1. Define a broken Raviart-Thomas polynomial $\tau_h \in \mathcal{RT}_{p-1}(\mathcal{T}_h)$, on each mesh element $K$, via

$$ \tau_h|_K := \arg \min_{\substack{w_h \in \mathcal{RT}_{p-1}(K) \\ \nabla \cdot w_h = 0}} \|w - w_h\|_K \quad \forall K \in \mathcal{T}_h. \quad (7.1) $$

2. Define a Raviart-Thomas polynomial $\sigma_h^a \in \mathcal{RT}_p(\mathcal{T}_a) \cap \mathcal{H}_0(\text{div}, \omega_a)$, on each vertex patch $\mathcal{T}_a$, via

$$ \sigma_h^a := \arg \min_{\substack{w_h \in \mathcal{RT}_p(\mathcal{T}_a) \cap \mathcal{H}_0(\text{div}, \omega_a) \\ \nabla \cdot w_h = \Pi_h^p(\nabla \psi^a \cdot w)}} \|\psi^a \tau_h - w_h\|_{\omega_a} \quad \forall a \in \mathcal{V}_h. \quad (7.2) $$

3. Extending $\sigma_h^a$ by zero outside of the patch subdomain $\omega_a$, define $\sigma_h \in \mathcal{RT}_p(\mathcal{T}_h) \cap \mathcal{H}_{0,N}(\text{div}, \Omega)$ via

$$ \sigma_h := \sum_{a \in \mathcal{V}_h} \sigma_h^a. \quad (7.3) $$

Definition 7.1 takes the same form as [26, Definition 5.2], up to the fact that the divergence constraint in (7.2) is modified in that we use $\Pi_h^p(\nabla \psi^a \cdot w)$ in place of $\nabla \psi^a \cdot \tau_h$. Similarly to Remark 3.2, we now inspect the (beginning of) the proof of [26, Lemma 5.3]. Here a term $(g_a, \varphi)_{\omega_a} = (\Pi_h^p(\psi^a \nabla \cdot v) + \nabla \psi^a \cdot \tau_T)_{\omega_a}$ appears (notation from the proof of [26, Lemma 5.3]), which has to be replaced by $(\Pi_h^p(\psi^a \nabla \cdot v + \nabla \psi^a \cdot v), \varphi)_{\omega_a}$. This makes appear the same supplementary term and its bound as in (3.5). Thus, [26, Lemma 5.3], recalling (2.11b) and $\nabla \cdot w = \nabla \cdot (\nabla \times v) = 0$ (current notation), so that there are no oscillation terms, lead to

$$ \|\psi^a \tau_h - \sigma_h^a\|_{\omega_a} \lesssim \left\{ \sum_{K \in \mathcal{T}_a} \|\nabla \times v - \Pi_{\mathcal{RT}}^{p-1}(\nabla \times v)\|_K^2 \right\}^{1/2}, \quad (7.4) $$

where crucially the hidden constant only depends on the shape-regularity parameter $\kappa_{\mathcal{T}_h}$. We now rework Definition 3.3.

**Definition 7.2** (Alternative of Definition 3.3). Let $v \in \mathcal{H}_{0,N}(\text{curl}, \Omega)$, a simplicial mesh $\mathcal{T}_h$ of $\Omega$, and a polynomial degree $p \ge 1$ be given:

1. For $w := \nabla \times v$ yielding $w \in \mathcal{H}_{0,N}(\text{div}, \Omega)$ with $\nabla \cdot w = 0$, define $\tau_h \in \mathcal{RT}_{p-1}(\mathcal{T}_h)$ by (7.1) and $\sigma_h^a \in \mathcal{RT}_p(\mathcal{T}_a) \cap \mathcal{H}_0(\text{div}, \omega_a)$ by (7.2) from Definition 7.1.

2. Define a broken Nédélec polynomial $\iota_h \in \mathcal{N}_{p-1}(\mathcal{T}_h)$, on each mesh element $K$, via

$$ \iota_h|_K := \arg \min_{\substack{v_h \in \mathcal{N}_{p-1}(K) \\ \nabla \times v_h = \tau_h}} \|v - v_h\|_K \quad \forall K \in \mathcal{T}_h. \quad (7.5) $$

3. Define a Raviart-Thomas polynomial $\theta_h^a \in \mathcal{RT}_p(\mathcal{T}_a) \cap \mathcal{H}_0(\text{div}, \omega_a)$, on each vertex patch $\mathcal{T}_a$, via

$$ \theta_h^a := \arg \min_{\substack{v_h \in \mathcal{RT}_p(\mathcal{T}_a) \cap \mathcal{H}_0(\text{div}, \omega_a) \\ \nabla \cdot v_h = \Pi_h^p(-\nabla \psi^a \cdot (\nabla \times v))}} \|\nabla \psi^a \times \iota_h - v_h\|_{\omega_a} \quad \forall a \in \mathcal{V}_h. \quad (7.6) $$

4. Define a Nédélec polynomial $h_h^a \in \mathcal{N}_p(\mathcal{T}_a) \cap \mathcal{H}_0(\text{curl}, \omega_a)$, on each vertex patch, via

$$ h_h^a := \arg \min_{\substack{v_h \in \mathcal{N}_p(\mathcal{T}_a) \cap \mathcal{H}_0(\text{curl}, \omega_a) \\ \nabla \times v_h = \sigma_h^a + \theta_h^a}} \|\psi^a \iota_h - v_h\|_{\omega_a} \quad \forall a \in \mathcal{V}_h. \quad (7.7) $$

5. Extending $h_h^a$ by zero outside of the patch subdomain $\omega_a$, define

$$ h_h := \sum_{a \in \mathcal{V}_h} h_h^a \in \mathcal{N}_p(\mathcal{T}_h) \cap \mathcal{H}_{0,N}(\text{curl}, \Omega). \quad (7.8) $$

Compared to Definition 3.3, Definition 7.2 avoids the construction of $\delta_h$ on Step 4 and of the correction terms $\delta_h^a$ on Step 5. As in (3.8), we can still set

$$ \delta_h := \sum_{a \in \mathcal{V}_h} \theta_h^a \in \mathcal{RT}_p(\mathcal{T}_h) \cap \mathcal{H}_{0,N}(\text{div}, \Omega) \quad \text{with} \quad \nabla \cdot \delta_h = 0, \quad (7.9) $$

but the commuting property $\nabla \times h_h = \sigma_h$ from (3.15) is here lost in that from (7.8), (7.7), (7.3), and (7.9)
subdomains $\omega_a$ are convex for all vertices $a \in \mathcal{V}_h$, we can employ [37, Theorem 3.9]. Thus, in this case, there also holds $\chi \in \mathbf{H}^1(\omega_a)$ with

$$ \|\nabla \chi\|_{\omega_a} \le \|\nabla \times \chi\|_{\omega_a} \stackrel{(7.16)}{=} \|\nabla \times \varphi\|_{\omega_a} \stackrel{(7.12)}{=} 1. \quad (7.18) $$

Let

$$ T := (\sigma_h^a + \theta_h^a, \varphi)_{\omega_a} - (\psi^a \iota_h, \nabla \times \varphi)_{\omega_a}. \quad (7.19) $$

As in (5.6), since $\sigma_h^a + \theta_h^a \in \mathcal{RT}_p(\mathcal{T}_a) \cap \mathbf{H}_0(\text{div}, \omega_a)$ is divergence-free, there exists $w_h \in \mathcal{N}_p(\mathcal{T}_a) \cap \mathbf{H}_0(\text{curl}, \omega_a)$ such that $\nabla \times w_h = \sigma_h^a + \theta_h^a$, and we can exchange $\varphi$ for $\chi$ in (7.19), i.e.,

$$ \begin{aligned} T &= (\nabla \times w_h, \varphi)_{\omega_a} - (\psi^a \iota_h, \nabla \times \varphi)_{\omega_a} = (w_h, \nabla \times \varphi)_{\omega_a} - (\psi^a \iota_h, \nabla \times \varphi)_{\omega_a} \\ &= (w_h, \nabla \times \chi)_{\omega_a} - (\psi^a \iota_h, \nabla \times \chi)_{\omega_a} = (\sigma_h^a + \theta_h^a, \chi)_{\omega_a} - (\psi^a \iota_h, \nabla \times \chi)_{\omega_a}. \end{aligned} $$

We will now work on this term.

Using the chain rule

$$ \nabla \times (\psi^a \chi) = \nabla \psi^a \times \chi + \psi^a \nabla \times \chi, \quad (7.20) $$

and since $(d \times e) \cdot f = e \cdot (f \times d)$ and $f \times d = -d \times f$ for vectors $d, e, f \in \mathbb{R}^3$, we arrive at

$$ \begin{aligned} T &= (\sigma_h^a + \theta_h^a, \chi)_{\omega_a} + (\iota_h, \nabla \psi^a \times \chi)_{\omega_a} - (\iota_h, \nabla \times (\psi^a \chi))_{\omega_a} \\ &= (\sigma_h^a + \theta_h^a - \nabla \psi^a \times \iota_h, \chi)_{\omega_a} - (\iota_h, \nabla \times (\psi^a \chi))_{\omega_a}. \end{aligned} $$

Moreover, the Green theorem gives

$$ (\nabla \times v, (\psi^a \chi))_{\omega_a} - (v, \nabla \times (\psi^a \chi))_{\omega_a} = 0, $$

so that altogether, also adding and subtracting $\psi^a \tau_h$,

$$ \begin{aligned} T &= (\sigma_h^a + \theta_h^a - \nabla \psi^a \times \iota_h - \psi^a \nabla \times v, \chi)_{\omega_a} - (\iota_h - v, \nabla \times (\psi^a \chi))_{\omega_a} \\ &= (\sigma_h^a - \psi^a \tau_h + \theta_h^a - \nabla \psi^a \times \iota_h, \chi)_{\omega_a} + (\tau_h - \nabla \times v, \psi^a \chi)_{\omega_a} \\ &\quad - (\iota_h - v, \nabla \times (\psi^a \chi))_{\omega_a}. \end{aligned} $$

In the remainder of the proof, we estimate the above terms separately, in four steps.

**Step 1.** The chain rule (7.20), the Cauchy–Schwarz inequality, mesh shape-regularity, and (7.17), (7.16), (7.12) give

$$ \|\nabla \times (\psi^a \chi)\|_{\omega_a} = \|\nabla \psi^a \times \chi + \psi^a \nabla \times \chi\|_{\omega_a} \le \|\nabla \psi^a\|_{\infty, \omega_a} \|\chi\|_{\omega_a} + \|\nabla \times \chi\|_{\omega_a} \lesssim 1, \quad (7.21) $$

so that the Cauchy–Schwarz inequality yields

$$ -(\iota_h - v, \nabla \times (\psi^a \chi))_{\omega_a} \lesssim \|\iota_h - v\|_{\omega_a}, $$

which is the first term in (7.11).

**Step 2.** In general

$$ (\tau_h - \nabla \times v, \psi^a \chi)_{\omega_a} \le \|\tau_h - \nabla \times v\|_{\omega_a} \|\psi^a \chi\|_{\omega_a} \stackrel{(7.17)}{\lesssim} \|\tau_h - \nabla \times v\|_{\omega_a} h_{\omega_a} $$

and

$$ \|\tau_h - \nabla \times v\|_{\omega_a}^2 = \sum_{K \in \mathcal{T}_a} \|\tau_h - \nabla \times v\|_K^2 \lesssim \sum_{K \in \mathcal{T}_a} \|\nabla \times v - \Pi_{\mathcal{RT}}^{p-1}(\nabla \times v)\|_K^2, \quad (7.22) $$

relying on (7.1) and the p-robust constrained–unconstrained equivalence of [26, Lemma A1] (recall that $\nabla \cdot (\nabla \times v) = 0$) (cf. (A.14) below). Since $h_{\omega_a} \approx h_K$ in the patch, this gives the second term in (7.11) with $\delta = 0$.

If $\Gamma_D = \emptyset$ and the patch subdomains $\omega_a$ are convex for all vertices $a \in \mathcal{V}_h$, we rather use (7.17) and (7.18), which give, together with the chain rule,

$$ \|\nabla (\psi^a \chi)\|_{\omega_a} = \|\nabla \psi^a \chi + \psi^a \nabla \chi\|_{\omega_a} \le \|\nabla \psi^a\|_{\infty, \omega_a} \|\chi\|_{\omega_a} + \|\nabla \chi\|_{\omega_a} \lesssim 1. $$
$$ \nabla \times h_h \stackrel{(7.8)}{=} \sum_{a \in \mathcal{V}_h} \nabla \times h_h^a \stackrel{(7.7)}{=} \sum_{a \in \mathcal{V}_h} (\sigma_h^a + \theta_h^a) \stackrel{(7.3)}{=} \sigma_h + \delta_h. \qquad (7.10) $$

This will be sufficient here, as in Theorem 3.11, the minimization does not have a constrained curl. It is also to be noted that the minimizations in (7.6) avoid the additional elementwise orthogonality constraint of (3.7). Commuting could be achieved by further lowering the degree $p-1$ to $p-2$ and constructing an appropriate $\delta_h^a$, but the above simplifications will be sufficient and actually helpful for us in deriving $p$-robust bounds.

Definition 7.2 is well-posed, which can be easily verified as in Section 4; namely, $\sigma_h^a + \theta_h^a \in \mathcal{RT}_p(\mathcal{T}_a) \cap \mathcal{H}_0(\text{div}, \omega_a)$ with $\nabla \cdot (\sigma_h^a + \theta_h^a) = 0$. We now modify the reasoning of Step 2b from the proof of Lemma 5.2, extending to the $\omega_a$-patch setting the hp-reasoning of Lemmas A.2 and A.3 from Appendix A below.

