$H(\text{curl})$-reconstruction of piecewise polynomial fields with application to $hp$-a posteriori nonconforming error analysis for Maxwell's equations

Zhaonan Dong, Alexandre Ern

► To cite this version:

Zhaonan Dong, Alexandre Ern. $H(\text{curl})$-reconstruction of piecewise polynomial fields with application to $hp$-a posteriori nonconforming error analysis for Maxwell's equations. 2025. hal-05213366

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https://inria.hal.science/hal-05213366v1

Preprint submitted on 18 Aug 2025

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# $\mathbf{H}(\text{curl})$-reconstruction of piecewise polynomial fields with application to hp-a posteriori nonconforming error analysis for Maxwell's equations

Z. Dong\*, A. Ern†

## Abstract

We devise and analyze a novel $\mathbf{H}(\text{curl})$-reconstruction operator for piecewise polynomial fields on shape-regular simplicial meshes. The (non-polynomial) reconstruction is devised over the mesh vertex patches using the partition of unity induced by hat basis functions in combination with local Helmholtz decompositions. Our main focus is on homogeneous tangential boundary conditions. We prove that the difference between the reconstructed $\mathbf{H}_0(\text{curl})$-field and the original, piecewise polynomial field, measured in the broken curl norm and in the $L^2$-norm, can be bounded in terms of suitable jump norms of the original field. The bounds are always $h$-optimal, and $p$-suboptimal by $\frac{1}{2}$-order for the broken curl norm and by $\frac{3}{2}$-order for the $L^2$-norm. An auxiliary result of independent interest is a novel broken-curl, divergence-preserving Poincaré inequality on vertex patches. Moreover, the $L^2$-norm estimate can be improved to $\frac{1}{2}$-order suboptimality under a (reasonable) assumption on the uniform elliptic regularity pickup for a Poisson problem with Neumann conditions over the vertex patches. We also discuss extensions of the $\mathbf{H}_0(\text{curl})$-reconstruction operator to the prescription of mixed boundary conditions, to agglomerated polytopal meshes, and to convex domains. Finally, we showcase an important application of the $\mathbf{H}(\text{curl})$-reconstruction operator to the hp-a posteriori nonconforming error analysis of Maxwell's equations. We focus on the (symmetric) interior penalty discontinuous Galerkin (dG) approximation of some simplified forms of Maxwell's equations.

**Keywords.** $\mathbf{H}(\text{curl})$-reconstruction, $h$-analysis, broken Poincaré inequality, a posteriori error analysis, nonconforming approximation, discontinuous Galerkin, Maxwell's equations

MSC. 65N30, 65N15, 78M10

## 1 Introduction

Given a piecewise polynomial field on a shape-regular simplicial mesh covering exactly a Lipschitz polyhedral domain, we show in the paper the existence of a (non-polynomial) $\mathbf{H}(\text{curl})$-conforming field whose distance to the original field in the broken curl norm and in the $L^2$-norm is bounded in terms of suitable jumps (and boundary values) of the original piecewise polynomial field, namely its tangential jumps (and boundary values) and the normal jumps (and boundary values) of its curl. All the estimates are local and $h$-optimal, and they are $p$-suboptimal by only $\frac{1}{2}$-order in the broken curl norm and by $\frac{3}{2}$-order in the $L^2$-norm. Here, $h$ denotes the mesh size and $p$ the polynomial degree. We emphasize that, as an auxiliary result to establish the $L^2$-norm estimate, we derive a novel broken-curl, divergence-preserving Poincaré inequality on the mesh vertex patches that is of independent interest. We also observe that the $L^2$-estimate can be improved to $\frac{1}{2}$-order $p$-suboptimality under a (reasonable) assumption

<footnote n="1">¹Inria, 48 rue Barrault, 75647 Paris, France and CERMICS, ENPC, Institut Polytechnique de Paris, 6 & 8 avenue B. Pascal, 77455 Marne-la-Vallée, France</footnote>

<footnote n="2">²CERMICS, ENPC, Institut Polytechnique de Paris, 6 & 8 avenue B. Pascal, 77455 Marne-la-Vallée, France and Inria, 48 rue Barrault, 75647 Paris, France</footnote>
on the uniform elliptic regularity pickup for a Poisson problem with Neumann conditions over
the mesh vertex patches.

We present an explicit construction of the (non-polynomial) **H**(*curl*)-conforming field over the mesh vertex patches using the partition of unity induced by hat basis functions in combination with local Helmholtz decompositions. The crucial advantage of resorting to a local construction over the mesh vertex patches is that the associated subdomains have a trivial topology, thereby circumventing any dependence of the estimates on the topology of the original domain. For simplicity, we devise the construction so that the (non-polynomial) **H**(*curl*)-conforming field satisfies homogeneous tangential boundary conditions, i.e., sits in **H**₀(*curl*; *D*), but we also discuss the extension to more general mixed boundary conditions. Another relevant extension is the case of agglomerated polytopal meshes, where the same *hp*-estimates as on simplicial meshes can be established by considering the underlying simplicial submesh under the assumption that this latter mesh has locally the same size as the original polytopal mesh. Finally, we also discuss a variant where the **H**(*curl*)-conforming field is defined globally without considering the mesh vertex patches. On convex domains, this leads to a *p*-optimal *L*²-estimate without any further assumption, but the estimate is only global and the constant depends on the topology of the global domain.

Reconstructing **H**(curl)-conforming fields from piecewise polynomial fields on a shape-
regular simplicial mesh has already been quite explored in the literature. A first possibility is
to devise a polynomial **H**(curl)-reconstruction using Nédélec (edge) finite elements defined by
prescribing the degrees of freedom as the averages of those of the original, piecewise polynomial
field. Such averaging operators have been considered in [8], and a systematic analysis can be
found in [21] with and without boundary prescription, leading to (local) *h*-optimal estimates
in the broken curl and *L*²-norms (see also [6, 35] for the seminal work in the *H*¹-context).
However, defining a reconstruction by prescription of degrees of freedom can hardly lead to *p*-optimal estimates because of the repeated use of discrete trace (and inverse) inequalities needed
in the analysis (see [7] for further insight in the *H*¹-context). A second possibility is to solve
local, discrete minimization problems using again Nédélec and Raviart–Thomas finite elements.
Following the seminal work in [5, 25, 26] in the *H*¹/*H*(div)-context, recent advances leading, in
particular, to full *p*-optimality in the **H**(curl)-context were accomplished in [13, 14]. [13] poses
local problems on edge patches, but does not achieve **H**(curl)-conformity. Instead, [14] poses
local problems on vertex patches and achieves **H**(curl)-conformity. However, in both works,
the original piecewise polynomial field must satisfy a certain orthogonality property, typically
associated with Galerkin's orthogonality when solving a curl-curl problem. We also refer the
reader to [11] for a simplified construction of [14], and to [15] for the use of the technique from
[14] to devise a stable, local, commuting operator in **H**(curl) with *hp*-optimal approximation
properties. The above discussion shows that a gap remains in the literature when it comes to
devising an **H**(curl)-conforming reconstruction from an arbitrary piecewise polynomial field
and ensuring *hp*-approximation properties.

The main application we have in mind for the present $H(\text{curl})$-reconstruction operator is the $hp$-a posteriori nonconforming error analysis of some simplified forms of Maxwell's equations involving the curl-curl operator and a zero-order term. In the case of elliptic problems, the idea of invoking an $H^1$-reconstruction operator based on a global Helmholtz decomposition to estimate the so-called nonconforming approximation error can be traced back to [17] in 2D and to [10] in 3D with Dirichlet boundary conditions (see also [4, 1] where the Helmholtz decomposition is also invoked). A further advance has been accomplished recently in [20] in the context of a discretization by the hybrid high-order (HHO) method introduced in [19] for elliptic problems including mixed Dirichlet–Neumann boundary conditions. Specifically, the $H^1$-reconstruction operator considered in [20] is devised locally on mesh vertex patches which all have a simple topology.

We consider the (symmetric) interior penalty discontinuous Galerkin (dG) method for space discretization. This approximation technique is classical for elliptic problems (see, e.g., [3, 18] and the references therein) and has been developed and analyzed in the context of Maxwell’s
equations in [36, 29]. Here, we perform the hp-a posteriori error analysis hinging on the present **H**(**curl**) -reconstruction operator, focusing on homogeneous Dirichlet boundary conditions. This analysis offers, to our knowledge, two ground-breaking features: it is the first time that a full hp-analysis is achieved, and it is the first time that the analysis hinges on local (vertex patch) Helmholtz decompositions to estimate the nonconforming error. We emphasize that, while the theoretical derivation of the error estimate relies on the **H**<sub>0</sub>(**curl**) -reconstruction operator, this reconstruction does not need to be computed in practice. Previous a posteriori error estimates bounded the nonconforming error using an averaging operator, as in [30, 31, 12], and were therefore only h-optimal. We notice, in passing, that [12] derived an hp-a posteriori bound on the conforming error, and addressed the frequency-dependence of the a posteriori estimates in the indefinite regime by using a dualty argument à la Schatz.

