$\underline{v}_h, \underline{\zeta}_h \in \underline{X}_{\text{curl},h}^k$ 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 $(\underline{H}, \underline{A}) \in \mathbf{H}(\text{curl}; \Omega) \times \mathbf{H}(\text{div}; \Omega)$ be the weak solution to (7.1) and assuming that $\underline{H} \in \mathbf{C}^0(\bar{\Omega}) \cap \mathbf{H}^{k+2}(\mathcal{T}_h)$ and $\underline{A} \in \mathbf{C}^0(\bar{\Omega}) \times \mathbf{H}^{k+2}(\mathcal{T}_h)$, we have

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

where the hidden constant in $\lesssim$ only depends on $\Omega$, $k$, the mesh regularity parameter, and $\mu_-$, $\mu_+$.
Proof As shown in the proof of [29, Theorem 10], the exactness of the rightmost part of the sequence (3.37), which holds owing to (3.48) and (3.45), and the Poincaré inequalities for $\underline{C}_h^k$ and $D_h^k$ (see Theorems 4 and 5) enable a reproduction of the arguments of the continuous inf-sup condition (see, e.g., [31, Section 2] or [2, Theorem 4.9]) to see that $\mathcal{A}_h$ satisfies a uniform inf-sup condition with respect to the norm on $\underline{X}_{\text{curl},h}^k \times \underline{X}_{\text{div},h}^k$ induced by $\|\cdot\|_{\mu,\text{curl},1,h}$ and $\|\cdot\|_{\text{div},1,h}$.

Using the Third Strang Lemma [27], we therefore obtain (7.3) provided we can prove that the consistency error

$$ \mathcal{E}_h((H, A); (\underline{\zeta}_h, \underline{v}_h)) := \mathcal{L}_h(\underline{\zeta}_h, \underline{v}_h) - \mathcal{A}_h((\underline{I}_{\text{curl},h}^k H, \underline{I}_{\text{div},h}^k A), (\underline{\zeta}_h, \underline{v}_h)) $$

satisfies, for all $(\underline{\zeta}_h, \underline{v}_h) \in \underline{X}_{\text{curl},h}^k \times \underline{X}_{\text{div},h}^k$,

$$ \begin{aligned} \mathcal{E}_h((H, A); (\underline{\zeta}_h, \underline{v}_h)) & \lesssim h^{k+1} \left( |\text{curl } H|_{H^{k+1}(\mathcal{T}_h)} + |H|_{H^{(k+1,2)}(\mathcal{T}_h)} + |A|_{H^{k+1}(\mathcal{T}_h)} + |A|_{H^{k+2}(\mathcal{T}_h)} \right) \quad (7.4) \\ & \times \left( \|\underline{\zeta}_h\|_{\mu,\text{curl},1,h} + \|\underline{v}_h\|_{\text{div},1,h} \right). \end{aligned} $$

Expanding according to the respective definitions $\mathcal{A}_h$, $\mathcal{L}_h$, $a_h$, $b_h$, and $c_h$, we have

$$ \mathcal{E}_{h,1}((H, A); (\underline{\zeta}_h, \underline{v}_h)) = \mathcal{E}_{h,1}((H, A); (\underline{\zeta}_h, \underline{v}_h)) \\ + \mathcal{E}_{h,2}((H, A); (\underline{\zeta}_h, \underline{v}_h)) + \mathcal{E}_{h,3}((H, A); (\underline{\zeta}_h, \underline{v}_h)), \quad (7.5) $$

with

$$ \mathcal{E}_{h,1}((H, A); (\underline{\zeta}_h, \underline{v}_h)) := \sum_{T \in \mathcal{T}_h} \left( \int_T J \cdot P_{\text{div},T}^k \underline{v}_T - (\underline{C}_T^k (\underline{I}_{\text{curl},T}^k H), \underline{v}_T)_{\text{div},T} \right), $$

$$ \mathcal{E}_{h,2}((H, A); (\underline{\zeta}_h, \underline{v}_h)) := - \sum_{T \in \mathcal{T}_h} \int_T D_T^k (\underline{I}_{\text{div},T}^k A) D_T^k \underline{v}_T $$

$$ \mathcal{E}_{h,3}((H, A); (\underline{\zeta}_h, \underline{v}_h)) := - \sum_{T \in \mathcal{T}_h} \left( \mu_T (\underline{I}_{\text{curl},T}^k H, \underline{\zeta}_T)_{\text{curl},T} - (\underline{C}_T^k \underline{\zeta}_T, \underline{I}_{\text{div},T}^k A)_{\text{div},T} \right). $$

Let us first estimate $\mathcal{E}_{h,1}$. Recalling that $J = \text{curl } H$, using the commutation formula (3.39), invoking the consistency (6.13) of the discrete L²-product on $\underline{X}_{\text{div},T}^k$ and applying a Cauchy–Schwarz inequality, we have

$$ \begin{aligned} \mathcal{E}_{h,1}((H, A); (\underline{\zeta}_h, \underline{v}_h)) &= \sum_{T \in \mathcal{T}_h} \left( \int_T \text{curl } H \cdot P_{\text{div},T}^k \underline{v}_T - (\underline{I}_{\text{div},T}^k (\text{curl } H), \underline{v}_T)_{\text{div},T} \right) \\ &\lesssim \sum_{T \in \mathcal{T}_h} h_T^{k+1} |\text{curl } H|_{H^{k+1}(T)} \|\underline{v}_T\|_{\text{div},T} \\ &\le h^{k+1} |\text{curl } H|_{H^{k+1}(\mathcal{T}_h)} \|\underline{v}_h\|_{\text{div},h}. \end{aligned} \quad (7.6) $$
Proof As shown in the proof of [29, Theorem 10], the exactness of the rightmost part of the sequence (3.37), which holds owing to (3.48) and (3.45), and the Poincaré inequalities for $\underline{C}_h^k$ and $D_h^k$ (see Theorems 4 and 5) enable a reproduction of the arguments of the continuous inf-sup condition (see, e.g., [31, Section 2] or [2, Theorem 4.9]) to see that $\mathcal{A}_h$ satisfies a uniform inf-sup condition with respect to the norm on $\underline{X}_{\text{curl},h}^k \times \underline{X}_{\text{div},h}^k$ induced by $\|\cdot\|_{\mu,\text{curl},1,h}$ and $\|\cdot\|_{\text{div},1,h}$.

Using the Third Strang Lemma [27], we therefore obtain (7.3) provided we can prove that the consistency error

$$ \mathcal{E}_h((H, A); (\underline{\zeta}_h, \underline{v}_h)) := \mathcal{L}_h(\underline{\zeta}_h, \underline{v}_h) - \mathcal{A}_h((\underline{I}_{\text{curl},h}^k H, \underline{I}_{\text{div},h}^k A), (\underline{\zeta}_h, \underline{v}_h)) $$

satisfies, for all $(\underline{\zeta}_h, \underline{v}_h) \in \underline{X}_{\text{curl},h}^k \times \underline{X}_{\text{div},h}^k$,

$$ \begin{aligned} \mathcal{E}_h((H, A); (\underline{\zeta}_h, \underline{v}_h)) & \lesssim h^{k+1} \left( |\text{curl } H|_{H^{k+1}(\mathcal{T}_h)} + |H|_{H^{(k+1,2)}(\mathcal{T}_h)} + |A|_{H^{k+1}(\mathcal{T}_h)} + |A|_{H^{k+2}(\mathcal{T}_h)} \right) \quad (7.4) \\ & \times \left( \|\underline{\zeta}_h\|_{\mu,\text{curl},1,h} + \|\underline{v}_h\|_{\text{div},1,h} \right). \end{aligned} $$

Expanding according to the respective definitions $\mathcal{A}_h$, $\mathcal{L}_h$, $a_h$, $b_h$, and $c_h$, we have

$$ \mathcal{E}_h((H, A); (\underline{\zeta}_h, \underline{v}_h)) = \mathcal{E}_{h,1}((H, A); (\underline{\zeta}_h, \underline{v}_h)) \\ + \mathcal{E}_{h,2}((H, A); (\underline{\zeta}_h, \underline{v}_h)) + \mathcal{E}_{h,3}((H, A); (\underline{\zeta}_h, \underline{v}_h)), \quad (7.5) $$

with

$$ \mathcal{E}_{h,1}((H, A); (\underline{\zeta}_h, \underline{v}_h)) := \sum_{T \in \mathcal{T}_h} \left( \int_T J \cdot P_{\text{div},T}^k \underline{v}_T - (\underline{C}_T^k (\underline{I}_{\text{curl},T}^k H), \underline{v}_T)_{\text{div},T} \right), $$

$$ \mathcal{E}_{h,2}((H, A); (\underline{\zeta}_h, \underline{v}_h)) := - \sum_{T \in \mathcal{T}_h} \int_T D_T^k (\underline{I}_{\text{div},T}^k A) D_T^k \underline{v}_T $$

$$ \mathcal{E}_{h,3}((H, A); (\underline{\zeta}_h, \underline{v}_h)) := - \sum_{T \in \mathcal{T}_h} \left( \mu_T (\underline{I}_{\text{curl},T}^k H, \underline{\zeta}_T)_{\text{curl},T} - (\underline{C}_T^k \underline{\zeta}_T, \underline{I}_{\text{div},T}^k A)_{\text{div},T} \right). $$

Let us first estimate $\mathcal{E}_{h,1}$. Recalling that $J = \text{curl } H$, using the commutation formula (3.39), invoking the consistency (6.13) of the discrete L²-product on $\underline{X}_{\text{div},T}^k$ and applying a Cauchy–Schwarz inequality, we have

$$ \begin{aligned} \mathcal{E}_{h,1}((H, A); (\underline{\zeta}_h, \underline{v}_h)) &= \sum_{T \in \mathcal{T}_h} \left( \int_T \text{curl } H \cdot P_{\text{div},T}^k \underline{v}_T - (\underline{I}_{\text{div},T}^k (\text{curl } H), \underline{v}_T)_{\text{div},T} \right) \\ &\lesssim \sum_{T \in \mathcal{T}_h} h_T^{k+1} |\text{curl } H|_{H^{k+1}(T)} \|\underline{v}_T\|_{\text{div},T} \\ &\le h^{k+1} |\text{curl } H|_{H^{k+1}(\mathcal{T}_h)} \|\underline{v}_h\|_{\text{div},h}. \end{aligned} \quad (7.6) $$
To handle $\mathcal{E}_{h,2}$, we use the commutation formula (3.40) to get $D_T^k(I_{\text{div},T}^k A) = \pi_{\mathcal{P},T}^k(\text{div} A) = 0$, and thus

$$ \mathcal{E}_{h,2}((H, A); (\underline{\zeta}_h, \underline{v}_h)) = 0. \qquad (7.7) $$

Finally, we turn to $\mathcal{E}_{h,3}$. Since $A \in \mathbf{H}_0(\text{curl}; \Omega)$, the adjoint consistency Theorem 10 enables us to replace, in $\mathcal{E}_{h,3}$, the term $(\underline{C}_T^k \underline{\zeta}_T, \underline{I}_{\text{div},T}^k A)_{\text{div},T}$ with $\int_T \text{curl} A \cdot \mathbf{P}_{\text{curl},T}^k \underline{\zeta}_T = \int_T \mu_T H \cdot \mathbf{P}_{\text{curl},T}^k \underline{\zeta}_T$ up to a term that is controlled, i.e.,

$$ \begin{aligned} \mathcal{E}_{h,3}((H, A); (\underline{\zeta}_h, \underline{v}_h)) & \le -\sum_{T \in \mathcal{T}_h} \left( \mu_T(\underline{I}_{\text{curl},T}^k H, \underline{\zeta}_T)_{\text{curl},T} - \mu_T \int_T H \cdot \mathbf{P}_{\text{curl},T}^k \underline{\zeta}_T \right) \\ & \quad + h^{k+1} \left( |A|_{\mathbf{H}^{k+1}(\mathcal{T}_h)} + |A|_{\mathbf{H}^{k+2}(\mathcal{T}_h)} \right) \|\underline{\zeta}_h\|_{\mu, \text{curl}, 1, h} \\ & \le -\sum_{T \in \mathcal{T}_h} \left( \mu_T \int_T [\mathbf{P}_{\text{curl},T}^k (\underline{I}_{\text{curl},T}^k H) - H] \cdot \mathbf{P}_{\text{curl},T}^k \underline{\zeta}_T + s_{\text{curl},T}(\underline{I}_{\text{curl},T}^k H, \underline{\zeta}_T) \right) \\ & \quad + h^{k+1} \left( |A|_{\mathbf{H}^{k+1}(\mathcal{T}_h)} + |A|_{\mathbf{H}^{k+2}(\mathcal{T}_h)} \right) \|\underline{\zeta}_h\|_{\mu, \text{curl}, 1, h}, \end{aligned} $$

where we have used $\|\underline{\zeta}_h\|_{\text{curl},h} + \|\underline{C}_h^k \underline{\zeta}_h\|_{\text{div},h} \le \|\underline{\zeta}_h\|_{\mu, \text{curl}, 1, h}$ and the second inequality comes from expanding $(\cdot, \cdot)_{\text{curl},T}$ according to its definition. Cauchy-Schwarz inequalities and the consistency properties (6.3) and (6.9) then lead to

$$ \begin{aligned} \mathcal{E}_{h,3}((H, A); (\underline{\zeta}_h, \underline{v}_h)) & \le h^{k+1} |H|_{\mathbf{H}^{(k+1,2)}(\mathcal{T}_h)} \|\underline{\zeta}_h\|_{\text{curl},h} \\ & \quad + h^{k+1} \left( |A|_{\mathbf{H}^{k+1}(\mathcal{T}_h)} + |A|_{\mathbf{H}^{k+2}(\mathcal{T}_h)} \right) \|\underline{\zeta}_h\|_{\mu, \text{curl}, 1, h}. \end{aligned} $$

Plugging this estimate together with (7.6) and (7.7) into (7.5), we infer that (7.4) holds, which concludes the proof. $\square$

## 7.3 Numerical tests

We present here the results of some numerical tests obtained with the DDR scheme (7.2) for the magnetostatics model (7.1), focusing on comparing outputs obtained using either the complements (2.5), hereafter denoted by (K), or the orthogonal complements of [31,29], denoted by (⊥). Both versions of the DDR complex, and related schemes, have been implemented in the HArDCore3D C++ framework (see https://github.com/jdroniou/HArDCore), using linear algebra facilities from the Eigen3 library (see http://eigen.tuxfamily.org) and the Intel MKL PARDISO library (see https://software.intel.com/en-us/mkl) for the resolution of the global sparse linear system. This solver proved to be the most efficient among those at our disposal. All tests were run on a 16-inch 2019 MacBook Pro equipped with an 8-core Intel Core i9 processor (I9-9980HK) and 32Gb of RAM, and running macOS Big Sur
To handle $\mathcal{E}_{h,2}$, we use the commutation formula (3.40) to get $D_T^k(I_{\text{div},T}^k A) = \pi_{\mathcal{P},T}^k(\text{div} A) = 0$, and thus

$$ \mathcal{E}_{h,2}((H, A); (\underline{\zeta}_h, \underline{v}_h)) = 0. \qquad (7.7) $$

Finally, we turn to $\mathcal{E}_{h,3}$. Since $A \in \mathbf{H}_0(\text{curl}; \Omega)$, the adjoint consistency Theorem 10 enables us to replace, in $\mathcal{E}_{h,3}$, the term $(\underline{C}_T^k \underline{\zeta}_T, \underline{I}_{\text{div},T}^k A)_{\text{div},T}$ with $\int_T \text{curl} A \cdot \mathbf{P}_{\text{curl},T}^k \underline{\zeta}_T = \int_T \mu_T H \cdot \mathbf{P}_{\text{curl},T}^k \underline{\zeta}_T$ up to a term that is controlled, i.e.,

$$ \begin{aligned} \mathcal{E}_{h,3}((H, A); (\underline{\zeta}_h, \underline{v}_h)) & \le - \sum_{T \in \mathcal{T}_h} \left( \mu_T (\underline{I}_{\text{curl},T}^k H, \underline{\zeta}_T)_{\text{curl},T} - \mu_T \int_T H \cdot \mathbf{P}_{\text{curl},T}^k \underline{\zeta}_T \right) \\ & \quad + h^{k+1} \left( |A|_{\mathbf{H}^{k+1}(\mathcal{T}_h)} + |A|_{\mathbf{H}^{k+2}(\mathcal{T}_h)} \right) \|\underline{\zeta}_h\|_{\mu, \text{curl}, 1, h} \\ & \le - \sum_{T \in \mathcal{T}_h} \left( \mu_T \int_T \left[ \mathbf{P}_{\text{curl},T}^k (\underline{I}_{\text{curl},T}^k H) - H \right] \cdot \mathbf{P}_{\text{curl},T}^k \underline{\zeta}_T + \mathbf{s}_{\text{curl},T} (\underline{I}_{\text{curl},T}^k H, \underline{\zeta}_T) \right) \\ & \quad + h^{k+1} \left( |A|_{\mathbf{H}^{k+1}(\mathcal{T}_h)} + |A|_{\mathbf{H}^{k+2}(\mathcal{T}_h)} \right) \|\underline{\zeta}_h\|_{\mu, \text{curl}, 1, h}, \end{aligned} $$

where we have used $\|\underline{\zeta}_h\|_{\text{curl},h} + \|\underline{C}_h^k \underline{\zeta}_h\|_{\text{div},h} \le \|\underline{\zeta}_h\|_{\mu, \text{curl}, 1, h}$ and the second inequality comes from expanding $(\cdot, \cdot)_{\text{curl},T}$ according to its definition. Cauchy-Schwarz inequalities and the consistency properties (6.3) and (6.9) then lead to

$$ \begin{aligned} \mathcal{E}_{h,3}((H, A); (\underline{\zeta}_h, \underline{v}_h)) & \le h^{k+1} |H|_{\mathbf{H}^{(k+1,2)}(\mathcal{T}_h)} \|\underline{\zeta}_h\|_{\text{curl},h} \\ & \quad + h^{k+1} \left( |A|_{\mathbf{H}^{k+1}(\mathcal{T}_h)} + |A|_{\mathbf{H}^{k+2}(\mathcal{T}_h)} \right) \|\underline{\zeta}_h\|_{\mu, \text{curl}, 1, h}. \end{aligned} $$

Plugging this estimate together with (7.6) and (7.7) into (7.5), we infer that (7.4) holds, which concludes the proof. $\square$

## 7.3 Numerical tests

We present here the results of some numerical tests obtained with the DDR scheme (7.2) for the magnetostatics model (7.1), focusing on comparing outputs obtained using either the complements (2.5), hereafter denoted by (K), or the orthogonal complements of [31,29], denoted by (⊥). Both versions of the DDR complex, and related schemes, have been implemented in the HArDCore3D C++ framework (see https://github.com/jdroniou/HArDCore), using linear algebra facilities from the Eigen3 library (see http://eigen.tuxfamily.org) and the Intel MKL PARDISO library (see https://software.intel.com/en-us/mkl) for the resolution of the global sparse linear system. This solver proved to be the most efficient among those at our disposal. All tests were run on a 16-inch 2019 MacBook Pro equipped with an 8-core Intel Core i9 processor (I9-9980HK) and 32Gb of RAM, and running macOS Big Sur
version 11.5.1. We consider a constant permeability $\mu = 1$, and the same exact smooth solution and mesh families as in [29, Section 4.4] for comparability.