**Lemma 7.3** ($p$-robust patch estimate). Let $v \in \mathcal{H}_{0,N}(\text{curl}, \Omega)$ and a vertex $a \in \mathcal{V}_h$ be given. Let $\sigma_h^a$ and $h_h^a$ be respectively given by Definitions 7.1 and 7.2. If $\Gamma_D = \emptyset$ and the patch subdomains $\omega_a$ are convex for all vertices $a \in \mathcal{V}_h$, set $\delta := 1$, otherwise set $\delta := 0$. Then

$$ \|\psi^a \iota_h - h_h^a\|_{\omega_a}^2 \lesssim \sum_{K \in \mathcal{T}_a} \left[ \min_{v_h \in \mathcal{N}_{p-1}(K)} \|v - v_h\|_K^2 + \left( \frac{h_K}{p^\delta} \|\nabla \times v - \Pi_{\mathcal{RT}}^{p-1}(\nabla \times v)\|_K \right)^2 \right], $$

where the hidden constant only depends on the shape-regularity parameter $\kappa_{\mathcal{T}_h}$ of the mesh $\mathcal{T}_h$.

*Proof.* Taking into account definition (7.5) and the constrained-unconstrained equivalence established in Lemma A.1 (employed with $p-1$ in place of $p$), we only need to show

$$ \|\psi^a \iota_h - h_h^a\|_{\omega_a} \lesssim \|v - \iota_h\|_{\omega_a} + \left\{ \sum_{K \in \mathcal{T}_a} \left( \frac{h_K}{p^\delta} \|\nabla \times v - \Pi_{\mathcal{RT}}^{p-1}(\nabla \times v)\|_K \right)^2 \right\}^{1/2}. \quad (7.11) $$

From definition (7.7), as in (5.3), [16, Theorem 3.3] gives

$$ \begin{aligned} \|\psi^a \iota_h - h_h^a\|_{\omega_a} &\lesssim \min_{\substack{\varphi \in \mathcal{H}_0(\text{curl}, \omega_a) \\ \nabla \times \varphi = \sigma_h^a + \theta_h^a}} \|\psi^a \iota_h - \varphi\|_{\omega_a} \\ &= \sup_{\substack{\varphi \in \mathcal{H}^\dagger(\text{curl}, \omega_a) \\ \|\nabla \times \varphi\|_{\omega_a} = 1}} \{ (\sigma_h^a + \theta_h^a, \varphi)_{\omega_a} - (\psi^a \iota_h, \nabla \times \varphi)_{\omega_a} \}, \end{aligned} $$

where we have, for the equality, employed the primal-dual equivalence as in (5.4). Fix now

$$ \varphi \in \mathcal{H}^\dagger(\text{curl}, \omega_a) \text{ with } \|\nabla \times \varphi\|_{\omega_a} = 1. \qquad (7.12) $$

Let $q \in H_*^1(\omega_a)$ be such that

$$ (\nabla q, \nabla v)_{\omega_a} = (\varphi, \nabla v)_{\omega_a} \quad \forall v \in H_*^1(\omega_a) \qquad (7.13) $$

and define

$$ \chi := \varphi - \nabla q. \qquad (7.14) $$

Recalling the notation (2.6) or (2.7), we have

$$ \chi \in \mathcal{H}^\dagger(\text{curl}, \omega_a) \cap \mathcal{H}_0(\text{div}, \omega_a) \text{ with } \nabla \cdot \chi = 0 \qquad (7.15) $$

together with

$$ \nabla \times \chi = \nabla \times \varphi. \qquad (7.16) $$

Moreover, from (7.15), (7.16), and (7.12), the Poincaré-Friedrichs-Weber inequality (2.18) gives

$$ \|\chi\|_{\omega_a} \stackrel{(2.18)}{\le} C_{\text{PFW}} h_{\omega_a} \|\nabla \times \chi\|_{\omega_a} \stackrel{(7.16)}{=} C_{\text{PFW}} h_{\omega_a} \|\nabla \times \varphi\|_{\omega_a} \stackrel{(7.12)}{\lesssim} h_{\omega_a}. \qquad (7.17) $$

In this way, we will be gaining the $h_K$ factor in the second term of (7.11), but not the factor $h_K/p$.

If $\Gamma_D = \emptyset$, we note that there is always zero normal trace prescribed on $\partial\omega_a$ in the space $\mathcal{H}_0(\text{div}, \omega_a)$, see (2.6d), and, congruently, $\mathcal{H}^\dagger(\text{curl}, \omega_a) = \mathcal{H}(\text{curl}, \omega_a)$, see (2.6c). Thus, if additionally the patch
Thus, employing Lemma A.3 for each $K \in \mathcal{T}_a$ and the Cauchy-Schwarz inequality, we infer

$$ (\tau_h - \nabla \times \mathbf{v}, \psi^a \chi)_{\omega_a} = \sum_{K \in \mathcal{T}_a} (\tau_h - \nabla \times \mathbf{v}, \psi^a \chi)_K \\ \lesssim \sum_{K \in \mathcal{T}_a} \left( \frac{h_K}{p} \|\nabla \times \mathbf{v} - \Pi_{RT}^{p-1}(\nabla \times \mathbf{v})\|_K \|\nabla(\psi^a \chi)\|_K \right) \\ \lesssim \left\{ \sum_{K \in \mathcal{T}_a} \left( \frac{h_K}{p} \|\nabla \times \mathbf{v} - \Pi_{RT}^{p-1}(\nabla \times \mathbf{v})\|_K \right)^2 \right\}^{1/2}, $$

which is the second term in (7.11) with $\delta = 1$.

Step 3. Relying on [16, Lemma A.3], as in (5.11), we have

$$ \|\boldsymbol{\theta}_h^a - \nabla \psi^a \times \boldsymbol{\iota}_h\|_{\omega_a} \lesssim h_{\omega_a}^{-1} \|\mathbf{v} - \boldsymbol{\iota}_h\|_{\omega_a} + h_{\omega_a}^{-1} \left\{ \sum_{K \in \mathcal{T}_a} \left( h_K \|\nabla \times \mathbf{v} - \Pi_{RT}^{p-1}(\nabla \times \mathbf{v})\|_K \right)^2 \right\}^{1/2}; \quad (7.23) $$

note that this bound is indeed p-robust, as the above oscillation term is not of hp type. Consequently,

$$ (\boldsymbol{\theta}_h^a - \nabla \psi^a \times \boldsymbol{\iota}_h, \chi)_{\omega_a} \le \|\boldsymbol{\theta}_h^a - \nabla \psi^a \times \boldsymbol{\iota}_h\|_{\omega_a} \|\chi\|_{\omega_a}, $$

and we have (7.11) with $\delta = 0$ from (7.17).

If $\Gamma_D = \emptyset$ and the patch subdomains $\omega_a$ are convex for all vertices $\mathbf{a} \in \mathcal{V}_h$, we again proceed more carefully. The Euler-Lagrange conditions of (7.6), where it is important that no additional elementwise orthogonality constraints as in (3.7) appear, give

$$ (\boldsymbol{\theta}_h^a - \nabla \psi^a \times \boldsymbol{\iota}_h, \chi)_{\omega_a} = (\boldsymbol{\theta}_h^a - \nabla \psi^a \times \boldsymbol{\iota}_h, \chi - \chi_h)_{\omega_a} $$

for

$$ \chi_h := \arg \min_{\mathbf{w}_h \in \mathcal{R}\mathcal{T}_p(\mathcal{T}_a) \cap \mathcal{H}_0(\text{div}, \omega_a) \atop \nabla \cdot \mathbf{w}_h = 0} \|\chi - \mathbf{w}_h\|_{\omega_a}. \quad (7.24) $$

We now crucially rely on the hp-approximation estimate of [26, Theorem 3.6] for $\Omega = \omega_a$, recalling that $\chi \in \mathcal{H}_0(\text{div}, \omega_a)$ has been prepared such that $\nabla \cdot \chi = 0$ and $\chi \in \mathcal{H}^1(\omega_a)$, so that we can take $s = 1$ in [26]. This gives

$$ \|\chi - \chi_h\|_{\omega_a} \lesssim \left\{ \sum_{K \in \mathcal{T}_a} \left( \frac{h_K}{p+1} (h_K^{-1} \|\chi\|_K^2 + \|\nabla \chi\|_K^2)^{1/2} \right)^2 \right\}^{1/2} \quad (7.25) \\ \lesssim \frac{h_{\omega_a}}{p} (h_{\omega_a}^{-1} \|\chi\|_{\omega_a}^2 + \|\nabla \chi\|_{\omega_a}^2)^{1/2} \lesssim \frac{h_{\omega_a}}{p}, $$

where we have employed (7.17) and (7.18). Thus,

$$ (\boldsymbol{\theta}_h^a - \nabla \psi^a \times \boldsymbol{\iota}_h, \chi)_{\omega_a} \lesssim \|\boldsymbol{\theta}_h^a - \nabla \psi^a \times \boldsymbol{\iota}_h\|_{\omega_a} \frac{h_{\omega_a}}{p} $$

and we conclude (7.11) with $\delta = 1$ using (7.23) (note that $p \ge 1$).

Step 4. We proceed as above in Step 3. We employ (7.4) and either estimate

$$ (\boldsymbol{\sigma}_h^a - \psi^a \tau_h, \chi)_{\omega_a} \le \|\boldsymbol{\sigma}_h^a - \psi^a \tau_h\|_{\omega_a} \|\chi\|_{\omega_a} $$

together with (7.17), or we employ $\chi_h$ from (7.24) and (7.25) together with the Euler-Lagrange conditions of (7.2) and the Cauchy-Schwarz inequality, which give

$$ (\boldsymbol{\sigma}_h^a - \psi^a \tau_h, \chi)_{\omega_a} = (\boldsymbol{\sigma}_h^a - \psi^a \tau_h, \chi - \chi_h)_{\omega_a} \le \|\boldsymbol{\sigma}_h^a - \psi^a \tau_h\|_{\omega_a} \|\chi - \chi_h\|_{\omega_a}. $$

This concludes the proof.

□

We are now ready to derive a p-robust variant of the inequality $\lesssim$ from (3.21b) with a lowered polynomial degree on the right-hand side.
subdomains $\omega_a$ are convex for all vertices $a \in \mathcal{V}_h$, we can employ [37, Theorem 3.9]. Thus, in this case, there also holds $\chi \in \mathbf{H}^1(\omega_a)$ with

$$ \|\nabla \chi\|_{\omega_a} \le \|\nabla \times \chi\|_{\omega_a} \stackrel{(7.16)}{=} \|\nabla \times \varphi\|_{\omega_a} \stackrel{(7.12)}{=} 1. \qquad (7.18) $$

Let

$$ T := (\sigma_h^a + \theta_h^a, \varphi)_{\omega_a} - (\psi^a \iota_h, \nabla \times \varphi)_{\omega_a}. \qquad (7.19) $$

As in (5.6), since $\sigma_h^a + \theta_h^a \in \mathcal{RT}_p(\mathcal{T}_a) \cap \mathbf{H}_0(\text{div}, \omega_a)$ is divergence-free, there exists $w_h \in \mathcal{N}_p(\mathcal{T}_a) \cap \mathbf{H}_0(\text{curl}, \omega_a)$ such that $\nabla \times w_h = \sigma_h^a + \theta_h^a$, and we can exchange $\varphi$ for $\chi$ in (7.19), i.e.,

$$ \begin{aligned} T &= (\nabla \times w_h, \varphi)_{\omega_a} - (\psi^a \iota_h, \nabla \times \varphi)_{\omega_a} = (w_h, \nabla \times \varphi)_{\omega_a} - (\psi^a \iota_h, \nabla \times \varphi)_{\omega_a} \\ &= (w_h, \nabla \times \chi)_{\omega_a} - (\psi^a \iota_h, \nabla \times \chi)_{\omega_a} = (\sigma_h^a + \theta_h^a, \chi)_{\omega_a} - (\psi^a \iota_h, \nabla \times \chi)_{\omega_a}. \end{aligned} $$

We will now work on this term.

Using the chain rule

$$ \nabla \times (\psi^a \chi) = \nabla \psi^a \times \chi + \psi^a \nabla \times \chi, \qquad (7.20) $$

and since $(d \times e) \cdot f = e \cdot (f \times d)$ and $f \times d = -d \times f$ for vectors $d, e, f \in \mathbb{R}^3$, we arrive at

$$ \begin{aligned} T &= (\sigma_h^a + \theta_h^a, \chi)_{\omega_a} + (\iota_h, \nabla \psi^a \times \chi)_{\omega_a} - (\iota_h, \nabla \times (\psi^a \chi))_{\omega_a} \\ &= (\sigma_h^a + \theta_h^a - \nabla \psi^a \times \iota_h, \chi)_{\omega_a} - (\iota_h, \nabla \times (\psi^a \chi))_{\omega_a}. \end{aligned} $$

Moreover, the Green theorem gives

$$ (\nabla \times v, (\psi^a \chi))_{\omega_a} - (v, \nabla \times (\psi^a \chi))_{\omega_a} = 0, $$

so that altogether, also adding and subtracting $\psi^a \tau_h$,

$$ \begin{aligned} T &= (\sigma_h^a + \theta_h^a - \nabla \psi^a \times \iota_h - \psi^a \nabla \times v, \chi)_{\omega_a} - (\iota_h - v, \nabla \times (\psi^a \chi))_{\omega_a} \\ &= (\sigma_h^a - \psi^a \tau_h + \theta_h^a - \nabla \psi^a \times \iota_h, \chi)_{\omega_a} + (\tau_h - \nabla \times v, \psi^a \chi)_{\omega_a} \\ &\quad - (\iota_h - v, \nabla \times (\psi^a \chi))_{\omega_a}. \end{aligned} $$

In the remainder of the proof, we estimate the above terms separately, in four steps.

**Step 1.** The chain rule (7.20), the Cauchy–Schwarz inequality, mesh shape-regularity, and (7.17), (7.16), (7.12) give

$$ \|\nabla \times (\psi^a \chi)\|_{\omega_a} = \|\nabla \psi^a \times \chi + \psi^a \nabla \times \chi\|_{\omega_a} \le \|\nabla \psi^a\|_{\infty, \omega_a} \|\chi\|_{\omega_a} + \|\nabla \times \chi\|_{\omega_a} \lesssim 1, \qquad (7.21) $$

so that the Cauchy–Schwarz inequality yields

$$ -(\iota_h - v, \nabla \times (\psi^a \chi))_{\omega_a} \lesssim \|\iota_h - v\|_{\omega_a}, $$

which is the first term in (7.11).

