}_T} h_F^{-1} \| \gamma_F^{k+1} \underline{q}_F - A_{q,F} \|_{\mathrm{L}^2(F)}^2 \\ & + \sum_{F \in \mathcal{F}_T} h_F^{-1} \| A_{q,F} - A_{q,\partial T} \|_{\mathrm{L}^2(F)}^2 \\ & + \sum_{F \in \mathcal{F}_T} h_F^{-1} \| P_{\mathrm{grad},T}^{k+1} \underline{q}_T - A_{q,\partial T} \|_{\mathrm{L}^2(F)}^2 =: \mathfrak{T}_1 + \mathfrak{T}_2 + \mathfrak{T}_3. \end{aligned} \quad (5.10) $$

The first term is estimated using a Poincaré-Wirtinger inequality on $\gamma_F^{k+1} \underline{q}_F$ and invoking (5.5) together with the definition (4.21) of $\|\cdot\|_{\mathrm{curl},T}$ to get

$$ \begin{aligned} \mathfrak{T}_1 & \le \sum_{F \in \mathcal{F}_T} h_F^{-1} \left( h_F \| \mathbf{grad}_F \gamma_F^{k+1} \underline{q}_F \|_{\mathrm{L}^2(F)} \right)^2 \\ & \le \sum_{F \in \mathcal{F}_T} h_F \| \underline{\mathbf{G}}_F^k \underline{q}_F \|_{\mathrm{curl},F}^2 \\ & \le \| \underline{\mathbf{G}}_T^k \underline{q}_T \|_{\mathrm{curl},T}^2. \end{aligned} \quad (5.11) $$

Let us turn to the second term in (5.10). Since $A_{q,\partial T}$ is a weighted average of all $(A_{q,F})_{F \in \mathcal{F}_T}$, the bound

$$ \mathfrak{T}_2 \le \| \underline{\mathbf{G}}_T^k \underline{q}_T \|_{\mathrm{curl},T}^2 \quad (5.12) $$
follows if we prove that, for all $F, F' \in \mathcal{F}_T$,

$$h_F^{-1} \|A_{q,F} - A_{q,F'}\|_{L^2(F)}^2 = h_F^{-1} |F| |A_{q,F} - A_{q,F'}|^2 \lesssim \|\underline{\mathbf{G}}_T^k \underline{q}_T\|_{\text{curl},T}^2. \quad (5.13)$$

Creating a sequence ($F = F_0, F_1, \dots, F_m = F'$) of faces in $\mathcal{F}_T$ such that, for all $i = 0, \dots, m-1$, the faces $F_i, F_{i+1}$ share an edge $E_i$, inserting

$$-\pi_{\mathcal{P},E_0}^0 q_{E_0} + \sum_{i=0}^{m-2} \left[ \left( \pi_{\mathcal{P},E_i}^0 q_{E_i} - A_{q,F_{i+1}} \right) - \left( \pi_{\mathcal{P},E_{i+1}}^0 q_{E_{i+1}} - A_{q,F_{i+1}} \right) \right] + \pi_{\mathcal{P},E_{m-1}}^0 q_{E_{m-1}} = 0$$

into $|A_{q,F} - A_{q,F'}|$, using triangle inequalities and the fact that $h_{F_i} \approx h_{F_{i+1}}$ and $|F_i| \approx |F_{i+1}|$ for all $i = 0, \dots, m-1$ by mesh regularity, and recalling the definition (4.21) of $\|\cdot\|_{\text{curl},F}$, (5.13) is a consequence of

$$\forall F \in \mathcal{F}_T, \quad h_F^{-1} |F| |A_{q,F} - \pi_{\mathcal{P},E}^0 q_E|^2 \lesssim h_F \|\underline{\mathbf{G}}_F^k \underline{q}_F\|_{\text{curl},F}^2 \quad \forall E \in \mathcal{E}_F. \quad (5.14)$$

To prove this relation, we write

$$\begin{aligned} \|A_{q,F} - \pi_{\mathcal{P},E}^0 q_E\|_{L^2(E)}^2 &\le \|A_{q,F} - q_E\|_{L^2(E)}^2 \\ &\lesssim h_F^{-1} \|A_{q,F} - \gamma_F^{k+1} \underline{q}_F\|_{L^2(F)}^2 + \|\gamma_F^{k+1} \underline{q}_F - q_E\|_{L^2(E)}^2 \\ &\lesssim h_F \|\underline{\mathbf{G}}_F^k \underline{q}_F\|_{\text{curl},F}^2 \end{aligned}$$

where the first inequality comes from the $L^2$-boundedness of $\pi_{\mathcal{P},E}^0$, the second inequality is obtained introducing $\gamma_F^{k+1} \underline{q}_F$ and using a triangle inequality together with a discrete trace inequality, while (5.5) together with the same arguments that lead to (5.11) yield the conclusion. The relation (5.14) follows noticing that $|F| \approx h_F |E|$, so that $|F| |A_{q,F} - \pi_{\mathcal{P},E}^0 q_E|^2 \approx h_F \|A_{q,F} - \pi_{\mathcal{P},E}^0 q_E\|_{L^2(E)}^2$. This concludes the proof of (5.13), hence of (5.12).

Finally, for $\mathfrak{Z}_3$, we apply the definition (4.1) of $P_{\text{grad},T}^{k+1}(\underline{q}_T - \underline{I}_{\text{grad},T}^k A_{q,\partial T})$ with $v_T \in \mathcal{R}^{c,k+2}(T)$ such that

$$\operatorname{div} v_T = P_{\text{grad},T}^{k+1} (\underline{q}_T - \underline{I}_{\text{grad},T}^k A_{q,\partial T})$$

and $\|v_T\|_{L^2(T)} \lesssim h_T \|P_{\text{grad},T}^{k+1}(\underline{q}_T - \underline{I}_{\text{grad},T}^k A_{q,\partial T})\|_{L^2(T)}$, see Lemma 9. Using the consistency properties (4.2) of $P_{\text{grad},T}^{k+1}$, (3.16) of $\mathbf{G}_T^k$ and (3.14) of $\gamma_F^{k+1}$, and a discrete trace inequality, this gives

$$\begin{aligned} \|P_{\text{grad},T}^{k+1} \underline{q}_T - A_{q,\partial T}\|_{L^2(T)} &\lesssim h_T \|\mathbf{G}_T^k \underline{q}_T\|_{L^2(T)} \\ &\quad + h_T \sum_{F \in \mathcal{F}_T} h_F^{-\frac{1}{2}} \|\gamma_F^{k+1} \underline{q}_F - A_{q,\partial T}\|_{L^2(F)} \\ &\lesssim h_T \|\underline{\mathbf{G}}_T^k \underline{q}_T\|_{\text{curl},T} + h_T (\mathfrak{Z}_1^{\frac{1}{2}} + \mathfrak{Z}_2^{\frac{1}{2}}), \end{aligned} \quad (5.15)$$
follows if we prove that, for all $F, F' \in \mathcal{F}_T$,

$$h_F^{-1} \|A_{q,F} - A_{q,F'}\|_{L^2(F)}^2 = h_F^{-1} |F| |A_{q,F} - A_{q,F'}|^2 \lesssim \|\underline{\mathbf{G}}_T^k \underline{q}_T\|_{\text{curl},T}^2. \quad (5.13)$$

Creating a sequence ($F = F_0, F_1, \dots, F_m = F'$) of faces in $\mathcal{F}_T$ such that, for all $i = 0, \dots, m-1$, the faces $F_i, F_{i+1}$ share an edge $E_i$, inserting

$$-\pi_{\mathcal{P},E_0}^0 q_{E_0} + \sum_{i=0}^{m-2} \left[ \left( \pi_{\mathcal{P},E_i}^0 q_{E_i} - A_{q,F_{i+1}} \right) - \left( \pi_{\mathcal{P},E_{i+1}}^0 q_{E_{i+1}} - A_{q,F_{i+1}} \right) \right] + \pi_{\mathcal{P},E_{m-1}}^0 q_{E_{m-1}} = 0$$

into $|A_{q,F} - A_{q,F'}|$, using triangle inequalities and the fact that $h_{F_i} \approx h_{F_{i+1}}$ and $|F_i| \approx |F_{i+1}|$ for all $i = 0, \dots, m-1$ by mesh regularity, and recalling the definition (4.21) of $\|\cdot\|_{\text{curl},F}$, (5.13) is a consequence of

$$\forall F \in \mathcal{F}_T, \quad h_F^{-1} |F| |A_{q,F} - \pi_{\mathcal{P},E}^0 q_E|^2 \lesssim h_F \|\underline{\mathbf{G}}_F^k \underline{q}_F\|_{\text{curl},F}^2 \quad \forall E \in \mathcal{E}_F. \quad (5.14)$$

To prove this relation, we write

$$\begin{aligned} \|A_{q,F} - \pi_{\mathcal{P},E}^0 q_E\|_{L^2(E)}^2 &\le \|A_{q,F} - q_E\|_{L^2(E)}^2 \\ &\lesssim h_F^{-1} \|A_{q,F} - \gamma_F^{k+1} \underline{q}_F\|_{L^2(F)}^2 + \|\gamma_F^{k+1} \underline{q}_F - q_E\|_{L^2(E)}^2 \\ &\lesssim h_F \|\underline{\mathbf{G}}_F^k \underline{q}_F\|_{\text{curl},F}^2 \end{aligned}$$

where the first inequality comes from the $L^2$-boundedness of $\pi_{\mathcal{P},E}^0$, the second inequality is obtained introducing $\gamma_F^{k+1} \underline{q}_F$ and using a triangle inequality together with a discrete trace inequality, while (5.5) together with the same arguments that lead to (5.11) yield the conclusion. The relation (5.14) follows noticing that $|F| \approx h_F|E|$, so that $|F| |A_{q,F} - \pi_{\mathcal{P},E}^0 q_E|^2 \approx h_F \|A_{q,F} - \pi_{\mathcal{P},E}^0 q_E\|_{L^2(E)}^2$. This concludes the proof of (5.13), hence of (5.12).

Finally, for $\mathfrak{T}_3$, we apply the definition (4.1) of $P_{\text{grad},T}^{k+1}(\underline{q}_T - \underline{I}_{\text{grad},T}^k A_{q,\partial T})$ with $v_T \in \mathcal{R}^{c,k+2}(T)$ such that

$$\text{div } v_T = P_{\text{grad},T}^{k+1}(\underline{q}_T - \underline{I}_{\text{grad},T}^k A_{q,\partial T})$$

and $\|v_T\|_{L^2(T)} \lesssim h_T \|P_{\text{grad},T}^{k+1}(\underline{q}_T - \underline{I}_{\text{grad},T}^k A_{q,\partial T})\|_{L^2(T)}$, see Lemma 9. Using the consistency properties (4.2) of $P_{\text{grad},T}^{k+1}$, (3.16) of $\mathbf{G}_T^k$ and (3.14) of $\gamma_F^{k+1}$, and a discrete trace inequality, this gives

$$\begin{aligned} \|P_{\text{grad},T}^{k+1} \underline{q}_T - A_{q,\partial T}\|_{L^2(T)} &\lesssim h_T \|\mathbf{G}_T^k \underline{q}_T\|_{L^2(T)} \\ &\quad + h_T \sum_{F \in \mathcal{F}_T} h_F^{-\frac{1}{2}} \|\gamma_F^{k+1} \underline{q}_F - A_{q,\partial T}\|_{L^2(F)} \\ &\lesssim h_T \|\underline{\mathbf{G}}_T^k \underline{q}_T\|_{\text{curl},T} + h_T (\mathfrak{T}_1^{\frac{1}{2}} + \mathfrak{T}_2^{\frac{1}{2}}), \end{aligned} \quad (5.15)$$
where the second inequality follows from (4.33) and a triangle inequality to write

$$ \sum_{F \in \mathcal{F}_T} h_F^{-\frac{1}{2}} \| \gamma_F^{k+1} \underline{q}_F - A_{q, \partial T} \|_{L^2(F)} \le \sum_{F \in \mathcal{F}_T} h_F^{-\frac{1}{2}} \| \gamma_F^{k+1} \underline{q}_F - A_{q, F} \|_{L^2(F)} \\ + \sum_{F \in \mathcal{F}_T} h_F^{-\frac{1}{2}} \| A_{q, F} - A_{q, \partial T} \|_{L^2(F)}. $$

Using discrete trace inequalities and the estimates (5.11) and (5.12) on $\mathfrak{I}_1$ and $\mathfrak{I}_2$, (5.15) leads to

$$ \mathfrak{I}_3 \lesssim h_T^{-2} \| P_{\text{grad},T}^{k+1} \underline{q}_T - A_{q, \partial T} \|_{L^2(T)}^2 \lesssim \| \underline{G}_T^k \underline{q}_T \|_{\text{curl},T}^2. $$

Plugging this bound together with the estimates on $\mathfrak{I}_1$ and $\mathfrak{I}_2$ into (5.10) concludes the proof of the bound on the second term in the right-hand side of (5.6). To bound the first term in the left-hand side of (5.6), we proceed as for $\text{grad}_F \gamma_F^{k+1} \underline{q}_F$ in Step 1 of this proof, using an integration by parts in the definition (4.1) of $P_{\text{grad},T}^{k+1} \underline{q}_T$ and selecting the test function $v_T = \text{grad}_F P_{\text{grad},T}^{k+1} \underline{q}_T$ (see Remark 17). $\square$

We are now ready to prove the discrete Poincaré inequality for the gradient.

**Proof (Theorem 3)** By the orthogonality condition (5.1), we can apply the discrete Poincaré–Wirtinger inequality in Hybrid High-Order spaces [28, Theorem 6.5] (with $p = q = 2$) to the vector of element- and face-polynomials $((P_{\text{grad},T}^{k+1} \underline{q}_T)_{T \in \mathcal{T}_h}, (\gamma_F^{k+1} \underline{q}_T)_{F \in \mathcal{F}_h})$ to get

$$ \sum_{T \in \mathcal{T}_h} \| P_{\text{grad},T}^{k+1} \underline{q}_T \|_{L^2(T)}^2 \\ \lesssim \sum_{T \in \mathcal{T}_h} \left( \| \text{grad} \, P_{\text{grad},T}^{k+1} \underline{q}_T \|_{L^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F^{-1} \| P_{\text{grad},T}^{k+1} \underline{q}_T - \gamma_F^{k+1} \underline{q}_T \|_{L^2(F)}^2 \right) \quad (5.16) \\ \lesssim \| \underline{G}_h^k \underline{q}_h \|_{\text{curl},h}^2, $$

where the conclusion is a consequence of (5.6) followed by the definition (4.21) of the $\|\cdot\|_{\text{curl},h}$-norm.

Let $T \in \mathcal{T}_h$. By definition (4.17) of $s_{\text{grad},T}$ we have

$$ \begin{aligned} s_{\text{grad},T}(\underline{q}_T, \underline{q}_T) &= \sum_{F \in \mathcal{F}_T} h_F \| P_{\text{grad},T}^{k+1} \underline{q}_T - \gamma_F^{k+1} \underline{q}_T \|_{L^2(F)}^2 \\ &\quad + \sum_{E \in \mathcal{E}_T} h_E^2 \| P_{\text{grad},T}^{k+1} \underline{q}_T - q_E \|_{L^2(E)}^2 \\ &\lesssim \sum_{F \in \mathcal{F}_T} h_F \| P_{\text{grad},T}^{k+1} \underline{q}_T - \gamma_F^{k+1} \underline{q}_T \|_{L^2(F)}^2 \quad (5.17) \\ &\quad + \sum_{F \in \mathcal{F}_T} \sum_{E \in \mathcal{E}_F} h_E^2 \| \gamma_F^{k+1} \underline{q}_T - q_E \|_{L^2(E)}^2 \\ &\lesssim h_T^2 \| \underline{G}_T^k \underline{q}_T \|_{\text{curl},T}^2, \end{aligned} $$
where the second inequality follows from (4.33) and a triangle inequality to write

$$ \sum_{F \in \mathcal{F}_T} h_F^{-\frac{1}{2}} \| \gamma_F^{k+1} \underline{q}_F - A_{q, \partial T} \|_{L^2(F)} \le \sum_{F \in \mathcal{F}_T} h_F^{-\frac{1}{2}} \| \gamma_F^{k+1} \underline{q}_F - A_{q, F} \|_{L^2(F)} \\ + \sum_{F \in \mathcal{F}_T} h_F^{-\frac{1}{2}} \| A_{q, F} - A_{q, \partial T} \|_{L^2(F)}. $$

Using discrete trace inequalities and the estimates (5.11) and (5.12) on $\mathfrak{I}_1$ and $\mathfrak{I}_2$, (5.15) leads to

$$ \mathfrak{I}_3 \lesssim h_T^{-2} \| P_{\text{grad},T}^{k+1} \underline{q}_T - A_{q, \partial T} \|_{L^2(T)}^2 \lesssim \| \underline{G}_T^k \underline{q}_T \|_{\text{curl},T}^2. $$

Plugging this bound together with the estimates on $\mathfrak{I}_1$ and $\mathfrak{I}_2$ into (5.10) concludes the proof of the bound on the second term in the right-hand side of (5.6). To bound the first term in the left-hand side of (5.6), we proceed as for $\text{grad}_F \gamma_F^{k+1} \underline{q}_F$ in Step 1 of this proof, using an integration by parts in the definition (4.1) of $P_{\text{grad},T}^{k+1} \underline{q}_T$ and selecting the test function $v_T = \text{grad}_F P_{\text{grad},T}^{k+1} \underline{q}_T$ (see Remark 17). $\square$

We are now ready to prove the discrete Poincaré inequality for the gradient.