Figure 1 presents the errors, for various values of $k$, computed in the relative discrete $\mathbf{H}(\text{curl}; \Omega) \times \mathbf{H}(\text{div}; \Omega)$ norm:

$$ \frac{(\|\underline{\mathbf{H}}_h - \underline{\mathbf{I}}_{\text{curl},h}^k \mathbf{H}\|_{\mu,\text{curl},1,h}^2 + \|\underline{\mathbf{A}}_h - \underline{\mathbf{I}}_{\text{div},h}^k \mathbf{A}\|_{\text{div},1,h}^2)^{1/2}}{(\|\underline{\mathbf{I}}_{\text{curl},h}^k \mathbf{H}\|_{\mu,\text{curl},1,h}^2 + \|\underline{\mathbf{I}}_{\text{div},h}^k \mathbf{A}\|_{\text{div},1,h}^2)^{1/2}} $$

In the case of the Koszul complements, Theorem 12 states that this error should decrease as $O(h^{k+1})$ with the mesh size. No such estimate is known for the DDR scheme using orthogonal complements and, due to the lack of key properties of these complements (hierarchical inclusions, structure of traces), it is not clear whether the analysis carried out in the rest of this paper could be adapted to such complements. Nonetheless, the graphs in Figure 1 show that both schemes converge with an order $k + 1$. The errors between (K) and $(\perp)$ are essentially indistinguishable, except for $k \ge 1$ on tetrahedral meshes, where $(\perp)$ leads to slightly larger errors than (K) – about twice as large on the finest mesh with $k = 3$.

The assembly of the $(\perp)$-DDR scheme requires, for any $Y \in \mathcal{T}_h \cup \mathcal{F}_h$, to compute bases for the $L^2$-orthogonal complements in $\mathcal{P}^l(Y)$ of $\mathcal{G}^l(Y)$ and $\mathcal{R}^l(Y)$, which is done by computing the kernels of local matrices through a full pivot LU algorithm [29, Section 5.1]. On the contrary, in the (K) version, explicit bases for $\mathcal{G}^{c,l}(Y)$ and $\mathcal{R}^{c,l}(Y)$ can be devised; even though these bases are then orthonormalised to ensure a better numerical stability of the scheme (especially on non-isotropic elements, see the discussion in [28, Section B.1.1] on this topic), the computational cost of creating the polynomial bases in $(\perp)$ can be expected to be larger than in (K). Figure 2 compares the processor times for the two DDR schemes required for (a) the creation of the bases for local polynomial spaces and (b) the model construction (computation of the discrete operators, potentials, and $L^2$-products, and global system assembly). We do not compare the linear system resolution times as they are very close for both schemes. In all the cases, the finest mesh of each sequence is considered; see Table 3. A profiling of the code shows that numerical integration is by far the most expensive operation. We therefore include in Figure 2 also a comparison between two integration strategies on general meshes: on one hand, the Homogeneous Numerical Integration of [20]; on the other hand, the use of standard quadratures on a simplicial subdivision of (nonsimplicial) elements. In the left column of Figure 2 we report the total CPU time, which constitutes the most reliable measure to assess performance. Since our code makes use of multi-threading, we also report, in the right column, wall-clock times, which are more representative of real-life performance on the selected architecture. Wall-clock times are subject to outside influences, such as the impact of other processes, and should therefore be regarded with caution.

As expected, when considering standard quadratures on element subdivisions, (K) polynomial bases are faster to create than $(\perp)$ polynomial bases, but not by a large factor (this factor however becomes very large when $P_{\text{grad},T}^{k+1}$
version 11.5.1. We consider a constant permeability $\mu = 1$, and the same exact smooth solution and mesh families as in [29, Section 4.4] for comparability.

Figure 1 presents the errors, for various values of $k$, computed in the relative discrete $\mathbf{H}(\text{curl}; \Omega) \times \mathbf{H}(\text{div}; \Omega)$ norm:

$$ \frac{(\|\underline{\mathbf{H}}_h - \underline{\mathbf{I}}_{\text{curl},h}^k \mathbf{H}\|_{\mu,\text{curl},1,h}^2 + \|\underline{\mathbf{A}}_h - \underline{\mathbf{I}}_{\text{div},h}^k \mathbf{A}\|_{\text{div},1,h}^2)^{1/2}}{(\|\underline{\mathbf{I}}_{\text{curl},h}^k \mathbf{H}\|_{\mu,\text{curl},1,h}^2 + \|\underline{\mathbf{I}}_{\text{div},h}^k \mathbf{A}\|_{\text{div},1,h}^2)^{1/2}} $$

In the case of the Koszul complements, Theorem 12 states that this error should decrease as $O(h^{k+1})$ with the mesh size. No such estimate is known for the DDR scheme using orthogonal complements and, due to the lack of key properties of these complements (hierarchical inclusions, structure of traces), it is not clear whether the analysis carried out in the rest of this paper could be adapted to such complements. Nonetheless, the graphs in Figure 1 show that both schemes converge with an order $k + 1$. The errors between (K) and $(\perp)$ are essentially indistinguishable, except for $k \ge 1$ on tetrahedral meshes, where $(\perp)$ leads to slightly larger errors than (K) – about twice as large on the finest mesh with $k = 3$.

The assembly of the $(\perp)$-DDR scheme requires, for any $Y \in \mathcal{T}_h \cup \mathcal{F}_h$, to compute bases for the $L^2$-orthogonal complements in $\mathcal{P}^l(Y)$ of $\mathcal{G}^l(Y)$ and $\mathcal{R}^l(Y)$, which is done by computing the kernels of local matrices through a full pivot LU algorithm [29, Section 5.1]. On the contrary, in the (K) version, explicit bases for $\mathcal{G}^{c,l}(Y)$ and $\mathcal{R}^{c,l}(Y)$ can be devised; even though these bases are then orthonormalised to ensure a better numerical stability of the scheme (especially on non-isotropic elements, see the discussion in [28, Section B.1.1] on this topic), the computational cost of creating the polynomial bases in $(\perp)$ can be expected to be larger than in (K). Figure 2 compares the processor times for the two DDR schemes required for (a) the creation of the bases for local polynomial spaces and (b) the model construction (computation of the discrete operators, potentials, and $L^2$-products, and global system assembly). We do not compare the linear system resolution times as they are very close for both schemes. In all the cases, the finest mesh of each sequence is considered; see Table 3. A profiling of the code shows that numerical integration is by far the most expensive operation. We therefore include in Figure 2 also a comparison between two integration strategies on general meshes: on one hand, the Homogeneous Numerical Integration of [20]; on the other hand, the use of standard quadratures on a simplicial subdivision of (nonsimplicial) elements. In the left column of Figure 2 we report the total CPU time, which constitutes the most reliable measure to assess performance. Since our code makes use of multi-threading, we also report, in the right column, wall-clock times, which are more representative of real-life performance on the selected architecture. Wall-clock times are subject to outside influences, such as the impact of other processes, and should therefore be regarded with caution.

As expected, when considering standard quadratures on element subdivisions, (K) polynomial bases are faster to create than $(\perp)$ polynomial bases, but not by a large factor (this factor however becomes very large when $P_{\text{grad},T}^{k+1}$
<figure><img src="image_3.png" /><figcaption>Fig. 1: Relative error estimates in discrete $H(\text{curl}; \Omega) \times H(\text{div}; \Omega)$ norm vs. $h$, for the Koszul complements of (2.5) [(K), continuous lines], and the orthogonal complements of [31][($\perp$), dashed lines].</figcaption></figure>

is required, which is not the case for the scheme (7.2), as the computation of $(\mathcal{R}^{k+2}(T))^{\perp}$ (see (4.1)) necessitates to integrate polynomials of degree $2k + 4$ over the elements). There is a more pronounced difference when comparing the time for model construction, which is mostly dedicated to the creation of the discrete vector calculus operators and potentials in $X_{\text{curl},h}^k$ and $X_{\text{div},h}^k$ (once these are created, assembling the global linear system itself takes only a small fraction of the total model construction time). Basis construction and model assembly times, on the other hand, basically even out between (K) and ($\perp$) when considering Homogeneous Numerical Integration, thereby showing the importance of efficient integral computation. Drawing more definitive conclusions is always difficult, as running times highly depend on specific implementation choices, and our implementation is designed for flexibility rather than for efficiency on one given model. The results presented in this section seem
<figure><img src="image_4.png" /><figcaption>Fig. 1: Relative error estimates in discrete $H(\text{curl}; \Omega) \times H(\text{div}; \Omega)$ norm vs. $h$, for the Koszul complements of (2.5) [(K), continuous lines], and the orthogonal complements of [31][($\perp$), dashed lines].</figcaption></figure>

is required, which is not the case for the scheme (7.2), as the computation of $(\mathcal{R}^{k+2}(T))^{\perp}$ (see (4.1)) necessitates to integrate polynomials of degree $2k + 4$ over the elements). There is a more pronounced difference when comparing the time for model construction, which is mostly dedicated to the creation of the discrete vector calculus operators and potentials in $X_{\text{curl},h}^k$ and $X_{\text{div},h}^k$ (once these are created, assembling the global linear system itself takes only a small fraction of the total model construction time). Basis construction and model assembly times, on the other hand, basically even out between (K) and ($\perp$) when considering Homogeneous Numerical Integration, thereby showing the importance of efficient integral computation. Drawing more definitive conclusions is always difficult, as running times highly depend on specific implementation choices, and our implementation is designed for flexibility rather than for efficiency on one given model. The results presented in this section seem
<table><caption>(a) Number of relevant mesh entities</caption><thead><tr><th>Mesh</th><th>card(Th)</th><th>card(Fh)</th><th>card(Eh)</th></tr></thead><tbody><tr><td>Cubic_Cells</td><td>4 096</td><td>13 056</td><td>13 872</td></tr><tr><td>Tetgen_Cube-0</td><td>2 925</td><td>6 228</td><td>3 965</td></tr><tr><td>Voro-small-0</td><td>2 197</td><td>15 969</td><td>27 546</td></tr><tr><td>Voro-small-1</td><td>356</td><td>2 376</td><td>4 042</td></tr></tbody></table>

<table><caption>(b) Dimension of the space $\mathbf{X}_{\text{curl},h}^k$ for $k \in \{0, \dots, 3\}$</caption><thead><tr><th>Mesh</th><th>dim(<i>X</i><sup>0</sup><sub>curl,h</sub>)</th><th>dim(<i>X</i><sup>1</sup><sub>curl,h</sub>)</th><th>dim(<i>X</i><sup>2</sup><sub>curl,h</sub>)</th><th>dim(<i>X</i><sup>3</sup><sub>curl,h</sub>)</th></tr></thead><tbody><tr><td>Cubic_Cells</td><td>13 872</td><td>83 296</td><td>207 504</td><td>398 784</td></tr><tr><td>Tetgen_Cube-0</td><td>3 956</td><td>38 314</td><td>105 594</td><td>214 580</td></tr><tr><td>Voro-small-0</td><td>27 546</td><td>111 787</td><td>243 345</td><td>—</td></tr><tr><td>Voro-small-1</td><td>4 042</td><td>16 636</td><td>36 474</td><td>64 624</td></tr></tbody></table>

<table><caption>(c) Dimension of the space $\mathbf{X}_{\text{div},h}^k$ for $k \in \{0, \dots, 3\}$</caption><thead><tr><th>Mesh</th><th>dim(<i>X</i><sup>0</sup><sub>div,h</sub>)</th><th>dim(<i>X</i><sup>1</sup><sub>div,h</sub>)</th><th>dim(<i>X</i><sup>2</sup><sub>div,h</sub>)</th><th>dim(<i>X</i><sup>3</sup><sub>div,h</sub>)</th></tr></thead><tbody><tr><td>Cubic_Cells</td><td>13 056</td><td>63 744</td><td>160 256</td><td>314 880</td></tr><tr><td>Tetgen_Cube-0</td><td>6 228</td><td>36 234</td><td>95 868</td><td>193 905</td></tr><tr><td>Voro-small-0</td><td>15 969</td><td>61 089</td><td>139 754</td><td>—</td></tr><tr><td>Voro-small-1</td><td>2 376</td><td>9 264</td><td>21 376</td><td>39 780</td></tr></tbody></table>

Table 3: Dimension of meshes and spaces considered for the evaluation of computational times in the numerical tests of Section 7.3.

to show, however, that the DDR complex using Koszul complements is not only theoretically better (as it allows for complete consistency analysis and error estimates), but also requires less computational resources, at least when efficient integration is not available in the codes at hand. The comparison of CPU times and wall clock times also confirms that the assembly step strongly benefits from parallel implementations.

To close this section, we briefly assess the evolution of construction and solution times with mesh refinement. On a linear problem such as the one considered here, it is expected that the solution time be larger than the construction time starting from a certain number of elements. To check whether this is the case, we consider the Voronoi mesh family “Voro-small-0” and the polynomial degree $k = 2$ for the sequence. This test is representative of a worst-case scenario for the construction time, since mesh elements are genuinely polyhedral and it is not possible to optimise the construction using standard (reference element) techniques. The plots in Figure 3 show that the asymptotic behaviour of the construction times (“Bases” and “Model”) scale linearly with the number of elements (with the former being essentially negligible with respect to the latter), whereas the solution time (“Solve”) has a quadratic scaling. The solution time exceeds the construction time starting from the fourth mesh in the sequence, which has 729 elements (a small number for a three-dimensional computation). For meshes of real-life geometries,
<table><caption>(a) Number of relevant mesh entities</caption><thead><tr><th>Mesh</th><th>card(Th)</th><th>card(Fh)</th><th>card(Eh)</th></tr></thead><tbody><tr><td>Cubic_Cells</td><td>4 096</td><td>13 056</td><td>13 872</td></tr><tr><td>Tetgen_Cube-0</td><td>2 925</td><td>6 228</td><td>3 965</td></tr><tr><td>Voro-small-0</td><td>2 197</td><td>15 969</td><td>27 546</td></tr><tr><td>Voro-small-1</td><td>356</td><td>2 376</td><td>4 042</td></tr></tbody></table>

<table><caption>(b) Dimension of the space $\mathbf{X}_{\text{curl},h}^k$ for $k \in \{0, \dots, 3\}$</caption><thead><tr><th>Mesh</th><th>dim(<i>X</i><sup>0</sup><sub>curl,h</sub>)</th><th>dim(<i>X</i><sup>1</sup><sub>curl,h</sub>)</th><th>dim(<i>X</i><sup>2</sup><sub>curl,h</sub>)</th><th>dim(<i>X</i><sup>3</sup><sub>curl,h</sub>)</th></tr></thead><tbody><tr><td>Cubic_Cells</td><td>13 872</td><td>83 296</td><td>207 504</td><td>398 784</td></tr><tr><td>Tetgen_Cube-0</td><td>3 956</td><td>38 314</td><td>105 594</td><td>214 580</td></tr><tr><td>Voro-small-0</td><td>27 546</td><td>111 787</td><td>243 345</td><td>—</td></tr><tr><td>Voro-small-1</td><td>4 042</td><td>16 636</td><td>36 474</td><td>64 624</td></tr></tbody></table>

<table><caption>(c) Dimension of the space $\mathbf{X}_{\text{div},h}^k$ for $k \in \{0, \dots, 3\}$</caption><thead><tr><th>Mesh</th><th>dim(<i>X</i><sup>0</sup><sub>div,h</sub>)</th><th>dim(<i>X</i><sup>1</sup><sub>div,h</sub>)</th><th>dim(<i>X</i><sup>2</sup><sub>div,h</sub>)</th><th>dim(<i>X</i><sup>3</sup><sub>div,h</sub>)</th></tr></thead><tbody><tr><td>Cubic_Cells</td><td>13 056</td><td>63 744</td><td>160 256</td><td>314 880</td></tr><tr><td>Tetgen_Cube-0</td><td>6 228</td><td>36 234</td><td>95 868</td><td>193 905</td></tr><tr><td>Voro-small-0</td><td>15 969</td><td>61 089</td><td>139 754</td><td>—</td></tr><tr><td>Voro-small-1</td><td>2 376</td><td>9 264</td><td>21 376</td><td>39 780</td></tr></tbody></table>

Table 3: Dimension of meshes and spaces considered for the evaluation of computational times in the numerical tests of Section 7.3.

to show, however, that the DDR complex using Koszul complements is not only theoretically better (as it allows for complete consistency analysis and error estimates), but also requires less computational resources, at least when efficient integration is not available in the codes at hand. The comparison of CPU times and wall clock times also confirms that the assembly step strongly benefits from parallel implementations.