The paper is organized as follows. In Section 2, we briefly present the continuous and discrete settings. In Section 3, we state our main results on the **H**<sub>0</sub>(curl)-reconstruction operator and comment on various possible extensions. The main result of this section is Theorem 3.7. In Section 4, we establish some preparatory results. In particular, we highlight the local broken-curl, divergence-preserving Poincaré inequality from Lemma 4.3. In Section 5, we prove our main result. Finally, in Section 6, we consider the approximation of Maxwell’s equations by the (symmetric) interior penalty dG method and develop an hp-version a posteriori error analysis based on the **H**<sub>0</sub>(curl)-reconstruction operator devised in Section 3. The main result of this section is Theorem 6.8.

## 2 Setting

In this section, we briefly describe the continuous and discrete settings.

### 2.1 Continuous setting

Let $D$ be an open, bounded, connected, Lipschitz set (domain) in $\mathbb{R}^d$ with boundary $\partial D$ and outward unit normal $n_D$. We assume that $D$ is a polyhedron so that it can be covered exactly by a simplicial mesh. To fix the ideas, we assume that $d = 3$, but all what follows readily extends to $d = 2$. We do not make any simplifying assumption on the topology of $D$, i.e., $D$ may be multiply connected and its boundary be composed of several connected components.

We use boldface font for vectors, vector fields, and functional spaces composed of such fields. We use standard notation for Lebesgue and Sobolev spaces, and fractional-order Sobolev spaces are equipped with the standard Sobolev-Slobodeckii seminorm based on the double integral (see, e.g., [22, Def. 2.14]). To alleviate the notation, the inner product and associated norm in the spaces $L^2(D)$ and $L^2(D)$ are denoted by $(\cdot, \cdot)$ and $\|\cdot\|$, respectively, whereas we add a subscript $\omega$ when we consider the Lebesgue spaces over a subdomain $\omega \subset D$. We consider the Hilbert Sobolev spaces

$$ \mathbf{H}(\mathrm{curl}; \omega) := \{v \in L^2(\omega) \mid \nabla \times v \in L^2(\omega)\}, \quad (2.1a) $$

$$ \mathbf{H}_0(\mathrm{curl}; \omega) := \{v \in \mathbf{H}(\mathrm{curl}; \omega) \mid \gamma_{\partial\omega}^c(v) = 0\}, \quad (2.1b) $$

$$ \mathbf{H}(\mathrm{div}; \omega) := \{v \in L^2(\omega) \mid \nabla \cdot v \in L^2(\omega)\}, \quad (2.1c) $$

$$ \mathbf{H}_0(\mathrm{div}; \omega) := \{v \in \mathbf{H}(\mathrm{div}; \omega) \mid \gamma_{\partial\omega}^d(v) = 0\}, \quad (2.1d) $$

where $\gamma_{\partial\omega}^c : \mathbf{H}(\mathrm{curl}; \omega) \to \mathbf{H}^{-\frac{1}{2}}(\partial\omega)$ (resp., $\gamma_{\partial\omega}^d : \mathbf{H}(\mathrm{div}; \omega) \to \mathbf{H}^{-\frac{1}{2}}(\partial\omega)$) is the extension by density of the tangential (resp., normal) trace operator such that $\gamma_{\partial\omega}^c(v) = v|_{\partial\omega} \cdot n_\omega$ (resp., $\gamma_{\partial\omega}^d(v) = v|_{\partial\omega} \cdot n_\omega$) for smooth fields. The subscript 0 is used for the nabla operator to indicate that the operator acts on functions or fields respecting suitable homogeneous Dirichlet conditions. Thus, $(\nabla, -\nabla_0 \cdot)$, $(\nabla \times, \nabla_0 \times)$ and $(\nabla \cdot, -\nabla_0)$ are pairs of adjoint operators; for instance, $(\nabla_0 \times v, w) = (v, \nabla \times w)$ for all $(v, w) \in \mathbf{H}_0(\mathrm{curl}; D) \times \mathbf{H}(\mathrm{curl}; D)$.
## 2.2 Mesh and polynomial spaces

Let $\mathcal{T}_h$ be an affine simplicial mesh covering $D$ exactly. A generic mesh cell is denoted by $K$, its diameter by $h_K$ and its outward unit normal by $n_K$. We write $\mathcal{F}_h$ for the set of mesh faces, $\mathcal{F}_h^{\text{int}}$ for the subset of mesh interfaces (shared by two distinct mesh cells, $K_l, K_r$), and $\mathcal{F}_h^{\text{bd}}$ for the subset of mesh boundary faces (shared by one mesh cell, $K_l$, and the boundary, $\partial D$). Every mesh interface $F \in \mathcal{F}_h^{\text{int}}$ is oriented by the unit normal, $n_F$, pointing from $K_l$ to $K_r$ (the orientation is arbitrary, but fixed). Every boundary face $F \in \mathcal{F}_h^{\text{bd}}$ is oriented by the unit normal $n_F := n_D|_F$. The diameter of a generic face $F \in \mathcal{F}_h$ is denoted by $h_F$.

The set of mesh vertices is denoted by $\mathcal{V}_h$ and is decomposed into the subset of interior vertices, $\mathcal{V}_h^{\text{int}}$, and the subset of boundary vertices, $\mathcal{V}_h^{\text{bd}}$. For all $\mathbf{a} \in \mathcal{V}_h$, $\mathcal{T}_a$ denotes the collection of the mesh cells which share $\mathbf{a}$ and $D_a$ the corresponding open subdomain (often called vertex patch). In addition, $\mathcal{F}_a$ denotes the collection of the mesh faces which share $\mathbf{a}$, and $\mathcal{F}_a^{\partial}$ denotes the collection of all the mesh faces lying on $\partial D_a$ and not containing $\mathbf{a}$; thus the two sets $\mathcal{F}_a$ and $\mathcal{F}_a^{\partial}$ are disjoint, and $\mathcal{F}_a^{\partial} = \emptyset$ whenever $\mathbf{a} \in \mathcal{V}_h^{\text{bd}}$. For all $\mathbf{a} \in \mathcal{V}_h$, we set $h_a := \text{diam}(D_a)$, and let $\psi_a$ be the hat basis function equal to 1 at $\mathbf{a}$ and having a support in $D_a$. Recall that the hat basis functions satisfy the following partition-of-unity property:

$$ \sum_{\mathbf{a} \in \mathcal{V}_h} \psi_{\mathbf{a}} = 1. \qquad (2.2) $$

For all $K \in \mathcal{T}_h$, $\mathcal{F}_K$ (resp. $\mathcal{V}_K$) is the collection of the mesh faces composing $\partial K$ (resp., mesh vertices in $\partial K$). For all $F \in \mathcal{F}_K$ and every piecewise smooth field $\mathbf{v}$ on $\mathcal{T}_h$, we define the local trace operators such that $\gamma_{K,F}^{\text{g}}(\mathbf{v})(\mathbf{x}) := \mathbf{v}|_K(\mathbf{x})$, $\gamma_{K,F}^{\text{c}}(\mathbf{v})(\mathbf{x}) := \mathbf{v}|_K(\mathbf{x}) \times \mathbf{n}_F$, $\gamma_{K,F}^{\text{d}}(\mathbf{v})(\mathbf{x}) := \mathbf{v}_h|_K(\mathbf{x}) \cdot \mathbf{n}_F$, for a.e. $\mathbf{x} \in F$. Then, for all $F \in \mathcal{F}_h^{\text{int}}$ and $x \in \{\text{g}, \text{c}, \text{d}\}$, we define the jump and average operators such that

$$ [\mathbf{v}]_F^x := \gamma_{K_l,F}^x(\mathbf{v}) - \gamma_{K_r,F}^x(\mathbf{v}), \quad \{[\mathbf{v}]_F^x\} := \frac{1}{2}(\gamma_{K_l,F}^x(\mathbf{v}) + \gamma_{K_r,F}^x(\mathbf{v})). \qquad (2.3) $$

We also set $[\mathbf{v}]_F^x := \{[\mathbf{v}]_F^x\} := \gamma_{K_l,F}^x(\mathbf{v})$ for all $F \in \mathcal{F}_h^{\text{bd}}$. For all $K \in \mathcal{T}_h$, we define the jump across the boundary of $K$ as $[\bullet]_{\partial K}^x|_F := \iota_{K,F}[\bullet]_F^x$ for all $F \in \mathcal{F}_K$, with $\iota_{K,F} := \mathbf{n}_K \cdot \mathbf{n}_F = \pm 1$.