*Proof (Theorem 3)* By the orthogonality condition (5.1), we can apply the discrete Poincaré–Wirtinger inequality in Hybrid High-Order spaces [28, Theorem 6.5] (with $p = q = 2$) to the vector of element- and face-polynomials $((P_{\text{grad},T}^{k+1} \underline{q}_T)_{T \in \mathcal{T}_h}, (\gamma_F^{k+1} \underline{q}_T)_{F \in \mathcal{F}_h})$ to get

$$ \sum_{T \in \mathcal{T}_h} \| P_{\text{grad},T}^{k+1} \underline{q}_T \|_{L^2(T)}^2 \\ \lesssim \sum_{T \in \mathcal{T}_h} \left( \| \text{grad} \, P_{\text{grad},T}^{k+1} \underline{q}_T \|_{L^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F^{-1} \| P_{\text{grad},T}^{k+1} \underline{q}_T - \gamma_F^{k+1} \underline{q}_T \|_{L^2(F)}^2 \right) \quad (5.16) \\ \lesssim \| \underline{G}_h^k \underline{q}_h \|_{\text{curl},h}^2, $$

where the conclusion is a consequence of (5.6) followed by the definition (4.21) of the $\|\cdot\|_{\text{curl},h}$-norm.

Let $T \in \mathcal{T}_h$. By definition (4.17) of $s_{\text{grad},T}$ we have

$$ \begin{aligned} s_{\text{grad},T}(\underline{q}_T, \underline{q}_T) &= \sum_{F \in \mathcal{F}_T} h_F \| P_{\text{grad},T}^{k+1} \underline{q}_T - \gamma_F^{k+1} \underline{q}_T \|_{L^2(F)}^2 \\ &\quad + \sum_{E \in \mathcal{E}_T} h_E^2 \| P_{\text{grad},T}^{k+1} \underline{q}_T - q_E \|_{L^2(E)}^2 \\ &\lesssim \sum_{F \in \mathcal{F}_T} h_F \| P_{\text{grad},T}^{k+1} \underline{q}_T - \gamma_F^{k+1} \underline{q}_T \|_{L^2(F)}^2 \quad (5.17) \\ &\quad + \sum_{F \in \mathcal{F}_T} \sum_{E \in \mathcal{E}_F} h_E^2 \| \gamma_F^{k+1} \underline{q}_T - q_E \|_{L^2(E)}^2 \\ &\lesssim h_T^2 \| \underline{G}_T^k \underline{q}_T \|_{\text{curl},T}^2, \end{aligned} $$
where the first inequality follows writing $\sum_{E \in \mathcal{E}_T} \bullet = \frac{1}{2} \sum_{F \in \mathcal{F}_T} \sum_{E \in \mathcal{E}_F} \bullet$, introducing $\pm \gamma_F^{k+1} \underline{q}_F$ in the norms and using triangle and discrete trace inequalities, while the conclusion is obtained invoking (5.5), (5.6), $h_E \approx h_F \approx h_T$ and the definition of $\|\underline{G}_T^k q_T\|_{\text{curl},T}$.

Using $h_T \le 1$, summing (5.17) over $T \in \mathcal{T}_h$, and adding the resulting estimate to (5.16) we infer that $\|\underline{q}_h\|_{\text{grad},h} \le \|\underline{G}_h^k q_h\|_{\text{curl},h}$. The Poincaré inequality (5.2) then follows from the norm equivalence (4.25). □

# 6 Consistency results

## 6.1 Primal consistency

In this section we state consistency results for the discrete potentials, vector calculus operators, stabilisation bilinear forms, and discrete L²-products. Because of the nature of the interpolator on $X_{\text{curl},T}^k$ (which requires higher regularity of functions), we introduce the following notation: For $T \in \mathcal{T}_h$ and $v \in \mathbf{H}^{\max(k+1,2)}(T)$,

$$|v|_{\mathbf{H}^{(k+1,2)}(T)} := \begin{cases} |v|_{\mathbf{H}^1(T)} + h_T |v|_{\mathbf{H}^2(T)} & \text{if } k=0, \\ |v|_{\mathbf{H}^{k+1}(T)} & \text{if } k \ge 1. \end{cases} \quad (6.1)$$

The corresponding global broken seminorm $|\cdot|_{\mathbf{H}^{(k+1,2)}(\mathcal{T}_h)}$ is such that, for all $v \in \mathbf{H}^{(k+1,2)}(\mathcal{T}_h)$,

$$|v|_{\mathbf{H}^{(k+1,2)}(\mathcal{T}_h)} := \left( \sum_{T \in \mathcal{T}_h} |v|^2_{\mathbf{H}^{(k+1,2)}(T)} \right)^{1/2}.$$

The proofs of the following theorems are postponed to Section 6.3.

**Theorem 6 (Consistency of the potential reconstructions)** It holds, for all $T \in \mathcal{T}_h$,

$$\|P_{\text{grad},T}^{k+1}(\underline{I}_{\text{grad},T}^k q) - q\|_{L^2(T)} \le h_T^{k+2} |q|_{\mathbf{H}^{k+2}(T)} \quad \forall q \in \mathbf{H}^{k+2}(T), \quad (6.2)$$

$$\|P_{\text{curl},T}^k (\underline{I}_{\text{curl},T}^k v) - v\|_{L^2(T)} \le h_T^{k+1} |v|_{\mathbf{H}^{(k+1,2)}(T)} \quad \forall v \in \mathbf{H}^{\max(k+1,2)}(T), \quad (6.3)$$

$$\|P_{\text{div},T}^k (\underline{I}_{\text{div},T}^k w) - w\|_{L^2(T)} \le h_T^{k+1} |w|_{\mathbf{H}^{k+1}(T)} \quad \forall w \in \mathbf{H}^{k+1}(T). \quad (6.4)$$

**Theorem 7 (Primal consistency of the discrete vector calculus operators)** It holds, for all $T \in \mathcal{T}_h$,

$$\|\underline{G}_T^k (\underline{I}_{\text{grad},T}^k q) - \text{grad } q\|_{L^2(T)} \le h_T^{k+1} |q|_{\mathbf{H}^{k+2}(T)} \quad \forall q \in \mathbf{H}^{k+2}(T), \quad (6.5)$$

$$\|\underline{C}_T^k (\underline{I}_{\text{curl},T}^k v) - \text{curl } v\|_{L^2(T)} \le h_T^{k+1} |\text{curl } v|_{\mathbf{H}^{k+1}(T)} \quad \forall v \in \mathbf{H}^2(T) \text{ s.t. } \text{curl } v \in \mathbf{H}^{k+1}(T), \quad (6.6)$$

$$\|D_T^k (\underline{I}_{\text{div},T}^k w) - \text{div } w\|_{L^2(T)} \le h_T^{k+1} |\text{div } w|_{\mathbf{H}^{k+1}(T)} \quad \forall w \in \mathbf{H}^1(T) \text{ s.t. } \text{div } w \in \mathbf{H}^{k+1}(T). \quad (6.7)$$
where the first inequality follows writing $\sum_{E \in \mathcal{E}_T} \bullet = \frac{1}{2} \sum_{F \in \mathcal{F}_T} \sum_{E \in \mathcal{E}_F} \bullet$, introducing $\pm \gamma_F^{k+1} \underline{q}_F$ in the norms and using triangle and discrete trace inequalities, while the conclusion is obtained invoking (5.5), (5.6), $h_E \approx h_F \approx h_T$ and the definition of $\|\underline{G}_T^k q_T\|_{\text{curl},T}$.

Using $h_T \le 1$, summing (5.17) over $T \in \mathcal{T}_h$, and adding the resulting estimate to (5.16) we infer that $\|\underline{q}_h\|_{\text{grad},h} \le \|\underline{G}_h^k q_h\|_{\text{curl},h}$. The Poincaré inequality (5.2) then follows from the norm equivalence (4.25). $\square$

# 6 Consistency results

## 6.1 Primal consistency

In this section we state consistency results for the discrete potentials, vector calculus operators, stabilisation bilinear forms, and discrete L²-products. Because of the nature of the interpolator on $X_{\text{curl},T}^k$ (which requires higher regularity of functions), we introduce the following notation: For $T \in \mathcal{T}_h$ and $v \in \mathbf{H}^{\max(k+1,2)}(T)$,

$$|v|_{\mathbf{H}^{(k+1,2)}(T)} := \begin{cases} |v|_{\mathbf{H}^1(T)} + h_T |v|_{\mathbf{H}^2(T)} & \text{if } k=0, \\ |v|_{\mathbf{H}^{k+1}(T)} & \text{if } k \ge 1. \end{cases} \quad (6.1)$$

The corresponding global broken seminorm $|\cdot|_{\mathbf{H}^{(k+1,2)}(\mathcal{T}_h)}$ is such that, for all $v \in \mathbf{H}^{(k+1,2)}(\mathcal{T}_h)$,

$$|v|_{\mathbf{H}^{(k+1,2)}(\mathcal{T}_h)} := \left( \sum_{T \in \mathcal{T}_h} |v|^2_{\mathbf{H}^{(k+1,2)}(T)} \right)^{1/2}.$$

The proofs of the following theorems are postponed to Section 6.3.

**Theorem 6 (Consistency of the potential reconstructions) It holds, for all $T \in \mathcal{T}_h$,**

$$\|P_{\text{grad},T}^{k+1}(\underline{I}_{\text{grad},T}^k q) - q\|_{\text{L}^2(T)} \le h_T^{k+2} |q|_{\text{H}^{k+2}(T)} \quad \forall q \in \text{H}^{k+2}(T), \quad (6.2)$$

$$\|P_{\text{curl},T}^k (\underline{I}_{\text{curl},T}^k v) - v\|_{\text{L}^2(T)} \le h_T^{k+1} |v|_{\text{H}^{(k+1,2)}(T)} \quad \forall v \in \text{H}^{\max(k+1,2)}(T), \quad (6.3)$$

$$\|P_{\text{div},T}^k (\underline{I}_{\text{div},T}^k w) - w\|_{\text{L}^2(T)} \le h_T^{k+1} |w|_{\text{H}^{k+1}(T)} \quad \forall w \in \text{H}^{k+1}(T). \quad (6.4)$$

**Theorem 7 (Primal consistency of the discrete vector calculus operators) It holds, for all $T \in \mathcal{T}_h$,**

$$\|\underline{G}_T^k (\underline{I}_{\text{grad},T}^k q) - \text{grad } q\|_{\text{L}^2(T)} \le h_T^{k+1} |q|_{\text{H}^{k+2}(T)} \quad \forall q \in \text{H}^{k+2}(T), \quad (6.5)$$

$$\|\underline{C}_T^k (\underline{I}_{\text{curl},T}^k v) - \text{curl } v\|_{\text{L}^2(T)} \le h_T^{k+1} |\text{curl } v|_{\text{H}^{k+1}(T)} \quad \forall v \in \text{H}^2(T) \text{ s.t. } \text{curl } v \in \text{H}^{k+1}(T), \quad (6.6)$$

$$\|\underline{D}_T^k (\underline{I}_{\text{div},T}^k w) - \text{div } w\|_{\text{L}^2(T)} \le h_T^{k+1} |\text{div } w|_{\text{H}^{k+1}(T)} \quad \forall w \in \text{H}^1(T) \text{ s.t. } \text{div } w \in \text{H}^{k+1}(T). \quad (6.7)$$
**Theorem 8 (Consistency of stabilisation forms)** For all $T \in \mathcal{T}_h$, the stabilisation forms defined by (4.17)–(4.19) satisfy the following consistency properties:

$$s_{\text{grad},T}(\underline{I}_{\text{grad},T}^k q, \underline{I}_{\text{grad},T}^k q)^{\frac{1}{2}} \leq h_T^{k+2} |q|_{H^{k+2}(T)} \quad \forall q \in H^{k+2}(T), \quad (6.8)$$

$$s_{\text{curl},T}(\underline{I}_{\text{curl},T}^k \nu, \underline{I}_{\text{curl},T}^k \nu)^{\frac{1}{2}} \leq h_T^{k+1} |\nu|_{H^{(k+1,2)}(T)} \quad \forall \nu \in \mathbf{H}^{\max(k+1,2)}(T), \quad (6.9)$$

$$s_{\text{div},T}(\underline{I}_{\text{div},T}^k \mathbf{w}, \underline{I}_{\text{div},T}^k \mathbf{w})^{\frac{1}{2}} \leq h_T^{k+1} |\mathbf{w}|_{H^{k+1}(T)} \quad \forall \mathbf{w} \in \mathbf{H}^{k+1}(T). \quad (6.10)$$

The following corollary is a straightforward consequence of Theorems 6 and 8, and of the definitions (4.14)–(4.16) of the discrete L²-products. Its proof is therefore omitted.

**Corollary 2 (Consistency of discrete L²-products)** It holds, for all $T \in \mathcal{T}_h$,

$$\left| \int_T q P_{\text{grad},T}^{k+1} \underline{r}_T - (\underline{I}_{\text{grad},T}^k q, \underline{r}_T)_{\text{grad},T} \right| \lesssim h_T^{k+2} |q|_{H^{k+2}(T)} \|\underline{r}_T\|_{\text{grad},T} \\ \forall q \in H^{k+2}(T), \forall \underline{r}_T \in \underline{X}_{\text{grad},T}^k, \quad (6.11)$$

$$\left| \int_T \nu \cdot \mathbf{P}_{\text{curl},T}^k \underline{\zeta}_T - (\underline{I}_{\text{curl},T}^k \nu, \underline{\zeta}_T)_{\text{curl},T} \right| \lesssim h_T^{k+1} |\nu|_{H^{(k+1,2)}(T)} \|\underline{\zeta}_T\|_{\text{curl},T} \\ \forall \nu \in \mathbf{H}^{\max(k+1,2)}(T), \forall \underline{\zeta}_T \in \underline{X}_{\text{curl},T}^k, \quad (6.12)$$

$$\left| \int_T \mathbf{w} \cdot \mathbf{P}_{\text{div},T}^k \underline{\xi}_T - (\underline{I}_{\text{div},T}^k \mathbf{w}, \underline{\xi}_T)_{\text{div},T} \right| \lesssim h_T^{k+1} |\mathbf{w}|_{H^{k+1}(T)} \|\underline{\xi}_T\|_{\text{div},T} \\ \forall \mathbf{w} \in \mathbf{H}^{k+1}(T), \forall \underline{\xi}_T \in \underline{X}_{\text{div},T}^k. \quad (6.13)$$

## 6.2 Adjoint consistency

Whenever a (formal) integration by parts is used to write the weak formulation of a PDE problem underpinning its discretisation, a form of adjoint consistency is required in the convergence analysis. We state here the adjoint consistency of the operators in the DDR sequence (3.37). Since this sequence does not incorporate boundary conditions, the corresponding adjoint consistency will be based on essential (homogeneous) boundary conditions. The regularity requirements will be expressed in terms of the broken Sobolev spaces and norms such that, for any $\ell \ge 1$,

$$H^\ell(\mathcal{T}_h) := \{g \in L^2(\Omega) : g|_T \in H^\ell(T) \text{ for all } T \in \mathcal{T}_h\}$$

$$\text{and } |g|_{H^\ell(\mathcal{T}_h)} := \left( \sum_{T \in \mathcal{T}_h} |g|_T |_{H^\ell(T)}^2 \right)^{\frac{1}{2}}.$$

The corresponding seminorms for vector-valued functions are denoted using boldface letters, as usual. We denote in what follows by $H_0^1(\Omega)$, $\mathbf{H}_0(\text{div};\Omega)$,
**Theorem 8 (Consistency of stabilisation forms)** For all $T \in \mathcal{T}_h$, the stabilisation forms defined by (4.17)–(4.19) satisfy the following consistency properties:

$$s_{\text{grad},T}(\underline{I}_{\text{grad},T}^k q, \underline{I}_{\text{grad},T}^k q)^{\frac{1}{2}} \le h_T^{k+2} |q|_{\mathbf{H}^{k+2}(T)} \quad \forall q \in \mathbf{H}^{k+2}(T), \quad (6.8)$$

$$s_{\text{curl},T}(\underline{I}_{\text{curl},T}^k \nu, \underline{I}_{\text{curl},T}^k \nu)^{\frac{1}{2}} \le h_T^{k+1} |\nu|_{\mathbf{H}^{(k+1,2)}(T)} \quad \forall \nu \in \mathbf{H}^{\max(k+1,2)}(T), \quad (6.9)$$

$$s_{\text{div},T}(\underline{I}_{\text{div},T}^k w, \underline{I}_{\text{div},T}^k w)^{\frac{1}{2}} \le h_T^{k+1} |w|_{\mathbf{H}^{k+1}(T)} \quad \forall w \in \mathbf{H}^{k+1}(T). \quad (6.10)$$

The following corollary is a straightforward consequence of Theorems 6 and 8, and of the definitions (4.14)–(4.16) of the discrete L²-products. Its proof is therefore omitted.