**Step 2.** In general

$$ (\tau_h - \nabla \times v, \psi^a \chi)_{\omega_a} \le \|\tau_h - \nabla \times v\|_{\omega_a} \|\psi^a \chi\|_{\omega_a} \stackrel{(7.17)}{\lesssim} \|\tau_h - \nabla \times v\|_{\omega_a} h_{\omega_a} $$

and

$$ \|\tau_h - \nabla \times v\|_{\omega_a}^2 = \sum_{K \in \mathcal{T}_a} \|\tau_h - \nabla \times v\|_K^2 \lesssim \sum_{K \in \mathcal{T}_a} \|\nabla \times v - \Pi_{\mathcal{RT}}^{p-1}(\nabla \times v)\|_K^2, \qquad (7.22) $$

relying on (7.1) and the p-robust constrained–unconstrained equivalence of [26, Lemma A1] (recall that $\nabla \cdot (\nabla \times v) = 0$) (cf. (A.14) below). Since $h_{\omega_a} \approx h_K$ in the patch, this gives the second term in (7.11) with $\delta = 0$.

If $\Gamma_D = \emptyset$ and the patch subdomains $\omega_a$ are convex for all vertices $a \in \mathcal{V}_h$, we rather use (7.17) and (7.18), which give, together with the chain rule,

$$ \|\nabla (\psi^a \chi)\|_{\omega_a} = \|\nabla \psi^a \chi + \psi^a \nabla \chi\|_{\omega_a} \le \|\nabla \psi^a\|_{\infty, \omega_a} \|\chi\|_{\omega_a} + \|\nabla \chi\|_{\omega_a} \lesssim 1. $$
**Proposition 7.4** (p-robust one-sided bound). Let $v \in H_{0,N}(\text{curl}, \Omega)$, a simplicial mesh $\mathcal{T}_h$ of $\Omega$, and a polynomial degree $p \ge 1$ be fixed. If $\Gamma_D = \emptyset$ and the patch subdomains $\omega_a$ are convex for all vertices $a \in \mathcal{V}_h$, set $\delta := 1$, otherwise set $\delta := 0$. Then

$$ \begin{aligned} & \min_{v_h \in \mathcal{N}_p(\mathcal{T}_h) \cap \mathcal{H}_{0,N}(\text{curl}, \Omega)} \left[ \|v - v_h\|^2 + \sum_{K \in \mathcal{T}_h} \left( \frac{h_K}{(p+1)^\delta} \|\nabla \times (v - v_h)\|_K \right)^2 \right] \\ & \lesssim \sum_{K \in \mathcal{T}_h} \left[ \min_{v_h \in \mathcal{N}_{p-1}(K)} \|v - v_h\|_K^2 + \left( \frac{h_K}{p^\delta} \|\nabla \times v - \Pi_{RT}^{p-1}(\nabla \times v)\|_K \right)^2 \right], \end{aligned} \quad (7.26) $$

where the hidden constant only depends on the shape-regularity parameter $\kappa_{\mathcal{T}_h}$ of the mesh $\mathcal{T}_h$.

*Proof.* We bound the left-hand side of (7.26) by

$$ \left[ \|v - h_h\|^2 + \sum_{K \in \mathcal{T}_h} \left( \frac{h_K}{(p+1)^\delta} \|\nabla \times (v - h_h)\|_K \right)^2 \right], $$

using $h_h$ from Definition 7.2. We now estimate the two above terms separately, in two steps.

*Step 1.* The triangle inequality gives, for $\iota_h$ given by (7.5),

$$ \|v - h_h\| \le \|v - \iota_h\| + \|\iota_h - h_h\|. $$

Thus, the constrained-unconstrained equivalence from Lemma A.1 (employed with $p-1$ in place of $p$) gives the desired result for the first term above (with $\delta = 1$ in any case). As for the second one, definition (7.8), the partition of unity (2.1), and an overlap estimate imply

$$ \|\iota_h - h_h\|^2 = \sum_{K \in \mathcal{T}_h} \left\| \sum_{a \in \mathcal{V}_K} (\psi^a \iota_h - h_h^a) \right\|_K^2 \le 4 \sum_{a \in \mathcal{V}_h} \|\psi^a \iota_h - h_h^a\|_{\omega_a}^2, $$

so that the desired bound follows from Lemma 7.3 [–].

*Step 2.* From (7.10), the partition of unity (2.1), (7.3), and adding and subtracting $\psi^a \nabla \times \iota_h$,

$$ \begin{aligned} \|\nabla \times (v - h_h)\|_K &= \|\nabla \times v - \sigma_h - \delta_h\|_K \\ &= \left\| \sum_{a \in \mathcal{V}_K} \psi^a (\nabla \times (v - \iota_h)) + \sum_{a \in \mathcal{V}_K} (\psi^a \nabla \times \iota_h - \sigma_h^a) - \delta_h \right\|_K \\ &\le \sum_{a \in \mathcal{V}_K} \|\psi^a (\nabla \times (v - \iota_h))\|_{\omega_a} + \sum_{a \in \mathcal{V}_K} \|\psi^a \nabla \times \iota_h - \sigma_h^a\|_{\omega_a} + \|\delta_h\|_K. \end{aligned} $$

Recalling that the constraint in (7.5) gives $\nabla \times \iota_h = \tau_h$, we estimate the three terms separately.

For the first term, we estimate by (7.22)

$$ \|\psi^a(\nabla \times (v - \iota_h))\|_{\omega_a} \le \|\tau_h - \nabla \times v\|_{\omega_a} \stackrel{(7.22)}{\lesssim} \left\{ \sum_{K \in \mathcal{T}_a} \|\nabla \times v - \Pi_{RT}^{p-1}(\nabla \times v)\|_K^2 \right\}^{1/2}. $$

For the second one, we use (7.4) to see that

$$ \|\psi^a \nabla \times \iota_h - \sigma_h^a\|_{\omega_a} \lesssim \left\{ \sum_{K \in \mathcal{T}_a} \|\nabla \times v - \Pi_{RT}^{p-1}(\nabla \times v)\|_K^2 \right\}^{1/2}. $$

For the third one, we infer from (7.9) and (2.1)

$$ \|\delta_h\|_K = \left\| \sum_{a \in \mathcal{V}_K} (\theta_h^a - \nabla \psi^a \times \iota_h) \right\|_K \le \sum_{a \in \mathcal{V}_K} \|\theta_h^a - \nabla \psi^a \times \iota_h\|_{\omega_a}. $$

For the arising term, we can now use (7.23). Thus, multiplying $\|\nabla \times (v - h_h)\|_K$ by $h_K/(p+1)^\delta$, using that $h_K/(p+1)^\delta \le h_K/p^\delta$ as well as $1/(p+1)^\delta \le 1$, and invoking the mesh shape-regularity, the proof is finished. $\square$
Thus, employing Lemma A.3 for each $K \in \mathcal{T}_a$ and the Cauchy–Schwarz inequality, we infer

$$ (\tau_h - \nabla \times v, \psi^a \chi)_{\omega_a} = \sum_{K \in \mathcal{T}_a} (\tau_h - \nabla \times v, \psi^a \chi)_K \\ \lesssim \sum_{K \in \mathcal{T}_a} \left( \frac{h_K}{p} \|\nabla \times v - \Pi_{RT}^{p-1}(\nabla \times v)\|_K \|\nabla(\psi^a \chi)\|_K \right) \\ \lesssim \left\{ \sum_{K \in \mathcal{T}_a} \left( \frac{h_K}{p} \|\nabla \times v - \Pi_{RT}^{p-1}(\nabla \times v)\|_K \right)^2 \right\}^{1/2}, $$

which is the second term in (7.11) with $\delta = 1$.

Step 3. Relying on [15, Lemma A.3], as in (5.11), we have

$$ \|\theta_h^a - \nabla \psi^a \times \iota_h\|_{\omega_a} \lesssim h_{\omega_a}^{-1} \|v - \iota_h\|_{\omega_a} + h_{\omega_a}^{-1} \left\{ \sum_{K \in \mathcal{T}_a} \left( h_K \|\nabla \times v - \Pi_{RT}^{p-1}(\nabla \times v) \|_K \right)^2 \right\}^{1/2}; \quad (7.23) $$

note that this bound is indeed p-robust, as the above oscillation term is not of hp type. Consequently,

$$ (\theta_h^a - \nabla \psi^a \times \iota_h, \chi)_{\omega_a} \le \|\theta_h^a - \nabla \psi^a \times \iota_h\|_{\omega_a} \|\chi\|_{\omega_a}, $$

and we have (7.11) with $\delta = 0$ from (7.17).

If $\Gamma_D = \emptyset$ and the patch subdomains $\omega_a$ are convex for all vertices $a \in V_h$, we again proceed more carefully. The Euler–Lagrange conditions of (7.6), where it is important that no additional elementwise orthogonality constraints as in (3.7) appear, give

$$ (\theta_h^a - \nabla \psi^a \times \iota_h, \chi)_{\omega_a} = (\theta_h^a - \nabla \psi^a \times \iota_h, \chi - \chi_h)_{\omega_a} $$

for

$$ \chi_h := \arg \min_{w_h \in \mathcal{R}\mathcal{T}_p(\mathcal{T}_a) \cap \mathcal{H}_0(\text{div}, \omega_a) \atop \nabla \cdot w_h = 0} \|\chi - w_h\|_{\omega_a}. \quad (7.24) $$

We now crucially rely on the hp-approximation estimate of [26, Theorem 3.6] for $\Omega = \omega_a$, recalling that $\chi \in \mathcal{H}_0(\text{div}, \omega_a)$ has been prepared such that $\nabla \cdot \chi = 0$ and $\chi \in \mathcal{H}^1(\omega_a)$, so that we can take $s = 1$ in [26]. This gives

$$ \begin{aligned} \|\chi - \chi_h\|_{\omega_a} & \lesssim \left\{ \sum_{K \in \mathcal{T}_a} \left( \frac{h_K}{p+1} (h_K^{-2} \|\chi\|_K^2 + \|\nabla \chi\|_K^2)^{1/2} \right)^2 \right\}^{1/2} \\ & \lesssim \frac{h_{\omega_a}}{p} (h_{\omega_a}^{-2} \|\chi\|_{\omega_a}^2 + \|\nabla \chi\|_{\omega_a}^2)^{1/2} \lesssim \frac{h_{\omega_a}}{p}, \end{aligned} \quad (7.25) $$

where we have employed (7.17) and (7.18). Thus,

$$ (\theta_h^a - \nabla \psi^a \times \iota_h, \chi)_{\omega_a} \lesssim \|\theta_h^a - \nabla \psi^a \times \iota_h\|_{\omega_a} \frac{h_{\omega_a}}{p} $$

and we conclude (7.11) with $\delta = 1$ using (7.23) (note that $p \ge 1$).

Step 4. We proceed as above in Step 3. We employ (7.4) and either estimate

$$ (\sigma_h^a - \psi^a \tau_h, \chi)_{\omega_a} \le \|\sigma_h^a - \psi^a \tau_h\|_{\omega_a} \|\chi\|_{\omega_a} $$

together with (7.17), or we employ $\chi_h$ from (7.24) and (7.25) together with the Euler–Lagrange conditions of (7.2) and the Cauchy–Schwarz inequality, which give

$$ (\sigma_h^a - \psi^a \tau_h, \chi)_{\omega_a} = (\sigma_h^a - \psi^a \tau_h, \chi - \chi_h)_{\omega_a} \le \|\sigma_h^a - \psi^a \tau_h\|_{\omega_a} \|\chi - \chi_h\|_{\omega_a}. $$

This concludes the proof.

□

We are now ready to derive a p-robust variant of the inequality $\lesssim$ from (3.21b) with a lowered polynomial degree on the right-hand side.
Denote

$$m^2 := \min_{\mathbf{v}_h \in \mathcal{N}_p(\mathcal{T}_h) \cap \mathcal{H}_{0,N}(\text{curl}, \Omega)} \left[ \|\mathbf{v} - \mathbf{v}_h\|^2 + \sum_{K \in \mathcal{T}_h} \left( \frac{h_K}{p+1} \|\nabla \times (\mathbf{v} - \mathbf{v}_h)\|_K \right)^2 \right] \quad (7.27)$$

and

$$[v_{K,q,s,t}(\mathbf{v})]^2 := \left( \frac{h_K^{\min\{q,s\}}}{q^s} \|\mathbf{v}\|_{\mathcal{H}^s(K)} \right)^2 + \left( \frac{h_K}{q} \frac{h_K^{\min\{q,t\}}}{q^t} \|\nabla \times \mathbf{v}\|_{\mathcal{H}^t(K)} \right)^2. \quad (7.28)$$

From (3.21b) in Theorem 3.9 and the characterization (2.11b), we have

$$m^2 \lesssim \sum_{K \in \mathcal{T}_h} \left[ \min_{\mathbf{v}_h \in \mathcal{N}_p(K)} \|\mathbf{v} - \mathbf{v}_h\|_K^2 + \left( \frac{h_K}{p+1} \min_{w_h \in \mathcal{R}\mathcal{T}_p(K)} \|\nabla \times \mathbf{v} - \mathbf{w}_h\|_K \right)^2 \right], \quad (7.29)$$

where the hidden constant can depend on $\kappa_{\mathcal{T}_h}$ and the polynomial degree $p$. From Proposition 7.4, in turn,

$$m^2 \lesssim \sum_{K \in \mathcal{T}_h} \left[ \min_{\mathbf{v}_h \in \mathcal{N}_{p-1}(K)} \|\mathbf{v} - \mathbf{v}_h\|_K^2 + \left( \frac{h_K}{p^\delta} \min_{w_h \in \mathcal{R}\mathcal{T}_{p-1}(K)} \|\nabla \times \mathbf{v} - \mathbf{w}_h\|_K \right)^2 \right], \quad (7.30)$$

where $p$ has to be greater or equal to 1 but the hidden constant is independent of $p$. These are the two ingredients necessary for the proof of Theorem 3.11 in the spirit of the proof of Theorem 3.6 in [26, Section 5.2]:

*Proof of Theorem 3.11, case (ii) (convex patches and tangential boundary conditions).* Recall that the parameter $\delta$ from Proposition 7.4 is one here. Thus, employing the notations (7.27)–(7.28) and the claim (3.27), we need to show that

$$m^2 \lesssim \sum_{K \in \mathcal{T}_h} [v_{K,p+1,s,t}(\mathbf{v})]^2, \quad (7.31)$$

where the hidden constant only depends on $\kappa_{\mathcal{T}_h}$, $s$, and $t$. We distinguish two cases that we later combine together.