To close this section, we briefly assess the evolution of construction and solution times with mesh refinement. On a linear problem such as the one considered here, it is expected that the solution time be larger than the construction time starting from a certain number of elements. To check whether this is the case, we consider the Voronoi mesh family “Voro-small-0” and the polynomial degree $k = 2$ for the sequence. This test is representative of a worst-case scenario for the construction time, since mesh elements are genuinely polyhedral and it is not possible to optimise the construction using standard (reference element) techniques. The plots in Figure 3 show that the asymptotic behaviour of the construction times (“Bases” and “Model”) scale linearly with the number of elements (with the former being essentially negligible with respect to the latter), whereas the solution time (“Solve”) has a quadratic scaling. The solution time exceeds the construction time starting from the fourth mesh in the sequence, which has 729 elements (a small number for a three-dimensional computation). For meshes of real-life geometries,
one can thus expect that the solution time will be the dominating cost (even if more efficient linear solvers become available at some point).

## A Results on local spaces

This section collects miscellaneous results on the Koszul complements defined in (2.3) and (2.5), as well as on the trimmed spaces (2.12) obtained from the latter. The first result on traces of Raviart–Thomas and Nédélec functions is known on simplices, see e.g. [10, Proposition 2.3.3]; we however provide a proof on general polyhedra for sake of completeness.

**Proposition 8 (Traces of Nédélec and Raviart–Thomas functions) It holds, for all $F \in \mathcal{T}_h$,**

$$ \forall E \in \mathcal{E}_F \quad (v_F)|_E \cdot t_E \in \mathcal{P}^{\ell-1}(E) \quad \forall v_F \in \mathcal{N}^\ell(F), \quad (A.1) $$

$$ \forall E \in \mathcal{E}_F \quad (w_F)|_E \cdot n_{FE} \in \mathcal{P}^{\ell-1}(E) \quad \forall w_F \in \mathcal{RT}^\ell(F) \quad (A.2) $$

and, for all $T \in \mathcal{T}_h$,

$$ \forall E \in \mathcal{E}_T \quad (v_T)|_E \cdot t_E \in \mathcal{P}^{\ell-1}(E) \quad \forall v_T \in \mathcal{N}^\ell(T), \quad (A.3) $$

$$ \forall F \in \mathcal{T}_T \quad (w_T)|_F \cdot n_F \in \mathcal{P}^{\ell-1}(F) \quad \forall w_T \in \mathcal{RT}^\ell(T), \quad (A.4) $$

$$ \forall F \in \mathcal{T}_T \quad (v_T)|_F \times n_F \in \mathcal{RT}^\ell(F) \quad \forall v_T \in \mathcal{N}^\ell(T). \quad (A.5) $$

**Proof 1.** *Proof of (A.1) and (A.3).* The tangent edge traces of functions in $\mathcal{G}^{\ell-1}(F)$ (resp. $\mathcal{G}^{\ell-1}(T)$) are in $\mathcal{P}^{\ell-1}(E)$ for all $E \in \mathcal{E}_F$ (resp. $E \in \mathcal{E}_T$). Let, for any $E \in \mathcal{E}_F$, $x_E$ denote the middle point of $E$. To prove (A.1), it then suffices to recall the definition (2.3a) and observe that the quantity $(x - x_F)^\perp \cdot t_E = (x - x_E)^\perp \cdot t_E + (x_E - x_F)^\perp \cdot t_E$ is constant over $E$, the cancellation coming from the fact that $(x - x_E)$ and $t_E$ are parallel for all $x \in E$. To prove (A.3), recall the definition (2.5a) of $\mathcal{G}^{c,\ell}(T)$ and observe that, for all $v \in \mathcal{P}^{\ell-1}(T)$ and all $x \in E$, $[(x - x_T) \times v] \cdot t_E = [(x - x_E) \times v] \cdot t_E + [(x_E - x_T) \times v] \cdot t_E \in \mathcal{P}^{\ell-1}(E)$, where the cancellation follows observing, as before, that the vectors $(x - x_E)$ and $t_E$ are parallel.

**2.** *Proof of (A.2) and (A.4).* The normal traces of functions in $\mathcal{R}^{\ell-1}(F)$ (resp. $\mathcal{R}^{\ell-1}(T)$) are in $\mathcal{P}^{\ell-1}(E)$ (resp. $\mathcal{P}^{\ell-1}(F)$) for all $E \in \mathcal{E}_F$ (resp. $F \in \mathcal{T}_T$). To conclude, recall the definition (2.3b) (resp. (2.5b)) of $\mathcal{R}^{c,\ell}(F)$ (resp. $\mathcal{R}^{c,\ell}(T)$) and observe, using similar arguments as above, that the quantity $(x - x_F) \cdot n_{FE}$ (resp. $(x - x_T) \cdot n_F$) is constant for any $x \in E$ (resp. $x \in F$). This implies, in particular, for all $T \in \mathcal{T}_h$,

$$ \forall F \in \mathcal{T}_T, \quad (z_T)|_F \cdot n_F \in \mathcal{P}^{\ell-1}(F) \quad \forall z_T \in \mathcal{R}^{c,\ell}(T). $$

**3.** *Proof of (A.5).* For all $v_T \in \mathcal{G}^{\ell-1}(T) \subset \mathcal{P}^{\ell-1}(T)$ we have $(v_T)|_F \times n_F \in \mathcal{P}^{\ell-1}(F) \subset \mathcal{RT}^\ell(F)$ since $n_F$ is constant. It therefore suffices to prove (A.5)
one can thus expect that the solution time will be the dominating cost (even if more efficient linear solvers become available at some point).

## A Results on local spaces

This section collects miscellaneous results on the Koszul complements defined in (2.3) and (2.5), as well as on the trimmed spaces (2.12) obtained from the latter. The first result on traces of Raviart-Thomas and Nédélec functions is known on simplices, see e.g. [10, Proposition 2.3.3]; we however provide a proof on general polyhedra for sake of completeness.

**Proposition 8 (Traces of Nédélec and Raviart-Thomas functions) It holds, for all $F \in \mathcal{T}_h$,**

$$ \forall E \in \mathcal{E}_F \quad (v_F)|_E \cdot t_E \in \mathcal{P}^{\ell-1}(E) \quad \forall v_F \in \mathcal{N}^\ell(F), \quad (A.1) $$

$$ \forall E \in \mathcal{E}_F \quad (w_F)|_E \cdot n_{FE} \in \mathcal{P}^{\ell-1}(E) \quad \forall w_F \in \mathcal{RT}^\ell(F) \quad (A.2) $$

and, for all $T \in \mathcal{T}_h$,

$$ \forall E \in \mathcal{E}_T \quad (v_T)|_E \cdot t_E \in \mathcal{P}^{\ell-1}(E) \quad \forall v_T \in \mathcal{N}^\ell(T), \quad (A.3) $$

$$ \forall F \in \mathcal{T}_T \quad (w_T)|_F \cdot n_F \in \mathcal{P}^{\ell-1}(F) \quad \forall w_T \in \mathcal{RT}^\ell(T), \quad (A.4) $$

$$ \forall F \in \mathcal{T}_T \quad (v_T)|_F \times n_F \in \mathcal{RT}^\ell(F) \quad \forall v_T \in \mathcal{N}^\ell(T). \quad (A.5) $$

**Proof 1.** *Proof of (A.1) and (A.3).* The tangent edge traces of functions in $\mathcal{G}^{\ell-1}(F)$ (resp. $\mathcal{G}^{\ell-1}(T)$) are in $\mathcal{P}^{\ell-1}(E)$ for all $E \in \mathcal{E}_F$ (resp. $E \in \mathcal{E}_T$). Let, for any $E \in \mathcal{E}_F$, $x_E$ denote the middle point of $E$. To prove (A.1), it then suffices to recall the definition (2.3a) and observe that the quantity $(x - x_F)^\perp \cdot t_E = (x - x_E)^\perp \cdot t_E + (x_E - x_F)^\perp \cdot t_E$ is constant over $E$, the cancellation coming from the fact that $(x - x_E)$ and $t_E$ are parallel for all $x \in E$. To prove (A.3), recall the definition (2.5a) of $\mathcal{G}^{c,\ell}(T)$ and observe that, for all $v \in \mathcal{P}^{\ell-1}(T)$ and all $x \in E$, $[(x - x_T) \times v] \cdot t_E = [(x - x_E) \times v] \cdot t_E + [(x_E - x_T) \times v] \cdot t_E \in \mathcal{P}^{\ell-1}(E)$, where the cancellation follows observing, as before, that the vectors $(x - x_E)$ and $t_E$ are parallel.

**2.** *Proof of (A.2) and (A.4).* The normal traces of functions in $\mathcal{R}^{\ell-1}(F)$ (resp. $\mathcal{R}^{\ell-1}(T)$) are in $\mathcal{P}^{\ell-1}(E)$ (resp. $\mathcal{P}^{\ell-1}(F)$) for all $E \in \mathcal{E}_F$ (resp. $F \in \mathcal{T}_T$). To conclude, recall the definition (2.3b) (resp. (2.5b)) of $\mathcal{R}^{c,\ell}(F)$ (resp. $\mathcal{R}^{c,\ell}(T)$) and observe, using similar arguments as above, that the quantity $(x - x_F) \cdot n_{FE}$ (resp. $(x - x_T) \cdot n_F$) is constant for any $x \in E$ (resp. $x \in F$). This implies, in particular, for all $T \in \mathcal{T}_h$,

$$ \forall F \in \mathcal{T}_T, \quad (z_T)|_F \cdot n_F \in \mathcal{P}^{\ell-1}(F) \quad \forall z_T \in \mathcal{R}^{c,\ell}(T). $$

**3.** *Proof of (A.5).* For all $v_T \in \mathcal{G}^{\ell-1}(T) \subset \mathcal{P}^{\ell-1}(T)$ we have $(v_T)|_F \times n_F \in \mathcal{P}^{\ell-1}(F) \subset \mathcal{RT}^\ell(F)$ since $n_F$ is constant. It therefore suffices to prove (A.5)
<figure><img src="image_4.png" /><figcaption>Fig. 2: CPU (left column) and wall times (right column), both measured in seconds, for the computation of the DDR bases ("Bases") and of the model construction ("Model") for Koszul (K) and orthogonal (⊥) complements using Homogeneous Numerical Integration (HNI) or quadratures on an element subdivision (quad.) on the finest mesh of each sequence; see Table 3.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 2: CPU (left column) and wall times (right column), both measured in seconds, for the computation of the DDR bases ("Bases") and of the model construction ("Model") for Koszul (K) and orthogonal (⊥) complements using Homogeneous Numerical Integration (HNI) or quadratures on an element subdivision (quad.) on the finest mesh of each sequence; see Table 3.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>(a) Homogeneous Numerical Integration (b) Quadratures on element subdivisions</figcaption></figure>

<caption>Fig. 3: Comparison of the wall times, all measured in seconds, for the computation of the DDR bases ("Bases"), the model construction ("Model") and the resolution of the linear system using the PARDISO direct solver ("Solve") for the Voro-small-0 mesh sequence for $k = 2$.</caption>

for $v_T \in \mathcal{G}^{c,l}(T)$. Recalling (2.5a), there is $z_T \in \mathcal{P}^{l-1}(T)$ such that $v_T = (x - x_T) \times z_T$. Thus, we can write

$$ \begin{aligned} (v_T)_F \times n_F &= ((x - x_T) \times z_T)_F \times n_F \\ &= ((x - x_T)_F \cdot n_F)(z_T)_F + ((z_T)_F \cdot n_F)(x_T - x)_F \\ &= ((x_F - x_T)_F \cdot n_F)(z_T)_F + ((z_T)_F \cdot n_F)(x_F - x)_F \\ &\quad - ((z_T)_F \cdot n_F)(x_F - x_T) \\ &= ((x_F - x_T) \times z_T)_F \times n_F + \underbrace{((z_T)_F \cdot n_F)(x_F - x)_F}_{\in \mathcal{P}^{l-1}(F)} \\ &\quad + \underbrace{((z_T)_F \cdot n_F)(x_F - x)_F}_{\in \mathcal{R}^{c,l}(F)}, \end{aligned} $$

where we have used the vector algebra identity

$$ (A \times B) \times C = (A \cdot C)B - (B \cdot C)A \quad \forall A, B, C \in \mathbb{R}^3 \quad (A.6) $$

with $A = x - x_T$, $B = z_T$, and $C = n_F$ to pass to the second line; to pass to the third line, we have noticed that $(x - x_T) \cdot n_F$ is constant on $F$ for the first term, and we have added $\pm x_F$ inside the last parentheses and developed; the last line follows from an application of (A.6) with $A = x_F - x_T$, $B = z_T$, and $C = n_F$. Since $\mathcal{P}^{l-1}(F) = \mathcal{R}^{l-1}(F) \oplus \mathcal{R}^{c,l-1}(F) \subset \mathcal{R}^{l-1}(F) \oplus \mathcal{R}^{c,l}(F) = \mathcal{R}\mathcal{T}^l(F)$, this concludes the proof. $\square$

**Lemma 9 (Norms of the inverses of local differential isomorphisms)**
The norms of the inverses of the isomorphisms defined in (2.8)-(2.10) satisfy, for all $F \in \mathcal{F}_h$ or $T \in \mathcal{T}_h$,

$$ \| (\mathrm{rot}_F)^{-1} \| \le h_F, \quad \| (\mathrm{div}_F)^{-1} \| \le h_F, \quad \| (\mathrm{div})^{-1} \| \le h_T, \quad \text{and} \quad \| (\mathrm{curl})^{-1} \| \le h_T $$
<figure><img src="image_6.png" /><figcaption>(a) Homogeneous Numerical Integration (b) Quadratures on element subdivisions</figcaption></figure>

<caption>Fig. 3: Comparison of the wall times, all measured in seconds, for the computation of the DDR bases ("Bases"), the model construction ("Model") and the resolution of the linear system using the PARDISO direct solver ("Solve") for the Voro-small-0 mesh sequence for $k = 2$.</caption>

for $v_T \in \mathcal{G}^{c,l}(T)$. Recalling (2.5a), there is $z_T \in \mathcal{P}^{l-1}(T)$ such that $v_T = (\mathbf{x} - \mathbf{x}_T) \times \mathbf{z}_T$. Thus, we can write

$$ \begin{aligned} (v_T)_F \times n_F &= ((\mathbf{x} - \mathbf{x}_T) \times z_T)_F \times n_F \\ &= ((\mathbf{x} - \mathbf{x}_T)_F \cdot n_F)(z_T)_F + ((z_T)_F \cdot n_F)(\mathbf{x}_T - \mathbf{x})_F \\ &= ((\mathbf{x}_F - \mathbf{x}_T)_F \cdot n_F)(z_T)_F + ((z_T)_F \cdot n_F)(\mathbf{x}_F - \mathbf{x})_F \\ &\quad - ((z_T)_F \cdot n_F)(\mathbf{x}_F - \mathbf{x}_T) \\ &= ((\mathbf{x}_F - \mathbf{x}_T) \times z_T)_F \times n_F + \underbrace{((z_T)_F \cdot n_F)(\mathbf{x}_F - \mathbf{x})_F}_{\in \mathcal{P}^{l-1}(F)}, \\ &\quad + \underbrace{((z_T)_F \cdot n_F)(\mathbf{x}_F - \mathbf{x})_F}_{\in \mathcal{R}^{c,l}(F)}, \end{aligned} $$

where we have used the vector algebra identity

$$ (A \times B) \times C = (A \cdot C)B - (B \cdot C)A \quad \forall A, B, C \in \mathbb{R}^3 \quad (A.6) $$

with $A = \mathbf{x} - \mathbf{x}_T$, $B = z_T$, and $C = n_F$ to pass to the second line; to pass to the third line, we have noticed that $(\mathbf{x} - \mathbf{x}_T) \cdot n_F$ is constant on $F$ for the first term, and we have added $\pm \mathbf{x}_F$ inside the last parentheses and developed; the last line follows from an application of (A.6) with $A = \mathbf{x}_F - \mathbf{x}_T$, $B = z_T$, and $C = n_F$. Since $\mathcal{P}^{l-1}(F) = \mathcal{R}^{l-1}(F) \oplus \mathcal{R}^{c,l-1}(F) \subset \mathcal{R}^{l-1}(F) \oplus \mathcal{R}^{c,l}(F) = \mathcal{R}\mathcal{T}^l(F)$, this concludes the proof. $\square$

**Lemma 9 (Norms of the inverses of local differential isomorphisms)**
The norms of the inverses of the isomorphisms defined in (2.8)-(2.10) satisfy, for all $F \in \mathcal{F}_h$ or $T \in \mathcal{T}_h$,

$$ \| (\mathrm{rot}_F)^{-1} \| \le h_F, \quad \| (\mathrm{div}_F)^{-1} \| \le h_F, \quad \| (\mathrm{div})^{-1} \| \le h_T, \quad \text{and} \quad \| (\mathrm{curl})^{-1} \| \le h_T $$
where, above, $\|\cdot\|$ denotes the norm of the corresponding isomorphism when its
domain and co-domains are endowed with their $L^2$-norms, and $a \le b$ means
that $a \le Cb$ with $C$ depending only on the polynomial degree $\ell$ and on the mesh
regularity parameter.