**Corollary 2 (Consistency of discrete L²-products)** It holds, for all $T \in \mathcal{T}_h$,

$$\left| \int_T q P_{\text{grad},T}^{k+1} \underline{r}_T - (\underline{I}_{\text{grad},T}^k q, \underline{r}_T)_{\text{grad},T} \right| \le h_T^{k+2} |q|_{\mathbf{H}^{k+2}(T)} \|\underline{r}_T\|_{\text{grad},T} \\ \forall q \in \mathbf{H}^{k+2}(T), \forall \underline{r}_T \in \underline{X}_{\text{grad},T}^k, \quad (6.11)$$

$$\left| \int_T \nu \cdot \mathbf{P}_{\text{curl},T}^k \underline{\zeta}_T - (\underline{I}_{\text{curl},T}^k \nu, \underline{\zeta}_T)_{\text{curl},T} \right| \le h_T^{k+1} |\nu|_{\mathbf{H}^{(k+1,2)}(T)} \|\underline{\zeta}_T\|_{\text{curl},T} \\ \forall \nu \in \mathbf{H}^{\max(k+1,2)}(T), \forall \underline{\zeta}_T \in \underline{X}_{\text{curl},T}^k, \quad (6.12)$$

$$\left| \int_T w \cdot \mathbf{P}_{\text{div},T}^k \underline{\xi}_T - (\underline{I}_{\text{div},T}^k w, \underline{\xi}_T)_{\text{div},T} \right| \le h_T^{k+1} |w|_{\mathbf{H}^{k+1}(T)} \|\underline{\xi}_T\|_{\text{div},T} \\ \forall w \in \mathbf{H}^{k+1}(T), \forall \underline{\xi}_T \in \underline{X}_{\text{div},T}^k. \quad (6.13)$$

## 6.2 Adjoint consistency

Whenever a (formal) integration by parts is used to write the weak formulation of a PDE problem underpinning its discretisation, a form of adjoint consistency is required in the convergence analysis. We state here the adjoint consistency of the operators in the DDR sequence (3.37). Since this sequence does not incorporate boundary conditions, the corresponding adjoint consistency will be based on essential (homogeneous) boundary conditions. The regularity requirements will be expressed in terms of the broken Sobolev spaces and norms such that, for any $\ell \ge 1$,

$$H^\ell(\mathcal{T}_h) := \{g \in L^2(\Omega) : g|_T \in H^\ell(T) \text{ for all } T \in \mathcal{T}_h\}$$

$$\text{and } |g|_{H^\ell(\mathcal{T}_h)} := \left( \sum_{T \in \mathcal{T}_h} |g|_T |_{H^\ell(T)}^2 \right)^{\frac{1}{2}}.$$

The corresponding seminorms for vector-valued functions are denoted using boldface letters, as usual. We denote in what follows by $H_0^1(\Omega)$, $\mathbf{H}_0(\text{div}; \Omega)$,
and $\mathbf{H}_0(\text{curl}; \Omega)$ the subspaces of $\mathrm{H}^1(\Omega)$, $\mathbf{H}(\text{div}; \Omega)$, and $\mathbf{H}(\text{curl}; \Omega)$ spanned by functions whose trace, normal trace, and tangential trace vanish on the boundary $\partial\Omega$ of $\Omega$, respectively.

**Theorem 9 (Adjoint consistency for the gradient)** *Define the gradient adjoint consistency error $\tilde{\mathcal{E}}_{\text{grad},h} : (\mathbf{C}^0(\bar{\Omega}) \cap \mathbf{H}_0(\text{div}; \Omega)) \times \bar{\mathbf{X}}_{\text{grad},h}^k \to \mathbb{R}$ by: For all $(\mathbf{v}, \underline{q}_h) \in (\mathbf{C}^0(\bar{\Omega}) \cap \mathbf{H}_0(\text{div}; \Omega)) \times \bar{\mathbf{X}}_{\text{grad},h}^k$*

$$ \tilde{\mathcal{E}}_{\text{grad},h}(\mathbf{v}, \underline{q}_h) := \sum_{T \in \mathcal{T}_h} \left[ (\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}|_T, \underline{\mathbf{G}}_T^k \underline{q}_T)_{\text{curl},T} + \int_T \text{div} \, \mathbf{v} \, P_{\text{grad},T}^{k+1} \underline{q}_T \right]. $$

Then, it holds, for all $\mathbf{v} \in \mathbf{C}^0(\bar{\Omega}) \cap \mathbf{H}_0(\text{div}; \Omega)$ such that $\mathbf{v} \in \mathbf{H}^{\max(k+1,2)}(\mathcal{T}_h)$ and all $\underline{q}_h \in \bar{\mathbf{X}}_{\text{grad},h}^k$,

$$ |\tilde{\mathcal{E}}_{\text{grad},h}(\mathbf{v}, \underline{q}_h)| \le h^{k+1} |\mathbf{v}|_{\mathbf{H}^{(k+1,2)}(\mathcal{T}_h)} \|\underline{\mathbf{G}}_h^k \underline{q}_h\|_{\text{curl},h}. \quad (6.14) $$

*Proof* See Section 6.4.

**Theorem 10 (Adjoint consistency for the curl)** *Define the curl adjoint consistency error $\tilde{\mathcal{E}}_{\text{curl},h} : (\mathbf{C}^0(\bar{\Omega}) \cap \mathbf{H}_0(\text{curl}; \Omega)) \times \bar{\mathbf{X}}_{\text{curl},h}^k \to \mathbb{R}$ by: For all $(\mathbf{w}, \underline{\mathbf{v}}_h) \in (\mathbf{C}^0(\bar{\Omega}) \cap \mathbf{H}_0(\text{curl}; \Omega)) \times \bar{\mathbf{X}}_{\text{curl},h}^k$*

$$ \tilde{\mathcal{E}}_{\text{curl},h}(\mathbf{w}, \underline{\mathbf{v}}_h) := \sum_{T \in \mathcal{T}_h} \left[ (\underline{\mathbf{I}}_{\text{div},T}^k \mathbf{w}|_T, \underline{\mathbf{G}}_T^k \underline{\mathbf{v}}_T)_{\text{div},T} - \int_T \text{curl} \, \mathbf{w} \cdot \mathbf{P}_{\text{curl},T}^k \underline{\mathbf{v}}_T \right]. \quad (6.15) $$

Then, for all $\mathbf{w} \in \mathbf{C}^0(\bar{\Omega}) \cap \mathbf{H}_0(\text{curl}; \Omega)$ such that $\mathbf{w} \in \mathbf{H}^{k+2}(\mathcal{T}_h)$ and all $\underline{\mathbf{v}}_h \in \bar{\mathbf{X}}_{\text{curl},h}^k$,

$$ |\tilde{\mathcal{E}}_{\text{curl},h}(\mathbf{w}, \underline{\mathbf{v}}_h)| \le h^{k+1} \left( |\mathbf{w}|_{\mathbf{H}^{k+1}(\mathcal{T}_h)} + |\mathbf{w}|_{\mathbf{H}^{k+2}(\mathcal{T}_h)} \right) \left( \|\underline{\mathbf{v}}_h\|_{\text{curl},h} + \|\underline{\mathbf{C}}_h^k \underline{\mathbf{v}}_h\|_{\text{div},h} \right). \quad (6.16) $$

*Proof* See Section 6.5.

**Theorem 11 (Adjoint consistency for the divergence)** *Define the divergence adjoint consistency error $\tilde{\mathcal{E}}_{\text{div},h} : (\mathbf{C}^0(\bar{\Omega}) \cap \mathbf{H}_0^1(\Omega)) \times \bar{\mathbf{X}}_{\text{div},h}^k \to \mathbb{R}$ by: For all $(\mathbf{q}, \underline{\mathbf{v}}_h) \in (\mathbf{C}^0(\bar{\Omega}) \cap \mathbf{H}_0^1(\Omega)) \times \bar{\mathbf{X}}_{\text{div},h}^k$*

$$ \tilde{\mathcal{E}}_{\text{div},h}(\mathbf{q}, \underline{\mathbf{v}}_h) := \int_{\Omega} \pi_{\mathcal{P},h}^k \mathbf{q} \, D_h^k \underline{\mathbf{v}}_h + \sum_{T \in \mathcal{T}_h} \int_{\Omega} \text{grad} \, \mathbf{q} \cdot \mathbf{P}_{\text{div},T}^k \underline{\mathbf{v}}_T. \quad (6.17) $$

Then, for all $\mathbf{q} \in \mathbf{C}^0(\bar{\Omega}) \cap \mathbf{H}_0^1(\Omega)$ such that $\mathbf{q} \in \mathbf{H}^{k+2}(\mathcal{T}_h)$ and all $\underline{\mathbf{v}}_h \in \bar{\mathbf{X}}_{\text{div},h}^k$,

$$ |\tilde{\mathcal{E}}_{\text{div},h}(\mathbf{q}, \underline{\mathbf{v}}_h)| \le h^{k+1} |\mathbf{q}|_{\mathbf{H}^{k+2}(\mathcal{T}_h)} \|\underline{\mathbf{v}}_h\|_{\text{div},h}. \quad (6.18) $$

*Proof* See Section 6.6.
and $\mathbf{H}_0(\text{curl}; \Omega)$ the subspaces of $\mathrm{H}^1(\Omega)$, $\mathbf{H}(\text{div}; \Omega)$, and $\mathbf{H}(\text{curl}; \Omega)$ spanned by functions whose trace, normal trace, and tangential trace vanish on the boundary $\partial\Omega$ of $\Omega$, respectively.

**Theorem 9 (Adjoint consistency for the gradient)** Define the gradient adjoint consistency error $\tilde{\mathcal{E}}_{\text{grad},h} : (\mathbf{C}^0(\bar{\Omega}) \cap \mathbf{H}_0(\text{div}; \Omega)) \times \underline{\mathbf{X}}_{\text{grad},h}^k \to \mathbb{R}$ by: For all $(\mathbf{v}, \underline{q}_h) \in (\mathbf{C}^0(\bar{\Omega}) \cap \mathbf{H}_0(\text{div}; \Omega)) \times \underline{\mathbf{X}}_{\text{grad},h}^k$

$$ \tilde{\mathcal{E}}_{\text{grad},h}(\mathbf{v}, \underline{q}_h) := \sum_{T \in \mathcal{T}_h} \left[ (\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}|_T, \underline{\mathbf{G}}_T^k \underline{q}_T)_{\text{curl},T} + \int_T \text{div} \, \mathbf{v} \, P_{\text{grad},T}^{k+1} \underline{q}_T \right]. $$

Then, it holds, for all $\mathbf{v} \in \mathbf{C}^0(\bar{\Omega}) \cap \mathbf{H}_0(\text{div}; \Omega)$ such that $\mathbf{v} \in \mathbf{H}^{\max(k+1,2)}(\mathcal{T}_h)$ and all $\underline{q}_h \in \underline{\mathbf{X}}_{\text{grad},h}^k$,

$$ |\tilde{\mathcal{E}}_{\text{grad},h}(\mathbf{v}, \underline{q}_h)| \lesssim h^{k+1} |\mathbf{v}|_{\mathbf{H}^{(k+1,2)}(\mathcal{T}_h)} \|\underline{\mathbf{G}}_h^k \underline{q}_h\|_{\text{curl},h}. \quad (6.14) $$

*Proof* See Section 6.4.

**Theorem 10 (Adjoint consistency for the curl)** Define the curl adjoint consistency error $\tilde{\mathcal{E}}_{\text{curl},h} : (\mathbf{C}^0(\bar{\Omega}) \cap \mathbf{H}_0(\text{curl}; \Omega)) \times \underline{\mathbf{X}}_{\text{curl},h}^k \to \mathbb{R}$ by: For all $(\mathbf{w}, \underline{\mathbf{v}}_h) \in (\mathbf{C}^0(\bar{\Omega}) \cap \mathbf{H}_0(\text{curl}; \Omega)) \times \underline{\mathbf{X}}_{\text{curl},h}^k$,

$$ \tilde{\mathcal{E}}_{\text{curl},h}(\mathbf{w}, \underline{\mathbf{v}}_h) := \sum_{T \in \mathcal{T}_h} \left[ (\underline{\mathbf{I}}_{\text{div},T}^k \mathbf{w}|_T, \underline{\mathbf{G}}_T^k \underline{\mathbf{v}}_T)_{\text{div},T} - \int_T \text{curl} \, \mathbf{w} \cdot \mathbf{P}_{\text{curl},T}^k \underline{\mathbf{v}}_T \right]. \quad (6.15) $$

Then, for all $\mathbf{w} \in \mathbf{C}^0(\bar{\Omega}) \cap \mathbf{H}_0(\text{curl}; \Omega)$ such that $\mathbf{w} \in \mathbf{H}^{k+2}(\mathcal{T}_h)$ and all $\underline{\mathbf{v}}_h \in \underline{\mathbf{X}}_{\text{curl},h}^k$,

$$ |\tilde{\mathcal{E}}_{\text{curl},h}(\mathbf{w}, \underline{\mathbf{v}}_h)| \lesssim h^{k+1} \left( |\mathbf{w}|_{\mathbf{H}^{k+1}(\mathcal{T}_h)} + |\mathbf{w}|_{\mathbf{H}^{k+2}(\mathcal{T}_h)} \right) \left( \|\underline{\mathbf{v}}_h\|_{\text{curl},h} + \|\underline{\mathbf{C}}_h^k \underline{\mathbf{v}}_h\|_{\text{div},h} \right). \quad (6.16) $$

*Proof* See Section 6.5.