Let $p \ge 1$ be the polynomial degree (see Remark 3.8 below for the case $p = 0$). Let $\mathbb{P}_{p,d}$ be the space composed of $d$-variate polynomials of total degree at most $p$ and set $\mathbb{P}_{p,d} := [\mathbb{P}_{p,d}]^d$. We consider the following broken polynomial space:

$$ \mathbf{P}_p^b(\mathcal{T}_h) := \{\mathbf{v}_h \in \mathbb{L}^2(D) \mid \mathbf{v}_h|_K \in \mathbb{P}_{p,d}, \forall K \in \mathcal{T}_h\}, \qquad (2.4) $$

and we denote the $\mathbb{L}^2$-orthogonal projection onto $\mathbf{P}_p^b(\mathcal{T}_h)$ as

$$ \Pi_h^b : \mathbb{L}^2(D) \to \mathbf{P}_p^b(\mathcal{T}_h). \qquad (2.5) $$

We use the notation $\mathbf{P}_p^b(\mathcal{T}_a)$ for the broken polynomial space based on the local mesh $\mathcal{T}_a$ for a mesh vertex $\mathbf{a} \in \mathcal{V}_h$.

## 2.3 hp-approximation tools

Let $\kappa_{\mathcal{T}_h}$ denote the shape-regularity parameter of the mesh $\mathcal{T}_h$, and $C(\kappa_{\mathcal{T}_h})$ a generic constant solely depending on $\kappa_{\mathcal{T}_h}$ (but independent of the mesh size and the polynomial degree), and whose value can change at each occurrence. Sometimes, we also abbreviate as $A \lesssim B$ the inequality $A \le C(\kappa_{\mathcal{T}_h})B$ for positive numbers $A$ and $B$.

**Lemma 2.1** (Discrete trace inequality). There exists $C(\kappa_{\mathcal{T}_h})$ so that, for all $v \in \mathbb{P}_{p,d}(K)$, all $K \in \mathcal{T}_h$, and all $p \ge 1$,

$$ \|v\|_{\partial K} \le C(\kappa_{\mathcal{T}_h}) \left(\frac{p^2}{h_K}\right)^{\frac{1}{2}} \|v\|_K. \qquad (2.6) $$
Proof. A proof can be found in [39] (with explicit constant in terms of d). □

**Lemma 2.2** (Local $L^2$-orthogonal projection). Let $\delta \in (0, \frac{1}{2}]$. There exists $C(\kappa_{\mathcal{T}_h})$ so that, for all $v \in H^{\frac{1}{2}+\delta}(K)$, all $K \in \mathcal{T}_h$, and all $p \ge 1$,

$$ \|v - \Pi_K^p(v)\|_{\partial K} \le C(\kappa_{\mathcal{T}_h}) \left(\frac{h_K}{p}\right)^\delta |v|_{H^{\frac{1}{2}+\delta}(K)}, \quad (2.7) $$

where $\Pi_K^p$ denotes the $L^2$-orthogonal projection onto $\mathbb{P}_{p,d}$.

Proof. A proof can be found in [34, Corollary 1.2]. □

**Lemma 2.3** (Local modified hp-Karkulik–Melenk operator). There exists $C(\kappa_{\mathcal{T}_h})$ such that, for all $\mathbf{a} \in \mathcal{V}_h$, there is a local interpolation operator $\mathcal{I}_{\mathbf{a}}^g : H^1(D_{\mathbf{a}}) \to P_p^b(\mathcal{T}_{\mathbf{a}}) \cap H^1(D_{\mathbf{a}})$ such that, for all $v \in H^1(D_{\mathbf{a}})$, all $K \in \mathcal{T}_{\mathbf{a}}$, and all $p \ge 1$,

$$ \left(\frac{p}{h_K}\right) \|v - \mathcal{I}_{\mathbf{a}}^g(v)\|_K + \left(\frac{p}{h_K}\right)^{\frac{1}{2}} \|v - \mathcal{I}_{\mathbf{a}}^g(v)\|_{\partial K} + \|\nabla \mathcal{I}_{\mathbf{a}}^g(v)\|_K \le C(\kappa_{\mathcal{T}_h}) \|\nabla v\|_{D_{\mathbf{a}}}. \quad (2.8) $$

Proof. See [33, 32] (see also [20, Corollary 2.5] for using the $H^1$-seminorm on the right-hand side). □

## 2.4 Jump lifting operator

For every field $v_h \in P_{p+1}^b(\mathcal{T}_h)$, $\nabla_h \times v_h$ denotes the broken curl of $v_h$ (evaluated cellwise). The reason we consider the space $P_{p+1}^b(\mathcal{T}_h)$ is that, in what follows, we shall consider the product of a hat basis function times a field in $P_p^b(\mathcal{T}_h)$. We define the jump lifting operator $L_h^p : P_{p+1}^b(\mathcal{T}_h) \to P_p^b(\mathcal{T}_h)$ such that

$$ (L_h^p(v_h), \phi_h) := \sum_{F \in \mathcal{F}_h} (\|v_h\|_F^c, \|\{\phi_h\}\|_F^g) L^2(F), \quad \forall \phi_h \in P_p^b(\mathcal{T}_h). \quad (2.9) $$

**Lemma 2.4** (Bound on lifting operator). There exists $C(\kappa_{\mathcal{T}_h})$ so that, for all $v_h \in P_{p+1}^b(\mathcal{T}_h)$, all $K \in \mathcal{T}_h$, and all $p \ge 1$,

$$ \|L_h^p(v_h)\|_K \le C(\kappa_{\mathcal{T}_h}) \left\{ \sum_{F \in \mathcal{F}_K} \left(\frac{p^2}{h_K}\right) \|\|v_h\|_F^c\|_F^2 \right\}^{\frac{1}{2}}. \quad (2.10) $$

Proof. Apply the Cauchy–Schwarz inequality and the discrete trace inequality (2.6). □

# 3 Main results on $H(\text{curl})$-reconstruction

This section presents, and comments on extensions of, our main result on $H_0(\text{curl})$-reconstruction. Specifically, we construct an operator

$$ \mathbf{R}^c : P_p^b(\mathcal{T}_h) \to H_0(\text{curl}; D), \quad (3.1) $$

such that, for all $E_h \in P_p^b(\mathcal{T}_h)$, the error $E_h - \mathbf{R}^c(E_h)$ is bounded in the broken curl norm and in the $L^2$-norm by suitable jumps of $E_h$ and $\nabla_h \times E_h$ across the mesh interfaces and by suitable traces on the mesh boundary faces.

The devising of $\mathbf{R}^c(E_h)$ combines local (non-polynomial) constructions on the vertex patches associated with each mesh vertex $\mathbf{a} \in \mathcal{V}_h$. For all $\mathbf{a} \in \mathcal{V}_h$, we define the following functional spaces:

$$ \mathbf{X}_0^c(D_{\mathbf{a}}) := H_0(\text{curl}; D_{\mathbf{a}}) \cap H(\text{div} = 0; D_{\mathbf{a}}), \quad (3.2a) $$

$$ \mathbf{X}^c(D_{\mathbf{a}}) := H(\text{curl}; D_{\mathbf{a}}) \cap H_0(\text{div} = 0; D_{\mathbf{a}}), \quad (3.2b) $$
where $H(\mathrm{div} = 0; D_a)$ (resp., $H_0(\mathrm{div} = 0; D_a)$) is the subspace of $H(\mathrm{div}; D_a)$ (resp., $H_0(\mathrm{div}; D_a)$) composed of divergence-free fields. We also set

$$X_0^g(D_a) := H_0^1(D_a), \qquad (3.3a)$$

$$X^g(D_a) := H^1(D_a)/\mathbb{R}. \qquad (3.3b)$$

**Definition 3.1** (Patchwise and global $H_0(\mathrm{curl})$-reconstruction). Let $E_h \in \mathbf{P}_p^b(\mathcal{T}_h)$. For all $\mathbf{a} \in \mathcal{V}_h$, let $U_\mathbf{a} \in \mathbf{X}_0^c(D_\mathbf{a})$ solve the following well-posed problem:

$$(\nabla_0 \times U_\mathbf{a}, \nabla_0 \times W_\mathbf{a})_{D_\mathbf{a}} = (\nabla_h \times (\psi_\mathbf{a} E_h) + L_h^p(\psi_\mathbf{a} E_h), \nabla_0 \times W_\mathbf{a})_{D_\mathbf{a}} \quad \forall W_\mathbf{a} \in \mathbf{X}_0^c(D_\mathbf{a}), \quad (3.4)$$

and let $\theta_\mathbf{a} \in X_0^g(D_\mathbf{a})$ solve the following well-posed problem:

$$(\nabla \theta_\mathbf{a}, \nabla v_\mathbf{a})_{D_\mathbf{a}} = (\psi_\mathbf{a} E_h, \nabla v_\mathbf{a})_{D_\mathbf{a}} \quad \forall v_\mathbf{a} \in X_0^g(D_\mathbf{a}). \qquad (3.5)$$