Proof We only prove the estimate on $\|(\mathbf{curl})^{-1}\|$, since the other ones follow from similar arguments. The idea is to use the transport $T \ni \mathbf{x} \mapsto \hat{\mathbf{x}} = h_T^{-1}(\mathbf{x} - \mathbf{x}_T) \in \hat{T}$ as in the proof of Lemma 2. We recall that $B(\rho) \subset \hat{T} \subset B(1)$, where $\rho$ is the mesh regularity parameter and $B(r) := \{\mathbf{y} \in \mathbb{R}^3 : |\mathbf{y}| < r\}$.

Let $\nu \in \mathcal{R}^{\ell-1}(T)$ and set $\hat{\nu}(\hat{\mathbf{x}}) := \nu(\mathbf{x})$. Given the definition of the change of variable $\mathbf{x} \mapsto \hat{\mathbf{x}}$, $\hat{\nu}$ belongs to $\mathcal{R}^{\ell-1}(\hat{T})$, and can be considered as a polynomial in $\mathcal{R}^{\ell-1}(\mathbb{R}^3)$. As $\mathbf{curl} : \hat{\mathbf{x}} \times \mathcal{P}^{\ell-1}(\mathbb{R}^3) \to \mathcal{R}^{\ell-1}(\mathbb{R}^3)$ is an isomorphism, it has a continuous inverse for any pair of norms we choose on the domain and co-domain; we endow $\hat{\mathbf{x}} \times \mathcal{P}^{\ell-1}(\mathbb{R}^3)$ with the $L^2(B(1))$-norm and $\mathcal{R}^{\ell-1}(\mathbb{R}^3)$ with the $L^2(B(\rho))$-norm. The continuity of the inverse of this $\mathbf{curl}$ operator gives $\hat{\mathbf{w}} \in \hat{\mathbf{x}} \times \mathcal{P}^{\ell-1}(\mathbb{R}^3)$ such that $\mathbf{curl} \hat{\mathbf{w}} = \hat{\nu}$ on $\mathbb{R}^3$ and $\|\hat{\mathbf{w}}\|_{L^2(B(1))} \lesssim \|\hat{\nu}\|_{L^2(B(\rho))}$, where the hidden constant depends only on the spaces and their norms, that is, on $\ell$ and $\rho$. Since $B(\rho) \subset \hat{T} \subset B(1)$, this shows that $\mathbf{curl} \hat{\mathbf{w}} = \hat{\nu}$ on $\hat{T}$ and $\|\hat{\mathbf{w}}\|_{L^2(\hat{T})} \lesssim \|\hat{\nu}\|_{L^2(\hat{T})}$.

For $\mathbf{x} \in T$, define $\mathbf{w}(\mathbf{x}) := h_T \hat{\mathbf{w}}(\hat{\mathbf{x}})$. Then, $\mathbf{w} \in (\mathbf{x} - \mathbf{x}_T) \times \mathcal{P}^{\ell-1}(T)$, $\mathbf{curl} \mathbf{w} = \mathbf{v}$ (the scaling by $h_T$ cancels out the factor $h_T^{-1}$ which appears when differentiating $\mathbf{x} \mapsto \hat{\mathbf{w}}(h_T^{-1}(\mathbf{x} - \mathbf{x}_T))$), and, denoting by $\mathbf{J}_T$ the Jacobian of the transport $\hat{T} \to T$, we have

$$
\|\mathbf{w}\|_{L^2(T)} = h_T |\mathbf{J}_T|^{\frac{1}{2}} \|\hat{\mathbf{w}}\|_{L^2(\hat{T})} \lesssim h_T |\mathbf{J}_T|^{\frac{1}{2}} \|\hat{\mathbf{v}}\|_{L^2(\hat{T})} = h_T \|\mathbf{v}\|_{L^2(T)},
$$

which concludes the proof. $\square$

B Curl lifting

We prove here that the face $\mathbf{R}_{\mathbf{curl},F}$ and element $\mathbf{R}_{\mathbf{curl},T}$ liftings, detailed in
Section 6.5, are well defined and satisfy the key properties (6.26) and (6.27).

B.1 Face lifting $\mathbf{R}_{\mathbf{curl},F}$

B.1.1 Existence of $\phi_{\nu_F}$

Owing to (6.22b), we look for $\phi_{\nu_F} = \operatorname{rot}_F q_F$ for some $q_F \in H^1(F)$. Using the property $\operatorname{rot}_F(\operatorname{rot}_F) = -\operatorname{div}_F(\operatorname{grad}_F) = -\Delta_F$ (which stems from (2.1)) and that $\operatorname{rot}_F q_F$ (resp. $t_E$) is $\operatorname{grad}_F q_F$ (resp. $n_{FE}$) rotated by $-\pi/2$ in the plane spanned by $F$, we see that (6.22) reduces to the following Neumann problem on $q_F$:

$$
\begin{equation}
\begin{aligned}
& -\Delta_F q_F = C_F^k \nu_F && \text{in } F, \\
& \operatorname{grad}_F q_F \cdot (\omega_{FE} n_{FE}) = \omega_{FE} \nu_E && \forall E \in \mathcal{E}_F.
\end{aligned}
\tag{B.1}
\end{equation}
$$
where, above, $\|\cdot\|$ denotes the norm of the corresponding isomorphism when its domain and co-domains are endowed with their $L^2$-norms, and $a \le b$ means that $a \le Cb$ with $C$ depending only on the polynomial degree $\ell$ and on the mesh regularity parameter.

*Proof* We only prove the estimate on $\|(\text{curl})^{-1}\|$, since the other ones follow from similar arguments. The idea is to use the transport $T \ni \mathbf{x} \mapsto \hat{\mathbf{x}} = h_T^{-1}(\mathbf{x} - \mathbf{x}_T) \in \hat{T}$ as in the proof of Lemma 2. We recall that $B(\rho) \subset \hat{T} \subset B(1)$, where $\rho$ is the mesh regularity parameter and $B(r) := \{\mathbf{y} \in \mathbb{R}^3 : |\mathbf{y}| < r\}$.

Let $\nu \in \mathcal{R}^{\ell-1}(T)$ and set $\hat{\nu}(\hat{\mathbf{x}}) := \nu(\mathbf{x})$. Given the definition of the change of variable $\mathbf{x} \mapsto \hat{\mathbf{x}}$, $\hat{\nu}$ belongs to $\mathcal{R}^{\ell-1}(\hat{T})$, and can be considered as a polynomial in $\mathcal{R}^{\ell-1}(\mathbb{R}^3)$. As **curl**: $\hat{\mathbf{x}} \times \mathcal{P}^{\ell-1}(\mathbb{R}^3) \to \mathcal{R}^{\ell-1}(\mathbb{R}^3)$ is an isomorphism, it has a continuous inverse for any pair of norms we choose on the domain and co-domain; we endow $\hat{\mathbf{x}} \times \mathcal{P}^{\ell-1}(\mathbb{R}^3)$ with the $L^2(B(1))$-norm and $\mathcal{R}^{\ell-1}(\mathbb{R}^3)$ with the $L^2(B(\rho))$-norm. The continuity of the inverse of this **curl** operator gives $\hat{\mathbf{w}} \in \hat{\mathbf{x}} \times \mathcal{P}^{\ell-1}(\mathbb{R}^3)$ such that **curl** $\hat{\mathbf{w}} = \hat{\nu}$ on $\mathbb{R}^3$ and $\|\hat{\mathbf{w}}\|_{L^2(B(1))} \lesssim \|\hat{\nu}\|_{L^2(B(\rho))}$, where the hidden constant depends only on the spaces and their norms, that is, on $\ell$ and $\rho$. Since $B(\rho) \subset \hat{T} \subset B(1)$, this shows that $\text{curl} \hat{\mathbf{w}} = \hat{\nu}$ on $\hat{T}$ and $\|\hat{\mathbf{w}}\|_{L^2(\hat{T})} \lesssim \|\hat{\nu}\|_{L^2(\hat{T})}$.

For $\mathbf{x} \in T$, define $\mathbf{w}(\mathbf{x}) := h_T \hat{\mathbf{w}}(\hat{\mathbf{x}})$. Then, $\mathbf{w} \in (\mathbf{x} - \mathbf{x}_T) \times \mathcal{P}^{\ell-1}(T)$, $\text{curl} \mathbf{w} = \mathbf{v}$ (the scaling by $h_T$ cancels out the factor $h_T^{-1}$ which appears when differentiating $\mathbf{x} \mapsto \hat{\mathbf{w}}(h_T^{-1}(\mathbf{x} - \mathbf{x}_T))$), and, denoting by $\mathbf{J}_T$ the Jacobian of the transport $\hat{T} \to T$, we have

$$ \|\mathbf{w}\|_{L^2(\hat{T})} = h_T |\mathbf{J}_T|^{\frac{1}{2}} \|\hat{\mathbf{w}}\|_{L^2(\hat{T})} \lesssim h_T |\mathbf{J}_T|^{\frac{1}{2}} \|\hat{\mathbf{v}}\|_{L^2(\hat{T})} = h_T \|\mathbf{v}\|_{L^2(\hat{T})}, $$

which concludes the proof. $\square$

## B Curl lifting

We prove here that the face $\mathcal{R}_{\text{curl},F}$ and element $\mathcal{R}_{\text{curl},T}$ liftings, detailed in Section 6.5, are well defined and satisfy the key properties (6.26) and (6.27).

### B.1 Face lifting $\mathcal{R}_{\text{curl},F}$

#### B.1.1 Existence of $\phi_{\nu_F}$

Owing to (6.22b), we look for $\phi_{\nu_F} = \text{rot}_F q_F$ for some $q_F \in H^1(F)$. Using the property $\text{rot}_F(\text{rot}_F) = -\text{div}_F(\text{grad}_F) = -\Delta_F$ (which stems from (2.1)) and that $\text{rot}_F q_F$ (resp. $t_E$) is $\text{grad}_F q_F$ (resp. $n_{FE}$) rotated by $-\pi/2$ in the plane spanned by $F$, we see that (6.22) reduces to the following Neumann problem on $q_F$:

$$ \begin{array}{ll} -\Delta_F q_F = C_F^k \nu_F & \text{in } F, \\ \text{grad}_F q_F \cdot (\omega_{FE} n_{FE}) = \omega_{FE} \nu_E & \forall E \in \mathcal{E}_F. \end{array} \qquad (B.1) $$
Recalling that $\omega_{FE}n_{FE}$ is the outer normal, in the plane spanned by $F$, to $F$ on $E$, we see that the compatibility condition of this Neumann problem simply amounts to the definition (3.19) of $C_F^k$ with $r_F = 1$. There exists therefore a unique $q_F \in H^1(F)$ solution of this problem with $\int_F q_F = 0$. Using $q_F$ as a test function in the weak formulation and applying Cauchy–Schwarz inequalities leads to

$$ \begin{aligned} \| \mathrm{grad}_F q_F \|_{L^2(F)}^2 &\le \| C_F^k \underline{v}_F \|_{L^2(F)} \| q_F \|_{L^2(F)} + \sum_{E \in \mathcal{E}_F} \| v_E \|_{L^2(E)} \| q_F \|_{L^2(E)} \\ &\lesssim h_F \| C_F^k \underline{v}_F \|_{L^2(F)} \| \mathrm{grad}_F q_F \|_{L^2(F)} + \left( \sum_{E \in \mathcal{E}_F} h_E \| v_E \|_{L^2(E)}^2 \right)^{\frac{1}{2}} \| \mathrm{grad}_F q_F \|_{L^2(F)}, \end{aligned} $$

where the second line follows from the Poincaré–Wirtinger inequality

$$ \| q_F \|_{L^2(F)} \lesssim h_F \| \mathrm{grad}_F q_F \|_{L^2(F)} $$

together with the continuous trace inequality (see [28, Remark 1.46 and Lemma 1.31])

$$ \| q_F \|_{L^2(E)} \lesssim h_E^{-1/2} \| q_F \|_{L^2(F)} + h_E^{1/2} \| \mathrm{grad}_F q_F \|_{L^2(F)}. $$

As a consequence,

$$ \| \phi_{\underline{v}_F} \|_{L^2(F)} \lesssim \| C_F^k \underline{v}_F \|_{L^2(F)} + \| \underline{v}_F \|_{\mathrm{curl},F}. \quad (B.2) $$

### B.1.2 Existence of $\psi_{\underline{v}_F}$

Fix $\omega_F \in C_c^\infty(F)$ such that $\omega_F = 1$ on a ball $B_F \subset F$ of radius $\approx h_F$ (the existence of such a ball follows from the mesh regularity assumption) and $0 \le \omega_F \le 1$. We look for $\psi_{\underline{v}_F}$ under the form $\omega_F r_F$ with $r_F \in \mathcal{P}^k(F)$. Since $\mathrm{div}_F : \mathcal{R}^{c,k+1}(F) \to \mathcal{P}^k(F)$ is an isomorphism, denoting as in Lemma 9 its inverse by $(\mathrm{div}_F)^{-1}$, the relation (6.23) is equivalent to

$$ \int_F \omega_F r_F w_F = \int_F (\gamma_{t,F}^k \underline{v}_F - \phi_{\underline{v}_F}) \cdot (\mathrm{div}_F)^{-1} w_F \quad \forall w_F \in \mathcal{P}^k(F). $$

Since $\omega_F \ge 0$ is strictly positive on a ball, the mapping $(r_F, w_F) \mapsto \int_F \omega_F r_F w_F$ is an inner product on $\mathcal{P}^k(F)$ and there exists therefore a unique $r_F \in \mathcal{P}^k(F)$ that satisfies this property. This establishes the existence of $\psi_{\underline{v}_F}$.

Moreover, since $\omega_F = 1$ on $B_F$ and $\|\cdot\|_{L^2(B_F)}$ and $\|\cdot\|_{L^2(F)}$ are uniformly equivalent on $\mathcal{P}^k(F)$ (see the proof of [28, Lemma 1.25]), using $w_F = r_F$ above leads to

$$ \begin{aligned} \| r_F \|_{L^2(F)}^2 &\lesssim \int_F \omega_F r_F^2 \le \| \gamma_{t,F}^k \underline{v}_F - \phi_{\underline{v}_F} \|_{L^2(F)} \| (\mathrm{div}_F)^{-1} r_F \|_{L^2(F)} \\ &\lesssim (\| \underline{v}_F \|_{\mathrm{curl},F} + \| C_F^k \underline{v}_F \|_{L^2(F)}) h_F \| r_F \|_{L^2(F)}, \end{aligned} $$

where the conclusion follows from a triangle inequality along with the boundedness (4.23) of $\gamma_{t,F}^k$ and the estimate (B.2) for the first factor, and Lemma 9 for the second factor. Simplifying, we obtain

$$ \| r_F \|_{L^2(F)} \lesssim h_F (\| \underline{v}_F \|_{\mathrm{curl},F} + \| C_F^k \underline{v}_F \|_{L^2(F)}). \quad (B.3) $$
Recalling that $\omega_{FE}n_{FE}$ is the outer normal, in the plane spanned by $F$, to $F$ on $E$, we see that the compatibility condition of this Neumann problem simply amounts to the definition (3.19) of $C_F^k$ with $r_F = 1$. There exists therefore a unique $q_F \in H^1(F)$ solution of this problem with $\int_F q_F = 0$. Using $q_F$ as a test function in the weak formulation and applying Cauchy–Schwarz inequalities leads to

$$ \begin{aligned} \| \mathbf{grad}_F q_F \|_{L^2(F)}^2 &\le \| C_F^k \underline{\mathbf{v}}_F \|_{L^2(F)} \| q_F \|_{L^2(F)} + \sum_{E \in \mathcal{E}_F} \| v_E \|_{L^2(E)} \| q_F \|_{L^2(E)} \\ &\lesssim h_F \| C_F^k \underline{\mathbf{v}}_F \|_{L^2(F)} \| \mathbf{grad}_F q_F \|_{L^2(F)} + \left( \sum_{E \in \mathcal{E}_F} h_E \| v_E \|_{L^2(E)}^2 \right)^{\frac{1}{2}} \| \mathbf{grad}_F q_F \|_{L^2(F)}, \end{aligned} $$

where the second line follows from the Poincaré–Wirtinger inequality

$$ \|q_F\|_{L^2(F)} \lesssim h_F \| \mathbf{grad}_F q_F \|_{L^2(F)} $$

together with the continuous trace inequality (see [28, Remark 1.46 and Lemma 1.31])

$$ \|q_F\|_{L^2(E)} \lesssim h_E^{-1/2} \|q_F\|_{L^2(F)} + h_E^{1/2} \| \mathbf{grad}_F q_F \|_{L^2(F)}. $$

As a consequence,

$$ \| \boldsymbol{\phi}_{\underline{\mathbf{v}}_F} \|_{L^2(F)} \lesssim \| C_F^k \underline{\mathbf{v}}_F \|_{L^2(F)} + \| \underline{\mathbf{v}}_F \|_{\mathrm{curl},F}. \quad (B.2) $$

### B.1.2 Existence of $\psi_{\underline{\mathbf{v}}_F}$

Fix $\omega_F \in C_c^\infty(F)$ such that $\omega_F = 1$ on a ball $B_F \subset F$ of radius $\approx h_F$ (the existence of such a ball follows from the mesh regularity assumption) and $0 \le \omega_F \le 1$. We look for $\psi_{\underline{\mathbf{v}}_F}$ under the form $\omega_F r_F$ with $r_F \in \mathcal{P}^k(F)$. Since $\mathrm{div}_F : \mathcal{R}^{c,k+1}(F) \to \mathcal{P}^k(F)$ is an isomorphism, denoting as in Lemma 9 its inverse by $(\mathrm{div}_F)^{-1}$, the relation (6.23) is equivalent to

$$ \int_F \omega_F r_F w_F = \int_F (\gamma_{t,F}^k \underline{\mathbf{v}}_F - \boldsymbol{\phi}_{\underline{\mathbf{v}}_F}) \cdot (\mathrm{div}_F)^{-1} w_F \quad \forall w_F \in \mathcal{P}^k(F). $$

Since $\omega_F \ge 0$ is strictly positive on a ball, the mapping $(r_F, w_F) \mapsto \int_F \omega_F r_F w_F$ is an inner product on $\mathcal{P}^k(F)$ and there exists therefore a unique $r_F \in \mathcal{P}^k(F)$ that satisfies this property. This establishes the existence of $\psi_{\underline{\mathbf{v}}_F}$.