*Case 1.* Suppose $p \le \max\{s,t\}$. On each tetrahedron $K \in \mathcal{T}_h$, we have from (2.8) and (2.9) that $[\mathcal{P}_p(K)]^3 \subset \mathcal{N}_p(K)$ and $[\mathcal{P}_p(K)]^3 \subset \mathcal{R}\mathcal{T}_p(K)$. Thus the elementwise regularity assumptions (3.24) and (3.26) together with [4, Lemma 4.1] give

$$\min_{\mathbf{v}_h \in \mathcal{N}_p(K)} \|\mathbf{v} - \mathbf{v}_h\|_K^2 + \left( \frac{h_K}{p+1} \min_{w_h \in \mathcal{R}\mathcal{T}_p(K)} \|\nabla \times \mathbf{v} - \mathbf{w}_h\|_K \right)^2 \lesssim [v_{K,p+1,s,t}(\mathbf{v})]^2,$$

where the hidden constant only depends on $\kappa_{\mathcal{T}_h}$, $s$, and $t$. Thus, from (7.29), there exists a constant $C_{\kappa_{\mathcal{T}_h}, p, s, t}$ only depending on $\kappa_{\mathcal{T}_h}$, $p$, $s$, and $t$ such that

$$m^2 \le C_{\kappa_{\mathcal{T}_h}, p, s, t} \sum_{K \in \mathcal{T}_h} [v_{K,p+1,s,t}(\mathbf{v})]^2.$$

Defining $C_{\kappa_{\mathcal{T}_h}, p, s, t}^* := \max_{0 \le p \le \max\{s,t\}} C_{\kappa_{\mathcal{T}_h}, p, s, t}$, only depending on $\kappa_{\mathcal{T}_h}$, $s$, and $t$, there holds, for all $0 \le p \le \max\{s,t\}$,

$$m^2 \le C_{\kappa_{\mathcal{T}_h}, p, s, t}^* \sum_{K \in \mathcal{T}_h} [v_{K,p+1,s,t}(\mathbf{v})]^2. \quad (7.32)$$

*Case 2.* Suppose $p > \max\{s,t\}$. Since $p$ is an integer, this implies that $p \ge 1$, so that we will be able to apply (7.30). As above, reducing $p+1$ to $p$ on the right-hand side,

$$\min_{\mathbf{v}_h \in \mathcal{N}_{p-1}(K)} \|\mathbf{v} - \mathbf{v}_h\|_K^2 + \left( \frac{h_K}{p} \min_{w_h \in \mathcal{R}\mathcal{T}_{p-1}(K)} \|\nabla \times \mathbf{v} - \mathbf{w}_h\|_K \right)^2 \lesssim [v_{K,p,s,t}(\mathbf{v})]^2,$$

where the hidden constant only depends on $\kappa_{\mathcal{T}_h}$, $s$, and $t$. Now, actually, $\min\{p, s\} = s = \min\{p+1, s\}$ and $p+1 \le 2p$, so that $1/p^s \le 2^s/(p+1)^s$. Similarly, $[-] \min\{p, t\} = t = \min\{p+1, t\}$ and $1/p^t \le 2^t/(p+1)^t$. As a consequence, we can rise $p$ back to $p+1$ on the right-hand side,

$$\min_{\mathbf{v}_h \in \mathcal{N}_{p-1}(K)} \|\mathbf{v} - \mathbf{v}_h\|_K^2 + \left( \frac{h_K}{p} \min_{w_h \in \mathcal{R}\mathcal{T}_{p-1}(K)} \|\nabla \times \mathbf{v} - \mathbf{w}_h\|_K \right)^2 \lesssim [v_{K,p+1,s,t}(\mathbf{v})]^2,$$
**Proposition 7.4** (p-robust one-sided bound). Let $v \in H_{0,N}(\text{curl}, \Omega)$, a simplicial mesh $\mathcal{T}_h$ of $\Omega$, and a polynomial degree $p \ge 1$ be fixed. If $\Gamma_D = \emptyset$ and the patch subdomains $\omega_a$ are convex for all vertices $a \in \mathcal{V}_h$, set $\delta := 1$, otherwise set $\delta := 0$. Then

$$ \begin{aligned} & \min_{v_h \in \mathcal{N}_p(\mathcal{T}_h) \cap \mathcal{H}_{0,N}(\text{curl}, \Omega)} \left[ \|v - v_h\|^2 + \sum_{K \in \mathcal{T}_h} \left( \frac{h_K}{(p+1)^\delta} \|\nabla \times (v - v_h)\|_K \right)^2 \right] \\ & \lesssim \sum_{K \in \mathcal{T}_h} \left[ \min_{v_h \in \mathcal{N}_{p-1}(K)} \|v - v_h\|_K^2 + \left( \frac{h_K}{p^\delta} \|\nabla \times v - \Pi_{\mathcal{RT}}^{p-1} (\nabla \times v)\|_K \right)^2 \right], \end{aligned} \quad (7.26) $$

where the hidden constant only depends on the shape-regularity parameter $\kappa_{\mathcal{T}_h}$ of the mesh $\mathcal{T}_h$.

*Proof.* We bound the left-hand side of (7.26) by

$$ \left[ \|v - h_h\|^2 + \sum_{K \in \mathcal{T}_h} \left( \frac{h_K}{(p+1)^\delta} \|\nabla \times (v - h_h)\|_K \right)^2 \right], $$

using $h_h$ from Definition 7.2. We now estimate the two above terms separately, in two steps.

*Step 1.* The triangle inequality gives, for $\iota_h$ given by (7.5),

$$ \|v - h_h\| \le \|v - \iota_h\| + \|\iota_h - h_h\|. $$

Thus, the constrained-unconstrained equivalence from Lemma A.1 (employed with $p-1$ in place of $p$) gives the desired result for the first term above (with $\delta = 1$ in any case). As for the second one, definition (7.8), the partition of unity (2.1), and an overlap estimate imply

$$ \|\iota_h - h_h\|^2 = \sum_{K \in \mathcal{T}_h} \left\| \sum_{a \in \mathcal{V}_K} (\psi^a \iota_h - h_h^a) \right\|_K^2 \le 4 \sum_{a \in \mathcal{V}_h} \|\psi^a \iota_h - h_h^a\|_{\omega_a}^2, $$

so that the desired bound follows from Lemma 7.3.

*Step 2.* From (7.10), the partition of unity (2.1), (7.3), and adding and subtracting $\psi^a \nabla \times \iota_h$,

$$ \begin{aligned} \|\nabla \times (v - h_h)\|_K &= \|\nabla \times v - \sigma_h - \delta_h\|_K \\ &= \left\| \sum_{a \in \mathcal{V}_K} \psi^a (\nabla \times (v - \iota_h)) + \sum_{a \in \mathcal{V}_K} (\psi^a \nabla \times \iota_h - \sigma_h^a) - \delta_h \right\|_K \\ &\le \sum_{a \in \mathcal{V}_K} \|\psi^a (\nabla \times (v - \iota_h))\|_{\omega_a} + \sum_{a \in \mathcal{V}_K} \|\psi^a \nabla \times \iota_h - \sigma_h^a\|_{\omega_a} + \|\delta_h\|_K. \end{aligned} $$

Recalling that the constraint in (7.5) gives $\nabla \times \iota_h = \tau_h$, we estimate the three terms separately.

For the first term, we estimate by (7.22)

$$ \|\psi^a(\nabla \times (v - \iota_h))\|_{\omega_a} \le \|\tau_h - \nabla \times v\|_{\omega_a} \stackrel{(7.22)}{\lesssim} \left\{ \sum_{K \in \mathcal{T}_a} \|\nabla \times v - \Pi_{\mathcal{RT}}^{p-1}(\nabla \times v)\|_K^2 \right\}^{1/2}. $$

For the second one, we use (7.4) to see that

$$ \|\psi^a \nabla \times \iota_h - \sigma_h^a\|_{\omega_a} \lesssim \left\{ \sum_{K \in \mathcal{T}_a} \|\nabla \times v - \Pi_{\mathcal{RT}}^{p-1}(\nabla \times v)\|_K^2 \right\}^{1/2}. $$

For the third one, we infer from (7.9) and (2.1)

$$ \|\delta_h\|_K = \left\| \sum_{a \in \mathcal{V}_K} (\theta_h^a - \nabla \psi^a \times \iota_h) \right\|_K \le \sum_{a \in \mathcal{V}_K} \|\theta_h^a - \nabla \psi^a \times \iota_h\|_{\omega_a}. $$

For the arising term, we can now use (7.23). Thus, multiplying $\|\nabla \times (v - h_h)\|_K$ by $h_K/(p+1)^\delta$, using that $h_K/(p+1)^\delta \le h_K/p^\delta$ as well as $1/(p+1)^\delta \le 1$, and invoking the mesh shape-regularity, the proof is finished. $\square$
where the constant beyond ≲ still only depends on $\kappa_{\mathcal{T}_h}$, $s$, and $t$. Thus, using (7.30) (recall that $\delta = 1$ here),

$$m^2 \le C_{\kappa_{\mathcal{T}_h}, s, t}^{\sharp} \sum_{K \in \mathcal{T}_h} [v_{K, p+1, s, t}(\mathbf{v})]^2 \quad (7.33)$$

for $C_{\kappa_{\mathcal{T}_h}, s, t}^{\sharp}$ only depending on $\kappa_{\mathcal{T}_h}$, $s$, and $t$ for all $p > \max\{s, t\}$.

**Conclusion.** The claim (7.31) and consequently (3.27) follows from (7.32) and (7.33) for the constant $\max\{C_{\kappa_{\mathcal{T}_h}, s, t}^{\star}, C_{\kappa_{\mathcal{T}_h}, s, t}^{\sharp}\}$ only depending on $\kappa_{\mathcal{T}_h}$, $s$, and $t$. □

*Proof of Theorem 3.11, case (i) (piecewise polynomial curl).* The proof is as above, replacing (7.28) by

$$[v_{K,q,s}(\mathbf{v})]^2 := \left( \frac{h_K^{\min\{q,s\}}}{q^s} \|\mathbf{v}\|_{H^s(K)} \right)^2$$

and considering the two cases $p \le s$ and $p > s$. Since one assumes $\nabla \times \mathbf{v} \in [\mathcal{P}_{p-1}(\mathcal{T}_h)]^3$, the second terms in (7.29) and (7.30) vanish. □

*Proof of Theorem 3.11, case (iii) (general case).* The proof is as above, modulo two adjustments. First, we need to replace (7.28) by

$$[v_{K,q,s,t}(\mathbf{v})]^2 := \left( \frac{h_K^{\min\{q,s\}}}{q^s} \|\mathbf{v}\|_{H^s(K)} \right)^2 + \left( h_K \frac{h_K^{\min\{q,t\}}}{q^t} \|\nabla \times \mathbf{v}\|_{H^t(K)} \right)^2.$$

Second, we use $h_K$ in place of $h_K/(p+1)$ in (7.27) and (7.29); this is available from (3.21b) in Theorem 3.9, noting that $1/(p+1) \le 1$. Third, we need to employ (7.30) from Proposition 7.4 with $\delta = 0$. □

# A $p$-robust equivalence of constrained and unconstrained best-approximation in $H(\text{curl})$ on a tetrahedron

In this Appendix, we extend [14, Lemma 1] to functions $\mathbf{v}$ with nonpolynomial curl, in the spirit of [26, Lemma A1]. This is a consequence of the breakthrough result of Costabel and McIntosh in [20, Proposition 4.2]. Interestingly enough, the constant hidden in the inequality is here independent of the polynomial degree $p$.

**Lemma A.1 (Equivalence of constrained and unconstrained best-approximation on a tetrahedron).** Let a polynomial degree $p \ge 0$, a tetrahedron $K$, and an arbitrary $\mathbf{v} \in H(\text{curl}, K)$ be fixed. Let

$$\tau_h := \arg \min_{\substack{\mathbf{w}_h \in \mathcal{RT}_p(K) \\ \nabla \cdot \mathbf{w}_h = 0}} \|\nabla \times \mathbf{v} - \mathbf{w}_h\|_K. \quad (A.1)$$

Then

$$\begin{align}
\min_{\mathbf{v}_h \in \mathcal{N}_p(K)} \|\mathbf{v} - \mathbf{v}_h\|_K &\le \min_{\substack{\mathbf{v}_h \in \mathcal{N}_p(K) \\ \nabla \times \mathbf{v}_h = \tau_h}} \|\mathbf{v} - \mathbf{v}_h\|_K \nonumber \\
&\lesssim \left( \min_{\mathbf{v}_h \in \mathcal{N}_p(K)} \|\mathbf{v} - \mathbf{v}_h\|_K + \frac{h_K}{p+1} \|\nabla \times \mathbf{v} - \mathbf{\Pi}_{\mathcal{RT}}^n (\nabla \times \mathbf{v})\|_K \right), \tag{A.2}
\end{align}$$

where the hidden constant only depends on the shape-regularity $\kappa_K := h_K/\rho_K$ of $K$.