Set

$$E_\mathbf{a} := U_\mathbf{a} + \nabla \theta_\mathbf{a}, \qquad (3.6)$$

and extending $E_\mathbf{a}$ by zero to $D$, set

$$\mathbf{R}^c(E_h) := \sum_{\mathbf{a} \in \mathcal{V}_h} E_\mathbf{a}. \qquad (3.7)$$

**Remark 3.2** (Local problems). As the vertex patch $D_\mathbf{a}$ is a simply connected Lipschitz domain with connected boundary, the local problems (3.4) and (3.5) are well-posed. Moreover, an equivalent definition is

$$U_\mathbf{a} := \arg \min_{\rho_\mathbf{a} \in \mathbf{X}_0^c(D_\mathbf{a})} \| \nabla_0 \times \rho_\mathbf{a} - \{ \nabla_h \times (\psi_\mathbf{a} E_h) + L_h^p(\psi_\mathbf{a} E_h) \} \|_{D_\mathbf{a}},$$

$$\theta_\mathbf{a} := \arg \min_{\xi_\mathbf{a} \in X_0^g(D_\mathbf{a})} \| \nabla \xi_\mathbf{a} - \psi_\mathbf{a} E_h \|_{D_\mathbf{a}}.$$

A simple but crucial observation is that $E_\mathbf{a} \in H_0(\mathrm{curl}; D_\mathbf{a})$ so that

$$\mathbf{R}^c(E_h) \in H_0(\mathrm{curl}; D). \qquad (3.8)$$

We now set

$$\delta_\mathbf{a} := E_\mathbf{a} - \psi_\mathbf{a} E_h \quad \forall \mathbf{a} \in \mathcal{V}_h, \qquad (3.9)$$

and estimate $\delta_\mathbf{a}$ in the broken curl norm and in the $L^2$-norm.

**Lemma 3.3** (Local bound in broken curl norm). There exists $C(\kappa_{\mathcal{T}_h})$ so that, for all $E_h \in \mathbf{P}_p^b(\mathcal{T}_h)$ and all $\mathbf{a} \in \mathcal{V}_h$,

$$\| \nabla_h \times \delta_\mathbf{a} \|_{D_\mathbf{a}} \le C(\kappa_{\mathcal{T}_h}) \left\{ \sum_{F \in \mathcal{F}_\mathbf{a}} \left( \frac{h_F}{p} \right) \| \llbracket \nabla_h \times E_h \rrbracket_F^d \|_F^2 + \left( \frac{p^2}{h_F} \right) \| \llbracket E_h \rrbracket_F^c \|_F^2 \right\}^{\frac{1}{2}}. \qquad (3.10)$$

*Proof.* The proof can be found in Section 5.1. $\square$

**Lemma 3.4** (Local $L^2$-bound using broken-curl, divergence-preserving Poincaré inequality). There exists $C(\kappa_{\mathcal{T}_h})$ so that, for all $E_h \in \mathbf{P}_p^b(\mathcal{T}_h)$ and all $\mathbf{a} \in \mathcal{V}_h$,

$$\| \delta_\mathbf{a} \|_{D_\mathbf{a}} \le C(\kappa_{\mathcal{T}_h}) p \left\{ \sum_{F \in \mathcal{F}_\mathbf{a}} \left( \frac{h_F}{p} \right)^3 \| \llbracket \nabla_h \times E_h \rrbracket_F^d \|_F^2 + h_F \| \llbracket E_h \rrbracket_F^c \|_F^2 \right\}^{\frac{1}{2}}. \qquad (3.11)$$

*Proof.* The proof can be found in Section 5.2. $\square$
The $L^2$-estimate from Lemma 3.4 can be improved by invoking the elliptic regularity pickup of the Poisson problem with homogeneous Neumann boundary conditions on the vertex patches. We need, however, a uniform estimate of the underlying constants over all the vertex patches in the mesh. It is natural to expect that these constants can be bounded in terms of the mesh shape-regularity. As we did not find a precise result in the literature supporting this claim, we formulate it as an assumption. We hope that this will stimulate further research in this direction to prove (or disprove) the assumption.

**Assumption 3.5** (Elliptic regularity pickup for Neumann problem). There exist $C(\kappa_{\mathcal{T}_h})$ and $\delta := \delta(\kappa_{\mathcal{T}_h}) > 0$ so that, for all $\mathbf{a} \in \mathcal{V}_h$ and all $\mathbf{g} \in \mathbf{X}^g(D_{\mathbf{a}}) := [X^g(D_{\mathbf{a}})]^d$, the unique solution $\zeta_{\mathbf{g}} \in X^g(D_{\mathbf{a}})$ of the Neumann problem $(\nabla\zeta_{\mathbf{g}}, \nabla\rho)_{D_{\mathbf{a}}} = (\mathbf{g}, \nabla\rho)_{D_{\mathbf{a}}}$ for all $\rho \in X^g(D_{\mathbf{a}})$ (i.e., $\Delta\zeta_{\mathbf{g}} = \nabla \cdot \mathbf{g}$ in $D_{\mathbf{a}}$ and $\mathbf{n}_{D_{\mathbf{a}}} \cdot \nabla\zeta_{\mathbf{g}} = \mathbf{n}_{D_{\mathbf{a}}} \cdot \mathbf{g}$ on $\partial D_{\mathbf{a}}$) satisfies the estimate

$$ \|\zeta_{\mathbf{g}}\|_{D_{\mathbf{a}}} + h_{\mathbf{a}}^{\frac{3}{2}+\delta} |\nabla\zeta_{\mathbf{g}}|_{H^{\frac{1}{2}+\delta}(D_{\mathbf{a}})} \le C(\kappa_{\mathcal{T}_h}) h_{\mathbf{a}}^2 \|\nabla\mathbf{g}\|_{D_{\mathbf{a}}}. \quad (3.12) $$

**Lemma 3.6** (Local $L^2$-bound using Sobolev embedding on vertex patches). Under Assumption 3.5, there exists $C(\kappa_{\mathcal{T}_h})$ so that, for all $\mathbf{E}_h \in \mathbf{P}_p^b(\mathcal{T}_h)$ and all $\mathbf{a} \in \mathcal{V}_h$,

$$ \|\delta_{\mathbf{a}}\|_{D_{\mathbf{a}}} \le C(\kappa_{\mathcal{T}_h}) \left\{ \sum_{F \in \mathcal{F}_{\mathbf{a}}} h_F \|\|\mathbf{E}_h\|\|_F^c \|\|_F^2 \right\}^{\frac{1}{2}}. \quad (3.13) $$

*Proof.* The proof can be found in Section 5.3. □

We are now ready to state global estimates on $\mathbf{R}^c(\mathbf{E}_h) - \mathbf{E}_h$.

**Theorem 3.7** (Bound on $\mathbf{R}^c(\mathbf{E}_h) - \mathbf{E}_h$). There exists $C(\kappa_{\mathcal{T}_h})$ so that, for all $\mathbf{E}_h \in \mathbf{P}_p^b(\mathcal{T}_h)$,

$$ \|\nabla_h \times (\mathbf{R}^c(\mathbf{E}_h) - \mathbf{E}_h)\| \le C(\kappa_{\mathcal{T}_h}) \left\{ \sum_{K \in \mathcal{T}_h} \left(\frac{h_K}{p}\right) \|\|\nabla_h \times \mathbf{E}_h\|\|_{\partial K}^d \|\|_{\partial K}^2 + \left(\frac{p^2}{h_K}\right) \|\|\mathbf{E}_h\|\|_{\partial K}^c \|\|_{\partial K}^2 \right\}^{\frac{1}{2}}, \quad (3.14a) $$

$$ \|\mathbf{R}^c(\mathbf{E}_h) - \mathbf{E}_h\| \le C(\kappa_{\mathcal{T}_h}) p \left\{ \sum_{K \in \mathcal{T}_h} \left(\frac{h_K}{p}\right)^3 \|\|\nabla_h \times \mathbf{E}_h\|\|_{\partial K}^d \|\|_{\partial K}^2 + \sum_{K \in \mathcal{T}_h} h_K \|\|\mathbf{E}_h\|\|_{\partial K}^c \|\|_{\partial K}^2 \right\}^{\frac{1}{2}}. \quad (3.14b) $$

Moreover, under Assumption 3.5, we have

$$ \|\mathbf{R}^c(\mathbf{E}_h) - \mathbf{E}_h\| \le C(\kappa_{\mathcal{T}_h}) \left\{ \sum_{K \in \mathcal{T}_h} h_K \|\|\mathbf{E}_h\|\|_{\partial K}^c \|\|_{\partial K}^2 \right\}^{\frac{1}{2}}. \quad (3.14c) $$

*Proof.* The bounds (3.14) readily follow from the local estimates established in Lemmas 3.3, 3.4, and 3.6 once we observe that, for all $K \in \mathcal{T}_h$, $(\mathbf{R}^c(\mathbf{E}_h) - \mathbf{E}_h)|_K = \sum_{\mathbf{a} \in \mathcal{V}_K} \delta_{\mathbf{a}}|_K$ owing to the partition-of-unity property (2.2). Invoking the triangle inequality and the mesh shape-regularity concludes the proof. □

**Remark 3.8** ($p=0$). The operator $\mathbf{R}^c$ can also be employed on $\mathbf{P}_0^b(\mathcal{T}_h) \subset \mathbf{P}_1^b(\mathcal{T}_h)$.