Moreover, since $\omega_F = 1$ on $B_F$ and $\|\cdot\|_{L^2(B_F)}$ and $\|\cdot\|_{L^2(F)}$ are uniformly equivalent on $\mathcal{P}^k(F)$ (see the proof of [28, Lemma 1.25]), using $w_F = r_F$ above leads to

$$ \begin{aligned} \|r_F\|_{L^2(F)}^2 &\lesssim \int_F \omega_F r_F^2 \le \|\gamma_{t,F}^k \underline{\mathbf{v}}_F - \boldsymbol{\phi}_{\underline{\mathbf{v}}_F}\|_{L^2(F)} \|(\mathrm{div}_F)^{-1} r_F\|_{L^2(F)} \\ &\lesssim (\|\underline{\mathbf{v}}_F\|_{\mathrm{curl},F} + \|C_F^k \underline{\mathbf{v}}_F\|_{L^2(F)}) h_F \|r_F\|_{L^2(F)}, \end{aligned} $$

where the conclusion follows from a triangle inequality along with the boundedness (4.23) of $\gamma_{t,F}^k$ and the estimate (B.2) for the first factor, and Lemma 9 for the second factor. Simplifying, we obtain

$$ \|r_F\|_{L^2(F)} \lesssim h_F (\|\underline{\mathbf{v}}_F\|_{\mathrm{curl},F} + \|C_F^k \underline{\mathbf{v}}_F\|_{L^2(F)}). \quad (B.3) $$
### B.1.3 Orthogonality property of $\mathbf{R}_{\text{curl},F}$

We prove here (6.26). Notice first that, since $\psi_{\underline{v}_F}$ vanishes on $\partial F$ and $\text{rot}_F \text{grad}_F = 0$, by (6.22) it holds

$$ \text{rot}_F(\mathbf{R}_{\text{curl},F} \underline{v}_F) = C_F^k \underline{v}_F \quad \text{and} \quad (\mathbf{R}_{\text{curl},F} \underline{v}_F) \cdot \mathbf{t}_E = v_E \quad \forall E \in \mathcal{E}_F. \quad (B.4) $$

Let $z_F \in \mathcal{R}^k(F)$ and write $z_F = \text{rot}_F r_F$ with $r_F \in \mathcal{P}^{0,k+1}(F)$. By (3.23) and Remark 11, we have

$$ \begin{aligned} \int_F \gamma_{t,F}^k \underline{v}_F \cdot z_F &= \int_F C_F^k \underline{v}_F r_F + \sum_{E \in \mathcal{F}_E} \omega_{FE} \int_E v_E r_F \\ &= \int_F \text{rot}_F (\mathbf{R}_{\text{curl},F} \underline{v}_F) r_F + \sum_{E \in \mathcal{F}_E} \omega_{FE} \int_E (\mathbf{R}_{\text{curl},F} \underline{v}_F) \cdot \mathbf{t}_E r_F \\ &= \int_F \mathbf{R}_{\text{curl},F} \underline{v}_F \cdot z_F, \end{aligned} $$

where the second equality follows from (B.4), and the conclusion has been obtained using an integration by parts. This proves that (6.26) holds for $z_F \in \mathcal{R}^k(F)$.

Let us now take $z_F \in \mathcal{R}^{c,k+1}(F)$. Integrating the left-hand side of (6.23) by parts yields

$$ \int_F \text{grad}_F \psi_{\underline{v}_F} \cdot z_F = \int_F (\gamma_{t,F}^k \underline{v}_F - \phi_{\underline{v}_F}) \cdot z_F. $$

Since $\mathbf{R}_{\text{curl},F} \underline{v}_F = \phi_{\underline{v}_F} + \text{grad}_F \psi_{\underline{v}_F}$, this establishes that (6.26) also holds for $z_F \in \mathcal{R}^{c,k+1}(F)$, and completes the proof of this orthogonality relation since $\mathcal{R}\mathcal{T}^{k+1}(F) = \mathcal{R}^k(F) \oplus \mathcal{R}^{c,k+1}(F)$.

## B.2 Element lifting $\mathbf{R}_{\text{curl},T}$

### B.2.1 Existence of $\delta_T \underline{v}_T$

Owing to (6.24b), we look for $\delta_T \underline{v}_T$ under the form of a potential gradient $\text{grad} \, q_T$ with $q_T \in \mathbb{H}^1(T)$. Equations (6.24a) and (6.24c) then show that $q_T$ must solve the Neumann problem

$$ \begin{array}{ll} \Delta q_T = - \operatorname{div} \mathbf{C}_T^k \underline{v}_T & \text{in } T, \\ \text{grad} \, q_T \cdot (\omega_{TF} \mathbf{n}_F) = \omega_{TF} (C_F^k \underline{v}_F - \mathbf{C}_T^k \underline{v}_T \cdot \mathbf{n}_F) & \forall F \in \mathcal{F}_T, \end{array} \quad (B.5) $$

where we recall that $\omega_{TF} \mathbf{n}_F$ is the outer normal to $T$ on $F$. The compatibility condition of this problem is

$$ \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F (C_F^k \underline{v}_F - \mathbf{C}_T^k \underline{v}_T \cdot \mathbf{n}_F) = - \int_T \operatorname{div} \mathbf{C}_T^k \underline{v}_T = - \sum_{T \in \mathcal{T}_T} \omega_{TF} \int_F \mathbf{C}_T^k \underline{v}_T \cdot \mathbf{n}_F, $$
### B.1.3 Orthogonality property of $R_{\text{curl},F}$

We prove here (6.26). Notice first that, since $\psi_{\underline{v}_F}$ vanishes on $\partial F$ and $\text{rot}_F \text{grad}_F = 0$, by (6.22) it holds

$$ \text{rot}_F(\mathbf{R}_{\text{curl},F} \underline{\mathbf{v}}_F) = C_F^k \underline{\mathbf{v}}_F \quad \text{and} \quad (\mathbf{R}_{\text{curl},F} \underline{\mathbf{v}}_F) \cdot \mathbf{t}_E = v_E \quad \forall E \in \mathcal{E}_F. \quad (B.4) $$

Let $z_F \in \mathcal{R}^k(F)$ and write $z_F = \text{rot}_F r_F$ with $r_F \in \mathcal{P}^{0,k+1}(F)$. By (3.23) and Remark 11, we have

$$ \begin{aligned} \int_F \gamma_{t,F}^k \underline{\mathbf{v}}_F \cdot z_F &= \int_F C_F^k \underline{\mathbf{v}}_F r_F + \sum_{E \in \mathcal{F}_E} \omega_{FE} \int_E v_E r_F \\ &= \int_F \text{rot}_F (\mathbf{R}_{\text{curl},F} \underline{\mathbf{v}}_F) r_F + \sum_{E \in \mathcal{F}_E} \omega_{FE} \int_E (\mathbf{R}_{\text{curl},F} \underline{\mathbf{v}}_F) \cdot \mathbf{t}_E r_F \\ &= \int_F \mathbf{R}_{\text{curl},F} \underline{\mathbf{v}}_F \cdot z_F, \end{aligned} $$

where the second equality follows from (B.4), and the conclusion has been obtained using an integration by parts. This proves that (6.26) holds for $z_F \in \mathcal{R}^k(F)$.

Let us now take $z_F \in \mathcal{R}^{c,k+1}(F)$. Integrating the left-hand side of (6.23) by parts yields

$$ \int_F \text{grad}_F \psi_{\underline{v}_F} \cdot z_F = \int_F (\gamma_{t,F}^k \underline{\mathbf{v}}_F - \phi_{\underline{v}_F}) \cdot z_F. $$

Since $\mathbf{R}_{\text{curl},F} \underline{\mathbf{v}}_F = \phi_{\underline{v}_F} + \text{grad}_F \psi_{\underline{v}_F}$, this establishes that (6.26) also holds for $z_F \in \mathcal{R}^{c,k+1}(F)$, and completes the proof of this orthogonality relation since $\mathcal{R}\mathcal{T}^{k+1}(F) = \mathcal{R}^k(F) \oplus \mathcal{R}^{c,k+1}(F)$.

## B.2 Element lifting $R_{\text{curl},T}$

### B.2.1 Existence of $\delta_T \underline{\mathbf{v}}_T$

Owing to (6.24b), we look for $\delta_T \underline{\mathbf{v}}_T$ under the form of a potential gradient $\text{grad} \, q_T$ with $q_T \in \mathbb{H}^1(T)$. Equations (6.24a) and (6.24c) then show that $q_T$ must solve the Neumann problem

$$ \begin{gathered} \Delta q_T = - \operatorname{div} \mathbf{C}_T^k \underline{\mathbf{v}}_T \quad \text{in } T, \qquad (B.5) \\ \operatorname{grad} q_T \cdot (\omega_{TF} \mathbf{n}_F) = \omega_{TF} (C_F^k \underline{\mathbf{v}}_F - \mathbf{C}_T^k \underline{\mathbf{v}}_T \cdot \mathbf{n}_F) \quad \forall F \in \mathcal{F}_T, \end{gathered} $$

where we recall that $\omega_{TF} \mathbf{n}_F$ is the outer normal to $T$ on $F$. The compatibility condition of this problem is

$$ \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F (C_F^k \underline{\mathbf{v}}_F - \mathbf{C}_T^k \underline{\mathbf{v}}_T \cdot \mathbf{n}_F) = - \int_T \operatorname{div} \mathbf{C}_T^k \underline{\mathbf{v}}_T = - \sum_{T \in \mathcal{T}_T} \omega_{TF} \int_F \mathbf{C}_T^k \underline{\mathbf{v}}_T \cdot \mathbf{n}_F, $$
which holds true owing to (3.30) with $r_T = 1$. There exists therefore a unique $q_T \in H^1(T)$ with $\int_T q_T = 0$ solution to (B.5). Using $q_T$ as a test function in the weak formulation of (B.5) yields

$$ \| \mathbf{grad} \, q_T \|_{L^2(T)}^2 \leq \| \mathbf{C}_T^k \mathbf{v}_T \|_{L^2(T)} \| \mathbf{grad} \, q_T \|_{L^2(T)} + \sum_{F \in \mathcal{F}_T} \| \mathbf{C}_F^k \mathbf{v}_F \|_{L^2(F)} \| q_T \|_{L^2(F)}. $$

Using the Poincaré–Wirtinger and continuous trace inequalities as we did to obtain (B.2), and recalling that $\delta_T \mathbf{v}_T = \mathbf{grad} \, q_T$, we infer

$$ \| \delta_T \mathbf{v}_T \|_{L^2(T)} \lesssim \| \mathbf{C}_T^k \mathbf{v}_T \|_{L^2(T)} + \left( \sum_{F \in \mathcal{F}_T} h_F \| \mathbf{C}_F^k \mathbf{v}_F \|_{L^2(F)}^2 \right)^{\frac{1}{2}} \lesssim \| \underline{\mathbf{C}}_T^k \mathbf{v}_T \|_{\mathrm{div},T}, \quad (B.6) $$

where the conclusion follows from (4.34).

### B.2.2 Existence of $\mathbf{R}_{\mathrm{curl},T} \mathbf{v}_T$

The equation (6.25b) suggests to look for $\mathbf{R}_{\mathrm{curl},T} \mathbf{v}_T = \mathbf{curl} \, z_T$. Since adding a gradient to $z_T$ does not change its curl, we can look for $z_T$ in the space

$$ z_T \in (\mathbf{grad} \, H^1(T))^{\perp} := \left\{ \mathbf{w} \in \mathbf{H}(\mathrm{curl}; T) : \int_T \mathbf{w} \cdot \mathbf{grad} \, r = 0 \quad \forall r \in H^1(T) \right\}. \quad (B.7) $$

The equations (6.25a) and (6.25c) then lead to a curl-curl problem on $z_T$, whose variational form is: Find $z_T \in (\mathbf{grad} \, H^1(T))^{\perp}$ such that

$$ \int_T \mathrm{curl} \, z_T \cdot \mathrm{curl} \, \mathbf{w} = \int_T (\mathbf{C}_T^k \mathbf{v}_T + \delta_T \mathbf{v}_T) \cdot \mathbf{w} \\ - \langle \omega_{T, \partial T} \mathbf{R}_{\mathrm{curl}, \partial T} \mathbf{v}_{\partial T}, \mathbf{w} \times \mathbf{n}_{\partial T} \rangle_{H_{\parallel}^{1/2}(\partial T), H_{\parallel}^{-1/2}(\partial T)} \\ \forall \mathbf{w} \in (\mathbf{grad} \, H^1(T))^{\perp}, \quad (B.8) $$

where $\omega_{T, \partial T} \mathbf{R}_{\mathrm{curl}, \partial T} \mathbf{v}_{\partial T}$ and $\mathbf{w} \times \mathbf{n}_{\partial T}$ are the functions defined on $\partial T$ by setting, respectively, $(\omega_{T, \partial T} \mathbf{R}_{\mathrm{curl}, \partial T} \mathbf{v}_{\partial T})|_F := (\omega_{TF} \mathbf{R}_{\mathrm{curl},F} \mathbf{v}_F)_{T,F}$ and $(\mathbf{w} \times \mathbf{n}_{\partial T})|_F := \mathbf{w}|_F \times \mathbf{n}_F$ for all $F \in \mathcal{F}_T$, $H_{\parallel}^{1/2}(\partial T)$ is the set of functions on $\partial T$ whose restriction to each face $F \in \mathcal{F}_T$ belongs to $H^{1/2}(F)$, and whose tangential traces on the edges are weakly continuous (see [3, Definition 3.1.2] for details), and $H_{\parallel}^{-1/2}(\partial T)$ is its dual space. Since the solution to (B.1) belongs to $H^{3/2}(F)$ (see [25, Corollary 23.5]), the edge tangential trace property in (B.4) ensures that $\omega_{T, \partial T} \mathbf{R}_{\mathrm{curl}, \partial T} \mathbf{v}_{\partial T}$ indeed belongs to $H_{\parallel}^{1/2}(\partial T)$.

Owing to the Poincaré inequality (B.15) and to the fact that $(\mathbf{grad} \, H^1(T))^{\perp}$ is a closed subspace of $\mathbf{H}(\mathrm{curl}; T)$, there exists a unique solution to (B.8). We now prove that $z_T$ satisfies (B.8) for all $\mathbf{w} \in \mathbf{H}(\mathrm{curl}; T) = \mathbf{grad} \, H^1(T) \oplus (\mathbf{grad} \, H^1(T))^{\perp}$, which amounts to showing that the right-hand side vanishes whenever $\mathbf{w} = \mathbf{grad} \, r$ for some $r \in H^1(T)$. By density of smooth functions in $H^1(T)$, we only need to prove this result for $r \in C^{\infty}(\bar{T})$. Plugging $\mathbf{w} = \mathbf{grad} \, r$ in
which holds true owing to (3.30) with $r_T = 1$. There exists therefore a unique $q_T \in H^1(T)$ with $\int_T q_T = 0$ solution to (B.5). Using $q_T$ as a test function in the weak formulation of (B.5) yields

$$ \| \operatorname{grad} q_T \|_{L^2(T)}^2 \le \| \mathbf{C}_T^k \mathbf{v}_T \|_{L^2(T)} \| \operatorname{grad} q_T \|_{L^2(T)} + \sum_{F \in \mathcal{F}_T} \| \mathbf{C}_F^k \mathbf{v}_F \|_{L^2(F)} \| q_T \|_{L^2(F)}. $$

Using the Poincaré–Wirtinger and continuous trace inequalities as we did to obtain (B.2), and recalling that $\delta_T \mathbf{v}_T = \operatorname{grad} q_T$, we infer

$$ \| \delta_T \mathbf{v}_T \|_{L^2(T)} \lesssim \| \mathbf{C}_T^k \mathbf{v}_T \|_{L^2(T)} + \left( \sum_{F \in \mathcal{F}_T} h_F \| \mathbf{C}_F^k \mathbf{v}_F \|_{L^2(F)}^2 \right)^{\frac{1}{2}} \lesssim \| \underline{\mathbf{C}}_T^k \mathbf{v}_T \|_{\operatorname{div}, T}, \quad (B.6) $$

where the conclusion follows from (4.34).