**Theorem 11 (Adjoint consistency for the divergence)** Define the divergence adjoint consistency error $\tilde{\mathcal{E}}_{\text{div},h} : (\mathbf{C}^0(\bar{\Omega}) \cap \mathbf{H}_0^1(\Omega)) \times \underline{\mathbf{X}}_{\text{div},h}^k \to \mathbb{R}$ by: For all $(\mathbf{q}, \underline{\mathbf{v}}_h) \in (\mathbf{C}^0(\bar{\Omega}) \cap \mathbf{H}_0^1(\Omega)) \times \underline{\mathbf{X}}_{\text{div},h}^k$,

$$ \tilde{\mathcal{E}}_{\text{div},h}(\mathbf{q}, \underline{\mathbf{v}}_h) := \int_{\Omega} \pi_{\mathcal{P},h}^k \mathbf{q} \, D_h^k \underline{\mathbf{v}}_h + \sum_{T \in \mathcal{T}_h} \int_{\Omega} \text{grad} \, \mathbf{q} \cdot \mathbf{P}_{\text{div},T}^k \underline{\mathbf{v}}_T. \quad (6.17) $$

Then, for all $\mathbf{q} \in \mathbf{C}^0(\bar{\Omega}) \cap \mathbf{H}_0^1(\Omega)$ such that $\mathbf{q} \in \mathbf{H}^{k+2}(\mathcal{T}_h)$ and all $\underline{\mathbf{v}}_h \in \underline{\mathbf{X}}_{\text{div},h}^k$,

$$ |\tilde{\mathcal{E}}_{\text{div},h}(\mathbf{q}, \underline{\mathbf{v}}_h)| \lesssim h^{k+1} |\mathbf{q}|_{\mathbf{H}^{k+2}(\mathcal{T}_h)} \|\underline{\mathbf{v}}_h\|_{\text{div},h}. \quad (6.18) $$

*Proof* See Section 6.6.
## 6.3 Proof of the primal consistency

*Proof (Theorem 6)* Let us start with (6.2). Since $H^2(T) \subset C^0(\bar{T})$, the mapping $P_{\text{grad},T}^{k+1} \circ I_{\text{grad},T}^k : H^2(T) \to \mathcal{P}^{k+1}(T)$ is well-defined and, owing to (4.2), it is a projector. Moreover, combining (4.26) and (4.22), it satisfies the $L^2(T)$-boundedness

$$ \| P_{\text{grad},T}^{k+1} (I_{\text{grad},T}^k q) \|_{L^2(T)} \lesssim \|q\|_{L^2(T)} + h_T |q|_{H^1(T)} + h_T^2 |q|_{H^2(T)} \quad \forall q \in H^2(T). $$

The approximation property (6.2) is thus a direct consequence of [28, Lemma 1.43]. The proofs of (6.3) (for $k \ge 1$) and (6.4) are similar, using the fact that the considered operators are projectors onto $\mathcal{P}^k(T)$ (see (4.7) and (4.12)) and invoking Proposition 6 and Lemma 6 to establish their $L^2$-boundedness. In the case $k=0$, since $P_{\text{curl},T}^0 \circ I_{\text{curl},T}^0$ requires the $H^2$-regularity of its argument, with $2 > k+1$, (6.3) cannot be deduced directly from [28, Lemma 1.43]. However, using the bounds (4.23) and (4.27) a direct proof can be done by introducing $\pi_{\mathcal{P},T}^0 \nu = P_{\text{curl},T}^0 (I_{\text{curl},T}^0 \pi_{\mathcal{P},T}^0 \nu)$:

$$ \begin{aligned} & \| P_{\text{curl},T}^0 (I_{\text{curl},T}^0 \nu) - \nu \|_{L^2(T)} \\ & \le \| P_{\text{curl},T}^0 [I_{\text{curl},T}^0 (\nu - \pi_{\mathcal{P},T}^0 \nu)] \|_{L^2(T)} + \| \pi_{\mathcal{P},T}^0 \nu - \nu \|_{L^2(T)} \\ & \lesssim \| \nu - \pi_{\mathcal{P},T}^0 \nu \|_{L^2(T)} + h_T |\nu - \pi_{\mathcal{P},T}^0 \nu|_{H^1(T)} + h_T^2 |\nu - \pi_{\mathcal{P},T}^0 \nu|_{H^2(T)}, \end{aligned} $$

and (6.3) follows using the approximation properties of $\pi_{\mathcal{P},T}^0$, the fact that the $H^1(T)$- and $H^2(T)$-seminorms of $\pi_{\mathcal{P},T}^0 \nu$ vanish, and the definition (6.1) of $|\cdot|_{H^{(k+1,2)}(T)}$. $\square$

*Proof (Theorem 7)* Let us prove (6.5). For any $q_T \in X_{\text{grad},T}^k$, taking $w_T = G_T^k q_T$ in (3.11) and using Cauchy-Schwarz inequalities along with discrete inverse and trace inequalities, it is inferred, after simplification,

$$ \| G_T^k q_T \|_{L^2(T)} \lesssim h_T^{-1} \| q_T \|_{L^2(T)} + \sum_{F \in \mathcal{F}_T} h_F^{-1/2} \| \gamma_F^{k+1} q_F \|_{L^2(F)} \lesssim h_T^{-1} \| q_T \|_{\text{grad},T}, $$

where the conclusion follows from the estimate on $\gamma_F^{k+1} q_F$ in (4.22) and from the definition of $\|\cdot\|_{\text{grad},T}$. As a result, for any $r \in H^2(T)$, making $q_T = I_{\text{grad},T}^k r$ and invoking (4.26), we infer

$$ \| G_T^k (I_{\text{grad},T}^k r) \|_{L^2(T)} \lesssim h_T^{-1} \|r\|_{L^2(T)} + |r|_{H^1(T)} + h_T |r|_{H^2(T)}. \quad (6.19) $$

Letting now $q \in H^{k+2}(T)$, we use the polynomial consistency (3.16) of $G_T^k$ followed by a triangle inequality to write

$$ \begin{aligned} & \| G_T^k (I_{\text{grad},T}^k q) - \text{grad} q \|_{L^2(T)} \\ & \le \| G_T^k [I_{\text{grad},T}^k (q - \pi_{\mathcal{P},T}^{k+1} q)] \|_{L^2(T)} + \| \text{grad} (\pi_{\mathcal{P},T}^{k+1} q - q) \|_{L^2(T)} \end{aligned} $$
## 6.3 Proof of the primal consistency

*Proof (Theorem 6)* Let us start with (6.2). Since $H^2(T) \subset C^0(\bar{T})$, the mapping $P_{\text{grad},T}^{k+1} \circ I_{\text{grad},T}^k : H^2(T) \to \mathcal{P}^{k+1}(T)$ is well-defined and, owing to (4.2), it is a projector. Moreover, combining (4.26) and (4.22), it satisfies the $L^2(T)$-boundedness

$$ \| P_{\text{grad},T}^{k+1} (I_{\text{grad},T}^k q) \|_{L^2(T)} \lesssim \|q\|_{L^2(T)} + h_T |q|_{H^1(T)} + h_T^2 |q|_{H^2(T)} \quad \forall q \in H^2(T). $$

The approximation property (6.2) is thus a direct consequence of [28, Lemma 1.43]. The proofs of (6.3) (for $k \ge 1$) and (6.4) are similar, using the fact that the considered operators are projectors onto $\mathcal{P}^k(T)$ (see (4.7) and (4.12)) and invoking Proposition 6 and Lemma 6 to establish their $L^2$-boundedness. In the case $k=0$, since $P_{\text{curl},T}^0 \circ I_{\text{curl},T}^0$ requires the $H^2$-regularity of its argument, with $2 > k+1$, (6.3) cannot be deduced directly from [28, Lemma 1.43]. However, using the bounds (4.23) and (4.27) a direct proof can be done by introducing $\pi_{\mathcal{P},T}^0 \nu = P_{\text{curl},T}^0 (I_{\text{curl},T}^0 \pi_{\mathcal{P},T}^0 \nu)$:

$$ \begin{aligned} & \| P_{\text{curl},T}^0 (I_{\text{curl},T}^0 \nu) - \nu \|_{L^2(T)} \\ & \le \| P_{\text{curl},T}^0 [I_{\text{curl},T}^0 (\nu - \pi_{\mathcal{P},T}^0 \nu)] \|_{L^2(T)} + \| \pi_{\mathcal{P},T}^0 \nu - \nu \|_{L^2(T)} \\ & \lesssim \| \nu - \pi_{\mathcal{P},T}^0 \nu \|_{L^2(T)} + h_T |\nu - \pi_{\mathcal{P},T}^0 \nu|_{H^1(T)} + h_T^2 |\nu - \pi_{\mathcal{P},T}^0 \nu|_{H^2(T)}, \end{aligned} $$

and (6.3) follows using the approximation properties of $\pi_{\mathcal{P},T}^0$, the fact that the $H^1(T)$- and $H^2(T)$-seminorms of $\pi_{\mathcal{P},T}^0 \nu$ vanish, and the definition (6.1) of $|\cdot|_{H^{(k+1,2)}(T)}$. $\square$

*Proof (Theorem 7)* Let us prove (6.5). For any $q_T \in X_{\text{grad},T}^k$, taking $w_T = G_T^k q_T$ in (3.11) and using Cauchy-Schwarz inequalities along with discrete inverse and trace inequalities, it is inferred, after simplification,

$$ \| G_T^k q_T \|_{L^2(T)} \lesssim h_T^{-1} \| q_T \|_{L^2(T)} + \sum_{F \in \mathcal{F}_T} h_F^{-1/2} \| \gamma_F^{k+1} q_F \|_{L^2(F)} \lesssim h_T^{-1} \| q_T \|_{\text{grad},T}, $$

where the conclusion follows from the estimate on $\gamma_F^{k+1} q_F$ in (4.22) and from the definition of $\|\cdot\|_{\text{grad},T}$. As a result, for any $r \in H^2(T)$, making $q_T = I_{\text{grad},T}^k r$ and invoking (4.26), we infer

$$ \| G_T^k (I_{\text{grad},T}^k r) \|_{L^2(T)} \lesssim h_T^{-1} \|r\|_{L^2(T)} + |r|_{H^1(T)} + h_T |r|_{H^2(T)}. \quad (6.19) $$

Letting now $q \in H^{k+2}(T)$, we use the polynomial consistency (3.16) of $G_T^k$ followed by a triangle inequality to write

$$ \begin{aligned} & \| G_T^k (I_{\text{grad},T}^k q) - \text{grad} q \|_{L^2(T)} \\ & \le \| G_T^k [I_{\text{grad},T}^k (q - \pi_{\mathcal{P},T}^{k+1} q)] \|_{L^2(T)} + \| \text{grad} (\pi_{\mathcal{P},T}^{k+1} q - q) \|_{L^2(T)} \end{aligned} $$
and conclude using (6.19) with $r = q - \pi_{\mathcal{P},T}^{k+1}q$ for the first term in the right-hand side followed by the approximation properties of $\pi_{\mathcal{P},T}^{k+1}$ (see [28, Theorem 1.45]).

To prove (6.6), we notice that

$$ \mathbf{C}_T^k(\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}) = \mathbf{P}_{\text{div},T}^k[\underline{\mathbf{C}}_T^k(\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v})] = \mathbf{P}_{\text{div},T}^k[\underline{\mathbf{I}}_{\text{div},T}^k(\text{curl} \mathbf{v})] $$

owing to (4.30) along with the commutation property (3.39), and conclude using the approximation properties (6.4) with $\mathbf{w} = \text{curl} \mathbf{v}$.

Finally, (6.7) is a straightforward consequence of the commutation property $D_T^k(\underline{\mathbf{I}}_{\text{div},T}^k \mathbf{w}) = \pi_{\mathcal{P},T}^k(\text{div} \mathbf{w})$ stated in (3.40) together with [28, Theorem 1.45]. □

**Remark 23 (Alternative proof of (6.5))** When $q \in C^1(\bar{T})$ is such that $\text{grad} q \in \mathbf{H}^{\max(k+1,2)}(T)$, the proof of (6.5) can be done following similar arguments as for (6.6), i.e., we write

$$ \mathbf{G}_T^k(\underline{\mathbf{I}}_{\text{grad},T}^k q) = \mathbf{P}_{\text{curl},T}^k[\underline{\mathbf{G}}_T^k(\underline{\mathbf{I}}_{\text{grad},T}^k q)] = \mathbf{P}_{\text{curl},T}^k[\underline{\mathbf{I}}_{\text{curl},T}^k(\text{grad} q)] $$

using (4.29) followed by (3.38), and conclude using the approximation properties (6.3) with $\mathbf{v} = \text{grad} q$. This argument, however, requires additional regularity on $q$ with respect to the one used above.

**Proof (Theorem 8)** We only prove (6.9), the other consistency properties being established in a similar way. Let $\mathbf{v} \in \mathbf{H}^{\max(k+1,2)}(T)$. By the polynomial consistency (3.25) of $\gamma_{t,F}^k$ and (4.7) of $\mathbf{P}_{\text{curl},T}^k$, it is easily checked that, for all $z_T \in \mathcal{P}^k(T)$ and all $\underline{\mathbf{w}}_T \in \underline{\mathbf{X}}_{\text{curl},T}^k$, it holds $s_{\text{curl},T}(\underline{\mathbf{I}}_{\text{curl},T}^k z_T, \underline{\mathbf{w}}_T) = 0$. Applying this with $z_T = \pi_{\mathcal{P},T}^k \mathbf{v}$ we infer

$$ s_{\text{curl},T}(\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}, \underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}) = s_{\text{curl},T}(\underline{\mathbf{I}}_{\text{curl},T}^k (\mathbf{v} - \pi_{\mathcal{P},T}^k \mathbf{v}), \underline{\mathbf{I}}_{\text{curl},T}^k (\mathbf{v} - \pi_{\mathcal{P},T}^k \mathbf{v})) \\ \lesssim \|\underline{\mathbf{I}}_{\text{curl},T}^k (\mathbf{v} - \pi_{\mathcal{P},T}^k \mathbf{v})\|_{\text{curl},T}^2 $$

the conclusion following from the definition of $\|\cdot\|_{\text{curl},T}$ and the norm equivalence (4.25). Invoking then (4.27) we infer

$$ s_{\text{curl},T}(\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}, \underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v})^{\frac{1}{2}} \lesssim \|\mathbf{v} - \pi_{\mathcal{P},T}^k \mathbf{v}\|_{\mathbf{L}^2(T)} \\ + h_T |\mathbf{v} - \pi_{\mathcal{P},T}^k \mathbf{v}|_{\mathbf{H}^1(T)} + h_T^2 |\mathbf{v} - \pi_{\mathcal{P},T}^k \mathbf{v}|_{\mathbf{H}^2(T)}, $$

and the estimate (6.9) follows from the approximation properties of $\pi_{\mathcal{P},T}^k$, see [28, Theorem 1.45], and the definition (6.1) of $|\cdot|_{\mathbf{H}^{(k+1,2)}(T)}$, using in the case $k=0$ the same arguments as in the proof of Theorem 6. □
and conclude using (6.19) with $r = q - \pi_{\mathcal{P},T}^{k+1}q$ for the first term in the right-hand side followed by the approximation properties of $\pi_{\mathcal{P},T}^{k+1}$ (see [28, Theorem 1.45]).

To prove (6.6), we notice that

$$ \mathbf{C}_T^k(\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}) = \mathbf{P}_{\text{div},T}^k[\underline{\mathbf{C}}_T^k(\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v})] = \mathbf{P}_{\text{div},T}^k[\underline{\mathbf{I}}_{\text{div},T}^k(\text{curl} \mathbf{v})] $$

owing to (4.30) along with the commutation property (3.39), and conclude using the approximation properties (6.4) with $\mathbf{w} = \text{curl} \mathbf{v}$.

Finally, (6.7) is a straightforward consequence of the commutation property $D_T^k(\underline{\mathbf{I}}_{\text{div},T}^k \mathbf{w}) = \pi_{\mathcal{P},T}^k(\text{div} \mathbf{w})$ stated in (3.40) together with [28, Theorem 1.45]. □

**Remark 23 (Alternative proof of (6.5))** When $q \in C^1(\bar{T})$ is such that $\text{grad} q \in \mathbf{H}^{\max(k+1,2)}(T)$, the proof of (6.5) can be done following similar arguments as for (6.6), i.e., we write

$$ \mathbf{G}_T^k(\underline{\mathbf{I}}_{\text{grad},T}^k q) = \mathbf{P}_{\text{curl},T}^k[\underline{\mathbf{G}}_T^k(\underline{\mathbf{I}}_{\text{grad},T}^k q)] = \mathbf{P}_{\text{curl},T}^k[\underline{\mathbf{I}}_{\text{curl},T}^k(\text{grad} q)] $$

using (4.29) followed by (3.38), and conclude using the approximation properties (6.3) with $\mathbf{v} = \text{grad} q$. This argument, however, requires additional regularity on $q$ with respect to the one used above.