**Remark 3.9** ($p$-suboptimality). The $\frac{1}{2}$-order suboptimality in (3.10) stems from the bound (2.10) on the jump lifting operator (which, itself, follows from the discrete trace inequality (2.6)). The $\frac{3}{2}$-order suboptimality in (3.11) inherits the $\frac{1}{2}$-order suboptimality from (3.10) to which a one-order suboptimality originating from the broken-curl, divergence-preserving Poincaré inequality is added. As this latter inequality does not involve discrete functions, no gain in the polynomial degree can be achieved. Finally, the $\frac{1}{2}$-order suboptimality in (3.13) stems from a Sobolev embedding invoked in a duality argument, which circumvents the need to invoke the broken curl estimate (3.10).
<figure><img src="image_2.png" /><figcaption>Figure 1: Triangular mesh $\mathcal{T}_h$ consisting of 2048 elements (left); agglomerated polygonal mesh $\tilde{\mathcal{T}}_H$ consisting of 514 elements (right).</figcaption></figure>

**Remark 3.10** (Agglomerated polytopal meshes). Let $\mathcal{T}_h$ be a simplicial background mesh covering $D$ exactly and let $\tilde{\mathcal{T}}_H$ be an agglomerated polytopal mesh built from $\mathcal{T}_h$ in such a way that every $\tilde{K} \in \tilde{\mathcal{T}}_H$ is the union of a uniformly bounded number of simplicial cells $K \in \mathcal{T}_{\tilde{K}} \subset \mathcal{T}_h$; see Figure 1 for an illustration. The above assumption on the mesh implies that there exists a constant $C_{\text{sh}} \ge 1$ such that

$$h_K \le h_{\tilde{K}} \le C_{\text{sh}}h_K \quad \forall K \in \mathcal{T}_{\tilde{K}}. \qquad (3.15)$$

Since $\mathbf{P}_p^b(\tilde{\mathcal{T}}_H) \subset \mathbf{P}_p^b(\mathcal{T}_h)$, an $\mathbf{H}_0(\text{curl})$-reconstruction of any field $\mathbf{E}_h \in \mathbf{P}_p^b(\tilde{\mathcal{T}}_H)$ can be devised by considering the $\mathbf{H}_0(\text{curl})$-reconstruction operator defined on $\mathbf{P}_p^b(\mathcal{T}_h)$. Since $\mathbf{E}_h \in \mathbf{P}_p^b(\tilde{\mathcal{T}}_H)$ and its broken curl only jump across the interfaces on $\mathcal{T}_h$ that are subsets of interfaces of $\tilde{\mathcal{T}}_H$, the error $\mathbf{R}^c(\mathbf{E}_h) - \mathbf{E}_h$ can still be bounded as in (3.14), both in the broken curl norm and in the $L^2$-norm, where the summations on the right-hand side run over the mesh cells in $\tilde{\mathcal{T}}_H$ and the local mesh size $h_{\tilde{K}}$ can be used owing to (3.15). Notice that the condition (3.15) does not accommodate polytopal meshes with tiny faces. This case is left to future work. (We mention [9] which derives a posteriori estimates for elliptic problems on polytopal meshes with tiny faces, but this work assumes that the domain is simply connected and does not rely on a partition of unity based on a simplicial background mesh.)

**Remark 3.11** (Mixed boundary conditions). The $\mathbf{H}_0(\text{curl})$-reconstruction from Definition 3.1 can be adapted with minor modifications to handle mixed boundary conditions. Let $\Gamma_D \subset \partial D$ and $\Gamma_N := \partial D \setminus \Gamma_D$, with $|\Gamma_N| \ne 0$, be Lipschitz subsets with connected interior that can be covered exactly by mesh faces. Let $\mathbf{H}_{\Gamma_D}(\text{curl}; D)$ be the subspace of $\mathbf{H}(\text{curl}; D)$ prescribing a homogeneous Dirichlet boundary condition only on $\Gamma_D$. To devise a global $\mathbf{H}_{\Gamma_D}(\text{curl})$-reconstruction, the local problems (3.4) and (3.5) defined on each vertex patch are now posed on the function spaces $\mathbf{X}_{\Gamma_D}^c(D_a) := \mathbf{H}_{\partial D_a \setminus \Gamma_N}(\text{curl}; D_a) \cap \mathbf{H}_{\partial D_a \cap \Gamma_N}(\text{div} = 0; D_a)$ (where the subscripts indicate where the tangential and normal boundary condition is prescribed, respectively) and $\mathbf{X}_{\Gamma_D}^g(D_a) := \mathbf{H}_{\partial D_a \setminus \Gamma_N}^1(D_a)$, for all $\mathbf{a} \in \mathcal{V}_h$. Then one sets again $\mathbf{E}_a := \mathbf{U}_a + \nabla \theta_a$, and extending $\mathbf{E}_a$ by zero to $D$, one sets $\mathbf{R}^c(\mathbf{E}_h) := \sum_{a \in \mathcal{V}_h} \mathbf{E}_a$, so that $\mathbf{R}^c(\mathbf{E}_h) \in \mathbf{H}_{\Gamma_D}(\text{curl}; D)$. The local broken curl estimate now relies on the local Helmholtz decomposition with mixed boundary conditions discussed in Remark 4.2 below. The estimate reads as follows:

$$\|\nabla_h \times \delta_a\|_{D_a} \le C(\kappa_{\mathcal{T}_h}) \left\{ \sum_{F \in \mathcal{F}_a \setminus \mathcal{F}_h^N} \left(\frac{h_F}{p}\right) \|\|\nabla_h \times \mathbf{E}_h\|\|_F^d\|_F^2 + \left(\frac{p^2}{h_F}\right) \|\|\mathbf{E}_h\|\|_F^c\|_F^2 \right\}^{\frac{1}{2}},$$

where $\mathcal{F}_h^N$ denotes the subset of mesh boundary faces allocated on $\Gamma_N$. The local $L^2$-error estimate relies on the broken Poincaré with mixed boundary conditions discussed in Remark
4.5 below (which requires one more conjecture, unless pure Neumann conditions are enforced). The estimate reads as follows:

$$ \|\delta_a\|_{D_a} \le C(\kappa_{\mathcal{T}_h}) p \left\{ \sum_{F \in \mathcal{F}_a \setminus \mathcal{F}_h^N} \left(\frac{h_F}{p}\right)^3 \|\llbracket \nabla_h \times \mathbf{E}_h \rrbracket_F^d \|_F^2 + h_F \|\llbracket \mathbf{E}_h \rrbracket_F^c \|_F^2 \right\}^{\frac{1}{2}}. $$

Finally, an extension of the bound from Lemma 3.6 to mixed boundary conditions is not expected to hold because the elliptic regularity pickup stated in Assumption 3.5 is not expected to hold.

**Remark 3.12** (Globally defined $H_0(\text{curl})$-reconstruction and convex domains). One can also consider a globally defined $H_0(\text{curl})$-reconstruction operator, say $\tilde{\mathbf{R}}^c : \mathbf{P}_p^b(\mathcal{T}_h) \to H_0(\text{curl}; D)$, without posing auxiliary problems on vertex patches exploiting the partition of unity. Assume first that $D$ has trivial topology. Then, one considers $U_D \in \mathbf{X}_0^c(D) := H_0(\text{curl}; D) \cap H(\text{div} = 0; D)$ solving the following well-posed problem:

$$ (\nabla_0 \times U_D, \nabla_0 \times W_D) = (\nabla_h \times E_h + L_h^p(E_h), \nabla_0 \times W_D) \quad \forall W_D \in \mathbf{X}_0^c(D), $$

and $\theta_D \in H_0^1(D)$ solving the following well-posed problem:

$$ (\nabla \theta_D, \nabla v_D) = (E_h, \nabla v_D) \quad \forall v_D \in H_0^1(D), $$

and one sets $\tilde{\mathbf{R}}^c(E_h) := U_D + \nabla \theta_D$. The error $\tilde{\mathbf{R}}^c(E_h) - E_h$ can again be bounded by the right-hand sides in (3.14), both in the broken curl norm and in the $L^2$-norm. The proofs, which follow similar arguments to those presented in Section 5 are omitted for brevity. One advantage of the globally defined $H_0(\text{curl})$-reconstruction operator is achieved when the domain $D$ is convex. Indeed, in this case, full elliptic regularity pickup can be invoked when deriving the $L^2$-error bound, leading to the hp-optimal estimate

$$ \|\tilde{\mathbf{R}}^c(E_h) - E_h\| \le C(\kappa_{\mathcal{T}_h}) \left\{ \sum_{K \in \mathcal{T}_h} \left(\frac{h_K}{p}\right) \|\llbracket \mathbf{E}_h \rrbracket_{\partial K}^c \|_{\partial K}^2 \right\}^{\frac{1}{2}}. $$

However, the broken curl norm bound cannot be improved and remains p-suboptimal by $\frac{1}{2}$-order. Finally, the above global construction can also be deployed if the domain $D$ has a more general topology upon posing the problem defining $U_D$ in $H_0(\text{curl}; D) \cap H_0(\text{curl} = 0; D)^\perp$, where $H_0(\text{curl} = 0; D)$ is the subspace of $H_0(\text{curl}; D)$ composed of curl-free fields and orthogonalities are meant in $L^2$. However, the constants in the bounds on $\tilde{\mathbf{R}}^c(E_h) - E_h$ now depend on the topology of $D$.