### B.2.2 Existence of $\mathbf{R}_{\operatorname{curl},T} \mathbf{v}_T$

The equation (6.25b) suggests to look for $\mathbf{R}_{\operatorname{curl},T} \mathbf{v}_T = \operatorname{curl} z_T$. Since adding a gradient to $z_T$ does not change its curl, we can look for $z_T$ in the space

$$ z_T \in (\operatorname{grad} H^1(T))^{\perp} := \left\{ \mathbf{w} \in \mathbf{H}(\operatorname{curl}; T) : \int_T \mathbf{w} \cdot \operatorname{grad} r = 0 \quad \forall r \in H^1(T) \right\}. \quad (B.7) $$

The equations (6.25a) and (6.25c) then lead to a curl-curl problem on $z_T$, whose variational form is: Find $z_T \in (\operatorname{grad} H^1(T))^{\perp}$ such that

$$ \int_T \operatorname{curl} z_T \cdot \operatorname{curl} \mathbf{w} = \int_T (\mathbf{C}_T^k \mathbf{v}_T + \delta_T \mathbf{v}_T) \cdot \mathbf{w} \\ - \langle \omega_{T, \partial T} \mathbf{R}_{\operatorname{curl}, \partial T} \mathbf{v}_{\partial T}, \mathbf{w} \times \mathbf{n}_{\partial T} \rangle_{H_{\parallel}^{1/2}(\partial T), H_{\parallel}^{-1/2}(\partial T)} \\ \forall \mathbf{w} \in (\operatorname{grad} H^1(T))^{\perp}, \quad (B.8) $$

where $\omega_{T, \partial T} \mathbf{R}_{\operatorname{curl}, \partial T} \mathbf{v}_{\partial T}$ and $\mathbf{w} \times \mathbf{n}_{\partial T}$ are the functions defined on $\partial T$ by setting, respectively, $(\omega_{T, \partial T} \mathbf{R}_{\operatorname{curl}, \partial T} \mathbf{v}_{\partial T})|_F := (\omega_{TF} \mathbf{R}_{\operatorname{curl}, F} \mathbf{v}_F)_{T, F}$ and $(\mathbf{w} \times \mathbf{n}_{\partial T})|_F := \mathbf{w}|_F \times \mathbf{n}_F$ for all $F \in \mathcal{F}_T$, $H_{\parallel}^{1/2}(\partial T)$ is the set of functions on $\partial T$ whose restriction to each face $F \in \mathcal{F}_T$ belongs to $H^{1/2}(F)$, and whose tangential traces on the edges are weakly continuous (see [3, Definition 3.1.2] for details), and $H_{\parallel}^{-1/2}(\partial T)$ is its dual space. Since the solution to (B.1) belongs to $H^{3/2}(F)$ (see [25, Corollary 23.5]), the edge tangential trace property in (B.4) ensures that $\omega_{T, \partial T} \mathbf{R}_{\operatorname{curl}, \partial T} \mathbf{v}_{\partial T}$ indeed belongs to $H_{\parallel}^{1/2}(\partial T)$.

Owing to the Poincaré inequality (B.15) and to the fact that $(\operatorname{grad} H^1(T))^{\perp}$ is a closed subspace of $\mathbf{H}(\operatorname{curl}; T)$, there exists a unique solution to (B.8). We now prove that $z_T$ satisfies (B.8) for all $\mathbf{w} \in \mathbf{H}(\operatorname{curl}; T) = \operatorname{grad} H^1(T) \oplus (\operatorname{grad} H^1(T))^{\perp}$, which amounts to showing that the right-hand side vanishes whenever $\mathbf{w} = \operatorname{grad} r$ for some $r \in H^1(T)$. By density of smooth functions in $H^1(T)$, we only need to prove this result for $r \in C^{\infty}(\bar{T})$. Plugging $\mathbf{w} = \operatorname{grad} r$ in
the right-hand side of (B.8), the duality product can be written as standard integrals (since $R_{\text{curl},F} \underline{v}_F \in L^2(F)$ for all $F \in \mathcal{F}_T$) and, integrating by parts, we obtain

$$
\begin{align*}
\int_T (\mathbf{C}_T^k \underline{v}_T + \delta_T \underline{v}_T) \cdot \operatorname{grad} r - \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \mathbf{R}_{\text{curl},F} \underline{v}_F \cdot (\operatorname{grad} r \times \mathbf{n}_F) \\
&= - \int_T \overline{\operatorname{div}(\mathbf{C}_T^k \underline{v}_T + \delta_T \underline{v}_T)} r + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F (\mathbf{C}_T^k \underline{v}_T + \delta_T \underline{v}_T) \cdot \mathbf{n}_F r \\
&\quad - \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \mathbf{R}_{\text{curl},F} \underline{v}_F \cdot \operatorname{rot}_F (r|_F) \\
&= \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F C_F^k \underline{v}_F r - \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \operatorname{rot}_F (\mathbf{R}_{\text{curl},F} \underline{v}_F) r|_F \\
&\quad - \sum_{F \in \mathcal{F}_T} \sum_{E \in \mathcal{F}_E} \omega_{TF} \omega_{FE} \int_E (\mathbf{R}_{\text{curl},F} \underline{v}_F \cdot \mathbf{t}_E) r|_F,
\end{align*}
$$

where we have used (6.24a) to cancel the term in the first equality, and (6.24c) together with integrations by parts on each face in the second equality. Re- calling (B.4) and that $\omega_{TF_1}\omega_{F_1E} + \omega_{TF_2}\omega_{F_2E} = 0$ if $F_1, F_2$ are the two faces of $T$ that share the edge $E$, the right-hand side above vanishes, which shows that (B.8) indeed holds for $w = \operatorname{grad} r$, and thus for all $w \in \mathbf{H}(\operatorname{curl}; T)$.

Since $R_{\text{curl},T} \underline{v}_T = \text{curl} z_T$, applying this relation to a generic $w \in C_c^\infty(T)$ and integrating by parts yields (6.25a); using then a generic $w \in C^\infty(\bar{T})$ and again integrating by parts, we infer (6.25c).

### B.2.3 Bound on $R_{\text{curl},T}$

We prove here the estimate (6.27). The estimate on $\text{curl} R_{\text{curl},T} \underline{v}_T$ follows from (6.25a), (4.34) and (B.6). It remains to bound the $L^2$-norm of $R_{\text{curl},T} \underline{v}_T$. To do so, we use $g_{\underline{v}_T}$ provided by Lemma 10 below and an integration by parts [3, Eq. (2.27)] to re-cast (B.8) as

$$ \int_T \text{curl} z_T \cdot \text{curl} w = \int_T (\mathbf{C}_T^k \underline{v}_T + \delta_T \underline{v}_T) \cdot w + \int_T \text{curl} w \cdot g_{\underline{v}_T} - \int_T w \cdot \text{curl} g_{\underline{v}_T}. $$

Making $w = z_T$, we deduce

$$ \| \text{curl} z_T \|_{L^2(T)}^2 \\ \lesssim \| \underline{\mathcal{C}}_T^k \underline{v}_T \|_{\text{div},T} h_T \| \text{curl} z_T \|_{L^2(T)} + \| \text{curl} z_T \|_{L^2(T)} \left( \| \underline{v}_T \|_{\text{curl},T} + \| \underline{\mathcal{C}}_T^k \underline{v}_T \|_{\text{div},T} \right), $$

where we have invoked (4.34), (B.6), the Poincaré inequality (B.15), and (B.9) below. Simplifying, using the norm equivalences (4.25), and recalling that $R_{\text{curl},T} \underline{v}_T = \text{curl} z_T$ concludes the proof of the $L^2$-estimate on $R_{\text{curl},T} \underline{v}_T$ stated in (6.27).
the right-hand side of (B.8), the duality product can be written as standard integrals (since $R_{\text{curl},F} \underline{v}_F \in L^2(F)$ for all $F \in \mathcal{F}_T$) and, integrating by parts, we obtain

$$
\begin{align*}
\int_T (C_T^k \underline{v}_T + \delta_T \underline{v}_T) \cdot \text{grad } r - \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F R_{\text{curl},F} \underline{v}_F \cdot (\text{grad } r \times n_F) \\
&= - \int_T \overline{\text{div}(C_T^k \underline{v}_T + \delta_T \underline{v}_T)} r + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F (C_T^k \underline{v}_T + \delta_T \underline{v}_T) \cdot n_F r \\
&\quad - \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F R_{\text{curl},F} \underline{v}_F \cdot \text{rot}_F (r|_F) \\
&= \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F C_F^k \underline{v}_F r - \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \text{rot}_F (R_{\text{curl},F} \underline{v}_F) r|_F \\
&\quad - \sum_{F \in \mathcal{F}_T} \sum_{E \in \mathcal{F}_E} \omega_{TF} \omega_{FE} \int_E (R_{\text{curl},F} \underline{v}_F \cdot \mathbf{t}_E) r|_F,
\end{align*}
$$

where we have used (6.24a) to cancel the term in the first equality, and (6.24c) together with integrations by parts on each face in the second equality. Re- calling (B.4) and that $\omega_{TF_1}\omega_{F_1E} + \omega_{TF_2}\omega_{F_2E} = 0$ if $F_1, F_2$ are the two faces of $T$ that share the edge $E$, the right-hand side above vanishes, which shows that (B.8) indeed holds for $w = \text{grad } r$, and thus for all $w \in \mathbf{H}(\text{curl}; T)$.

Since $R_{\text{curl},T} \underline{v}_T = \text{curl} z_T$, applying this relation to a generic $w \in C_c^\infty(T)$ and integrating by parts yields (6.25a); using then a generic $w \in C^\infty(\bar{T})$ and again integrating by parts, we infer (6.25c).

### B.2.3 Bound on $R_{\text{curl},T}$

We prove here the estimate (6.27). The estimate on $\text{curl} R_{\text{curl},T} \underline{v}_T$ follows from (6.25a), (4.34) and (B.6). It remains to bound the $L^2$-norm of $R_{\text{curl},T} \underline{v}_T$. To do so, we use $g_{\underline{v}_T}$ provided by Lemma 10 below and an integration by parts [3, Eq. (2.27)] to re-cast (B.8) as

$$ \int_T \text{curl} z_T \cdot \text{curl} w = \int_T (C_T^k \underline{v}_T + \delta_T \underline{v}_T) \cdot w + \int_T \text{curl} w \cdot g_{\underline{v}_T} - \int_T w \cdot \text{curl} g_{\underline{v}_T}. $$

Making $w = z_T$, we deduce

$$ \begin{align*}
\| \text{curl} z_T \|_{L^2(T)}^2 \\
\lesssim \| \underline{C}_T^k \underline{v}_T \|_{\text{div},T} h_T \| \text{curl} z_T \|_{L^2(T)} + \| \text{curl} z_T \|_{L^2(T)} ( \| \underline{v}_T \|_{\text{curl},T} + \| \underline{C}_T^k \underline{v}_T \|_{\text{div},T} ),
\end{align*} $$

where we have invoked (4.34), (B.6), the Poincaré inequality (B.15), and (B.9) below. Simplifying, using the norm equivalences (4.25), and recalling that $R_{\text{curl},T} \underline{v}_T = \text{curl} z_T$ concludes the proof of the $L^2$-estimate on $R_{\text{curl},T} \underline{v}_T$ stated in (6.27).
**Lemma 10 (Lifting in $H^1(T)$)** There exists $g_{\underline{v}_T} \in H^1(T)$ such that the tangential trace of $g_{\underline{v}_T}$ on $\partial T$ is $R_{\text{curl},\partial T} \underline{v}_{\partial T}$, and

$$ \|g_{\underline{v}_T}\|_{L^2(T)} + h_T \|\text{curl } g_{\underline{v}_T}\|_{L^2(T)} \lesssim \|\underline{v}_T\|_{\text{curl},T} + \|\underline{C}_T^k \underline{v}_T\|_{\text{div},T}. \quad (B.9) $$

*Proof* Recalling that

$$ R_{\text{curl},\partial T} \underline{v}_{\partial T} = \phi_{\underline{v}_{\partial T}} + \text{grad}_{\partial T} \psi_{\underline{v}_{\partial T}}, \quad (B.10) $$

with obvious notations (each of these functions, restricted to a face $F \in \mathcal{F}_T$, corresponds to the function obtained replacing $\partial T$ by $F$), we construct $g_{\underline{v}_T} = g_{\underline{v}_T, \phi} + g_{\underline{v}_T, \psi}$, each addend corresponding to the addends in the decomposition (B.10) of $R_{\text{curl},\partial T} \underline{v}_{\partial T}$.

1. **Construction of $g_{\underline{v}_T, \phi}$.** We assume, for the moment, that $h_T = 1$. By [25, Corollary 23.5] and inverse inequalities on the polynomials $C_F^k \underline{v}_F$ and $(v_E)_{E \in \mathcal{F}_E}$ (recalling that $1 = h_T \approx h_F \approx h_E$ for all $F \in \mathcal{F}_T$ and $E \in \mathcal{E}_T$), there exists $\epsilon \in (0, 1/2)$ such that $\text{grad}_F q_F \in H^{1/2+\epsilon}(F)$ and

$$ \begin{aligned} \| \text{grad}_F q_F \|_{H^{1/2+\epsilon}(F)} &\lesssim \| C_F^k \underline{v}_F \|_{L^2(F)} + \sum_{E \in \mathcal{E}_F} \| v_E \|_{L^2(E)} \\ &\lesssim \| C_F^k \underline{v}_F \|_{L^2(F)} + \| \underline{v}_F \|_{\text{curl},F}. \end{aligned} $$

Above, when invoking [25, Corollary 23.5], we have used the fact that, since $\epsilon < 1/2$, the $H^\epsilon(\partial F)$-norm is equivalent to the sum of the $H^\epsilon(E)$-norms over $E \in \mathcal{F}_E$. By construction, $\phi_{\underline{v}_{\partial T}}$ has strongly continuous tangential traces on the edges of $T$ so

$$ \begin{aligned} |\phi_{\underline{v}_{\partial T}}|^2_{H_{\parallel}^{1/2}(\partial T)} &\lesssim \sum_{F \in \mathcal{F}_T} |\phi_{\underline{v}_F}|^2_{H^{1/2}(F)} = \sum_{F \in \mathcal{F}_T} |\text{grad}_F q_F|^2_{H^{1/2}(F)} \\ &\lesssim \sum_{F \in \mathcal{F}_T} \| \text{grad}_F q_F \|_{H^{1/2+\epsilon}(F)}^2 \lesssim \sum_{F \in \mathcal{F}_T} \left( \| C_F^k \underline{v}_F \|_{L^2(F)} + \| \underline{v}_F \|_{\text{curl},F} \right)^2. \end{aligned} $$

Combined with (B.2) and recalling that the local length scales are $\approx 1$, this leads to

$$ \| \phi_{\underline{v}_{\partial T}} \|_{L^2(\partial T)} + |\phi_{\underline{v}_{\partial T}}|_{H_{\parallel}^{1/2}(\partial T)} \lesssim \| \underline{C}_T^k \underline{v}_T \|_{\text{div},T} + \| \underline{v}_T \|_{\text{curl},T}. $$

Since $\phi_{\underline{v}_{\partial T}}$ belongs to $H_{\parallel}^{1/2}(\partial T)$, by [3, Theorem 3.1.3] there exists $g_{\underline{v}_T, \phi} \in H^1(T)$ such that the tangential trace of $g_{\underline{v}_T, \phi}$ is $\phi_{\underline{v}_{\partial T}}$ and

$$ \begin{aligned} \|g_{\underline{v}_T, \phi}\|_{L^2(T)} + \|\text{curl } g_{\underline{v}_T, \phi}\|_{L^2(T)} &\lesssim \|\phi_{\underline{v}_{\partial T}}\|_{L^2(\partial T)} + |\phi_{\underline{v}_{\partial T}}|_{H_{\parallel}^{1/2}(\partial T)} \\ &\lesssim \|\underline{C}_T^k \underline{v}_T\|_{\text{div},T} + \|\underline{v}_T\|_{\text{curl},T}. \end{aligned} $$

This was done under the assumption that $h_T = 1$. Using a scaling argument, we infer from the estimate above that, for an element $T$ of generic diameter $h_T$,

$$ \|g_{\underline{v}_T, \phi}\|_{L^2(T)} + h_T \|\text{curl } g_{\underline{v}_T, \phi}\|_{L^2(T)} \lesssim \|\underline{C}_T^k \underline{v}_T\|_{\text{div},T} + \|\underline{v}_T\|_{\text{curl},T}. \quad (B.11) $$
**Lemma 10 (Lifting in $H^1(T)$)** There exists $g_{\underline{v}_T} \in H^1(T)$ such that the tangential trace of $g_{\underline{v}_T}$ on $\partial T$ is $R_{\text{curl},\partial T} \underline{v}_{\partial T}$, and

$$ \|g_{\underline{v}_T}\|_{L^2(T)} + h_T \|\text{curl } g_{\underline{v}_T}\|_{L^2(T)} \lesssim \|\underline{v}_T\|_{\text{curl},T} + \|\underline{C}_T^k \underline{v}_T\|_{\text{div},T}. \quad (B.9) $$

*Proof* Recalling that

$$ R_{\text{curl},\partial T} \underline{v}_{\partial T} = \phi_{\underline{v}_{\partial T}} + \text{grad}_{\partial T} \psi_{\underline{v}_{\partial T}}, \quad (B.10) $$

with obvious notations (each of these functions, restricted to a face $F \in \mathcal{F}_T$, corresponds to the function obtained replacing $\partial T$ by $F$), we construct $g_{\underline{v}_T} = g_{\underline{v}_T, \phi} + g_{\underline{v}_T, \psi}$, each addend corresponding to the addends in the decomposition (B.10) of $R_{\text{curl},\partial T} \underline{v}_{\partial T}$.