**Proof (Theorem 8)** We only prove (6.9), the other consistency properties being established in a similar way. Let $\mathbf{v} \in \mathbf{H}^{\max(k+1,2)}(T)$. By the polynomial consistency (3.25) of $\gamma_{t,F}^k$ and (4.7) of $\mathbf{P}_{\text{curl},T}^k$, it is easily checked that, for all $z_T \in \mathcal{P}^k(T)$ and all $\underline{\mathbf{w}}_T \in \underline{\mathbf{X}}_{\text{curl},T}^k$, it holds $s_{\text{curl},T}(\underline{\mathbf{I}}_{\text{curl},T}^k z_T, \underline{\mathbf{w}}_T) = 0$. Applying this with $z_T = \pi_{\mathcal{P},T}^k \mathbf{v}$ we infer

$$ s_{\text{curl},T}(\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}, \underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}) = s_{\text{curl},T}(\underline{\mathbf{I}}_{\text{curl},T}^k (\mathbf{v} - \pi_{\mathcal{P},T}^k \mathbf{v}), \underline{\mathbf{I}}_{\text{curl},T}^k (\mathbf{v} - \pi_{\mathcal{P},T}^k \mathbf{v})) \\ \leq \|\underline{\mathbf{I}}_{\text{curl},T}^k (\mathbf{v} - \pi_{\mathcal{P},T}^k \mathbf{v})\|_{\text{curl},T}^2 $$

the conclusion following from the definition of $\|\cdot\|_{\text{curl},T}$ and the norm equivalence (4.25). Invoking then (4.27) we infer

$$ s_{\text{curl},T}(\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}, \underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v})^{\frac{1}{2}} \leq \|\mathbf{v} - \pi_{\mathcal{P},T}^k \mathbf{v}\|_{\mathbf{L}^2(T)} \\ + h_T |\mathbf{v} - \pi_{\mathcal{P},T}^k \mathbf{v}|_{\mathbf{H}^1(T)} + h_T^2 |\mathbf{v} - \pi_{\mathcal{P},T}^k \mathbf{v}|_{\mathbf{H}^2(T)}, $$

and the estimate (6.9) follows from the approximation properties of $\pi_{\mathcal{P},T}^k$, see [28, Theorem 1.45], and the definition (6.1) of $|\cdot|_{\mathbf{H}^{(k+1,2)}(T)}$, using in the case $k=0$ the same arguments as in the proof of Theorem 6. □
## 6.4 Proof of the adjoint consistency for the gradient

*Proof (Theorem 9.)* It holds, by definition (4.15) of the local discrete L²-product in $\underline{\mathbf{X}}_{\text{curl},h}^k$ and (4.29),

$$ \tilde{\mathcal{E}}_{\text{grad},h}(\mathbf{v}, \underline{q}_h) = \sum_{T \in \mathcal{T}_h} \left[ \int_T \mathbf{P}_{\text{curl},T}^k (\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}) \cdot \mathbf{G}_T^k \underline{q}_T + \text{s}_{\text{curl},T} (\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}|_T, \underline{\mathbf{G}}_T^k \underline{q}_T) + \int_T \text{div} \, \mathbf{v} \, P_{\text{grad},T}^{k+1} \underline{q}_T \right]. \quad (6.20) $$

Using Remark 17, we have, for all $\mathbf{w}_T \in \mathcal{P}^k(T)$,

$$ \int_T P_{\text{grad},T}^{k+1} \underline{q}_T \, \text{div} \, \mathbf{w}_T + \int_T \mathbf{G}_T^k \underline{q}_T \cdot \mathbf{w}_T - \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \gamma_F^{k+1} \underline{q}_F (\mathbf{w}_T \cdot \mathbf{n}_F) = 0. $$

Subtracting this quantity from (6.20), we obtain

$$ \tilde{\mathcal{E}}_{\text{grad},h}(\mathbf{v}, \underline{q}_h) = \sum_{T \in \mathcal{T}_h} \left[ \int_T (\mathbf{P}_{\text{curl},T}^k (\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}) - \mathbf{w}_T) \cdot \mathbf{G}_T^k \underline{q}_T + \text{s}_{\text{curl},T} (\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}|_T, \underline{\mathbf{G}}_T^k \underline{q}_T) \right] \\ + \sum_{T \in \mathcal{T}_h} \left[ \int_T \text{div}(\mathbf{v} - \mathbf{w}_T) P_{\text{grad},T}^{k+1} \underline{q}_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F (\mathbf{w}_T - \mathbf{v}) \cdot \mathbf{n}_F \gamma_F^{k+1} \underline{q}_F \right], $$

where $\mathbf{v}$ is introduced into the boundary term by single-valuedness of the discrete trace, and using $\mathbf{v}|_F \cdot \mathbf{n}_F = 0$ whenever $F \subset \partial\Omega$. Integrating by parts the third term in the right-hand side of the above expression, we obtain

$$ \tilde{\mathcal{E}}_{\text{grad},h}(\mathbf{v}, \underline{q}_h) = \sum_{T \in \mathcal{T}_h} \left[ \int_T (\mathbf{P}_{\text{curl},T}^k (\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}) - \mathbf{w}_T) \cdot \mathbf{G}_T^k \underline{q}_T + \text{s}_{\text{curl},T} (\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}|_T, \underline{\mathbf{G}}_T^k \underline{q}_T) \right] \\ + \sum_{T \in \mathcal{T}_h} \left[ - \int_T (\mathbf{v} - \mathbf{w}_T) \cdot \text{grad} \, P_{\text{grad},T}^{k+1} \underline{q}_T \right. \\ \left. + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F (\mathbf{w}_T - \mathbf{v}) \cdot \mathbf{n}_F (\gamma_F^{k+1} \underline{q}_F - P_{\text{grad},T}^{k+1} \underline{q}_T) \right]. \quad (6.21) $$

We set $\mathbf{w}_T = \boldsymbol{\pi}_{\mathcal{P},T}^k \mathbf{v}$ and use (6.3) and the approximation properties of $\boldsymbol{\pi}_{\mathcal{P},T}^k$ stated in [28, Theorem 1.45] to see that

$$ \| \mathbf{P}_{\text{curl},T}^k (\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}) - \boldsymbol{\pi}_{\mathcal{P},T}^k \mathbf{v} \|_{\mathbf{L}^2(T)} + \| \mathbf{v} - \boldsymbol{\pi}_{\mathcal{P},T}^k \mathbf{v} \|_{\mathbf{L}^2(T)} \\ + \sum_{F \in \mathcal{F}_T} h_F^{\frac{1}{2}} \| \mathbf{v} - \boldsymbol{\pi}_{\mathcal{P},T}^k \mathbf{v} \|_{\mathbf{L}^2(F)} \lesssim h_T^{k+1} |\mathbf{v}|_{\mathbf{H}^{(k+1,2)}(T)}. $$

Using Cauchy-Schwarz inequalities on the integrals and on the stabilisation bilinear form in (6.21), the bound (4.33) together with the norm equivalence
6.4 Proof of the adjoint consistency for the gradient

Proof (Theorem 9.) It holds, by definition (4.15) of the local discrete L²-product in $\underline{\mathbf{X}}_{\text{curl},h}^k$ and (4.29),

$$ \tilde{\mathcal{E}}_{\text{grad},h}(\mathbf{v}, \underline{q}_h) = \sum_{T \in \mathcal{T}_h} \left[ \int_T \mathbf{P}_{\text{curl},T}^k (\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}) \cdot \mathbf{G}_T^k \underline{q}_T + \text{s}_{\text{curl},T} (\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}|_T, \underline{\mathbf{G}}_T^k \underline{q}_T) + \int_T \text{div} \, \mathbf{v} \, P_{\text{grad},T}^{k+1} \underline{q}_T \right]. \quad (6.20) $$

Using Remark 17, we have, for all $\mathbf{w}_T \in \mathcal{P}^k(T)$,

$$ \int_T P_{\text{grad},T}^{k+1} \underline{q}_T \, \text{div} \, \mathbf{w}_T + \int_T \mathbf{G}_T^k \underline{q}_T \cdot \mathbf{w}_T - \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \gamma_F^{k+1} \underline{q}_F (\mathbf{w}_T \cdot \mathbf{n}_F) = 0. $$

Subtracting this quantity from (6.20), we obtain

$$ \tilde{\mathcal{E}}_{\text{grad},h}(\mathbf{v}, \underline{q}_h) = \sum_{T \in \mathcal{T}_h} \left[ \int_T (\mathbf{P}_{\text{curl},T}^k (\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}) - \mathbf{w}_T) \cdot \mathbf{G}_T^k \underline{q}_T + \text{s}_{\text{curl},T} (\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}|_T, \underline{\mathbf{G}}_T^k \underline{q}_T) \right] \\ + \sum_{T \in \mathcal{T}_h} \left[ \int_T \text{div}(\mathbf{v} - \mathbf{w}_T) P_{\text{grad},T}^{k+1} \underline{q}_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F (\mathbf{w}_T - \mathbf{v}) \cdot \mathbf{n}_F \gamma_F^{k+1} \underline{q}_F \right], $$

where $\mathbf{v}$ is introduced into the boundary term by single-valuedness of the discrete trace, and using $\mathbf{v}|_F \cdot \mathbf{n}_F = 0$ whenever $F \subset \partial\Omega$. Integrating by parts the third term in the right-hand side of the above expression, we obtain

$$ \tilde{\mathcal{E}}_{\text{grad},h}(\mathbf{v}, \underline{q}_h) = \sum_{T \in \mathcal{T}_h} \left[ \int_T (\mathbf{P}_{\text{curl},T}^k (\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}) - \mathbf{w}_T) \cdot \mathbf{G}_T^k \underline{q}_T + \text{s}_{\text{curl},T} (\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}|_T, \underline{\mathbf{G}}_T^k \underline{q}_T) \right] \\ + \sum_{T \in \mathcal{T}_h} \left[ - \int_T (\mathbf{v} - \mathbf{w}_T) \cdot \text{grad} \, P_{\text{grad},T}^{k+1} \underline{q}_T \right. \\ \left. + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F (\mathbf{w}_T - \mathbf{v}) \cdot \mathbf{n}_F (\gamma_F^{k+1} \underline{q}_F - P_{\text{grad},T}^{k+1} \underline{q}_T) \right]. \quad (6.21) $$

We set $\mathbf{w}_T = \pi_{\mathcal{P},T}^k \mathbf{v}$ and use (6.3) and the approximation properties of $\pi_{\mathcal{P},T}^k$ stated in [28, Theorem 1.45] to see that

$$ \| \mathbf{P}_{\text{curl},T}^k (\underline{\mathbf{I}}_{\text{curl},T}^k \mathbf{v}) - \pi_{\mathcal{P},T}^k \mathbf{v} \|_{\mathbf{L}^2(T)} + \| \mathbf{v} - \pi_{\mathcal{P},T}^k \mathbf{v} \|_{\mathbf{L}^2(T)} \\ + \sum_{F \in \mathcal{F}_T} h_F^{\frac{1}{2}} \| \mathbf{v} - \pi_{\mathcal{P},T}^k \mathbf{v} \|_{\mathbf{L}^2(F)} \lesssim h_T^{k+1} |\mathbf{v}|_{\mathbf{H}^{(k+1,2)}(T)}. $$

Using Cauchy-Schwarz inequalities on the integrals and on the stabilisation bilinear form in (6.21), the bound (4.33) together with the norm equivalence
(4.25), and the consistency property (6.9) of the stabilisation term, we arrive at

$$
\begin{aligned}
|\tilde{\mathcal{E}}_{\text{grad},h}(\mathbf{v}, \underline{q}_h)| &\le \sum_{T \in \mathcal{T}_h} h_T^{k+1} |\mathbf{v}|_{\mathbf{H}^{(k+1,2)}(T)} \|\underline{\mathbf{G}}_T^k \underline{q}_T\|_{\text{curl},T} \\
&\quad + \sum_{T \in \mathcal{T}_h} h_T^{k+1} |\mathbf{v}|_{\mathbf{H}^{(k+1,2)}(T)} \|\text{grad } \mathbf{P}_{\text{grad},T}^{k+1} \underline{q}_T\|_{\mathbf{L}^2(T)} \\
&\quad + \sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} h_T^{k+1} |\mathbf{v}|_{\mathbf{H}^{(k+1,2)}(T)} h_F^{-\frac{1}{2}} \|\gamma_F^{k+1} \underline{q}_F - \mathbf{P}_{\text{grad},T}^{k+1} \underline{q}_T\|_{\mathbf{L}^2(F)}.
\end{aligned}
$$

The conclusion follows from the estimate (5.6), and Cauchy–Schwarz inequal-
ities on the sums. $\square$

6.5 Proof of the adjoint consistency for the curl

The proof of the adjoint consistency for the curl hinges on liftings defined as
solutions of local problems. For any $F \in \mathcal{F}_h$, the face lifting $\mathbf{R}_{\text{curl},F} : \underline{\mathbf{X}}_{\text{curl},F}^k \to$
$\mathbf{H}(\text{rot};F) \cap \mathbf{H}(\text{div};F)$ is such that, for all $\underline{\mathbf{v}}_F \in \underline{\mathbf{X}}_{\text{curl},F}^k$, $\mathbf{R}_{\text{curl},F} \underline{\mathbf{v}}_F = \boldsymbol{\phi}_{\underline{\mathbf{v}}_F} +$
$\text{grad}_F \psi_{\underline{\mathbf{v}}_F}$ with $\boldsymbol{\phi}_{\underline{\mathbf{v}}_F} \in \mathbf{H}(\text{rot};F) \cap \mathbf{H}(\text{div};F)$ such that

$$
\mathrm{rot}_F \boldsymbol{\phi}_{\underline{v}_F} = C_F^k \underline{v}_F \quad \text{in } F, \qquad (6.22\text{a})
$$

$$
\operatorname{div}_F \boldsymbol{\phi}_{\underline{v}_F} = 0 \quad \text{in } F, \qquad (6.22\text{b})
$$

$$
\boldsymbol{\phi}_{\underline{v}_F} \cdot \mathbf{t}_E = v_E \quad \text{on all } E \in \mathcal{E}_F, \qquad (6.22c)
$$

while $\psi_{\underline{v}_F} \in C_c^\infty(F)$ is such that

$$
- \int_F \psi_{\underline{v}_F} \operatorname{div}_F z_F = \int_F (\gamma_{t,F}^k \underline{v}_F - \boldsymbol{\phi}_{\underline{v}_F}) \cdot z_F \quad \forall z_F \in \mathcal{R}^{c,k+1}(F). \quad (6.23)
$$

Let now $T \in \mathcal{T}_h$. The *curl correction* $\delta_T : \underline{\mathbf{X}}_{\text{curl},T}^k \to \mathbf{H}(\text{curl};T) \cap \mathbf{H}(\text{div};T)$ is such that, for all $\underline{\mathbf{v}}_T \in \underline{\mathbf{X}}_{\text{curl},T}^k$,

$$
\operatorname{div} \delta_T \underline{v}_T = - \operatorname{div} \mathbf{C}_T^k \underline{v}_T \quad \text{in } T, \qquad (6.24\text{a})
$$

$$
\text{curl} \delta_T \underline{v}_T = \mathbf{0} \qquad \text{in } T, \tag{6.24b}
$$

$$
\delta_T \underline{v}_T \cdot \mathbf{n}_F = C_F^k \underline{v}_F - \mathbf{C}_T^k \underline{v}_T \cdot \mathbf{n}_F \quad \text{on all } F \in \mathcal{F}_T. \qquad (6.24c)
$$