# 4 Preparatory results

In this section, we collect three preparatory results. The first result is a (classical) local Helmholtz decomposition to be invoked in the proof of Lemma 3.3. The second one is a local broken-curl, divergence-preserving Poincaré inequality to be invoked in the proof of Lemma 3.4. Finally, the third one is a local Sobolev embedding to be invoked in the proof of Lemma 3.6.

**Lemma 4.1** (Local Helmholtz decomposition). For all $\mathbf{a} \in \mathcal{V}_h$ and all $\mathbf{v} \in L^2(D_a)$, there exist $\psi \in \mathbf{X}_0^c(D_a)$ (see (3.2a)) and $\xi \in X^g(D_a)$ (see (3.3)), so that

$$ \mathbf{v} = \nabla_0 \times \psi + \nabla \xi \quad \text{in } D_a, \qquad (4.1a) $$

$$ \|\mathbf{v}\|_{D_a}^2 = \|\nabla_0 \times \psi\|_{D_a}^2 + \|\nabla \xi\|_{D_a}^2. \qquad (4.1b) $$

*Proof.* For completeness, we sketch a short proof. We solve the Neumann problem $(\nabla \xi, \nabla w)|_{D_a} = (\mathbf{v}, \nabla w)|_{D_a}$ in $X^g(D_a)$, and observe that $\nabla \xi - \mathbf{v} \in H_0(\text{div} = 0; D_a)$. Using [28, Theorem 3], we infer that there exists $\zeta \in H_0(\text{curl}; D_a)$ such that $\nabla_0 \times \zeta = \nabla \xi - \mathbf{v}$. Then, we solve the Dirichlet problem $(\nabla \gamma, \nabla w)|_{D_a} = (\zeta, \nabla w)|_{D_a}$ in $X_0^g(D_a)$ and set $\psi = \zeta - \nabla \gamma$. The pair $(\psi, \xi)$ satisfies all the required properties. $\square$
**Remark 4.2** (Mixed boundary conditions). For mixed boundary conditions, Lemma 4.1 is adapted as follows: For all $\mathbf{a} \in \mathcal{V}_h$ and all $\mathbf{v} \in \mathcal{L}^2(D_\mathbf{a})$, there exist $\psi \in \mathcal{X}_{\Gamma_D}^c(D_\mathbf{a})$ and $\xi \in H_{\partial D_\mathbf{a} \cap \Gamma_N}^1(D_\mathbf{a})$, so that (4.1) still holds true. To prove this result, one solves Poisson problems with mixed boundary conditions.

**Lemma 4.3** (Local broken-curl, divergence-preserving Poincaré inequality). There exists a constant $C(\kappa_{\mathcal{T}_h})$ so that, for all $\mathbf{a} \in \mathcal{V}_h$ and all $\mathbf{v} \in \mathcal{H}^1(\mathcal{T}_h) := \{\mathbf{w} \in \mathcal{L}^2(D_\mathbf{a}) | \nabla_h \mathbf{w} \in \mathcal{L}^2(D_\mathbf{a})\}$, there exists $\mathbf{v}_\mathbf{a}^c \in \mathcal{H}_0(\text{curl}; D_\mathbf{a})$ satisfying $\mathbf{v}_\mathbf{a}^c - \mathbf{v} \in \mathcal{H}(\text{div} = 0; D_\mathbf{a})$ such that

$$ \|\mathbf{v}_\mathbf{a}^c - \mathbf{v}\|_{D_\mathbf{a}} \le C(\kappa_{\mathcal{T}_h}) h_\mathbf{a} \left\{ \|\nabla_h \times (\mathbf{v}_\mathbf{a}^c - \mathbf{v})\|_{D_\mathbf{a}}^2 + \sum_{F \in \mathcal{F}_\mathbf{a}} h_F^{-1} \|\llbracket \mathbf{v} \rrbracket_F^c \|_F^2 + \sum_{F \in \mathcal{F}_\mathbf{a}^\partial} h_F^{-1} \|\gamma_{K(F),F}^c(\mathbf{v})\|_F^2 \right\}^{\frac{1}{2}}, \quad (4.2) $$

where $K(F)$ is the unique mesh cell in $\mathcal{T}_\mathbf{a}$ of which $F \in \mathcal{F}_\mathbf{a}^\partial$ is a face.

*Proof.* Let $\mathbf{a} \in \mathcal{V}_h$ and let $\mathbf{v} \in \mathcal{H}^1(\mathcal{T}_\mathbf{a})$. Recall that $X_0^g(D_\mathbf{a}) := H_0^1(D_\mathbf{a})$ and $X^g(D_\mathbf{a}) := H^1(D_\mathbf{a})/\mathbb{R}$.

(1) Let $\rho_\mathbf{a} \in X_0^g(D_\mathbf{a})$ solve the following well-posed problem:

$$ (\nabla \rho_{\mathbf{a}}, \nabla w_{\mathbf{a}})_{D_{\mathbf{a}}} = (\mathbf{v}, \nabla w_{\mathbf{a}})_{D_{\mathbf{a}}} \quad \forall w_{\mathbf{a}} \in X_0^g(D_{\mathbf{a}}), \quad (4.3) $$

so that $\nabla \rho_{\mathbf{a}} - \mathbf{v} \in \mathcal{H}(\text{div} = 0; D_\mathbf{a})$. Setting $\mathbf{v}_\mathbf{a}^c := \nabla \rho_{\mathbf{a}} + \Theta_\mathbf{a}$ for any $\Theta_\mathbf{a} \in X_0^c(D_\mathbf{a})$, we obtain $\mathbf{v}_\mathbf{a}^c \in \mathcal{H}_0(\text{curl}; D_\mathbf{a})$ with $\mathbf{v}_\mathbf{a}^c - \mathbf{v} \in \mathcal{H}(\text{div} = 0; D_\mathbf{a})$.

(2) Since $D_\mathbf{a}$ is a simply connected, Lipschitz domain, we infer from [2, Lemma 3.5] that there exists $\phi \in \mathcal{X}^g(D_\mathbf{a}) := [X^g(D_\mathbf{a})]^d$ such that $\nabla \times \phi = \mathbf{v}_\mathbf{a}^c - \mathbf{v}$ in $D_\mathbf{a}$ (notice that $\nabla \cdot \phi = 0$ in $D_\mathbf{a}$ as well, but this additional property is not needed here). However, the stability constant on $\|\nabla \phi\|_{D_\mathbf{a}}$ is not known explicitly in terms of the geometry of $D_\mathbf{a}$. Therefore, we invoke instead [27, Theorem 35] which gives $\phi \in \mathcal{X}^g(D_\mathbf{a})$ such that $\nabla \times \phi = \mathbf{v}_\mathbf{a}^c - \mathbf{v}$ in $D_\mathbf{a}$ and a stability constant $C(\kappa_{\mathcal{T}_h})$ only depending on the mesh shape-regularity so that the following bound holds:

$$ \|\nabla \phi\|_{D_\mathbf{a}} \le C(\kappa_{\mathcal{T}_h}) \|\nabla \times \phi\|_{D_\mathbf{a}} = C(\kappa_{\mathcal{T}_h}) \|\mathbf{v}_\mathbf{a}^c - \mathbf{v}\|_{D_\mathbf{a}}. \quad (4.4) $$

(3) Integrating by parts the curl operator in each cell $K \in \mathcal{T}_\mathbf{a}$, we infer that

$$ \begin{aligned} \|\mathbf{v}_\mathbf{a}^c - \mathbf{v}\|_{D_\mathbf{a}}^2 &= (\mathbf{v}_\mathbf{a}^c - \mathbf{v}, \nabla \times \phi)_{D_\mathbf{a}} \\ &= (\mathbf{v}_\mathbf{a}^c, \nabla \times \phi)_{D_\mathbf{a}} - (\mathbf{v}, \nabla \times \phi)_{D_\mathbf{a}} \\ &= -(\nabla_h \times (\mathbf{v}_\mathbf{a}^c - \mathbf{v}), \tilde{\phi})_{D_\mathbf{a}} - \sum_{K \in \mathcal{T}_\mathbf{a}} (\mathbf{v}, n_K \times \phi)_{\partial K} =: T_1 + T_2, \end{aligned} $$

and it remains to bound $T_1$ and $T_2$.