*Proof.* The first inequality is obvious, since the second minimization set has an additional curl constraint. In order to show the second one, denote respectively by

$$\iota_h := \arg \min_{\substack{\mathbf{v}_h \in \mathcal{N}_p(K) \\ \nabla \times \mathbf{v}_h = \tau_h}} \|\mathbf{v} - \mathbf{v}_h\|_K \quad (A.3)$$

and

$$\tilde{\iota}_h := \arg \min_{\substack{\mathbf{v}_h \in \mathcal{N}_p(K) \\ \nabla \times \mathbf{v}_h = \tau_h}} \|\mathbf{v} - \mathbf{v}_h\|_K \quad (A.4)$$
Denote

$$m^2 := \min_{\mathbf{v}_h \in \mathcal{N}_p(\mathcal{T}_h) \cap \mathcal{H}_{0,N}(\text{curl}, \Omega)} \left[ \|\mathbf{v} - \mathbf{v}_h\|^2 + \sum_{K \in \mathcal{T}_h} \left( \frac{h_K}{p+1} \|\nabla \times (\mathbf{v} - \mathbf{v}_h)\|_K \right)^2 \right] \quad (7.27)$$

and

$$[v_{K,q,s,t}(\mathbf{v})]^2 := \left( \frac{h_K^{\min\{q,s\}}}{q^s} \|\mathbf{v}\|_{\mathcal{H}^s(K)} \right)^2 + \left( \frac{h_K}{q} \frac{h_K^{\min\{q,t\}}}{q^t} \|\nabla \times \mathbf{v}\|_{\mathcal{H}^t(K)} \right)^2. \quad (7.28)$$

From (3.21b) in Theorem 3.9 and the characterization (2.11b), we have

$$m^2 \lesssim \sum_{K \in \mathcal{T}_h} \left[ \min_{\mathbf{v}_h \in \mathcal{N}_p(K)} \|\mathbf{v} - \mathbf{v}_h\|_K^2 + \left( \frac{h_K}{p+1} \min_{w_h \in \mathcal{R}\mathcal{T}_p(K)} \|\nabla \times \mathbf{v} - \mathbf{w}_h\|_K \right)^2 \right], \quad (7.29)$$

where the hidden constant can depend on $\kappa_{\mathcal{T}_h}$ and the polynomial degree $p$. From Proposition 7.4, in turn,

$$m^2 \lesssim \sum_{K \in \mathcal{T}_h} \left[ \min_{\mathbf{v}_h \in \mathcal{N}_{p-1}(K)} \|\mathbf{v} - \mathbf{v}_h\|_K^2 + \left( \frac{h_K}{p^\delta} \min_{w_h \in \mathcal{R}\mathcal{T}_{p-1}(K)} \|\nabla \times \mathbf{v} - \mathbf{w}_h\|_K \right)^2 \right], \quad (7.30)$$

where $p$ has to be greater or equal to 1 but the hidden constant is independent of $p$. These are the two ingredients necessary for the proof of Theorem 3.11 in the spirit of the proof of Theorem 3.6 in [26, Section 5.2]:

*Proof of Theorem 3.11, case (ii) (convex patches and tangential boundary conditions).* Recall that the parameter $\delta$ from Proposition 7.4 is one here. Thus, employing the notations (7.27)–(7.28) and the claim (3.27), we need to show that

$$m^2 \lesssim \sum_{K \in \mathcal{T}_h} [v_{K,p+1,s,t}(\mathbf{v})]^2, \quad (7.31)$$

where the hidden constant only depends on $\kappa_{\mathcal{T}_h}$, $s$, and $t$. We distinguish two cases that we later combine together.

*Case 1.* Suppose $p \le \max\{s,t\}$. On each tetrahedron $K \in \mathcal{T}_h$, we have from (2.8) and (2.9) that $[\mathcal{P}_p(K)]^3 \subset \mathcal{N}_p(K)$ and $[\mathcal{P}_p(K)]^3 \subset \mathcal{R}\mathcal{T}_p(K)$. Thus the elementwise regularity assumptions (3.24) and (3.26) together with [4, Lemma 4.1] give

$$\min_{\mathbf{v}_h \in \mathcal{N}_p(K)} \|\mathbf{v} - \mathbf{v}_h\|_K^2 + \left( \frac{h_K}{p+1} \min_{w_h \in \mathcal{R}\mathcal{T}_p(K)} \|\nabla \times \mathbf{v} - \mathbf{w}_h\|_K \right)^2 \lesssim [v_{K,p+1,s,t}(\mathbf{v})]^2,$$

where the hidden constant only depends on $\kappa_{\mathcal{T}_h}$, $s$, and $t$. Thus, from (7.29), there exists a constant $C_{\kappa_{\mathcal{T}_h},p,s,t}$ only depending on $\kappa_{\mathcal{T}_h}$, $p$, $s$, and $t$ such that

$$m^2 \le C_{\kappa_{\mathcal{T}_h}, p, s, t} \sum_{K \in \mathcal{T}_h} [v_{K,p+1,s,t}(\mathbf{v})]^2.$$

Defining $C_{\kappa_{\mathcal{T}_h}, p, s, t}^* := \max_{0 \le p \le \max\{s,t\}} C_{\kappa_{\mathcal{T}_h}, p, s, t}$, only depending on $\kappa_{\mathcal{T}_h}$, $s$, and $t$, there holds, for all $0 \le p \le \max\{s,t\}$,

$$m^2 \le C_{\kappa_{\mathcal{T}_h}, p, s, t}^* \sum_{K \in \mathcal{T}_h} [v_{K,p+1,s,t}(\mathbf{v})]^2. \quad (7.32)$$

*Case 2.* Suppose $p > \max\{s,t\}$. Since $p$ is an integer, this implies that $p \ge 1$, so that we will be able to apply (7.30). As above, reducing $p+1$ to $p$ on the right-hand side,

$$\min_{\mathbf{v}_h \in \mathcal{N}_{p-1}(K)} \|\mathbf{v} - \mathbf{v}_h\|_K^2 + \left( \frac{h_K}{p} \min_{w_h \in \mathcal{R}\mathcal{T}_{p-1}(K)} \|\nabla \times \mathbf{v} - \mathbf{w}_h\|_K \right)^2 \lesssim [v_{K,p,s,t}(\mathbf{v})]^2,$$

where the hidden constant only depends on $\kappa_{\mathcal{T}_h}$, $s$, and $t$. Now, actually, $\min\{p,s\} = s = \min\{p+1,s\}$ and $p+1 \le 2p$, so that $1/p^s \le 2^s/(p+1)^s$. Similarly, $\min\{p,t\} = t = \min\{p+1,t\}$ and $1/p^t \le 2^t/(p+1)^t$. As a consequence, we can rise $p$ back to $p+1$ on the right-hand side,

$$\min_{\mathbf{v}_h \in \mathcal{N}_{p-1}(K)} \|\mathbf{v} - \mathbf{v}_h\|_K^2 + \left( \frac{h_K}{p} \min_{w_h \in \mathcal{R}\mathcal{T}_{p-1}(K)} \|\nabla \times \mathbf{v} - \mathbf{w}_h\|_K \right)^2 \lesssim [v_{K,p+1,s,t}(\mathbf{v})]^2,$$
the constrained and unconstrained minimizers. We then need to show

$$ \|v - \iota_h\|_K \lesssim \|v - \tilde{\iota}_h\|_K + \frac{h_K}{p+1} \|\nabla \times v - \Pi_{\mathcal{RT}}^p (\nabla \times v)\|_K, \quad (A.5) $$

where $\lesssim$ means inequality up to a constant only depending on the shape-regularity $\kappa_K$.

Using (A.1) and (4.2), $\tau_h - \nabla \times \tilde{\iota}_h \in \{w_h \in \mathcal{RT}_p(K); \nabla \cdot w_h = 0\}$. Thus, we can use [20, Proposition 4.2], cf. also the reformulation in [12, Theorem 2], stipulating the existence of $\varphi_h \in \mathcal{N}_p(K)$ with $\nabla \times \varphi_h = \tau_h - \nabla \times \tilde{\iota}_h$ such that

$$ \|\varphi_h\|_K \lesssim \min_{\substack{\varphi \in \mathcal{H}(\text{curl}, K) \\ \nabla \times \varphi = \tau_h - \nabla \times \tilde{\iota}_h}} \|\varphi\|_K. \quad (A.6) $$

Shifting now the right-hand side of (A.6) by $\tilde{\iota}_h$, we arrive at

$$ \|\varphi_h\|_K \lesssim \min_{\substack{\varphi \in \mathcal{H}(\text{curl}, K) \\ \nabla \times \varphi = \tau_h}} \|\varphi - \tilde{\iota}_h\|_K. $$

A primal-dual equivalence as in, e.g., [13, Lemma 5.5] implies (as in Section 2.3, $H_0(\text{curl}, K)$ is composed of those $\varphi \in \mathcal{H}(\text{curl}, K)$ that verify $\varphi \times n_K = 0$ on $\partial K$ in the weak sense (2.3))

$$ \begin{aligned} \min_{\substack{\varphi \in \mathcal{H}(\text{curl}, K) \\ \nabla \times \varphi = \tau_h}} \|\varphi - \tilde{\iota}_h\|_K &= \sup_{\substack{\varphi \in \mathcal{H}_0(\text{curl}, K) \\ \|\nabla \times \varphi\|_K = 1}} \{(\tau_h, \varphi)_K - (\tilde{\iota}_h, \nabla \times \varphi)_K\} \\ &\le \sup_{\substack{\varphi \in \mathcal{H}_0(\text{curl}, K) \\ \|\nabla \times \varphi\|_K = 1}} \{(\nabla \times v, \varphi)_K - (\tilde{\iota}_h, \nabla \times \varphi)_K\} \\ &\quad + \sup_{\substack{\varphi \in \mathcal{H}_0(\text{curl}, K) \\ \|\nabla \times \varphi\|_K = 1}} (\tau_h - \nabla \times v, \varphi)_K \\ &\lesssim \min_{\substack{\varphi \in \mathcal{H}(\text{curl}, K) \\ \nabla \times \varphi = \nabla \times v}} \|\varphi - \tilde{\iota}_h\|_K + \frac{h_K}{p+1} \|\nabla \times v - \Pi_{\mathcal{RT}}^p (\nabla \times v)\|_K, \end{aligned} $$

where, to estimate the second term on the middle line, we have used the technical result of Lemma A.2 below.

Consequently, since $v \in \mathcal{H}(\text{curl}, K)$ satisfies the curl constraint above,

$$ \|\varphi_h\|_K \lesssim \|v - \tilde{\iota}_h\|_K + \frac{h_K}{p+1} \|\nabla \times v - \Pi_{\mathcal{RT}}^p (\nabla \times v)\|_K. \quad (A.7) $$

Now note that $(\varphi_h + \tilde{\iota}_h) \in \mathcal{N}_p(K)$ with $\nabla \times (\varphi_h + \tilde{\iota}_h) = \tau_h$. Thus, $\varphi_h + \tilde{\iota}_h$ belongs to the minimization set in (A.3), and the minimization property (A.3) of $\iota_h$ implies $\|v - \iota_h\|_K \le \|v - (\varphi_h + \tilde{\iota}_h)\|_K$. Thus, by virtue of the triangle inequality and using (A.7), we altogether infer

$$ \begin{aligned} \|v - \iota_h\|_K &\le \|v - (\varphi_h + \tilde{\iota}_h)\|_K \le \|v - \tilde{\iota}_h\|_K + \|\varphi_h\|_K \\ &\lesssim \|v - \tilde{\iota}_h\|_K + \frac{h_K}{p+1} \|\nabla \times v - \Pi_{\mathcal{RT}}^p (\nabla \times v)\|_K, \end{aligned} $$

i.e., (A.5), and the proof is finished.

□

**Lemma A.2 (hp data oscillation).** Let the assumptions of Lemma A.1 be verified. Then

$$ \sup_{\substack{\varphi \in \mathcal{H}_0(\text{curl}, K) \\ \|\nabla \times \varphi\|_K = 1}} (\tau_h - \nabla \times v, \varphi)_K \lesssim \frac{h_K}{p+1} \|\nabla \times v - \Pi_{\mathcal{RT}}^p (\nabla \times v)\|_K, $$

where the hidden constant only depends on the shape-regularity $\kappa_K := h_K/\rho_K$ of K.

*Proof.* Fix $\varphi \in \mathcal{H}_0(\text{curl}, K)$ with $\|\nabla \times \varphi\|_K = 1$. From (2.15), there exists $\psi \in \mathcal{H}^1(K) \cap \mathcal{H}_0(\text{curl}, K)$ such that $\nabla \times \psi = \nabla \times \varphi$ and

$$ \|\nabla \psi\|_K \le \|\nabla \times \varphi\|_K = 1. $$

Since $\tau_h \in \mathcal{RT}_p(K)$ with $\nabla \cdot \tau_h = 0$, there exists $w_h \in \mathcal{N}_p(K)$ such that $\nabla \times w_h = \tau_h$. Thus, by the Green theorem,

$$ \begin{aligned} (\tau_h - \nabla \times v, \varphi)_K &= (\nabla \times (w_h - v), \varphi)_K = (w_h - v, \nabla \times \varphi)_K \\ &= (w_h - v, \nabla \times \psi)_K = (\tau_h - \nabla \times v, \psi)_K, \end{aligned} $$

and we conclude by the following Lemma A.3.

□
where the constant beyond ≲ still only depends on $κ_{T_h}$, s, and t. Thus, using (7.30) (recall that $δ = 1$ here),

$$m^2 \le C_{\kappa_{T_h}, s, t}^{\#} \sum_{K \in \mathcal{T}_h} [v_{K, p+1, s, t}(\mathbf{v})]^2 \quad (7.33)$$

for $C_{\kappa_{T_h}, s, t}^{\#}$ only depending on $κ_{T_h}$, s, and t for all $p > \max\{s, t\}$.