The curl correction lifts the difference between the face curl $C_F^k \underline{v}_F$ and the
normal component of the element curl $\mathbf{C}_T^k \underline{v}_T$ as a function defined over $T$. Its
role is to ensure the well-posedness of the problem defining the *element lifting*
$\mathbf{R}_{\text{curl},T} : \underline{\mathbf{X}}_{\text{curl},T}^k \to \mathbf{H}(\text{curl};T) \cap \mathbf{H}(\text{div};T)$ such that, for all $\underline{\mathbf{v}}_T \in \underline{\mathbf{X}}_{\text{curl},T}^k$,

$$
\text{curl } \mathbf{R}_{\text{curl},T} \underline{\mathbf{v}}_T = \mathbf{C}_T^k \underline{\mathbf{v}}_T + \delta_T \underline{\mathbf{v}}_T \quad \text{in } T, \qquad (6.25\text{a})
$$

$$
\operatorname{div} \mathbf{R}_{\text{curl},T} \underline{\mathbf{v}}_T = 0 \qquad \text{in } T, \qquad (6.25\text{b})
$$

$$
(\mathbf{R}_{\text{curl},T} \underline{\mathbf{v}}_T)_{t,F} = \mathbf{R}_{\text{curl},F} \underline{\mathbf{v}}_F \quad \text{on all } F \in \mathcal{F}_T. \qquad (6.25c)
$$
(4.25), and the consistency property (6.9) of the stabilisation term, we arrive at

$$
\begin{aligned}
|\tilde{\mathcal{E}}_{\text{grad},h}(\mathbf{v}, \underline{q}_h)| & \le \sum_{T \in \mathcal{T}_h} h_T^{k+1} |\mathbf{v}|_{\mathbf{H}^{(k+1,2)}(T)} \|\underline{\mathbf{G}}_T^k \underline{q}_T\|_{\text{curl},T} \\
& \quad + \sum_{T \in \mathcal{T}_h} h_T^{k+1} |\mathbf{v}|_{\mathbf{H}^{(k+1,2)}(T)} \|\text{grad } \mathbf{P}_{\text{grad},T}^{k+1} \underline{q}_T\|_{\mathbf{L}^2(T)} \\
& \quad + \sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} h_T^{k+1} |\mathbf{v}|_{\mathbf{H}^{(k+1,2)}(T)} h_F^{-\frac{1}{2}} \|\gamma_F^{k+1} \underline{q}_F - \mathbf{P}_{\text{grad},T}^{k+1} \underline{q}_T\|_{\mathbf{L}^2(F)}.
\end{aligned}
$$

The conclusion follows from the estimate (5.6), and Cauchy–Schwarz inequal-
ities on the sums. $\square$

6.5 Proof of the adjoint consistency for the curl

The proof of the adjoint consistency for the curl hinges on liftings defined as
solutions of local problems. For any $F \in \mathcal{F}_h$, the face lifting $\mathbf{R}_{\text{curl},F} : \underline{\mathbf{X}}_{\text{curl},F}^k \to$
$\mathbf{H}(\text{rot}; F) \cap \mathbf{H}(\text{div}; F)$ is such that, for all $\underline{\mathbf{v}}_F \in \underline{\mathbf{X}}_{\text{curl},F}^k$, $\mathbf{R}_{\text{curl},F} \underline{\mathbf{v}}_F = \boldsymbol{\phi}_{\underline{\mathbf{v}}_F} +$
$\text{grad}_F \psi_{\underline{\mathbf{v}}_F}$ with $\boldsymbol{\phi}_{\underline{\mathbf{v}}_F} \in \mathbf{H}(\text{rot}; F) \cap \mathbf{H}(\text{div}; F)$ such that

$$
\mathrm{rot}_F \boldsymbol{\phi}_{\underline{\mathbf{v}}_F} = C_F^k \underline{\mathbf{v}}_F \quad \text{in } F, \qquad (6.22\text{a})
$$

$$
\operatorname{div}_F \boldsymbol{\phi}_{\underline{\mathbf{v}}_F} = 0 \quad \text{in } F, \qquad (6.22\text{b})
$$

$$
\boldsymbol{\phi}_{\underline{\mathbf{v}}_F} \cdot \mathbf{t}_E = v_E \quad \text{on all } E \in \mathcal{E}_F, \qquad (6.22c)
$$

while $\psi_{\underline{\mathbf{v}}_F} \in C_c^\infty(F)$ is such that

$$
- \int_F \psi_{\underline{\mathbf{v}}_F} \operatorname{div}_F z_F = \int_F (\gamma_{t,F}^k \underline{\mathbf{v}}_F - \boldsymbol{\phi}_{\underline{\mathbf{v}}_F}) \cdot z_F \quad \forall z_F \in \mathcal{R}^{c,k+1}(F). \quad (6.23)
$$

Let now $T \in \mathcal{T}_h$. The curl correction $\delta_T : \underline{\mathbf{X}}_{\text{curl},T}^k \to \mathbf{H}(\text{curl};T) \cap \mathbf{H}(\text{div};T)$ is such that, for all $\underline{\mathbf{v}}_T \in \underline{\mathbf{X}}_{\text{curl},T}^k$,

$$
\operatorname{div} \delta_T \underline{\mathbf{v}}_T = - \operatorname{div} \mathbf{C}_T^k \underline{\mathbf{v}}_T \quad \text{in } T, \qquad (6.24\text{a})
$$

$$
\text{curl} \delta_T \underline{\mathbf{v}}_T = \mathbf{0} \qquad \text{in } T, \qquad (6.24\text{b})
$$

$$
\delta_T \underline{\mathbf{v}}_T \cdot \mathbf{n}_F = C_F^k \underline{\mathbf{v}}_F - \mathbf{C}_T^k \underline{\mathbf{v}}_T \cdot \mathbf{n}_F \quad \text{on all } F \in \mathcal{F}_T. \qquad (6.24c)
$$

The curl correction lifts the difference between the face curl $C_F^k \underline{\mathbf{v}}_F$ and the
normal component of the element curl $\mathbf{C}_T^k \underline{\mathbf{v}}_T$ as a function defined over $T$. Its
role is to ensure the well-posedness of the problem defining the *element lifting*
$\mathbf{R}_{\text{curl},T}: \underline{\mathbf{X}}_{\text{curl},T}^k \to \mathbf{H}(\text{curl};T) \cap \mathbf{H}(\text{div};T)$ such that, for all $\underline{\mathbf{v}}_T \in \underline{\mathbf{X}}_{\text{curl},T}^k$,

$$
\mathbf{curl} \mathbf{R}_{\text{curl},T} \underline{\mathbf{v}}_T = \mathbf{C}_T^k \underline{\mathbf{v}}_T + \delta_T \underline{\mathbf{v}}_T \quad \text{in } T, \qquad (6.25\text{a})
$$

$$
\operatorname{div} \mathbf{R}_{\text{curl},T} \underline{\mathbf{v}}_T = 0 \qquad \text{in } T, \qquad (6.25\text{b})
$$

$$
(\mathbf{R}_{\text{curl},T} \underline{\mathbf{v}}_T)_{t,F} = \mathbf{R}_{\text{curl},F} \underline{\mathbf{v}}_F \quad \text{on all } F \in \mathcal{F}_T. \qquad (6.25c)
$$
In Appendix B we prove that these lifting operators are well-defined, and that they satisfy the following two key properties:

- Orthogonality of the face lifting: For all $F \in \mathcal{F}_h$,

$$ \int_F (\gamma_{\text{t},F}^k \underline{\nu}_F - \mathbf{R}_{\text{curl},F} \underline{\nu}_F) \cdot \underline{z}_F = 0 \quad \forall (\underline{\nu}_F, \underline{z}_F) \in \underline{\mathcal{X}}_{\text{curl},F}^k \times \mathcal{R} \mathcal{T}^{k+1}(F); \quad (6.26) $$

- Boundedness of the element lifting: For all $T \in \mathcal{T}_h$,

$$ \begin{gathered} \| \mathbf{R}_{\text{curl},T} \underline{\nu}_T \|_{\mathbf{L}^2(T)} + \| \text{curl} \mathbf{R}_{\text{curl},T} \underline{\nu}_T \|_{\mathbf{L}^2(T)} \lesssim \| \underline{\nu}_T \|_{\text{curl},T} + \| \underline{\mathcal{C}}_T^k \underline{\nu}_T \|_{\text{div},T} \\ \forall \underline{\nu}_T \in \underline{\mathcal{X}}_{\text{curl},T}^k. \end{gathered} \quad (6.27) $$

**Lemma 8 (Approximation properties of $\mathcal{N}^{k+1}(T)$ on polyhedral elements)** For all $T \in \mathcal{T}_h$ and all $\mathbf{w} \in \mathbf{H}^{k+2}(T)$, there exists $z_T \in \mathcal{N}^{k+1}(T)$ such that

$$ \| \mathbf{w} - z_T \|_{\mathbf{L}^2(T)} \lesssim h_T^{k+1} (\| \mathbf{w} \|_{\mathbf{H}^{k+1}(T)} + \| \mathbf{w} \|_{\mathbf{H}^{k+2}(T)}), \quad (6.28) $$

$$ \| \text{curl } \mathbf{w} - \text{curl } z_T \|_{\mathbf{L}^2(T)} \lesssim h_T^{k+1} \| \mathbf{w} \|_{\mathbf{H}^{k+2}(T)}. \quad (6.29) $$

*Proof* By the mesh regularity assumption, there is a simplex $S \subset T$ whose inradius is $\ge h_T$. Following the arguments in the proof of [28, Lemma 1.25], we infer the norm equivalence

$$ \|q\|_{\mathbf{L}^2(S)} \approx \|q\|_{\mathbf{L}^2(T)} \quad \forall q \in \mathcal{P}^{k+1}(T). \quad (6.30) $$

Let us take $z_T$ as the Nédélec interpolant in $\mathcal{N}^{k+1}(S)$ of $\mathbf{w}$; $z_T$ can be uniquely extended as an element of $\mathcal{N}^{k+1}(T)$. By the arguments in the proof of [40, Theorem 3.14 and Corollary 3.17], and since $S \subset T$, it holds

$$ \begin{gathered} \| \mathbf{w} - z_T \|_{\mathbf{L}^2(S)} \lesssim h_T^{k+1} (\| \mathbf{w} \|_{\mathbf{H}^{k+1}(T)} + \| \mathbf{w} \|_{\mathbf{H}^{k+2}(T)}), \quad (6.31) \\ \| \text{curl } \mathbf{w} - \text{curl } z_T \|_{\mathbf{L}^2(S)} \lesssim h_T^{k+1} \| \mathbf{w} \|_{\mathbf{H}^{k+2}(T)}. \end{gathered} $$

We then write, introducing $\pi_{\mathcal{P},T}^{k+1}\mathbf{w}$ and using triangle inequalities,

$$ \begin{align*} \| \mathbf{w} - z_T \|_{\mathbf{L}^2(T)} &\lesssim \| \mathbf{w} - \pi_{\mathcal{P},T}^{k+1} \mathbf{w} \|_{\mathbf{L}^2(T)} + \| \pi_{\mathcal{P},T}^{k+1} \mathbf{w} - z_T \|_{\mathbf{L}^2(T)} \\ &\lesssim h_T^{k+1} \| \mathbf{w} \|_{\mathbf{H}^{k+1}(T)} + \| \pi_{\mathcal{P},T}^{k+1} \mathbf{w} - z_T \|_{\mathbf{L}^2(S)} \\ &\lesssim h_T^{k+1} (\| \mathbf{w} \|_{\mathbf{H}^{k+1}(T)} + \| \mathbf{w} \|_{\mathbf{H}^{k+2}(T)}), \end{align*} $$

where we have used the approximation property of $\pi_{\mathcal{P},T}^{k+1}$ together with the norm equivalence (6.30) in the second line, and concluded by introducing $\mathbf{w}$ and invoking (6.31) to write

$$ \begin{align*} \| \pi_{\mathcal{P},T}^{k+1} \mathbf{w} - z_T \|_{\mathbf{L}^2(S)} &\lesssim \| \pi_{\mathcal{P},T}^{k+1} \mathbf{w} - \mathbf{w} \|_{\mathbf{L}^2(S)} + \| \mathbf{w} - z_T \|_{\mathbf{L}^2(S)} \\ &\lesssim h_T^{k+1} \| \mathbf{w} \|_{\mathbf{H}^{k+1}(T)} + h_T^{k+1} (\| \mathbf{w} \|_{\mathbf{H}^{k+1}(T)} + \| \mathbf{w} \|_{\mathbf{H}^{k+2}(T)}). \end{align*} $$

This concludes the proof of (6.28). The proof of (6.29) is done in a similar way, introducing $\text{curl}(\pi_{\mathcal{P},T}^{k+1}\mathbf{w})$ and using the approximation property $\| \text{curl } \mathbf{w} - \text{curl}(\pi_{\mathcal{P},T}^{k+1}\mathbf{w})\|_{\mathbf{L}^2(T)} \lesssim h_T^{k+1} \| \mathbf{w} \|_{\mathbf{H}^{k+2}(T)}$. $\square$
In Appendix B we prove that these lifting operators are well-defined, and that they satisfy the following two key properties:

- Orthogonality of the face lifting: For all $F \in \mathcal{F}_h$,

$$ \int_F (\gamma_{\text{t},F}^k \underline{v}_F - \mathbf{R}_{\text{curl},F} \underline{v}_F) \cdot \underline{z}_F = 0 \quad \forall (\underline{v}_F, \underline{z}_F) \in \underline{X}_{\text{curl},F}^k \times \mathcal{R} \mathcal{T}^{k+1}(F); \quad (6.26) $$

- Boundedness of the element lifting: For all $T \in \mathcal{T}_h$,

$$ \begin{gathered} \| \mathbf{R}_{\text{curl},T} \underline{v}_T \|_{\mathbf{L}^2(T)} + \| \text{curl} \mathbf{R}_{\text{curl},T} \underline{v}_T \|_{\mathbf{L}^2(T)} \lesssim \| \underline{v}_T \|_{\text{curl},T} + \| \underline{C}_T^k \underline{v}_T \|_{\text{div},T} \\ \forall \underline{v}_T \in \underline{X}_{\text{curl},T}^k. \end{gathered} \quad (6.27) $$

**Lemma 8 (Approximation properties of $\mathcal{N}^{k+1}(T)$ on polyhedral elements)** For all $T \in \mathcal{T}_h$ and all $\mathbf{w} \in \mathbf{H}^{k+2}(T)$, there exists $z_T \in \mathcal{N}^{k+1}(T)$ such that

$$ \| \mathbf{w} - z_T \|_{\mathbf{L}^2(T)} \lesssim h_T^{k+1} (|\mathbf{w}|_{\mathbf{H}^{k+1}(T)} + |\mathbf{w}|_{\mathbf{H}^{k+2}(T)}), \quad (6.28) $$

$$ \| \text{curl} \mathbf{w} - \text{curl} z_T \|_{\mathbf{L}^2(T)} \lesssim h_T^{k+1} |\mathbf{w}|_{\mathbf{H}^{k+2}(T)}. \quad (6.29) $$

*Proof* By the mesh regularity assumption, there is a simplex $S \subset T$ whose inradius is $\ge h_T$. Following the arguments in the proof of [28, Lemma 1.25], we infer the norm equivalence

$$ \|q\|_{\mathbf{L}^2(S)} \approx \|q\|_{\mathbf{L}^2(T)} \quad \forall q \in \mathcal{P}^{k+1}(T). \quad (6.30) $$

Let us take $z_T$ as the Nédélec interpolant in $\mathcal{N}^{k+1}(S)$ of $\mathbf{w}$; $z_T$ can be uniquely extended as an element of $\mathcal{N}^{k+1}(T)$. By the arguments in the proof of [40, Theorem 3.14 and Corollary 3.17], and since $S \subset T$, it holds

$$ \begin{gathered} \| \mathbf{w} - z_T \|_{\mathbf{L}^2(S)} \lesssim h_T^{k+1} (|\mathbf{w}|_{\mathbf{H}^{k+1}(T)} + |\mathbf{w}|_{\mathbf{H}^{k+2}(T)}), \quad (6.31) \\ \| \text{curl} \mathbf{w} - \text{curl} z_T \|_{\mathbf{L}^2(S)} \lesssim h_T^{k+1} |\mathbf{w}|_{\mathbf{H}^{k+2}(T)}. \end{gathered} $$