(3a) Bound on $T_1$. Using the Cauchy-Schwarz inequality, recalling that $\phi$ has zero mean-value over $D_\mathbf{a}$, and invoking the Poincaré inequality on the vertex patch (see [38, Theorem 3.2] and also [21, Lem. 5.7]), we infer that

$$ |T_1| \lesssim h_\mathbf{a} \|\nabla_h \times (\mathbf{v}_\mathbf{a}^c - \mathbf{v})\|_{D_\mathbf{a}} \|\nabla \phi\|_{D_\mathbf{a}} \lesssim h_\mathbf{a} \|\nabla_h \times (\mathbf{v}_\mathbf{a}^c - \mathbf{v})\|_{D_\mathbf{a}} \|\mathbf{v}_\mathbf{a}^c - \mathbf{v}\|_{D_\mathbf{a}}, $$

where the second bound follows from (4.4).

(3b) Bound on $T_2$. Re-organizing the summation over the mesh faces in $\mathcal{F}_\mathbf{a}$ and $\mathcal{F}_\mathbf{a}^\partial$ (recall that these two sets are disjoint) and observing that $\phi$ has continuous traces over all the faces in $\mathcal{F}_\mathbf{a}$ gives

$$ T_2 = \sum_{F \in \mathcal{F}_\mathbf{a}} (\|\mathbf{v}\|_F^c, n_F \times \phi)_F + \sum_{F \in \mathcal{F}_\mathbf{a}^\partial} (\gamma_{K(F),F}^c(\mathbf{v}), n_{K(F)} \times \phi)_F. $$
Invoking the same arguments as above together with the mesh shape-regularity and a multiplicative trace inequality (see, e.g., [22, Lem. 12.15]), we infer that

$$ egin{aligned} |T_2| &\le \left\{ \sum_{F \in \mathcal{F}_a} h_F^{-1} \|\llbracket \mathbf{v} \rrbracket_F^c \|_F^2 + \sum_{F \in \mathcal{F}_a^\partial} h_F^{-1} \|\gamma_{K(F),F}^c(\mathbf{v})\|_F^2 \right\}^{\frac{1}{2}} \left\{ \sum_{F \in \mathcal{F}_a \cup \mathcal{F}_a^\partial} h_F \|\phi\|_F^2 \right\}^{\frac{1}{2}} \\ &\lesssim h_a \left\{ \sum_{F \in \mathcal{F}_a} h_F^{-1} \|\llbracket \mathbf{v} \rrbracket_F^c \|_F^2 + \sum_{F \in \mathcal{F}_a^\partial} h_F^{-1} \|\gamma_{K(F),F}^c(\mathbf{v})\|_F^2 \right\}^{\frac{1}{2}} \|\mathbf{v}_a^c - \mathbf{v}\|_{D_a}. \end{aligned} $$

Putting the bounds on $T_1$ and $T_2$ together proves (4.2). □

**Remark 4.4 (Extension).** A simple inspection of the proof shows that Lemma 4.3 holds true for any $\mathbf{v} \in \mathcal{L}^2(D_a)$ with $\nabla_h \times \mathbf{v} \in \mathcal{L}^2(D_a)$ and $(\mathbf{n}_K \times \mathbf{v})|_K|_F \in \mathcal{L}^2(F)$ for all $K \in \mathcal{T}_a$ and all $F \in \mathcal{F}_K$.

**Remark 4.5 (Mixed boundary conditions).** For mixed boundary conditions, Lemma 4.3 is adapted as follows: There exist a constant $C(\kappa_{\mathcal{T}_h})$ so that, for all $\mathbf{a} \in \mathcal{V}_h$ and all $\mathbf{v} \in \mathcal{H}^1(\mathcal{T}_a)$, there exists $\mathbf{v}_a^c \in \mathcal{H}_{\partial D_a \setminus \Gamma_N}(\text{curl}; D_a)$ satisfying $\mathbf{v}_a^c - \mathbf{v} \in \mathcal{H}_{\partial D_a \cap \Gamma_N}(\text{div} = 0; D_a)$ such that

$$ \|\mathbf{v}_a^c - \mathbf{v}\|_{D_a} \le C(\kappa_{\mathcal{T}_h}) h_a \left\{ \|\nabla_h \times (\mathbf{v}_a^c - \mathbf{v})\|_{D_a}^2 + \sum_{F \in \mathcal{F}_a \setminus \mathcal{F}_h^N} h_F^{-1} \|\llbracket \mathbf{v} \rrbracket_F^c \|_F^2 \right. \\ \left. + \sum_{F \in \mathcal{F}_a^\partial \setminus \mathcal{F}_h^N} h_F^{-1} \|\gamma_{K(F),F}^c(\mathbf{v})\|_F^2 \right\}^{\frac{1}{2}}. $$

To prove this inequality, one adapts the above proof as follows: First, [2, Lemma 3.5] can be replaced by [28, Theorem 3] on regular decompositions with mixed boundary conditions, but the stability constant is not given explicitly in terms of the geometry of $D_a$. To have a stability constant only depending on the mesh shape-regularity, one needs to conjecture that the result of [27] can be extended to mixed boundary conditions. We notice that the extension holds for pure Neumann conditions, as shown in [27, Theorem 32].

**Remark 4.6 (Local broken-divergence, curl-preserving Poincaré inequality).** One can adapt the proof of Lemma 4.3 to establish the following local broken-divergence, curl-preserving Poincaré inequality: There exists a constant $C(\kappa_{\mathcal{T}_h})$ so that, for all $\mathbf{a} \in \mathcal{V}_h$ and all $\mathbf{v} \in \mathcal{H}^1(\mathcal{T}_a)$, there exist $\mathbf{v}_a^c \in \mathcal{H}_0(\text{div}; D_a)$ satisfying $\mathbf{v}_a^c - \mathbf{v} \in \mathcal{H}(\text{curl} = 0; D_a)$ such that,

$$ \|\mathbf{v}_a^c - \mathbf{v}\|_{D_a} \le C(\kappa_{\mathcal{T}_h}) h_a \left\{ \|\nabla_h \cdot (\mathbf{v}_a^c - \mathbf{v})\|_{D_a}^2 + \sum_{F \in \mathcal{F}_a} h_F^{-1} \|\llbracket \mathbf{v} \rrbracket_F^d \|_F^2 + \sum_{F \in \mathcal{F}_a^\partial} h_F^{-1} \|\gamma_{K(F),F}^d(\mathbf{v})\|_F^2 \right\}^{\frac{1}{2}}. $$

**Lemma 4.7 (Local Sobolev embedding).** Under Assumption 3.5, there exists a constant $C(\kappa_{\mathcal{T}_h})$ such that, for all $\mathbf{a} \in \mathcal{V}_h$ and all $\mathbf{v} \in \mathcal{X}^c(D_a)$ (see (3.2b)),

$$ |\mathbf{v}|_{H^{\frac{1}{2}+\delta}(K)} \le C(\kappa_{\mathcal{T}_h}) h_a^{\frac{1}{2}-\delta} \|\nabla \times \mathbf{v}\|_{D_a} \quad \forall K \in \mathcal{T}_a, \quad (4.5) $$

where the real number $\delta := \delta(\kappa_{\mathcal{T}_h}) \in (0, \frac{1}{2}]$ comes from Assumption 3.5.