1. Construction of $g_{\underline{v}_T, \phi}$. We assume, for the moment, that $h_T = 1$. By [25, Corollary 23.5] and inverse inequalities on the polynomials $C_F^k \underline{v}_F$ and $(v_E)_{E \in \mathcal{F}_E}$ (recalling that $1 = h_T \approx h_F \approx h_E$ for all $F \in \mathcal{F}_T$ and $E \in \mathcal{E}_T$), there exists $\epsilon \in (0, 1/2)$ such that $\text{grad}_F q_F \in H^{1/2+\epsilon}(F)$ and

$$ \begin{aligned} \| \text{grad}_F q_F \|_{H^{1/2+\epsilon}(F)} &\lesssim \| C_F^k \underline{v}_F \|_{L^2(F)} + \sum_{E \in \mathcal{E}_F} \| v_E \|_{L^2(E)} \\ &\lesssim \| C_F^k \underline{v}_F \|_{L^2(F)} + \| \underline{v}_F \|_{\text{curl},F}. \end{aligned} $$

Above, when invoking [25, Corollary 23.5], we have used the fact that, since $\epsilon < 1/2$, the $H^\epsilon(\partial F)$-norm is equivalent to the sum of the $H^\epsilon(E)$-norms over $E \in \mathcal{F}_E$. By construction, $\phi_{\underline{v}_{\partial T}}$ has strongly continuous tangential traces on the edges of $T$ so

$$ \begin{aligned} |\phi_{\underline{v}_{\partial T}}|^2_{H_{\parallel}^{1/2}(\partial T)} &\lesssim \sum_{F \in \mathcal{F}_T} |\phi_{\underline{v}_F}|^2_{H^{1/2}(F)} = \sum_{F \in \mathcal{F}_T} |\text{grad}_F q_F|^2_{H^{1/2}(F)} \\ &\lesssim \sum_{F \in \mathcal{F}_T} \| \text{grad}_F q_F \|_{H^{1/2+\epsilon}(F)}^2 \lesssim \sum_{F \in \mathcal{F}_T} \left( \| C_F^k \underline{v}_F \|_{L^2(F)} + \| \underline{v}_F \|_{\text{curl},F} \right)^2. \end{aligned} $$

Combined with (B.2) and recalling that the local length scales are $\approx 1$, this leads to

$$ \| \phi_{\underline{v}_{\partial T}} \|_{L^2(\partial T)} + |\phi_{\underline{v}_{\partial T}}|_{H_{\parallel}^{1/2}(\partial T)} \lesssim \| \underline{C}_T^k \underline{v}_T \|_{\text{div},T} + \| \underline{v}_T \|_{\text{curl},T}. $$

Since $\phi_{\underline{v}_{\partial T}}$ belongs to $H_{\parallel}^{1/2}(\partial T)$, by [3, Theorem 3.1.3] there exists $g_{\underline{v}_T, \phi} \in H^1(T)$ such that the tangential trace of $g_{\underline{v}_T, \phi}$ is $\phi_{\underline{v}_{\partial T}}$ and

$$ \begin{aligned} \| g_{\underline{v}_T, \phi} \|_{L^2(T)} + \| \text{curl } g_{\underline{v}_T, \phi} \|_{L^2(T)} &\lesssim \| \phi_{\underline{v}_{\partial T}} \|_{L^2(\partial T)} + |\phi_{\underline{v}_{\partial T}}|_{H_{\parallel}^{1/2}(\partial T)} \\ &\lesssim \| \underline{C}_T^k \underline{v}_T \|_{\text{div},T} + \| \underline{v}_T \|_{\text{curl},T}. \end{aligned} $$

This was done under the assumption that $h_T = 1$. Using a scaling argument, we infer from the estimate above that, for an element $T$ of generic diameter $h_T$,

$$ \| g_{\underline{v}_T, \phi} \|_{L^2(T)} + h_T \| \text{curl } g_{\underline{v}_T, \phi} \|_{L^2(T)} \lesssim \| \underline{C}_T^k \underline{v}_T \|_{\text{div},T} + \| \underline{v}_T \|_{\text{curl},T}. \quad (B.11) $$
2. Construction of $g_{\mathbf{v}_T, \psi}$. By definition, $g_{\mathbf{v}_T, \psi}$ is the lifting of $\mathrm{grad}_{\partial T} \psi_{\mathbf{v}_{\partial T}}$. Re-
calling the construction of each $\psi_{\mathbf{v}_F} = \omega_F r_F$, for $F \in \mathcal{F}_T$, we can extend $r_F$
into a polynomial $r_{TF} \in \mathcal{P}^k(T)$ (for example, by making $r_{TF}$ independent of
the coordinate perpendicular to $F$). We then have, by (B.3),

$$
\|r_{TF}\|_{L^2(T)} \lesssim h_T^{\frac{1}{2}} \|r_F\|_{L^2(F)} \lesssim h_T \left( h_F^{\frac{1}{2}} \|\underline{v}_F\|_{\mathrm{curl},F} + h_F^{\frac{1}{2}} \|C_F^k \underline{v}_F\|_{L^2(F)} \right). \quad (B.12)
$$

The smooth, compactly supported function $\omega_F$ can be extended in $T$ into $\omega_{TF}$ such that $0 \le \omega_{TF} \le 1$, $\omega_{TF}$ has a compact support in a ball of radius $\approx h_T$ that does not touch the faces in $\mathcal{F}_T \setminus \{F\}$, and $|\mathrm{grad} \omega_{TF}| \lesssim h_T^{-1}$. Then, for each $F \in \mathcal{F}_T$, the chain rule yields

$$
\begin{equation} \tag{B.13}
\begin{split}
\| \mathrm{grad}(\omega_{TF} r_{TF}) \|_{L^2(T)} &\lesssim \| \mathrm{grad} r_{TF} \|_{L^2(T)} + h_T^{-1} \| r_{TF} \|_{L^2(T)} \\
&\lesssim h_F^{\frac{1}{2}} \| \underline{v}_F \|_{\mathrm{curl},F} + h_F^{\frac{1}{2}} \| C_F^k \underline{v}_F \|_{L^2(F)},
\end{split}
\end{equation}
$$

where the second inequality follows from an inverse inequality and (B.12). We
then set

$$
g_{\mathbf{v}_T, \psi} = \sum_{F \in \mathcal{F}_T} \mathrm{grad}(\omega_{TF} r_{TF}) \in \mathbf{C}^{\infty}(\bar{T}).
$$

By choice of the supports of $(\omega_{TF})_{F \in \mathcal{F}_T}$, the tangential trace of $g_{\mathbf{v}_T, \psi}$ on each
face $F \in \mathcal{F}_T$ is $\mathrm{grad}_F(\omega_{TF} r_{TF})|_F = \mathrm{grad}_F \psi_{\mathbf{v}_F}$. Moreover, the estimate (B.13)
gives

$$
\begin{align}
\| g_{\mathbf{v}_T, \psi} \|_{L^2(T)} &\lesssim \left[ \sum_{F \in \mathcal{F}_T} \left( h_F \| \underline{v}_F \|_{\mathrm{curl},F}^2 + h_F \| C_F^k \underline{v}_F \|_{L^2(F)}^2 \right) \right]^{\frac{1}{2}} \tag{B.14} \\
&\lesssim \| \underline{v}_T \|_{\mathrm{curl},T} + \| \underline{C}_T^k \underline{v}_T \|_{\mathrm{div},T}. \nonumber
\end{align}
$$

Since $g_{\mathbf{v}_T, \psi}$ is a gradient, we also have $\mathrm{curl} g_{\mathbf{v}_T, \psi} = 0$ and thus, combining (B.11) and (B.14) yields the estimate (B.9) on $g_{\mathbf{v}_T} = g_{\mathbf{v}_T, \phi} + g_{\mathbf{v}_T, \psi}$. $\square$

**Lemma 11 (Local Poincaré inequality for curl)** With $(\mathrm{grad} H^1(T))^{\perp}$ defined by (B.7), it holds

$$
\|w\|_{L^2(T)} \lesssim h_T \| \mathrm{curl} w \|_{L^2(T)} \quad \forall w \in (\mathrm{grad} H^1(T))^{\perp}. \quad (B.15)
$$

Proof By [3, Theorem 3.4.1], for all $v \in H(\mathrm{div};T)$ such that $\mathrm{div} v = 0$ and
$\langle v \cdot n_T, 1 \rangle_{\partial T} = 0$ (where $\langle \cdot, \cdot \rangle_{\partial T}$ is the $H^{-\frac{1}{2}}(\partial T)$-$H^{\frac{1}{2}}(\partial T)$ duality product and
$n_T$ is the outer normal to $T$), there exists $z \in H(\mathrm{curl};T)$ such that $\int_T z = 0$
and $v = \mathrm{curl} z$. Moreover, $\|z\|_{L^2(T)} \le C_0 \|v\|_{L^2(T)} = C_0 \|\mathrm{curl} z\|_{L^2(T)}$ and an
inspection of the proof shows that $C_0 \lesssim h_T$ (this estimate is obtained via a
scaling argument, and noticing that, if $h_T = 1$, the constants appearing in the
proof of [3, Theorem 3.4.1] do not depend on $T$ under our mesh regularity
assumptions).
2. Construction of $g_{\mathbf{v}_T, \psi}$. By definition, $g_{\mathbf{v}_T, \psi}$ is the lifting of $\mathrm{grad}_{\partial T} \psi_{\mathbf{v}_{\partial T}}$. Re-
calling the construction of each $\psi_{\mathbf{v}_F} = \omega_F r_F$, for $F \in \mathcal{F}_T$, we can extend $r_F$
into a polynomial $r_{TF} \in \mathcal{P}^k(T)$ (for example, by making $r_{TF}$ independent of
the coordinate perpendicular to $F$). We then have, by (B.3),

$$
\|r_{TF}\|_{L^2(T)} \lesssim h_T^{\frac{1}{2}} \|r_F\|_{L^2(F)} \lesssim h_T \left( h_F^{\frac{1}{2}} \|\underline{v}_F\|_{\mathrm{curl},F} + h_F^{\frac{1}{2}} \|C_F^k \underline{v}_F\|_{L^2(F)} \right). \quad (B.12)
$$

The smooth, compactly supported function $\omega_F$ can be extended in $T$ into $\omega_{TF}$ such that $0 \le \omega_{TF} \le 1$, $\omega_{TF}$ has a compact support in a ball of radius $\approx h_T$ that does not touch the faces in $\mathcal{F}_T \setminus \{F\}$, and $|\mathrm{grad} \omega_{TF}| \lesssim h_T^{-1}$. Then, for each $F \in \mathcal{F}_T$, the chain rule yields

$$
\begin{align}
\| \mathrm{grad}(\omega_{TF} r_{TF}) \|_{L^2(T)} &\lesssim \| \mathrm{grad} r_{TF} \|_{L^2(T)} + h_T^{-1} \| r_{TF} \|_{L^2(T)} \nonumber \\
&\lesssim h_F^{\frac{1}{2}} \| \underline{v}_F \|_{\mathrm{curl},F} + h_F^{\frac{1}{2}} \| C_F^k \underline{v}_F \|_{L^2(F)}, \tag{B.13}
\end{align}
$$

where the second inequality follows from an inverse inequality and (B.12). We
then set

$$
g_{\mathbf{v}_T, \psi} = \sum_{F \in \mathcal{F}_T} \mathrm{grad}(\omega_{TF} r_{TF}) \in \mathbf{C}^{\infty}(\bar{T}).
$$

By choice of the supports of $(\omega_{TF})_{F \in \mathcal{F}_T}$, the tangential trace of $g_{\mathbf{v}_T, \psi}$ on each
face $F \in \mathcal{F}_T$ is $\mathrm{grad}_F(\omega_{TF} r_{TF})|_F = \mathrm{grad}_F \psi_{\mathbf{v}_F}$. Moreover, the estimate (B.13)
gives

$$
\begin{align}
\| g_{\mathbf{v}_T, \psi} \|_{L^2(T)} &\lesssim \left[ \sum_{F \in \mathcal{F}_T} \left( h_F \| \underline{v}_F \|_{\mathrm{curl},F}^2 + h_F \| C_F^k \underline{v}_F \|_{L^2(F)}^2 \right) \right]^{\frac{1}{2}} \tag{B.14} \\
&\lesssim \| \underline{v}_T \|_{\mathrm{curl},T} + \| C_T^k \underline{v}_T \|_{\mathrm{div},T}. \nonumber
\end{align}
$$

Since $g_{\mathbf{v}_T, \psi}$ is a gradient, we also have $\mathrm{curl} g_{\mathbf{v}_T, \psi} = 0$ and thus, combining (B.11) and (B.14) yields the estimate (B.9) on $g_{\mathbf{v}_T} = g_{\mathbf{v}_T, \phi} + g_{\mathbf{v}_T, \psi}$. $\square$

**Lemma 11 (Local Poincaré inequality for curl)** With $(\mathrm{grad} H^1(T))^{\perp}$ defined by (B.7), it holds

$$
\|w\|_{L^2(T)} \lesssim h_T \| \mathrm{curl} w \|_{L^2(T)} \quad \forall w \in (\mathrm{grad} H^1(T))^{\perp}. \quad (B.15)
$$

Proof By [3, Theorem 3.4.1], for all $\mathbf{v} \in \mathbf{H}(\mathrm{div};T)$ such that $\mathrm{div} \mathbf{v} = 0$ and $\langle \mathbf{v} \cdot \mathbf{n}_T, 1 \rangle_{\partial T} = 0$ (where $\langle \cdot, \cdot \rangle_{\partial T}$ is the $H^{-\frac{1}{2}}(\partial T) - H^{\frac{1}{2}}(\partial T)$ duality product and $\mathbf{n}_T$ is the outer normal to $T$), there exists $\mathbf{z} \in \mathbf{H}(\mathrm{curl};T)$ such that $\int_T \mathbf{z} = 0$ and $\mathbf{v} = \mathrm{curl} \mathbf{z}$. Moreover, $\|\mathbf{z}\|_{L^2(T)} \le C_0 \|\mathbf{v}\|_{L^2(T)} = C_0 \|\mathrm{curl} \mathbf{z}\|_{L^2(T)}$ and an inspection of the proof shows that $C_0 \lesssim h_T$ (this estimate is obtained via a scaling argument, and noticing that, if $h_T = 1$, the constants appearing in the proof of [3, Theorem 3.4.1] do not depend on $T$ under our mesh regularity assumptions).
Take $w \in (\mathrm{grad} H^1(T))^{\perp}$ and let $(w_m)_{m \in \mathbb{N}}$ be a sequence in $C^{\infty}(\bar{T})$ which converges to $w$ in $H(\mathrm{curl}; T)$, see [3, Proposition 2.2.12]. Apply the result above to $v = \mathrm{curl} w_m$, which satisfies the requirements since, on each $F \in \mathcal{F}_T$, we have $\mathrm{curl} w_m \cdot n_{TF} = \mathrm{rot}_F((w_m)_{t,F})$ (where $n_{TF} = (n_T)_{|F}$ and, as before, $(w_m)_{t,F}$ is the tangential trace of $w_m$ on $F$, oriented here according to $n_{TF}$), and $w_m$ is continuous on $\partial T$. This yields $z_m \in H(\mathrm{curl}; T)$ such that $\mathrm{curl}(w_m - z_m) = 0$ and $\|z_m\|_{L^2(T)} \lesssim h_T \|\mathrm{curl} w_m\|_{L^2(T)}$. In particular, since the second Betti number of $T$ is zero, $w_m - z_m \in \mathrm{grad} H^1(T)$, and thus $\int_T (w_m - z_m) \cdot w = 0$. Hence,

$$ \int_T w_m \cdot w = \int_T z_m \cdot w \lesssim \|w\|_{L^2(T)} \|z_m\|_{L^2(T)} \lesssim \|w\|_{L^2(T)} h_T \|\mathrm{curl} w_m\|_{L^2(T)}. $$

The conclusion follows by letting $m \to \infty$ and simplifying by $\|w\|_{L^2(T)}$. $\square$

## C Notations

The notations used in the paper follow these rules: polynomial spaces, subspaces and projections are in curly letters; functions and operators with values in $\mathbb{R}^2$ or $\mathbb{R}^3$ are in boldface; the exponents indicate the maximum polynomial degree of the space or operator; full discrete gradient and curl, which need to be projected to define the operators in the DDR sequence, are in sans serif (and boldface since they are $\mathbb{R}^3$-valued); spaces, vectors and operators made of components attached to mesh entities of different dimensions are underlined. Table 4 lists the main notations used in the design and analysis of the DDR complex.