We then write, introducing $\pi_{\mathcal{P},T}^{k+1}\mathbf{w}$ and using triangle inequalities,

$$ \begin{align*} \| \mathbf{w} - z_T \|_{\mathbf{L}^2(T)} &\lesssim \| \mathbf{w} - \pi_{\mathcal{P},T}^{k+1} \mathbf{w} \|_{\mathbf{L}^2(T)} + \| \pi_{\mathcal{P},T}^{k+1} \mathbf{w} - z_T \|_{\mathbf{L}^2(T)} \\ &\lesssim h_T^{k+1} |\mathbf{w}|_{\mathbf{H}^{k+1}(T)} + \| \pi_{\mathcal{P},T}^{k+1} \mathbf{w} - z_T \|_{\mathbf{L}^2(S)} \\ &\lesssim h_T^{k+1} (|\mathbf{w}|_{\mathbf{H}^{k+1}(T)} + |\mathbf{w}|_{\mathbf{H}^{k+2}(T)}), \end{align*} $$

where we have used the approximation property of $\pi_{\mathcal{P},T}^{k+1}$ together with the norm equivalence (6.30) in the second line, and concluded by introducing $\mathbf{w}$ and invoking (6.31) to write

$$ \begin{align*} \| \pi_{\mathcal{P},T}^{k+1} \mathbf{w} - z_T \|_{\mathbf{L}^2(S)} &\lesssim \| \pi_{\mathcal{P},T}^{k+1} \mathbf{w} - \mathbf{w} \|_{\mathbf{L}^2(S)} + \| \mathbf{w} - z_T \|_{\mathbf{L}^2(S)} \\ &\lesssim h_T^{k+1} |\mathbf{w}|_{\mathbf{H}^{k+1}(T)} + h_T^{k+1} (|\mathbf{w}|_{\mathbf{H}^{k+1}(T)} + |\mathbf{w}|_{\mathbf{H}^{k+2}(T)}). \end{align*} $$

This concludes the proof of (6.28). The proof of (6.29) is done in a similar way, introducing $\text{curl}(\pi_{\mathcal{P},T}^{k+1}\mathbf{w})$ and using the approximation property $\| \text{curl} \mathbf{w} - \text{curl}(\pi_{\mathcal{P},T}^{k+1}\mathbf{w})\|_{\mathbf{L}^2(T)} \lesssim h_T^{k+1} |\mathbf{w}|_{\mathbf{H}^{k+2}(T)}$. $\square$
Proof (Theorem 10) For all $T \in \mathcal{T}_h$, select $z_T \in \mathcal{N}^{k+1}(T)$ given by Lemma 8. Using (4.16) to expand $(\cdot, \cdot)_{\text{div},h}$ together with (4.30), and recalling (4.6), we see that it holds, for all $\underline{v}_h \in \underline{\mathcal{X}}_{\text{curl},h}^k$,

$$
\begin{align}
\tilde{\mathcal{E}}_{\text{curl},h}(\mathbf{w}, \underline{\mathbf{v}}_h) &= \sum_{T \in \mathcal{T}_h} \int_T (\mathbf{P}_{\text{div},T}^k (\underline{\mathbf{I}}_{\text{div},T}^k \mathbf{w})_T - z_T) \cdot \mathbf{C}_T^k \underline{\mathbf{v}}_T \nonumber \\
&\quad + \sum_{T \in \mathcal{T}_h} s_{\text{div},T} (\underline{\mathbf{I}}_{\text{div},T}^k \mathbf{w})_T + \sum_{T \in \mathcal{T}_h} \int_T \mathbf{P}_{\text{curl},T}^k \mathbf{v}_T \cdot \mathbf{v}_T \nonumber \\
&\quad + \sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F (z_T \times \mathbf{n}_F) \cdot \mathbf{\gamma}_{T,F}^k \underline{\mathbf{v}}_F \nonumber \\
&=: \mathfrak{I}_1 + \mathfrak{I}_2 + \mathfrak{I}_3 + \mathfrak{I}_4. \tag{6.32}
\end{align}
$$

Using Cauchy–Schwarz and triangle inequalities, it is readily inferred for
the first term

$$
\begin{align}
|\mathfrak{I}_1| &\lesssim \left[ \sum_{T \in \mathcal{T}_h} \left( \| \mathbf{P}_{\text{div},T}^k (\underline{\mathbf{I}}_{\text{div},T}^k \mathbf{w}) - \mathbf{w} \|_{\text{L}^2(T)}^2 + \| \mathbf{w} - z_T \|_{\text{L}^2(T)}^2 \right) \right]^{\frac{1}{2}} \nonumber \\
&\quad \times \left( \sum_{T \in \mathcal{T}_h} \| \mathbf{C}_T^k \underline{\mathbf{v}}_T \|_{\text{L}^2(T)}^2 \right)^{\frac{1}{2}} \tag{6.33} \\
&\lesssim h^{k+1} \left( |\mathbf{w}|_{\text{H}^{k+1}(\mathcal{T}_h)} + |\mathbf{w}|_{\text{H}^{k+2}(\mathcal{T}_h)} \right) \| \underline{\mathbf{C}}_h^k \underline{\mathbf{v}}_h \|_{\text{div},h}, \nonumber
\end{align}
$$

where the conclusion follows using the approximation properties (6.4) and
(6.28) to bound the first factor, and (4.34) along with the norm equivalence
(4.25) to bound the second.

For $\mathfrak{I}_2$, combining the consistency property (6.10) of $s_{\text{div},T}$ with discrete Cauchy–Schwarz inequalities and the definition of the $\|\cdot\|_{\text{div},h}$-norm readily gives

$$
|\mathfrak{I}_2| \lesssim h^{k+1} |\mathbf{w}|_{\mathbf{H}^{k+1}(\mathcal{T}_h)} \|\underline{\mathbf{C}}_h^k \underline{\mathbf{v}}_h\|_{\text{div},h}. \quad (6.34)
$$

For $\mathfrak{I}_3$, Cauchy–Schwarz inequalities, the approximation property (6.29), and the definition of the norm $\|\cdot\|_{\text{curl},h}$ yield

$$
\begin{align}
|\mathfrak{I}_3| &\le \left( \sum_{T \in \mathcal{T}_h} \| \mathrm{curl}(z_T - \mathbf{w}) \|_{\mathrm{L}^2(T)}^2 \right)^{\frac{1}{2}} \left( \sum_{T \in \mathcal{T}_h} \| \mathbf{P}_{\mathrm{curl},T}^k \underline{\mathbf{v}}_T \|_{\mathrm{L}^2(T)}^2 \right)^{\frac{1}{2}} \tag{6.35} \\
&\lesssim h^{k+1} |\mathbf{w}|_{\mathrm{H}^{k+2}(\mathcal{T}_h)} \| \underline{\mathbf{v}}_h \|_{\mathrm{curl},h}. \notag
\end{align}
$$

Let us now consider the last term in the right-hand side of (6.32). Since
$(z_T)|_F \times \mathbf{n}_F \in \mathcal{R}\mathcal{T}^{k+1}(F)$ as a consequence of (A.5) with $\ell = k+1$, by (6.26) we
can replace $\boldsymbol{\gamma}_{\text{t},F}^k \underline{\boldsymbol{v}}_T$ by $\mathbf{R}_{\text{curl},F} \underline{\boldsymbol{v}}_F$ in the boundary integral. Using the fact that
both $\mathbf{R}_{\text{curl},F} \underline{\boldsymbol{v}}_F$ and the (rotated) tangential trace of $\mathbf{w}$ are continuous across
Proof (Theorem 10) For all $T \in \mathcal{T}_h$, select $z_T \in \mathcal{N}^{k+1}(T)$ given by Lemma 8. Using (4.16) to expand $(\cdot, \cdot)_{\text{div},h}$ together with (4.30), and recalling (4.6), we see that it holds, for all $\underline{v}_h \in \underline{\mathcal{X}}_{\text{curl},h}^k$,

$$
\begin{align}
\tilde{\mathcal{E}}_{\text{curl},h}(\mathbf{w}, \underline{\mathbf{v}}_h) &= \sum_{T \in \mathcal{T}_h} \int_T (\mathbf{P}_{\text{div},T}^k (\underline{\mathbf{I}}_{\text{div},T}^k \mathbf{w})_T - z_T) \cdot \mathbf{C}_T^k \underline{\mathbf{v}}_T \nonumber \\
&\quad + \sum_{T \in \mathcal{T}_h} s_{\text{div},T} (\underline{\mathbf{I}}_{\text{div},T}^k \mathbf{w})_T + \sum_{T \in \mathcal{T}_h} \int_T \mathbf{P}_{\text{curl},T}^k \mathbf{v}_T \cdot \mathbf{v}_T \nonumber \\
&\quad + \sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F (z_T \times \mathbf{n}_F) \cdot \mathbf{\gamma}_{T,F}^k \underline{\mathbf{v}}_F \nonumber \\
&=: \mathfrak{I}_1 + \mathfrak{I}_2 + \mathfrak{I}_3 + \mathfrak{I}_4. \tag{6.32}
\end{align}
$$

Using Cauchy–Schwarz and triangle inequalities, it is readily inferred for
the first term

$$
\begin{align}
|\mathfrak{I}_1| &\lesssim \left[ \sum_{T \in \mathcal{T}_h} \left( \| \mathbf{P}_{\text{div},T}^k (\underline{\mathbf{I}}_{\text{div},T}^k \mathbf{w}) - \mathbf{w} \|_{\text{L}^2(T)}^2 + \| \mathbf{w} - z_T \|_{\text{L}^2(T)}^2 \right) \right]^{\frac{1}{2}} \nonumber \\
&\quad \times \left( \sum_{T \in \mathcal{T}_h} \| \mathbf{C}_T^k \underline{\mathbf{v}}_T \|_{\text{L}^2(T)}^2 \right)^{\frac{1}{2}} \tag{6.33} \\
&\lesssim h^{k+1} \left( |\mathbf{w}|_{\text{H}^{k+1}(\mathcal{T}_h)} + |\mathbf{w}|_{\text{H}^{k+2}(\mathcal{T}_h)} \right) \| \underline{\mathbf{C}}_h^k \underline{\mathbf{v}}_h \|_{\text{div},h}, \nonumber
\end{align}
$$

where the conclusion follows using the approximation properties (6.4) and
(6.28) to bound the first factor, and (4.34) along with the norm equivalence
(4.25) to bound the second.

For $\mathfrak{I}_2$, combining the consistency property (6.10) of $s_{\text{div},T}$ with discrete Cauchy–Schwarz inequalities and the definition of the $\|\cdot\|_{\text{div},h}$-norm readily gives

$$
|\mathfrak{I}_2| \lesssim h^{k+1} |\mathbf{w}|_{\mathbf{H}^{k+1}(\mathcal{T}_h)} \|\underline{\mathbf{C}}_h^k \underline{\mathbf{v}}_h\|_{\text{div},h}. \quad (6.34)
$$

For $\mathfrak{I}_3$, Cauchy–Schwarz inequalities, the approximation property (6.29), and the definition of the norm $\|\cdot\|_{\text{curl},h}$ yield

$$
\begin{align}
|\mathfrak{I}_3| &\le \left( \sum_{T \in \mathcal{T}_h} \| \mathrm{curl}(z_T - \mathbf{w}) \|_{\mathrm{L}^2(T)}^2 \right)^{\frac{1}{2}} \left( \sum_{T \in \mathcal{T}_h} \| \mathbf{P}_{\mathrm{curl},T}^k \underline{\mathbf{v}}_T \|_{\mathrm{L}^2(T)}^2 \right)^{\frac{1}{2}} \tag{6.35} \\
&\lesssim h^{k+1} |\mathbf{w}|_{\mathrm{H}^{k+2}(\mathcal{T}_h)} \| \underline{\mathbf{v}}_h \|_{\mathrm{curl},h}. \notag
\end{align}
$$

Let us now consider the last term in the right-hand side of (6.32). Since
$(z_T)|_F \times \mathbf{n}_F \in \mathcal{R}\mathcal{T}^{k+1}(F)$ as a consequence of (A.5) with $\ell = k+1$, by (6.26) we
can replace $\boldsymbol{\gamma}_{T,F}^k \underline{\boldsymbol{v}}_T$ by $\mathbf{R}_{\text{curl},F} \underline{\boldsymbol{v}}_F$ in the boundary integral. Using the fact that
both $\mathbf{R}_{\text{curl},F} \underline{\boldsymbol{v}}_F$ and the (rotated) tangential trace of $\mathbf{w}$ are continuous across
interfaces, along with the fact that $\omega_{T_1F} + \omega_{T_2F} = 0$ for all $F \in \mathcal{F}_h$ between two elements $T_1, T_2$, and $w|_F \times n_F = 0$ for all $F \subset \partial\Omega$, we then have

$$ \begin{aligned} \mathfrak{I}_4 &= \sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F (z_T - w) \times n_F \cdot \mathbf{R}_{\text{curl},F} \underline{v}_F \\ &= \sum_{T \in \mathcal{T}_h} \left( \int_T (z_T - w) \cdot \text{curl} \mathbf{R}_{\text{curl},T} \underline{v}_T - \int_T \text{curl}(z_T - w) \cdot \mathbf{R}_{\text{curl},T} \underline{v}_T \right), \end{aligned} $$

where the conclusion follows recalling that $\mathbf{R}_{\text{curl},F} \underline{v}_F = (\mathbf{R}_{\text{curl},T} \underline{v}_T)_{t,F}$ for all $T \in \mathcal{T}_h$ and all $F \in \mathcal{F}_T$ (see (6.25c)), and integrating by parts. Using Cauchy-Schwarz inequalities, it is inferred

$$ |\mathfrak{I}_4| \le \left[ \sum_{T \in \mathcal{T}_h} \left( \|z_T - w\|_{\mathcal{L}^2(T)}^2 + \|\text{curl}(z_T - w)\|_{\mathcal{L}^2(T)}^2 \right) \right]^{\frac{1}{2}} \times \left[ \sum_{T \in \mathcal{T}_h} \left( \|\text{curl} \mathbf{R}_{\text{curl},T} \underline{v}_T\|_{\mathcal{L}^2(T)}^2 + \|\mathbf{R}_{\text{curl},T} \underline{v}_T\|_{\mathcal{L}^2(T)}^2 \right) \right]^{\frac{1}{2}}. $$

The approximation properties (6.28)–(6.29) of $z_T$ along with the boundedness (6.27) of $\mathbf{R}_{\text{curl},T} \underline{v}_T$ yield

$$ |\mathfrak{I}_4| \lesssim h^{k+1} \left( |w|_{\mathbf{H}^{k+1}(\mathcal{T}_h)} + |w|_{\mathbf{H}^{k+2}(\mathcal{T}_h)} \right) \left( \|\underline{v}_h\|_{\text{curl},h} + \|\underline{C}_h^k \underline{v}_h\|_{\text{div},h} \right). \quad (6.36) $$

Plugging (6.33)–(6.36) into (6.32), (6.16) follows. $\square$

## 6.6 Proof of the adjoint consistency for the divergence

*Proof (Theorem 11)* Combining the definition (6.17) of the adjoint consistency error for the divergence with (4.11) summed over $T \in \mathcal{T}_h$, we infer that it holds, for all $(q, \underline{v}_h)$ as in the theorem and all $q_h \in \mathcal{P}^{k+1}(\mathcal{T}_h)$ with $q_T := (q_h)|_T$ for all $T \in \mathcal{T}_h$,

$$ \tilde{\mathcal{E}}_{\text{div},h}(q, \underline{v}_h) = \sum_{T \in \mathcal{T}_h} \left[ \int_T (\pi_{\mathcal{P},T}^k q - q_T) D_T^k \underline{v}_T + \int_T \text{grad}(q - q_T) \cdot \mathbf{P}_{\text{div},T}^k \underline{v}_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F (q_T - q) \underline{v}_F \right], $$

where the cancellation of $\pi_{\mathcal{P},T}^k$ is justified by its definition along with $D_T^k \underline{v}_T \in \mathcal{P}^k(T)$, while the insertion of $q$ into the boundary integral is possible thanks to its single-valuedness at interfaces along with the fact that it vanishes on $\partial\Omega$.
interfaces, along with the fact that $\omega_{T_1F} + \omega_{T_2F} = 0$ for all $F \in \mathcal{F}_h$ between two elements $T_1, T_2$, and $w|_F \times n_F = 0$ for all $F \subset \partial\Omega$, we then have

$$ \begin{aligned} \mathfrak{I}_4 &= \sum_{T \in \mathcal{T}_h} \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F (z_T - w) \times n_F \cdot \mathbf{R}_{\text{curl},F} \underline{v}_F \\ &= \sum_{T \in \mathcal{T}_h} \left( \int_T (z_T - w) \cdot \text{curl} \mathbf{R}_{\text{curl},T} \underline{v}_T - \int_T \text{curl}(z_T - w) \cdot \mathbf{R}_{\text{curl},T} \underline{v}_T \right), \end{aligned} $$

where the conclusion follows recalling that $\mathbf{R}_{\text{curl},F} \underline{v}_F = (\mathbf{R}_{\text{curl},T} \underline{v}_T)_{t,F}$ for all $T \in \mathcal{T}_h$ and all $F \in \mathcal{F}_T$ (see (6.25c)), and integrating by parts. Using Cauchy-Schwarz inequalities, it is inferred

$$ |\mathfrak{I}_4| \le \left[ \sum_{T \in \mathcal{T}_h} \left( \|z_T - w\|_{L^2(T)}^2 + \|\text{curl}(z_T - w)\|_{L^2(T)}^2 \right) \right]^{\frac{1}{2}} \times \left[ \sum_{T \in \mathcal{T}_h} \left( \|\text{curl} \mathbf{R}_{\text{curl},T} \underline{v}_T\|_{L^2(T)}^2 + \|\mathbf{R}_{\text{curl},T} \underline{v}_T\|_{L^2(T)}^2 \right) \right]^{\frac{1}{2}}. $$