*Conclusion.* The claim (7.31) and consequently (3.27) follows from (7.32) and (7.33) for the constant $\max\{C_{\kappa_{T_h}, s, t}^*, C_{\kappa_{T_h}, s, t}^{\#}\}$ only depending on $κ_{T_h}$, s, and t. □

*Proof of Theorem 3.11, case (i) (piecewise polynomial curl).* The proof is as above, replacing (7.28) by

$$[v_{K,q,s}(\mathbf{v})]^2 := \left( \frac{h_K^{\min\{q,s\}}}{q^s} \|\mathbf{v}\|_{H^s(K)} \right)^2$$

and considering the two cases $p \le s$ and $p > s$. Since one assumes $∇×\mathbf{v} \in [\mathcal{P}_{p-1}(T_h)]^3$, the second terms in (7.29) and (7.30) vanish. □

*Proof of Theorem 3.11, case (iii) (general case).* The proof is as above, modulo two adjustments. First, we need to replace (7.28) by

$$[v_{K,q,s,t}(\mathbf{v})]^2 := \left( \frac{h_K^{\min\{q,s\}}}{q^s} \|\mathbf{v}\|_{H^s(K)} \right)^2 + \left( h_K \frac{h_K^{\min\{q,t\}}}{q^t} \|\nabla \times \mathbf{v}\|_{H^t(K)} \right)^2.$$

Second, we use $h_K$ in place of $h_K/(p+1)$ in (7.27) and (7.29); this is available from (3.21b) in Theorem 3.9, noting that $1/(p+1) \le 1$. Third, we need to employ (7.30) from Proposition 7.4 with $δ = 0$. □

# A p-robust equivalence of constrained and unconstrained best-approximation in $H(\text{curl})$ on a tetrahedron

In this Appendix, we extend [14, Lemma 1] to functions $\mathbf{v}$ with nonpolynomial curl, in the spirit of [26, Lemma A1]. This is a consequence of the breakthrough result of Costabel and McIntosh in [20, Proposition 4.2]. Interestingly enough, the constant hidden in the inequality is here independent of the polynomial degree $p$.

**Lemma A.1 (Equivalence of constrained and unconstrained best-approximation on a tetrahedron).** Let a polynomial degree $p \ge 0$, a tetrahedron $K$, and an arbitrary $\mathbf{v} \in H(\text{curl}, K)$ be fixed. Let

$$\tau_h := \arg \min_{\substack{\mathbf{w}_h \in \mathcal{R}\mathcal{T}_p(K) \\ \nabla \cdot \mathbf{w}_h = 0}} \|\nabla \times \mathbf{v} - \mathbf{w}_h\|_K. \quad (A.1)$$

Then

$$\begin{align}
\min_{\mathbf{v}_h \in \mathcal{N}_p(K)} \|\mathbf{v} - \mathbf{v}_h\|_K &\le \min_{\substack{\mathbf{v}_h \in \mathcal{N}_p(K) \\ \nabla \times \mathbf{v}_h = \tau_h}} \|\mathbf{v} - \mathbf{v}_h\|_K \nonumber \\
&\lesssim \left( \min_{\mathbf{v}_h \in \mathcal{N}_p(K)} \|\mathbf{v} - \mathbf{v}_h\|_K + \frac{h_K}{p+1} \|\nabla \times \mathbf{v} - \Pi_{\mathcal{R}\mathcal{T}}^p (\nabla \times \mathbf{v})\|_K \right), \tag{A.2}
\end{align}$$

where the hidden constant only depends on the shape-regularity $κ_K := h_K/ρ_K$ of $K$.

*Proof.* The first inequality is obvious, since the second minimization set has an additional curl constraint. In order to show the second one, denote respectively by

$$\iota_h := \arg \min_{\substack{\mathbf{v}_h \in \mathcal{N}_p(K) \\ \nabla \times \mathbf{v}_h = \tau_h}} \|\mathbf{v} - \mathbf{v}_h\|_K \quad (A.3)$$

and

$$\tilde{\iota}_h := \arg \min_{\mathbf{v}_h \in \mathcal{N}_p(K)} \|\mathbf{v} - \mathbf{v}_h\|_K \quad (A.4)$$

the constrained and unconstrained minimizers. We then need to show

$$\|\mathbf{v} - \iota_h\|_K \lesssim \|\mathbf{v} - \tilde{\iota}_h\|_K + \frac{h_K}{p+1} \|\nabla \times \mathbf{v} - \Pi_{\mathcal{R}\mathcal{T}}^p (\nabla \times \mathbf{v})\|_K, \quad (A.5)$$
**Lemma A.3** (hp data estimate). Let the assumptions of Lemma A.1 be verified. Let $\psi \in H^1(K)$. Then

$$|(\tau_h - \nabla \times \mathbf{v}, \psi)_K| \lesssim \frac{h_K}{p+1} \|\nabla \times \mathbf{v} - \Pi_{\mathcal{RT}}^p (\nabla \times \mathbf{v})\|_K \|\nabla \psi\|_K,$$

where the hidden constant only depends on the shape-regularity $\kappa_K := h_K/\rho_K$ of $K$.

*Proof.* Let $q \in H_0^1(K)$ be such that

$$(\nabla q, \nabla v)_K = (\psi, \nabla v)_K \quad \forall v \in H_0^1(K). \qquad (A.8)$$

Defining $\chi := \psi - \nabla q$, we have

$$\chi \in \mathcal{H}(\mathrm{div}, K) \text{ with } \nabla \cdot \chi = 0. \qquad (A.9)$$

Moreover, by the Green theorem, the right-hand side in (A.8) can be equivalently written as $(\psi, \nabla v)_K = -(\nabla \cdot \psi, v)_K$. Thus, since $K$ is convex, the elliptic regularity shift, see, e.g., [38, Chapter 3], gives $q \in H^2(K)$ with

$$\|\nabla(\nabla q)\|_K \lesssim \|\nabla \cdot \psi\|_K \le \|\nabla \psi\|_K,$$

where $\lesssim$ means inequality up to a constant only depending on the shape-regularity $\kappa_K$. Thus, in addition to (A.9), there also holds $\chi \in H^1(K)$ with

$$\|\nabla \chi\|_K \le \|\nabla \psi\|_K + \|\nabla(\nabla q)\|_K \lesssim \|\nabla \psi\|_K. \qquad (A.10)$$

Now, since from (A.1) $\tau_h - \nabla \times \mathbf{v} \in \mathcal{H}(\mathrm{div}, K)$ with $\nabla \cdot (\tau_h - \nabla \times \mathbf{v}) = 0$, there follows by the Green theorem

$$(\tau_h - \nabla \times \mathbf{v}, \nabla q)_K = 0.$$

Consequently,

$$(\tau_h - \nabla \times \mathbf{v}, \psi)_K = (\tau_h - \nabla \times \mathbf{v}, \chi)_K. \qquad (A.11)$$

Let respectively

$$\chi_h := \arg \min_{\substack{w_h \in \mathcal{RT}_p(K) \\ \nabla \cdot w_h = 0}} \|\chi - w_h\|_K$$

and

$$\tilde{\chi}_h := \arg \min_{w_h \in \mathcal{RT}_p(K)} \|\chi - w_h\|_K$$

be the constrained and unconstrained Raviart–Thomas approximations of $\chi$. The Euler–Lagrange conditions of (A.1) allow us to subtract $\chi_h$ (but not $\tilde{\chi}_h$) in (A.11), so that the Cauchy–Schwarz inequality and, crucially, the p-robust constrained–unconstrained $\mathcal{H}(\mathrm{div})$ equivalence of [26, Lemma A1] (note that $\nabla \cdot \chi = 0$) lead to

$$(\tau_h - \nabla \times \mathbf{v}, \chi)_K = (\tau_h - \nabla \times \mathbf{v}, \chi - \chi_h)_K \le \|\tau_h - \nabla \times \mathbf{v}\|_K \|\chi - \chi_h\|_K \qquad (A.12) \\ \lesssim \|\tau_h - \nabla \times \mathbf{v}\|_K \|\chi - \tilde{\chi}_h\|_K.$$

Now, since $\mathcal{RT}_p(K)$ contains by (2.9) polynomials up to degree $p$ in each component and since the minimization for $\tilde{\chi}_h$ is unconstrained, the hp approximation bound (2.17) gives

$$\|\chi - \tilde{\chi}_h\|_K \lesssim \frac{h_K}{p+1} \|\nabla \chi\|_K. \qquad (A.13)$$

Finally, in addition to (A.1), let

$$\tilde{\tau}_h := \arg \min_{w_h \in \mathcal{RT}_p(K)} \|\nabla \times \mathbf{v} - w_h\|_K,$$

i.e., from (2.11b), $\tilde{\tau}_h = \Pi_{\mathcal{RT}}^p (\nabla \times \mathbf{v})$. It follows once again from the p-robust constrained–unconstrained $\mathcal{H}(\mathrm{div})$ equivalence of [26, Lemma A1] (note that $\nabla \cdot (\nabla \times \mathbf{v}) = 0$) that

$$\|\tau_h - \nabla \times \mathbf{v}\|_K \lesssim \|\tilde{\tau}_h - \nabla \times \mathbf{v}\|_K = \|\nabla \times \mathbf{v} - \Pi_{\mathcal{RT}}^p (\nabla \times \mathbf{v})\|_K. \qquad (A.14)$$

Thus, the desired result is a combination of (A.11), (A.12), (A.13), and (A.14) together with (A.10). $\square$
where $\lesssim$ means inequality up to a constant only depending on the shape-regularity $\kappa_K$.

Using (A.1) and (4.2), $\tau_h - \nabla \times \tilde{\iota}_h \in \{w_h \in \mathcal{RT}_p(K); \nabla \cdot w_h = 0\}$. Thus, we can use [20, Proposition 4.2], cf. also the reformulation in [12, Theorem 2], stipulating the existence of $\varphi_h \in \mathcal{N}_p(K)$ with $\nabla \times \varphi_h = \tau_h - \nabla \times \tilde{\iota}_h$ such that

$$ \|\varphi_h\|_K \lesssim \min_{\substack{\varphi \in \mathcal{H}(\text{curl},K) \\ \nabla \times \varphi = \tau_h - \nabla \times \tilde{\iota}_h}} \|\varphi\|_K. \quad (A.6) $$

Shifting now the right-hand side of (A.6) by $\tilde{\iota}_h$, we arrive at

$$ \|\varphi_h\|_K \lesssim \min_{\substack{\varphi \in \mathcal{H}(\text{curl},K) \\ \nabla \times \varphi = \tau_h}} \|\varphi - \tilde{\iota}_h\|_K. $$

A primal-dual equivalence as in, e.g., [13, Lemma 5.5] implies (as in Section 2.3, $H_0(\text{curl}, K)$ is composed of those $\varphi \in \mathcal{H}(\text{curl}, K)$ that verify $\varphi \times n_K = 0$ on $\partial K$ in the weak sense (2.3))

$$ \begin{aligned} \min_{\substack{\varphi \in \mathcal{H}(\text{curl},K) \\ \nabla \times \varphi = \tau_h}} \|\varphi - \tilde{\iota}_h\|_K &= \sup_{\substack{\varphi \in \mathcal{H}_0(\text{curl},K) \\ \|\nabla \times \varphi\|_K = 1}} \{(\tau_h, \varphi)_K - (\tilde{\iota}_h, \nabla \times \varphi)_K\} \\ &\le \sup_{\substack{\varphi \in \mathcal{H}_0(\text{curl},K) \\ \|\nabla \times \varphi\|_K = 1}} \{(\nabla \times v, \varphi)_K - (\tilde{\iota}_h, \nabla \times \varphi)_K\} \\ &\quad + \sup_{\substack{\varphi \in \mathcal{H}_0(\text{curl},K) \\ \|\nabla \times \varphi\|_K = 1}} (\tau_h - \nabla \times v, \varphi)_K \\ &\lesssim \min_{\substack{\varphi \in \mathcal{H}(\text{curl},K) \\ \nabla \times \varphi = \nabla \times v}} \|\varphi - \tilde{\iota}_h\|_K + \frac{h_K}{p+1} \|\nabla \times v - \Pi_{\mathcal{RT}}^p (\nabla \times v)\|_K, \end{aligned} $$

where, to estimate the second term on the middle line, we have used the technical result of Lemma A.2 below.

Consequently, since $v \in \mathcal{H}(\text{curl}, K)$ satisfies the curl constraint above,

$$ \|\varphi_h\|_K \lesssim \|v - \tilde{\iota}_h\|_K + \frac{h_K}{p+1} \|\nabla \times v - \Pi_{\mathcal{RT}}^p (\nabla \times v)\|_K. \quad (A.7) $$

Now note that $(\varphi_h + \tilde{\iota}_h) \in \mathcal{N}_p(K)$ with $\nabla \times (\varphi_h + \tilde{\iota}_h) = \tau_h$. Thus, $\varphi_h + \tilde{\iota}_h$ belongs to the minimization set in (A.3), and the minimization property (A.3) of $\tilde{\iota}_h$ implies $\|v - \tilde{\iota}_h\|_K \le \|v - (\varphi_h + \tilde{\iota}_h)\|_K$. Thus, by virtue of the triangle inequality and using (A.7), we altogether infer

$$ \begin{aligned} \|v - \tilde{\iota}_h\|_K &\le \|v - (\varphi_h + \tilde{\iota}_h)\|_K \le \|v - \tilde{\iota}_h\|_K + \|\varphi_h\|_K \\ &\lesssim \|v - \tilde{\iota}_h\|_K + \frac{h_K}{p+1} \|\nabla \times v - \Pi_{\mathcal{RT}}^p (\nabla \times v)\|_K, \end{aligned} $$

i.e., (A.5), and the proof is finished.

<figure><img src="image_6.png" /></figure>

**Lemma A.2 (hp data oscillation).** Let the assumptions of Lemma A.1 be verified. Then

$$ \sup_{\substack{\varphi \in \mathcal{H}_0(\text{curl},K) \\ \|\nabla \times \varphi\|_K = 1}} (\tau_h - \nabla \times v, \varphi)_K \lesssim \frac{h_K}{p+1} \|\nabla \times v - \Pi_{\mathcal{RT}}^p (\nabla \times v)\|_K, $$

where the hidden constant only depends on the shape-regularity $\kappa_K := h_K/\rho_K$ of K.