## References

1. J. Aghili, S. Boyaval, and D. A. Di Pietro. Hybridization of mixed high-order methods on general meshes and application to the Stokes equations. *Comput. Meth. Appl. Math.*, 15(2):111–134, 2015.

2. D. Arnold. *Finite Element Exterior Calculus*. SIAM, 2018.

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

4. L. Beirão da Veiga, F. Brezzi, F. Dassi, L. D. Marini, and A. Russo. A family of three-dimensional virtual elements with applications to magnetostatics. *SIAM J. Numer. Anal.*, 56(5):2940–2962, 2018.

5. L. Beirão da Veiga, F. Brezzi, F. Dassi, L. D. Marini, and A. Russo. Lowest order virtual element approximation of magnetostatic problems. *Comput. Methods Appl. Mech. Engrg.*, 332:343–362, 2018.

6. L. Beirão da Veiga, F. Brezzi, F. Dassi, L. D. Marini, and A. Russo. Serendipity virtual elements for general elliptic equations in three dimensions. *Chin. Ann. Math. Ser. B*, 39(2):315–334, 2018.

7. L. Beirão da Veiga, F. Brezzi, L. D. Marini, and A. Russo. $H(\mathrm{div})$ and $H(\mathrm{curl})$-conforming VEM. *Numer. Math.*, 133:303–332, 2016.

8. L. Beirão da Veiga, K. Lipnikov, and G. Manzini. *The mimetic finite difference method for elliptic problems*, volume 11 of *MS&A. Modeling, Simulation and Applications*. Springer, Cham, 2014.

9. L. Beirão da Veiga and L. Mascotto. Interpolation and stability properties of low order face and edge virtual element spaces. arXiv preprint, 2020.
Take $w \in (\mathrm{grad} H^1(T))^{\perp}$ and let $(w_m)_{m \in \mathbb{N}}$ be a sequence in $C^{\infty}(\bar{T})$ which converges to $w$ in $H(\mathrm{curl}; T)$, see [3, Proposition 2.2.12]. Apply the result above to $v = \mathrm{curl} w_m$, which satisfies the requirements since, on each $F \in \mathcal{F}_T$, we have $\mathrm{curl} w_m \cdot n_{TF} = \mathrm{rot}_F((w_m)_{t,F})$ (where $n_{TF} = (n_T)_{|F}$ and, as before, $(w_m)_{t,F}$ is the tangential trace of $w_m$ on $F$, oriented here according to $n_{TF}$), and $w_m$ is continuous on $\partial T$. This yields $z_m \in H(\mathrm{curl}; T)$ such that $\mathrm{curl}(w_m - z_m) = 0$ and $\|z_m\|_{L^2(T)} \lesssim h_T \|\mathrm{curl} w_m\|_{L^2(T)}$. In particular, since the second Betti number of $T$ is zero, $w_m - z_m \in \mathrm{grad} H^1(T)$, and thus $\int_T (w_m - z_m) \cdot w = 0$. Hence,

$$ \int_T w_m \cdot w = \int_T z_m \cdot w \lesssim \|w\|_{L^2(T)} \|z_m\|_{L^2(T)} \lesssim \|w\|_{L^2(T)} h_T \|\mathrm{curl} w_m\|_{L^2(T)}. $$

The conclusion follows by letting $m \to \infty$ and simplifying by $\|w\|_{L^2(T)}$. $\square$

## C Notations

The notations used in the paper follow these rules: polynomial spaces, subspaces and projections are in curly letters; functions and operators with values in $\mathbb{R}^2$ or $\mathbb{R}^3$ are in boldface; the exponents indicate the maximum polynomial degree of the space or operator; full discrete gradient and curl, which need to be projected to define the operators in the DDR sequence, are in sans serif (and boldface since they are $\mathbb{R}^3$-valued); spaces, vectors and operators made of components attached to mesh entities of different dimensions are underlined. Table 4 lists the main notations used in the design and analysis of the DDR complex.

## References

1. J. Aghili, S. Boyaval, and D. A. Di Pietro. Hybridization of mixed high-order methods on general meshes and application to the Stokes equations. *Comput. Meth. Appl. Math.*, 15(2):111–134, 2015.

2. D. Arnold. *Finite Element Exterior Calculus*. SIAM, 2018.

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

4. L. Beirão da Veiga, F. Brezzi, F. Dassi, L. D. Marini, and A. Russo. A family of three-dimensional virtual elements with applications to magnetostatics. *SIAM J. Numer. Anal.*, 56(5):2940–2962, 2018.

5. L. Beirão da Veiga, F. Brezzi, F. Dassi, L. D. Marini, and A. Russo. Lowest order virtual element approximation of magnetostatic problems. *Comput. Methods Appl. Mech. Engrg.*, 332:343–362, 2018.

6. L. Beirão da Veiga, F. Brezzi, F. Dassi, L. D. Marini, and A. Russo. Serendipity virtual elements for general elliptic equations in three dimensions. *Chin. Ann. Math. Ser. B*, 39(2):315–334, 2018.

7. L. Beirão da Veiga, F. Brezzi, L. D. Marini, and A. Russo. $H(\mathrm{div})$ and $H(\mathrm{curl})$-conforming VEM. *Numer. Math.*, 133:303–332, 2016.

8. L. Beirão da Veiga, K. Lipnikov, and G. Manzini. *The mimetic finite difference method for elliptic problems*, volume 11 of *MS&A. Modeling, Simulation and Applications*. Springer, Cham, 2014.

9. L. Beirão da Veiga and L. Mascotto. Interpolation and stability properties of low order face and edge virtual element spaces. arXiv preprint, 2020.
<table><caption>Table 4: Notations for spaces and operators.</caption><thead><tr><th>Notation</th><th>Meaning</th><th>Reference</th></tr></thead><tbody><tr><td><b>G</b><sup>l</sup>, <b>R</b><sup>l</sup>, <b>G</b><sup>c,l</sup>, <b>R</b><sup>c,l</sup></td><td>Polynomial ranges of gradient/curl, and complements</td><td>(2.3), (2.5)</td></tr><tr><td>π<sub>G,X</sub><sup>l</sup>, π<sub>R,X</sub><sup>l</sup>, π<sub>G,X</sub><sup>c,l</sup>, π<sub>R,X</sub><sup>c,l</sup></td><td>L<sup>2</sup>-projections on <b>G</b><sup>l</sup>(X), <b>R</b><sup>l</sup>(X), <b>G</b><sup>c,l</sup>(X), <b>R</b><sup>c,l</sup>(X)</td><td>Section 2.4</td></tr><tr><td><b>N</b><sup>l</sup>, <b>R</b><sup>l</sup></td><td>Nédélec and Raviart–Thomas spaces</td><td>(2.12)</td></tr><tr><td><b>Γ</b><sub>S,S<sup>c</sup></sub>(·,·), <b>Γ</b><sub>X,Y</sub><sup>l</sup>(·,·)</td><td>Recovery operator</td><td>(2.13), (2.17)</td></tr><tr><td><b>X</b><sup>k</sup><sub>grad,h</sub>, <b>X</b><sup>k</sup><sub>curl,h</sub>, <b>X</b><sup>k</sup><sub>div,h</sub></td><td>Spaces of the DDR sequence</td><td>(3.1)–(3.3)</td></tr><tr><td><b>I</b><sup>k</sup><sub>grad,h</sub>, <b>I</b><sup>k</sup><sub>curl,h</sub>, <b>I</b><sup>k</sup><sub>div,h</sub></td><td>Interpolators on the DDR spaces</td><td>(3.5)–(3.7)</td></tr><tr><td><b>G</b><sub>E</sub>, <b>G</b><sub>F</sub><sup>k</sup>, <b>G</b><sub>T</sub><sup>k</sup></td><td>Edge, face and element full gradients</td><td>(3.8), (3.9), (3.11)</td></tr><tr><td><b>C</b><sub>F</sub><sup>k</sup>, <b>C</b><sub>T</sub><sup>k</sup></td><td>Face and element full curl</td><td>(3.19), (3.28)</td></tr><tr><td><b>D</b><sub>T</sub><sup>k</sup></td><td>Element divergence</td><td>(3.32)</td></tr><tr><td><b>G</b><sub>h</sub><sup>k</sup>, <b>C</b><sub>h</sub><sup>k</sup>, <b>D</b><sub>h</sub><sup>k</sup></td><td>Global discrete gradient, curl and divergence</td><td>(3.18), (3.31), (3.33)</td></tr><tr><td>γ<sub>F</sub><sup>k+1</sup>, <b>P</b><sup>k+1</sup><sub>grad,T</sub></td><td>Scalar trace and potential on <b>X</b><sup>k</sup><sub>grad,T</sub></td><td>(3.10), (4.1)</td></tr><tr><td>γ<sub>t,F</sub><sup>k</sup>, <b>P</b><sup>k</sup><sub>curl,T</sub></td><td>Tangential trace and potential on <b>X</b><sup>k</sup><sub>curl,T</sub></td><td>(3.22), (4.4)</td></tr><tr><td><b>P</b><sup>k</sup><sub>div,T</sub></td><td>Potential on <b>X</b><sup>k</sup><sub>div,T</sub></td><td>(4.9)</td></tr><tr><td>(·,·)<sub>•,T</sub>, (·,·)<sub>•,h</sub>, ||·||<sub>•,h</sub></td><td>Discrete L<sup>2</sup>-products and norms (• ∈ {grad, curl, div})</td><td>Section 4.4</td></tr><tr><td>|||·|||<sub>•,h</sub></td><td>Components norm (• ∈ {grad, curl, div})</td><td>Section 4.5</td></tr></tbody></table>

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<table><caption>Table 4: Notations for spaces and operators.</caption><thead><tr><th>Notation</th><th>Meaning</th><th>Reference</th></tr></thead><tbody><tr><td><b>G</b><sup>l</sup>, <b>R</b><sup>l</sup>, <b>G</b><sup>c,l</sup>, <b>R</b><sup>c,l</sup></td><td>Polynomial ranges of gradient/curl, and complements</td><td>(2.3), (2.5)</td></tr><tr><td>π<sub>G,X</sub><sup>l</sup>, π<sub>R,X</sub><sup>l</sup>, π<sub>G,X</sub><sup>c,l</sup>, π<sub>R,X</sub><sup>c,l</sup></td><td>L<sup>2</sup>-projections on <b>G</b><sup>l</sup>(X), <b>R</b><sup>l</sup>(X), <b>G</b><sup>c,l</sup>(X), <b>R</b><sup>c,l</sup>(X)</td><td>Section 2.4</td></tr><tr><td><b>N</b><sup>l</sup>, <b>R</b><sup>l</sup></td><td>Nédélec and Raviart-Thomas spaces</td><td>(2.12)</td></tr><tr><td><b>Γ</b><sub>S,S<sup>c</sup></sub>(·,·), <b>Γ</b><sub>X,Y</sub><sup>l</sup>(·,·)</td><td>Recovery operator</td><td>(2.13), (2.17)</td></tr><tr><td><b>X</b><sup>k</sup><sub>grad,h</sub>, <b>X</b><sup>k</sup><sub>curl,h</sub>, <b>X</b><sup>k</sup><sub>div,h</sub></td><td>Spaces of the DDR sequence</td><td>(3.1)-(3.3)</td></tr><tr><td><b>I</b><sup>k</sup><sub>grad,h</sub>, <b>I</b><sup>k</sup><sub>curl,h</sub>, <b>I</b><sup>k</sup><sub>div,h</sub></td><td>Interpolators on the DDR spaces</td><td>(3.5)-(3.7)</td></tr><tr><td><b>G</b><sub>E</sub>, <b>G</b><sub>F</sub><sup>k</sup>, <b>G</b><sub>T</sub><sup>k</sup></td><td>Edge, face and element full gradients</td><td>(3.8), (3.9), (3.11)</td></tr><tr><td><b>C</b><sub>F</sub><sup>k</sup>, <b>C</b><sub>T</sub><sup>k</sup></td><td>Face and element full curl</td><td>(3.19), (3.28)</td></tr><tr><td><b>D</b><sub>T</sub><sup>k</sup></td><td>Element divergence</td><td>(3.32)</td></tr><tr><td><b>G</b><sub>h</sub><sup>k</sup>, <b>C</b><sub>h</sub><sup>k</sup>, <b>D</b><sub>h</sub><sup>k</sup></td><td>Global discrete gradient, curl and divergence</td><td>(3.18), (3.31), (3.33)</td></tr><tr><td>γ<sub>F</sub><sup>k+1</sup>, <b>P</b><sup>k+1</sup><sub>grad,T</sub></td><td>Scalar trace and potential on <b>X</b><sup>k</sup><sub>grad,T</sub></td><td>(3.10), (4.1)</td></tr><tr><td>γ<sub>t,F</sub><sup>k</sup>, <b>P</b><sup>k</sup><sub>curl,T</sub></td><td>Tangential trace and potential on <b>X</b><sup>k</sup><sub>curl,T</sub></td><td>(3.22), (4.4)</td></tr><tr><td><b>P</b><sup>k</sup><sub>div,T</sub></td><td>Potential on <b>X</b><sup>k</sup><sub>div,T</sub></td><td>(4.9)</td></tr><tr><td>(·,·)<sub>•,T</sub>, (·,·)<sub>•,h</sub>, ||·||<sub>•,h</sub></td><td>Discrete L<sup>2</sup>-products and norms (• ∈ {grad, curl, div})</td><td>Section 4.4</td></tr><tr><td>|||·|||<sub>•,h</sub></td><td>Components norm (• ∈ {grad, curl, div})</td><td>Section 4.5</td></tr></tbody></table>

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22. L. Codecasa, R. Specogna, and F. Trevisan. Symmetric positive-definite constitutive matrices for discrete eddy-current problems. *IEEE Transactions on Magnetics*, 43:510-515, 2007.

23. L. Codecasa, R. Specogna, and F. Trevisan. Base functions and discrete constitutive relations for staggered polyhedral grids. *Comput. Methods Appl. Mech. Engrg.*, 198(9-12):1117-1123, 2009.

24. L. Codecasa, R. Specogna, and F. Trevisan. A new set of basis functions for the discrete geometric approach. *J. Comput. Phys.*, 19(299):7401-7410, 2010.

25. M. Dauge. *Elliptic boundary value problems on corner domains*, volume 1341 of *Lecture Notes in Mathematics*. Springer-Verlag, Berlin, 1988. Smoothness and asymptotics of solutions.

26. P. R. B. Devloo, O. Durán, S. M. Gomes, and M. Ainsworth. High-order composite finite element exact sequences based on tetrahedral-hexahedral-prismatic-pyramidal partitions. *Comput. Methods Appl. Mech. Engrg.*, 355:952-975, 2019.

27. D. A. Di Pietro and J. Droniou. A third Strang lemma for schemes in fully discrete formulation. *Calcolo*, 55(40), 2018.

28. D. A. Di Pietro and J. Droniou. *The Hybrid High-Order method for polytopal meshes*. Number 19 in *Modeling, Simulation and Application*. Springer International Publishing, 2020.

29. D. A. Di Pietro and J. Droniou. An arbitrary-order method for magnetostatics on polyhedral meshes based on a discrete de Rham sequence. *J. Comput. Phys.*, 429(109991), 2021.

30. D. A. Di Pietro and J. Droniou. A DDR method for the Reissner-Mindlin plate bending problem on polygonal meshes, May 2021. https://hal.archives-ouvertes.fr/hal-03234088.

31. D. A. Di Pietro, J. Droniou, and F. Rapetti. Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra. *Math. Models Methods Appl. Sci.*, 30(9):1809-1855, 2020.

32. D. A. Di Pietro and A. Ern. Arbitrary-order mixed methods for heterogeneous anisotropic diffusion on general meshes. *IMA J. Numer. Anal.*, 37(1):40-63, 2017.

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

34. J. Droniou and R. Eymard. A mixed finite volume scheme for anisotropic diffusion problems on any grid. *Numer. Math.*, 105:35-71, 2006.

35. J. Droniou, R. Eymard, T. Gallouët, C. Guichard, and R. Herbin. *The gradient discretisation method*, volume 82 of *Mathematics & Applications*. Springer, 2018.

36. J. Droniou, R. Eymard, T. Gallouët, and R. Herbin. A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. *Math. Models Methods Appl. Sci. (M3AS)*, 20(2):1-31, 2010.

37. O. Durán, P. R. B. Devloo, S. M. Gomes, and F. Valentin. A multiscale hybrid method for Darcy's problems using mixed finite element local solvers. *Comput. Methods Appl. Mech. Engrg.*, 354:213-244, 2019.

38. R. Eymard, T. Gallouët, and R. Herbin. Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes. SUSHI: a scheme using stabilization and hybrid interfaces. *IMA J. Numer. Anal.*, 30(4):1009-1043, 2010.

39. A. Gillette, A. Rand, and C. Bajaj. Construction of scalar and vector finite element families on polygonal and polyhedral meshes. *Comput. Methods Appl. Math.*, 16(4):667-683, 2016.

40. R. Hiptmair. Finite elements in computational electromagnetism. *Acta Numer.*, 11:237-339, 2002.

41. Y. Kuznetsov, K. Lipnikov, and M. Shashkov. Mimetic finite difference method on polygonal meshes for diffusion-type problems. *Comput. Geosci.*, 8:301-324, 2004.

42. K. Lipnikov, M. Shashkov, and D. Svyatskiy. The mimetic finite difference discretization of diffusion problem on unstructured polyhedral meshes. *J. Comput. Phys.*, 211(2):473-491, 2006.
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