The approximation properties (6.28)–(6.29) of $z_T$ along with the boundedness (6.27) of $\mathbf{R}_{\text{curl},T} \underline{v}_T$ yield

$$ |\mathfrak{I}_4| \lesssim h^{k+1} \left( |w|_{H^{k+1}(\mathcal{T}_h)} + |w|_{H^{k+2}(\mathcal{T}_h)} \right) \left( \|\underline{v}_h\|_{\text{curl},h} + \|\underline{C}_h^k \underline{v}_h\|_{\text{div},h} \right). \quad (6.36) $$

Plugging (6.33)–(6.36) into (6.32), (6.16) follows. $\square$

## 6.6 Proof of the adjoint consistency for the divergence

*Proof (Theorem 11)* Combining the definition (6.17) of the adjoint consistency error for the divergence with (4.11) summed over $T \in \mathcal{T}_h$, we infer that it holds, for all $(q, \underline{v}_h)$ as in the theorem and all $q_h \in \mathcal{P}^{k+1}(\mathcal{T}_h)$ with $q_T := (q_h)|_T$ for all $T \in \mathcal{T}_h$,

$$ \tilde{\mathcal{E}}_{\text{div},h}(q, \underline{v}_h) = \sum_{T \in \mathcal{T}_h} \left[ \int_T (\pi_{\mathcal{P},T}^k q - q_T) D_T^k \underline{v}_T + \int_T \text{grad}(q - q_T) \cdot \mathbf{P}_{\text{div},T}^k \underline{v}_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F (q_T - q) \underline{v}_F \right], $$

where the cancellation of $\pi_{\mathcal{P},T}^k$ is justified by its definition along with $D_T^k \underline{v}_T \in \mathcal{P}^k(T)$, while the insertion of $q$ into the boundary integral is possible thanks to its single-valuedness at interfaces along with the fact that it vanishes on $\partial\Omega$.
Taking absolute values and using Cauchy–Schwarz inequalities in the right-hand side along with $h_F \approx h_T$ for all $T \in \mathcal{T}_h$ and all $F \in \mathcal{T}_T$, we infer

$$ \begin{aligned} |\tilde{\mathcal{E}}_{\text{div},h}(q, \underline{v}_h)| & \lesssim \left[ \sum_{T \in \mathcal{T}_h} (h_T^{-2} \|q - q_T\|_{\text{L}^2(T)}^2 + \|\text{grad}(q - q_T)\|_{\text{L}^2(T)}^2 + h_T^{-1} \|q_T - q\|_{\partial T}^2) \right]^{\frac{1}{2}} \\ & \quad \times \left[ \sum_{T \in \mathcal{T}_h} \left( h_T^2 \|D_{T}^k \underline{v}_T\|_{\text{L}^2(T)}^2 + \|\mathbf{P}_{\text{div},T}^k \underline{v}_T\|_{\text{L}^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F \|v_F\|_{\text{L}^2(F)}^2 \right) \right]^{\frac{1}{2}}. \end{aligned} \quad (6.37) $$

Taking $q_h$ such that $q_T = \pi_{\mathcal{P},T}^{k+1} q|_T$ for all $T \in \mathcal{T}_h$ and using the approximation properties of the L$^2$-orthogonal projector [28, Theorem 1.45], it is inferred that the first factor in the right-hand side of (6.37) is $\lesssim h^{k+1}|q|_{H^{k+2}(\mathcal{T}_h)}$. Moving to the second factor, we use, for all $T \in \mathcal{T}_h$, [32, Lemma 8] followed by the local seminorm equivalence (4.25) to write $h_T \|D_T^k \underline{v}_T\|_{\text{L}^2(T)} \lesssim \|\underline{v}_T\|_{\text{div},T} \lesssim \|\underline{v}_T\|_{\text{div},T}$. The same norm equivalence and the definition of the $\|\cdot\|_{\text{div},T}$-norm also yields $\|\mathbf{P}_{\text{div},T}^k \underline{v}_T\|_{\text{L}^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F \|v_F\|_{\text{L}^2(F)}^2 \lesssim \|\underline{v}_T\|_{\text{div},T}$. The second factor in the right-hand side of (6.37) is therefore $\lesssim \|\underline{v}_h\|_{\text{div},h}$, and the proof is complete. □

<figure><img src="image_2.png" /></figure>

## 7 Convergence analysis for a DDR discretisation of magnetostatics

We analyse in this section the DDR approximation of the following magnetostatics model, in which the unknowns are the magnetic field $H \in H(\text{curl}; \Omega)$ and the vector potential $A \in H(\text{div}; \Omega)$:

$$ \mu H - \text{curl } A = 0, \quad \text{curl } H = J, \quad \text{div } A = 0 \quad \begin{array}{ll} \text{in } \Omega, & (7.1) \\ \text{on } \partial \Omega. & \end{array} $$

The free current $J$ belongs to $\text{curl } H(\text{curl}; \Omega)$ and we assume, for the sake of simplicity, that the magnetic permeability $\mu$ is piecewise-constant on the considered meshes, with $\mu: \Omega \to [\mu_-, \mu_+]$ for some constant numbers $0 < \mu_- \le \mu_+$.

### 7.1 Scheme

As shown in [29], a scheme based on the discrete de Rham tools can be written by replacing, in the weak formulation of (7.1), the continuous L$^2$-products by discrete ones built on the local products. Denote by $\mu_T$ the constant value of $\mu$ over $T \in \mathcal{T}_h$ and define the bilinear forms $a_h: \underline{X}_{\text{curl},h}^k \times \underline{X}_{\text{curl},h}^k \to \mathbb{R}$, $b_h: \underline{X}_{\text{curl},h}^k \times \underline{X}_{\text{div},h}^k \to \mathbb{R}$, and $c_h: \underline{X}_{\text{div},h}^k \times \underline{X}_{\text{div},h}^k \to \mathbb{R}$ as follows: For all
Taking absolute values and using Cauchy–Schwarz inequalities in the right-hand side along with $h_F \approx h_T$ for all $T \in \mathcal{T}_h$ and all $F \in \mathcal{T}_T$, we infer

$$ \begin{aligned} |\tilde{\mathcal{E}}_{\text{div},h}(q, \underline{v}_h)| & \lesssim \left[ \sum_{T \in \mathcal{T}_h} (h_T^{-2} \|q - q_T\|_{\text{L}^2(T)}^2 + \|\text{grad}(q - q_T)\|_{\text{L}^2(T)}^2 + h_T^{-1} \|q_T - q\|_{\partial T}^2) \right]^{\frac{1}{2}} \\ & \times \left[ \sum_{T \in \mathcal{T}_h} \left( h_T^2 \|D_{T}^k \underline{v}_T\|_{\text{L}^2(T)}^2 + \|\mathbf{P}_{\text{div},T}^k \underline{v}_T\|_{\text{L}^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F \|v_F\|_{\text{L}^2(F)}^2 \right) \right]^{\frac{1}{2}}. \end{aligned} \quad (6.37) $$

Taking $q_h$ such that $q_T = \pi_{\mathcal{P},T}^{k+1} q|_T$ for all $T \in \mathcal{T}_h$ and using the approximation properties of the L$^2$-orthogonal projector [28, Theorem 1.45], it is inferred that the first factor in the right-hand side of (6.37) is $\lesssim h^{k+1}|q|_{H^{k+2}(\mathcal{T}_h)}$. Moving to the second factor, we use, for all $T \in \mathcal{T}_h$, [32, Lemma 8] followed by the local seminorm equivalence (4.25) to write $h_T \|D_T^k \underline{v}_T\|_{\text{L}^2(T)} \lesssim \|\underline{v}_T\|_{\text{div},T} \lesssim \|\underline{v}_T\|_{\text{div},T}$. The same norm equivalence and the definition of the $\|\cdot\|_{\text{div},T}$-norm also yields $\|\mathbf{P}_{\text{div},T}^k \underline{v}_T\|_{\text{L}^2(T)}^2 + \sum_{F \in \mathcal{F}_T} h_F \|v_F\|_{\text{L}^2(F)}^2 \lesssim \|\underline{v}_T\|_{\text{div},T}$. The second factor in the right-hand side of (6.37) is therefore $\lesssim \|\underline{v}_h\|_{\text{div},h}$, and the proof is complete. □

<figure><img src="image_3.png" /></figure>

## 7 Convergence analysis for a DDR discretisation of magnetostatics

We analyse in this section the DDR approximation of the following magneto-statics model, in which the unknowns are the magnetic field $H \in H(\text{curl}; \Omega)$ and the vector potential $A \in H(\text{div}; \Omega)$:

$$ \mu H - \text{curl } A = 0, \quad \text{curl } H = J, \quad \text{div } A = 0 \quad \begin{array}{ll} \text{in } \Omega, & (7.1) \\ \text{on } \partial \Omega. & \end{array} $$

The free current $J$ belongs to $\text{curl } H(\text{curl}; \Omega)$ and we assume, for the sake of simplicity, that the magnetic permeability $\mu$ is piecewise-constant on the considered meshes, with $\mu: \Omega \to [\mu_-, \mu_+]$ for some constant numbers $0 < \mu_- \le \mu_+$.

### 7.1 Scheme

As shown in [29], a scheme based on the discrete de Rham tools can be written by replacing, in the weak formulation of (7.1), the continuous L$^2$-products by discrete ones built on the local products. Denote by $\mu_T$ the constant value of $\mu$ over $T \in \mathcal{T}_h$ and define the bilinear forms $a_h: \underline{X}_{\text{curl},h}^k \times \underline{X}_{\text{curl},h}^k \to \mathbb{R}$, $b_h: \underline{X}_{\text{curl},h}^k \times \underline{X}_{\text{div},h}^k \to \mathbb{R}$, and $c_h: \underline{X}_{\text{div},h}^k \times \underline{X}_{\text{div},h}^k \to \mathbb{R}$ as follows: For all
$\underline{v}_h, \underline{\zeta}_h \in \underline{X}_{\text{curl},h}^k$ and all $\underline{w}_h, \underline{v}_h \in \underline{X}_{\text{div},h}^k$,

$$a_h(\underline{v}_h, \underline{\zeta}_h) := \sum_{T \in \mathcal{T}_h} \mu_T(\underline{v}_T, \underline{\zeta}_T)_{\text{curl},T}, \quad b_h(\underline{\zeta}_h, \underline{v}_h) := (\underline{C}_h^k \underline{\zeta}_h, \underline{v}_h)_{\text{div},h},$$

$$c_h(\underline{w}_h, \underline{v}_h) := \int_{\Omega} D_h^k \underline{w}_h D_h^k \underline{v}_h.$$

The discrete problem then reads: Find $\underline{H}_h \in \underline{X}_{\text{curl},h}^k$ and $\underline{A}_h \in \underline{X}_{\text{div},h}^k$ such that

$$\begin{align}
a_h(\underline{H}_h, \underline{\zeta}_h) - b_h(\underline{\zeta}_h, \underline{A}_h) &= 0 \qquad \forall \underline{\zeta}_h \in \underline{X}_{\text{curl},h}^k, \nonumber \\
b_h(\underline{H}_h, \underline{v}_h) + c_h(\underline{A}_h, \underline{v}_h) &= \sum_{T \in \mathcal{T}_h} \int_T \underline{J} \cdot \underline{P}_{\text{div},T}^k \underline{v}_T \qquad \forall \underline{v}_h \in \underline{X}_{\text{div},h}^k. \tag{7.2}
\end{align}$$

The equations of this problem can be recast in the standard variational form $\mathcal{A}_h((\underline{H}_h, \underline{A}_h), (\underline{\zeta}_h, \underline{v}_h)) = \mathcal{L}_h(\underline{\zeta}_h, \underline{v}_h)$, where $\mathcal{A}_h : (\underline{X}_{\text{curl},h}^k \times \underline{X}_{\text{div},h}^k)^2 \to \mathbb{R}$ and $\mathcal{L}_h : \underline{X}_{\text{curl},h}^k \times \underline{X}_{\text{div},h}^k \to \mathbb{R}$ are the bilinear and linear forms, respectively, such that

$$\mathcal{A}_h((\underline{v}_h, \underline{w}_h), (\underline{\zeta}_h, \underline{v}_h)) := a_h(\underline{v}_h, \underline{\zeta}_h) - b_h(\underline{\zeta}_h, \underline{w}_h) + b_h(\underline{v}_h, \underline{v}_h) + c_h(\underline{w}_h, \underline{v}_h),$$

$$\mathcal{L}_h(\underline{\zeta}_h, \underline{v}_h) := \sum_{T \in \mathcal{T}_h} \int_T \underline{J} \cdot \underline{P}_{\text{div},T}^k \underline{v}_T.$$

## 7.2 Error estimate

To measure the error, we introduce the following $\mathbf{H}(\text{curl}; \Omega)$- and $\mathbf{H}(\text{div}; \Omega)$-like (graph) norms on $\underline{X}_{\text{curl},h}^k$ and $\underline{X}_{\text{div},h}^k$, respectively:

$$\|\underline{\zeta}_h\|_{\mu, \text{curl}, 1, h} := \left( a_h(\underline{\zeta}_h, \underline{\zeta}_h) + \|\underline{C}_h^k \underline{\zeta}_h\|_{\text{div},h}^2 \right)^{\frac{1}{2}} \quad \forall \underline{\zeta}_h \in \underline{X}_{\text{curl},h}^k,$$

$$\|\underline{v}_h\|_{\text{div},1,h} := \left( \|\underline{v}_h\|_{\text{div},h}^2 + \|D_h^k \underline{v}_h\|_{L^2(\Omega)}^2 \right)^{\frac{1}{2}} \quad \forall \underline{v}_h \in \underline{X}_{\text{div},h}^k.$$

**Theorem 12 (Error estimate for the magnetostatics problem)** Assume that both the first and second Betti numbers of $\Omega$ are zero (i.e., $\Omega$ is not crossed by any tunnel and does not enclose any void). Then, there exists a unique solution $(\underline{H}_h, \underline{A}_h) \in \underline{X}_{\text{curl},h}^k \times \underline{X}_{\text{div},h}^k$ to (7.2). Moreover, letting $(\mathbf{H}, \mathbf{A}) \in \mathbf{H}(\text{curl}; \Omega) \times \mathbf{H}(\text{div}; \Omega)$ be the weak solution to (7.1) and assuming that $\mathbf{H} \in \mathbf{C}^0(\bar{\Omega}) \cap \mathbf{H}^{k+2}(\mathcal{T}_h)$ and $\mathbf{A} \in \mathbf{C}^0(\bar{\Omega}) \times \mathbf{H}^{k+2}(\mathcal{T}_h)$, we have

$$\|\underline{H}_h - \underline{I}_{\text{curl},h}^k \mathbf{H}\|_{\mu, \text{curl}, 1, h} + \|\underline{A}_h - \underline{I}_{\text{div},h}^k \mathbf{A}\|_{\text{div},1,h} \\ \lesssim h^{k+1} \left( |\text{curl } \mathbf{H}|_{\mathbf{H}^{k+1}(\mathcal{T}_h)} + |\mathbf{H}|_{\mathbf{H}^{(k+1,2)}(\mathcal{T}_h)} + |\mathbf{A}|_{\mathbf{H}^{k+1}(\mathcal{T}_h)} + |\mathbf{A}|_{\mathbf{H}^{k+2}(\mathcal{T}_h)} \right), \quad (7.3)$$

where the hidden constant in $\lesssim$ only depends on $\Omega$, $k$, the mesh regularity parameter, and $\mu_-$, $\mu_+$.