*Proof.* Fix $\varphi \in \mathcal{H}_0(\text{curl}, K)$ with $\|\nabla \times \varphi\|_K = 1$. From (2.15), there exists $\psi \in \mathcal{H}^1(K) \cap \mathcal{H}_0(\text{curl}, K)$ such that $\nabla \times \psi = \nabla \times \varphi$ and

$$ \|\nabla \psi\|_K \le \|\nabla \times \varphi\|_K = 1. $$

Since $\tau_h \in \mathcal{RT}_p(K)$ with $\nabla \cdot \tau_h = 0$, there exists $w_h \in \mathcal{N}_p(K)$ such that $\nabla \times w_h = \tau_h$. Thus, by the Green theorem,

$$ \begin{aligned} (\tau_h - \nabla \times v, \varphi)_K &= (\nabla \times (w_h - v), \varphi)_K = (w_h - v, \nabla \times \varphi)_K \\ &= (w_h - v, \nabla \times \psi)_K = (\tau_h - \nabla \times v, \psi)_K, \end{aligned} $$

and we conclude by the following Lemma A.3.

<figure><img src="image_7.png" /></figure>
B Broken regular decomposition in a patch

Let $\mathcal{T}_a$ be a vertex patch for a mesh vertex $\mathbf{a} \in \mathcal{V}_h$ as in Section 2.2. Recall the local spaces (2.6) and (2.7), including the notation $\gamma_D$ standing for the faces sharing the vertex $\mathbf{a}$ and lying in $\bar{\Gamma}_D$. Then there holds:

**Lemma B.1** (Broken regular decomposition in a patch). Let $\varphi \in \mathbf{H}^\dagger(\text{curl}, \omega_a)$. Then there exists $\psi \in \mathbf{H}^\dagger(\text{curl}, \omega_a)$ with

$$ \psi|_{K} \in \mathbf{H}^{1}(K) \quad \forall K \in \mathcal{T}_{a} \qquad (\text{B.1a}) $$

such that

$$ \nabla \times \psi = \nabla \times \varphi \qquad (\text{B.1b}) $$

and

$$ \|\psi\|_{\omega_a} \lesssim h_{\omega_a} \|\nabla \times \varphi\|_{\omega_a}, \qquad (\text{B.1c}) $$

$$ \left\{ \sum_{K \in \mathcal{T}_a} \|\nabla \psi\|_K^2 \right\}^{1/2} \lesssim \|\nabla \times \varphi\|_{\omega_a}, \qquad (\text{B.1d}) $$

where the hidden constants only depend on the shape-regularity parameter $\kappa_{\mathcal{T}_h}$ of the mesh $\mathcal{T}_h$.

*Proof.* Our proof involves a reference patch. For a fixed shape-regularity parameter $\kappa_{\mathcal{T}_h}$, there is a maximal number of tetrahedra $|\mathcal{T}_a|$ in each vertex patch $\mathcal{T}_a$. Therefore, there exists a finite number of ways these tetrahedra can be connected together in the patch, and we may, for each value of $\kappa_{\mathcal{T}_h}$, define a finite set $\mathbb{R}$ of references patches of unit diameter in such a way that each vertex patch $\mathcal{T}_a$ in the mesh is the image of an element $\tilde{\mathcal{T}} \in \mathbb{R}$ through a piecewise affine map that preserves connectivity.

Let $\tilde{\mathcal{T}} \in \mathbb{R}$ be the reference patch associated with the given $\mathcal{T}_a$, and denote by $\mathcal{A}: \tilde{\omega} \to \omega_a$ the corresponding piecewise affine map, where $\tilde{\omega}$ is the open subdomain corresponding to $\tilde{\mathcal{T}}$. Then, for each $K \in \mathcal{T}_a$, we denote by

$$ (\mathcal{A}^c \mathbf{v})|_K := \mathbb{J}_K^{-\mathrm{T}} \mathbf{v} \circ \mathcal{A}^{-1}, \quad (\mathcal{A}^d \mathbf{v})|_K := (\det \mathbb{J}_K)^{-1} \mathbb{J}_K \mathbf{v} \circ \mathcal{A}^{-1}, \qquad (\text{B.2}) $$

the curl- and divergence-preserving Piola mappings associated with $\mathcal{A}$, where $\mathbb{J}_K$ is the (constant) Jacobian matrix of $\mathcal{A}$ on $K$, see e.g. [29, Section 7.2]. It is standard that $\mathcal{A}^c$ maps $\mathbf{H}^\dagger(\text{curl}, \tilde{\omega})$ into $\mathbf{H}^\dagger(\text{curl}, \omega_a)$, where $\mathbf{H}^\dagger(\text{curl}, \tilde{\omega})$ embeds essential boundary conditions on $\mathcal{A}^{-1}(\gamma_D)$. $\mathcal{A}^c$ and $\mathcal{A}^d$ are bijective, and we denote by $(\mathcal{A}^c)^{-1}$ and $(\mathcal{A}^d)^{-1}$ their inverses. Finally, we have the commuting property $\nabla \times (\mathcal{A}^c \cdot) = \mathcal{A}^d(\nabla \times \cdot)$.

Consider $\varphi \in \mathbf{H}^\dagger(\text{curl}, \omega_a)$ and let $\tilde{\varphi} := (\mathcal{A}^c)^{-1}(\varphi) \in \mathbf{H}^\dagger(\text{curl}, \tilde{\omega})$. We can employ (2.15) (where we take $\mathbf{w}$ in $\mathbf{H}_{0,D}^1(\omega)$) on the reference patch $\tilde{\omega}$, where $C_{L,\tilde{\omega}}$ is now a generic constant only depending on $\kappa_{\mathcal{T}_h}$ due to the above discussion. Therefore, there exists $\tilde{\psi} \in \mathbf{H}^1(\tilde{\omega})$, with $\tilde{\psi} = 0$ on $\mathcal{A}^{-1}(\gamma_D)$ if $\gamma_D$ includes at least one face and of componentwise mean value zero on $\tilde{\omega}$ otherwise such that

$$ \nabla \times \tilde{\psi} = \nabla \times \tilde{\varphi} \quad \text{and} \quad \|\nabla \tilde{\psi}\|_{\tilde{\omega}} \le C_{L, \tilde{\omega}} \|\nabla \times \tilde{\varphi}\|_{\tilde{\omega}}. $$

We also note that we have

$$ \|\tilde{\psi}\|_{\tilde{\omega}} \lesssim \|\nabla \tilde{\psi}\|_{\tilde{\omega}} \lesssim \|\nabla \times \tilde{\psi}\|_{\tilde{\omega}} = \|\nabla \times \tilde{\varphi}\|_{\tilde{\omega}} \qquad (\text{B.3}) $$

by the Poincaré-Friedrichs inequality (2.16), since $\tilde{\omega}$ is of unit diameter.

We then introduce $\psi := \mathcal{A}^c(\tilde{\psi}) \in \mathbf{H}^\dagger(\text{curl}, \omega_a)$ and claim that it satisfies all the requirements in (B.1). Indeed, elementwise, the Piola mapping $\mathcal{A}^c$ amounts to an affine change of coordinates and a multiplication by a constant matrix. Therefore, it is clear that it preserves smoothness elementwise and (B.1a) follows from $\tilde{\psi} \in \mathbf{H}^1(\tilde{\omega})$. Moreover, (B.1b) holds true since

$$ \begin{aligned} \nabla \times \psi &= \nabla \times \mathcal{A}^c(\tilde{\psi}) = \mathcal{A}^d(\nabla \times \tilde{\psi}) = \mathcal{A}^d(\nabla \times \tilde{\varphi}) = \mathcal{A}^d(\nabla \times ((\mathcal{A}^c)^{-1}(\varphi))) \\ &= \mathcal{A}^d((\mathcal{A}^d)^{-1}(\nabla \times \varphi)) = \nabla \times \varphi. \end{aligned} $$

The estimates (B.1c) and (B.1d) then follow from (B.3) and the usual scaling arguments due to the structure (B.2) of the Piola mappings. Indeed, for each $K \in \mathcal{T}_a$, we have, on the one hand,

$$ \begin{aligned} \|\psi\|_K^2 &= \|\mathcal{A}^c(\tilde{\psi})\|_K^2 = \|\mathbb{J}_K^{-\mathrm{T}} \tilde{\psi} \circ \mathcal{A}^{-1}\|_K^2 \\ &\lesssim h_K^{-2} \|\tilde{\psi} \circ \mathcal{A}^{-1}\|_K^2 \lesssim h_K^{-2} |K| \|\tilde{\psi}\|_K^2 \lesssim h_K \|\tilde{\psi}\|_K^2, \end{aligned} $$
**Lemma A.3** (hp data estimate). Let the assumptions of Lemma A.1 be verified. Let $\psi \in H^1(K)$. Then

$$|(\tau_h - \nabla \times \mathbf{v}, \psi)_K| \lesssim \frac{h_K}{p+1} \|\nabla \times \mathbf{v} - \Pi_{\mathcal{RT}}^p (\nabla \times \mathbf{v})\|_K \|\nabla \psi\|_K,$$

where the hidden constant only depends on the shape-regularity $\kappa_K := h_K/\rho_K$ of $K$.

*Proof.* Let $q \in H_0^1(K)$ be such that

$$(\nabla q, \nabla v)_K = (\psi, \nabla v)_K \quad \forall v \in H_0^1(K). \qquad (A.8)$$

Defining $\chi := \psi - \nabla q$, we have

$$\chi \in \mathcal{H}(\mathrm{div}, K) \text{ with } \nabla \cdot \chi = 0. \qquad (A.9)$$

Moreover, by the Green theorem, the right-hand side in (A.8) can be equivalently written as $(\psi, \nabla v)_K = -(\nabla \cdot \psi, v)_K$. Thus, since $K$ is convex, the elliptic regularity shift, see, e.g., [38, Chapter 3], gives $q \in H^2(K)$ with

$$\|\nabla(\nabla q)\|_K \lesssim \|\nabla \cdot \psi\|_K \le \|\nabla \psi\|_K,$$

where $\lesssim$ means inequality up to a constant only depending on the shape-regularity $\kappa_K$. Thus, in addition to (A.9), there also holds $\chi \in H^1(K)$ with

$$\|\nabla \chi\|_K \le \|\nabla \psi\|_K + \|\nabla(\nabla q)\|_K \lesssim \|\nabla \psi\|_K. \qquad (A.10)$$

Now, since from (A.1) $\tau_h - \nabla \times \mathbf{v} \in \mathcal{H}(\mathrm{div}, K)$ with $\nabla \cdot (\tau_h - \nabla \times \mathbf{v}) = 0$, there follows by the Green theorem

$$(\tau_h - \nabla \times \mathbf{v}, \nabla q)_K = 0.$$

Consequently,

$$(\tau_h - \nabla \times \mathbf{v}, \psi)_K = (\tau_h - \nabla \times \mathbf{v}, \chi)_K. \qquad (A.11)$$

Let respectively

$$\chi_h := \arg \min_{\substack{w_h \in \mathcal{RT}_p(K) \\ \nabla \cdot w_h = 0}} \|\chi - w_h\|_K$$

and

$$\tilde{\chi}_h := \arg \min_{\substack{w_h \in \mathcal{RT}_p(K) \\ \nabla \cdot w_h = 0}} \|\chi - w_h\|_K$$

be the constrained and unconstrained Raviart–Thomas approximations of $\chi$. The Euler–Lagrange conditions of (A.1) allow us to subtract $\chi_h$ (but not $\tilde{\chi}_h$) in (A.11), so that the Cauchy–Schwarz inequality and, crucially, the p-robust constrained–unconstrained $\mathcal{H}(\mathrm{div})$ equivalence of [26, Lemma A1] (note that $\nabla \cdot \chi = 0$) lead to

$$\begin{aligned} (\tau_h - \nabla \times \mathbf{v}, \chi)_K &= (\tau_h - \nabla \times \mathbf{v}, \chi - \chi_h)_K \le \|\tau_h - \nabla \times \mathbf{v}\|_K \|\chi - \chi_h\|_K \\ &\lesssim \|\tau_h - \nabla \times \mathbf{v}\|_K \|\chi - \tilde{\chi}_h\|_K. \end{aligned} \qquad (A.12)$$

Now, since $\mathcal{RT}_p(K)$ contains by (2.9) polynomials up to degree $p$ in each component and since the minimization for $\tilde{\chi}_h$ is unconstrained, the hp approximation bound (2.17) gives

$$\|\chi - \tilde{\chi}_h\|_K \lesssim \frac{h_K}{p+1} \|\nabla \chi\|_K. \qquad (A.13)$$

Finally, in addition to (A.1), let

$$\tilde{\tau}_h := \arg \min_{\substack{w_h \in \mathcal{RT}_p(K) \\ \nabla \cdot w_h = 0}} \|\nabla \times \mathbf{v} - w_h\|_K,$$

i.e., from (2.11b), $\tilde{\tau}_h = \Pi_{\mathcal{RT}}^p (\nabla \times \mathbf{v})$. It follows once again from the p-robust constrained–unconstrained $\mathcal{H}(\mathrm{div})$ equivalence of [26, Lemma A1] (note that $\nabla \cdot (\nabla \times \mathbf{v}) = 0$) that

$$\|\tau_h - \nabla \times \mathbf{v}\|_K \lesssim \|\tilde{\tau}_h - \nabla \times \mathbf{v}\|_K = \|\nabla \times \mathbf{v} - \Pi_{\mathcal{RT}}^p (\nabla \times \mathbf{v})\|_K. \qquad (A.14)$$

Thus, the desired result is a combination of (A.11), (A.12), (A.13), and (A.14) together with (A.10). $\square$
where $\tilde{K} = \mathcal{A}^{-1}(K)$. On the other hand, it is true that

$$ \begin{aligned} \|\nabla \times \tilde{\psi}\|_{\tilde{K}}^2 &= \|(\mathcal{A}^d)^{-1}(\nabla \times \psi)\|_{\tilde{K}}^2 = \|(\det \mathbb{J}_K) \mathbb{J}_K^{-1}(\nabla \times \psi) \circ \mathcal{A}\|_{\tilde{K}}^2 \\ &\lesssim (\det \mathbb{J}_K)^2 h_{\tilde{K}}^{-2} \|(\nabla \times \psi) \circ \mathcal{A}\|_{\tilde{K}}^2 \lesssim \frac{(\det \mathbb{J}_K)^2 h_{\tilde{K}}^{-2}}{|K|} \|\nabla \times \psi\|_{\tilde{K}}^2 \quad (B.4) \\ &\lesssim h_K \|\nabla \times \psi\|_{\tilde{K}}^2, \end{aligned} $$

so that (B.1c) follows from (B.3).

Finally, we establish (B.1d) by directly differentiating in (B.2). Specifically, we have

$$ \partial_{\ell} \psi = \mathbb{J}_{K}^{-T} \partial_{\ell} (\tilde{\psi} \circ \mathcal{A}^{-1}) = \mathbb{J}_{K}^{-T} \partial_{\ell} (\mathcal{A}_{j}^{-1}) (\partial_{j} \tilde{\psi} \circ \mathcal{A}^{-1}), $$

so that

$$ |\partial_{\ell} \psi| \le h_{\tilde{K}}^{-2} |\nabla \tilde{\psi} \circ \mathcal{A}^{-1}| $$

and

$$ \|\partial_{\ell} \psi\|_{K}^{2} \lesssim h_{K}^{-4} |K| \|\nabla \tilde{\psi}\|_{\tilde{K}}^{2}. $$

We therefore arrive at

$$ \sum_{K \in \mathcal{T}_a} \|\nabla \psi\|_{K}^{2} \lesssim h_{\omega_a}^{-1} \|\nabla \tilde{\psi}\|_{\tilde{\omega}}^{2}, $$

and the conclusion follows from (B.3) and (B.4). □

## Acknowledgment

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 647134 GATIPOR).

## References

[1] Adams, R. A. Pure and Applied Mathematics, Vol. 65. Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.

[2] Arnold, D., and Guzmán, J. Local $L^2$-bounded commuting projections in FEEC. ESAIM Math. Model. Numer. Anal. 55 (2021), 2169-2184. https://doi.org/10.1051/m2an/2021054.

[3] Arnold, D. N., Falk, R. S., and Winther, R. Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15 (2006), 1-155. https://doi.org/10.1017/S0962492906210018.

[4] Babuška, I., and Suri, M. The h-p version of the finite element method with quasi-uniform meshes. RAIRO Modél. Math. Anal. Numér. 21 (1987), 199-238. http://dx.doi.org/10.1051/m2an/1987210201991.

[5] Bebendorf, M. A note on the Poincaré inequality for convex domains. Z. Anal. Anwendungen 22 (2003), 751-756. http://dx.doi.org/10.4171/ZAA/1170.

[6] Beck, R., Hiptmair, R., Hoppe, R. H. W., and Wohlmuth, B. Residual based a posteriori error estimators for eddy current computation. M2AN Math. Model. Numer. Anal. 34 (2000), 159-182. https://doi.org/10.1051/m2an:2000136.

[7] Bespalov, A., and Heuer, N. Optimal error estimation for H(curl)-conforming p-interpolation in two dimensions. SIAM J. Numer. Anal. 47 (2009), 3977-3989. https://doi-org.ezproxy.is.cuni.cz/10.1137/090753802.

[8] Bespalov, A., and Heuer, N. A new H(div)-conforming p-interpolation operator in two dimensions. ESAIM Math. Model. Numer. Anal. 45 (2011), 255-275. https://doi.org/10.1051/m2an/2010039.

[9] Boffi, D., Brezzi, F., and Fortin, M. Mixed finite element methods and applications, vol. 44 of Springer Series in Computational Mathematics. Springer, Heidelberg, 2013. https://doi.org/10.1007/978-3-642-36519-5.
B Broken regular decomposition in a patch

Let $\mathcal{T}_a$ be a vertex patch for a mesh vertex $\mathbf{a} \in \mathcal{V}_h$ as in Section 2.2. Recall the local spaces (2.6) and (2.7), including the notation $\gamma_D$ standing for the faces sharing the vertex $\mathbf{a}$ and lying in $\bar{\Gamma}_D$. Then there holds:

**Lemma B.1** (Broken regular decomposition in a patch). Let $\varphi \in \mathbf{H}^\dagger(\text{curl}, \omega_a)$. Then there exists $\psi \in \mathbf{H}^\dagger(\text{curl}, \omega_a)$ with

$$ \psi|_{K} \in \mathbf{H}^1(K) \quad \forall K \in \mathcal{T}_a \qquad (\text{B.1a}) $$

such that

$$ \nabla \times \psi = \nabla \times \varphi \qquad (\text{B.1b}) $$

and

$$ \|\psi\|_{\omega_a} \lesssim h_{\omega_a} \|\nabla \times \varphi\|_{\omega_a}, \qquad (\text{B.1c}) $$

$$ \left\{ \sum_{K \in \mathcal{T}_a} \|\nabla \psi\|_K^2 \right\}^{1/2} \lesssim \|\nabla \times \varphi\|_{\omega_a}, \qquad (\text{B.1d}) $$

where the hidden constants only depend on the shape-regularity parameter $\kappa_{\mathcal{T}_h}$ of the mesh $\mathcal{T}_h$.

*Proof.* Our proof involves a reference patch. For a fixed shape-regularity parameter $\kappa_{\mathcal{T}_h}$, there is a maximal number of tetrahedra $|\mathcal{T}_a|$ in each vertex patch $\mathcal{T}_a$. Therefore, there exists a finite number of ways these tetrahedra can be connected together in the patch, and we may, for each value of $\kappa_{\mathcal{T}_h}$, define a finite set $\mathbb{R}$ of reference patches of unit diameter in such a way that each vertex patch $\mathcal{T}_a$ in the mesh is the image of an element $\tilde{\mathcal{T}} \in \mathbb{R}$ through a piecewise affine map that preserves connectivity.

Let $\tilde{\mathcal{T}} \in \mathbb{R}$ be the reference patch associated with the given $\mathcal{T}_a$, and denote by $\mathcal{A}: \tilde{\omega} \to \omega_a$ the corresponding piecewise affine map, where $\tilde{\omega}$ is the open subdomain corresponding to $\tilde{\mathcal{T}}$. Then, for each $K \in \mathcal{T}_a$, we denote by

$$ (\mathcal{A}^c \mathbf{v})|_K := \mathbb{J}_K^{-\mathrm{T}} \mathbf{v} \circ \mathcal{A}^{-1}, \quad (\mathcal{A}^d \mathbf{v})|_K := (\det \mathbb{J}_K)^{-1} \mathbb{J}_K \mathbf{v} \circ \mathcal{A}^{-1}, \qquad (\text{B.2}) $$

the curl- and divergence-preserving Piola mappings associated with $\mathcal{A}$, where $\mathbb{J}_K$ is the (constant) Jacobian matrix of $\mathcal{A}$ on $K$, see e.g. [29, Section 7.2]. It is standard that $\mathcal{A}^c$ maps $\mathbf{H}^\dagger(\text{curl}, \tilde{\omega})$ into $\mathbf{H}^\dagger(\text{curl}, \omega_a)$, where $\mathbf{H}^\dagger(\text{curl}, \tilde{\omega})$ embeds essential boundary conditions on $\mathcal{A}^{-1}(\gamma_D)$. $\mathcal{A}^c$ and $\mathcal{A}^d$ are bijective, and we denote by $(\mathcal{A}^c)^{-1}$ and $(\mathcal{A}^d)^{-1}$ their inverses. Finally, we have the commuting property $\nabla \times (\mathcal{A}^c \cdot) = \mathcal{A}^d(\nabla \times \cdot)$.

Consider $\varphi \in \mathbf{H}^\dagger(\text{curl}, \omega_a)$ and let $\tilde{\varphi} := (\mathcal{A}^c)^{-1}(\varphi) \in \mathbf{H}^\dagger(\text{curl}, \tilde{\omega})$. We can employ (2.15) (where we take $\mathbf{w}$ in $\mathbf{H}_{0,D}^1(\omega)$) on the reference patch $\tilde{\omega}$, where $C_{L,\tilde{\omega}}$ is now a generic constant only depending on $\kappa_{\mathcal{T}_h}$ due to the above discussion. Therefore, there exists $\tilde{\psi} \in \mathbf{H}^1(\tilde{\omega})$, with $\tilde{\psi} = 0$ on $\mathcal{A}^{-1}(\gamma_D)$ if $\gamma_D$ includes at least one face and of componentwise mean value zero on $\tilde{\omega}$ otherwise such that

$$ \nabla \times \tilde{\psi} = \nabla \times \tilde{\varphi} \quad \text{and} \quad \|\nabla \tilde{\psi}\|_{\tilde{\omega}} \le C_{L,\tilde{\omega}} \|\nabla \times \tilde{\varphi}\|_{\tilde{\omega}}. $$

We also note that we have

$$ \|\tilde{\psi}\|_{\tilde{\omega}} \lesssim \|\nabla \tilde{\psi}\|_{\tilde{\omega}} \lesssim \|\nabla \times \tilde{\psi}\|_{\tilde{\omega}} = \|\nabla \times \tilde{\varphi}\|_{\tilde{\omega}} \qquad (\text{B.3}) $$

by the Poincaré-Friedrichs inequality (2.16), since $\tilde{\omega}$ is of unit diameter.

We then introduce $\psi := \mathcal{A}^c(\tilde{\psi}) \in \mathbf{H}^\dagger(\text{curl}, \omega_a)$ and claim that it satisfies all the requirements in (B.1). Indeed, elementwise, the Piola mapping $\mathcal{A}^c$ amounts to an affine change of coordinates and a multiplication by a constant matrix. Therefore, it is clear that it preserves smoothness elementwise and (B.1a) follows from $\tilde{\psi} \in \mathbf{H}^1(\tilde{\omega})$. Moreover, (B.1b) holds true since

$$ \begin{aligned} \nabla \times \psi &= \nabla \times \mathcal{A}^c(\tilde{\psi}) = \mathcal{A}^d(\nabla \times \tilde{\psi}) = \mathcal{A}^d(\nabla \times \tilde{\varphi}) = \mathcal{A}^d(\nabla \times ((\mathcal{A}^c)^{-1}(\varphi))) \\ &= \mathcal{A}^d((\mathcal{A}^d)^{-1}(\nabla \times \varphi)) = \nabla \times \varphi. \end{aligned} $$

The estimates (B.1c) and (B.1d) then follow from (B.3) and the usual scaling arguments due to the structure (B.2) of the Piola mappings. Indeed, for each $K \in \mathcal{T}_a$, we have, on the one hand,

$$ \begin{aligned} \|\psi\|_K^2 &= \|\mathcal{A}^c(\tilde{\psi})\|_K^2 = \|\mathbb{J}_K^{-\mathrm{T}} \tilde{\psi} \circ \mathcal{A}^{-1}\|_K^2 \\ &\lesssim h_K^{-2} \|\tilde{\psi} \circ \mathcal{A}^{-1}\|_K^2 \lesssim h_K^{-2} |K| \|\tilde{\psi}\|_K^2 \lesssim h_K \|\tilde{\psi}\|_K^2, \end{aligned} $$
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where $\tilde{K} = \mathcal{A}^{-1}(K)$. On the other hand, it is true that

$$ \begin{aligned} \|\nabla \times \tilde{\psi}\|_{\tilde{K}}^2 &= \|(\mathcal{A}^d)^{-1}(\nabla \times \psi)\|_{\tilde{K}}^2 = \|(\det \mathbb{J}_K) \mathbb{J}_K^{-1}(\nabla \times \psi) \circ \mathcal{A}\|_{\tilde{K}}^2 \\ &\lesssim (\det \mathbb{J}_K)^2 h_K^{-2} \|(\nabla \times \psi) \circ \mathcal{A}\|_{\tilde{K}}^2 \lesssim \frac{(\det \mathbb{J}_K)^2 h_K^{-2}}{|K|} \|\nabla \times \psi\|_K^2 \quad (B.4) \\ &\lesssim h_K \|\nabla \times \psi\|_K^2, \end{aligned} $$

so that (B.1c) follows from (B.3).

Finally, we establish (B.1d) by directly differentiating in (B.2). Specifically, we have

$$ \partial_{\ell} \psi = \mathbb{J}_{K}^{-T} \partial_{\ell} (\tilde{\psi} \circ \mathcal{A}^{-1}) = \mathbb{J}_{K}^{-T} \partial_{\ell} (\mathcal{A}_{j}^{-1}) (\partial_{j} \tilde{\psi} \circ \mathcal{A}^{-1}), $$

so that

$$ |\partial_{\ell} \psi| \le h_K^{-2} |\nabla \tilde{\psi} \circ \mathcal{A}^{-1}| $$

and

$$ \|\partial_{\ell} \psi\|_{K}^{2} \lesssim h_{K}^{-4} |K| \|\nabla \tilde{\psi}\|_{\tilde{K}}^{2}. $$

We therefore arrive at

$$ \sum_{K \in \mathcal{T}_a} \|\nabla \psi\|_K^2 \lesssim h_{\omega_a}^{-1} \|\nabla \tilde{\psi}\|_{\tilde{\omega}}^2, $$

and the conclusion follows from (B.3) and (B.4). □

## Acknowledgment

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 647134 GATIPOR).

## References

[1] Adams, R. A. Pure and Applied Mathematics, Vol. 65. Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.

[2] Arnold, D., and Guzmán, J. Local $L^2$-bounded commuting projections in FEEC. ESAIM Math. Model. Numer. Anal. 55 (2021), 2169-2184. https://doi.org/10.1051/m2an/2021054.

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[4] Babuška, I., and Suri, M. The h-p version of the finite element method with quasi-uniform meshes. RAIRO Modél. Math. Anal. Numér. 21 (1987), 199-238. http://dx.doi.org/10.1051/m2an/1987210201991.

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