**Proof.** (1) Let $\mathbf{a} \in \mathcal{V}_h$ and $\mathbf{v} \in \mathcal{X}^c(D_a)$. Invoking the same arguments as in Step (2) of the proof of Lemma 4.3, we infer that there exists $\phi \in \mathcal{X}^g(D_a)$ such that $\nabla \times \phi = \nabla \times \mathbf{v}$ and $\nabla \cdot \phi = 0$ in $D_a$, and

$$ \|\nabla \phi\|_{D_a} \le C(\kappa_{\mathcal{T}_h}) \|\nabla \times \phi\|_{D_a} = C(\kappa_{\mathcal{T}_h}) \|\nabla \times \mathbf{v}\|_{D_a}. \quad (4.6) $$
Since $\nabla \times (v - \phi) = 0$, there exists a function $p \in X^g(D_a)$ such that $\nabla p = v - \phi$. Altogether, this leads to the regular decomposition

$$v = \nabla p + \phi, \quad p \in X^g(D_a), \quad \phi \in X^g(D_a).$$

(2) Since $v \in H_0(\text{div} = 0; D)$ by assumption, we infer that $(\nabla p, \nabla \rho)_{D_a} = (\phi, \nabla \rho)_{D_a}$ for all $\rho \in X^d(D_a)$. Therefore, we can apply Assumption 3.5 to $g := \phi$, and observing that $\zeta_g = p$, we infer that

$$h_{\mathbf{a}}^{\frac{3}{2}+\delta} |\nabla p|_{H^{\frac{1}{2}+\delta}(D_a)} \lesssim h_{\mathbf{a}}^2 \|\nabla \phi\|_{D_a} \lesssim h_{\mathbf{a}}^2 \|\nabla \times \mathbf{v}\|_{D_a},$$

where the second bound follows from (4.6). Since $|\nabla p|_{H^{\frac{1}{2}+\delta}(K)} \le |\nabla p|_{H^{\frac{1}{2}+\delta}(D_a)}$ for all $K \in \mathcal{T}_a$, we infer that

$$|\nabla p|_{H^{\frac{1}{2}+\delta}(K)} \lesssim h_{\mathbf{a}}^{\frac{1}{2}-\delta} \|\nabla \times \mathbf{v}\|_{D_a}. \qquad (4.7)$$

(3) For all $K \in \mathcal{T}_a$, invoking the geometric mapping to the reference simplex $\tilde{K}$ and the embedding $|\hat{\phi}|_{H^{\frac{1}{2}}(\tilde{K})} \le \hat{C} \|\nabla \hat{\phi}\|_{\tilde{K}}$, we infer that

$$|\phi|_{H^{\frac{1}{2}+\delta}(K)} \lesssim h_K^{\frac{1}{2}-\delta} \|\nabla \phi\|_K \le h_K^{\frac{1}{2}-\delta} \|\nabla \phi\|_{D_a} \lesssim h_{\mathbf{a}}^{\frac{1}{2}-\delta} \|\nabla \times \mathbf{v}\|_{D_a},$$

where the last bound follows again from (4.6) and $h_K \le h_a$. Combining this bound with (4.7) and invoking the triangle inequality concludes the proof. $\square$

# 5 Proof of the main results from Section 3

This section contains the proofs of Lemmas 3.3, 3.4, and 3.6.

## 5.1 Proof of Lemma 3.3

Let $a \in \mathcal{V}_h$. Our goal is bound the error $\delta_a := E_a - \psi_a E_h$ defined in (3.9) in the broken curl norm. Specifically, our goal is to prove that

$$\|\nabla_h \times \delta_a\|_{D_a} \le C \left\{ \sum_{F \in \mathcal{F}_a} \left(\frac{h_F}{p}\right) \|\|\nabla \times E_h\|\|_F^d\|_F^2 + \sum_{F \in \mathcal{F}_a} \left(\frac{p^2}{h_F}\right) \|\|E_h\|\|_F^c\|_F^2 \right\}^{\frac{1}{2}}.$$

To this purpose, we apply the Helmholtz decomposition (4.1a) to $v := \nabla_h \times \delta_a$ in the vertex patch $D_a$. This gives $\psi \in X_0^c(D_a)$ and $\xi \in X^g(D_a)$ such that $\nabla_h \times \delta_a = \nabla_0 \times \psi + \nabla \xi$ in $D_a$. Taking the $L^2(D_a)$-inner product with $\nabla_h \times \delta_a$, we infer that

$$\|\nabla_h \times \delta_a\|_{D_a}^2 = (\nabla_h \times \delta_a, \nabla_0 \times \psi)_{D_a} + (\nabla_h \times \delta_a, \nabla \xi)_{D_a} =: I + \mathrm{II}, \qquad (5.1)$$

and it remains to bound I and II.

(1) Bound on I. Since $\nabla_h \times \delta_a = \nabla_0 \times E_a - \nabla_h \times (\psi_a E_h) = \nabla_0 \times U_a - \nabla_h \times (\psi_a E_h)$ owing to the definition (3.6) of $E_a$, we have

$$I = (\nabla_0 \times U_a - \nabla_h \times (\psi_a E_h), \nabla_0 \times \psi)_{D_a}.$$

Recalling the definition (3.4) of $U_a$ and since $\psi \in X_0^c(D_a)$, this gives

$$I = (L_h^p(\psi_a E_h), \nabla_0 \times \psi)_{D_a}.$$

Invoking the Cauchy-Schwarz inequality, the stability bound (4.1b) and the fact that $\psi_a$ is continuous and bounded by one, we infer that

$$\begin{aligned} I &\le \|L_h^p(\psi_a E_h)\|_{D_a} \|\nabla_0 \times \psi\|_{D_a} \\ &\lesssim \left\{ \sum_{F \in \mathcal{F}_a} \left(\frac{p^2}{h_F}\right) \|\|\psi_a E_h\|\|_F^c\|_F^2 \right\}^{\frac{1}{2}} \|\nabla_0 \times \psi\|_{D_a} \\ &\lesssim \left\{ \sum_{F \in \mathcal{F}_a} \left(\frac{p^2}{h_F}\right) \|\|E_h\|\|_F^c\|_F^2 \right\}^{\frac{1}{2}} \|\nabla_h \times \delta_a\|_{D_a}, \end{aligned} \qquad (5.2)$$
where we used (4.1b) in the last bound.

(2) Bound on II. Since $(\nabla_0 \times \mathbf{E}_a, \nabla \xi)_{D_a} = 0$, we have

$$ \mathrm{II} = -(\nabla_h \times (\psi_a \mathbf{E}_h), \nabla \xi)_{D_a}. $$

Let $\mathcal{I}_a^g$ be the modified hp-Karkulik-Melenk interpolation operator on the vertex patch $D_a$ from Lemma 2.3. We have

$$ \mathrm{II} = (\nabla_h \times (\psi_a \mathbf{E}_h), \nabla(\xi - \mathcal{I}_a^g(\xi)))_{D_a} + (\nabla_h \times (\psi_a \mathbf{E}_h), \nabla \mathcal{I}_a^g(\xi))_{D_a} =: \mathrm{II}_1 + \mathrm{II}_2, $$

and we bound the two terms on the right-hand side.

(2a) Bound on $\mathrm{II}_1$. We integrate by parts the gradient operator in all the cells composing $\mathcal{T}_a$. This gives

$$ \mathrm{II}_1 = \sum_{K \in \mathcal{T}_a} (\xi - \mathcal{I}_a^g(\xi), n_K \cdot (\nabla_h \times (\psi_a \mathbf{E}_h)))_{\partial K}. $$

All the mesh faces in $\cup_{K \in \mathcal{T}_a} \mathcal{F}_K$ are either in $\mathcal{F}_a$ or in $\mathcal{F}_a^\partial$. In the first case, we observe that $\xi - \mathcal{I}_a^g(\xi)$ is single-valued across every interface $F \in \mathcal{F}_a \cap \mathcal{F}_h^{\mathrm{int}}$ and that we have set conventionally $\llbracket \bullet \rrbracket_F^d := \bullet|_F \cdot n_D$ on every boundary face $F \in \mathcal{F}_a \cap \mathcal{F}_h^{\mathrm{bd}}$. In the second case, $\psi_a$ vanishes identically on the face in question, and so does the normal component of $\nabla_h \times (\psi_a \mathbf{E}_h)$. Altogether, this gives

$$ \mathrm{II}_1 = \sum_{F \in \mathcal{F}_a} (\xi - \mathcal{I}_a^g(\xi), \llbracket \nabla_h \times (\psi_a \mathbf{E}_h) \rrbracket_F^d)_F. $$

We have $\nabla_h \times (\psi_a \mathbf{E}_h) = \psi_a \nabla_h \times \mathbf{E}_h + (\nabla \psi_a) \times \mathbf{E}_h$ so that

$$ \llbracket \nabla_h \times \psi_a \mathbf{E}_h \rrbracket_F^d = \psi_a \llbracket \nabla_h \times \mathbf{E}_h \rrbracket_F^d + \nabla \psi_a \llbracket \mathbf{E}_h \rrbracket_F^c, $$

since $\psi_a$ is continuous and $(n_F \times \nabla \psi_a)_F$ is single-valued across every interface $F \in \mathcal{F}_a \cap \mathcal{F}_h^{\mathrm{int}}$. This gives

$$ \mathrm{II}_1 = - \sum_{F \in \mathcal{F}_a} (\xi - \mathcal{I}_a^g(\xi), \psi_a \llbracket \nabla_h \times \mathbf{E}_h \rrbracket_F^d)_F - \sum_{F \in \mathcal{F}_a} (\xi - \mathcal{I}_a^g(\xi), \nabla \psi_a \llbracket \mathbf{E}_h \rrbracket_F^c)_F. $$

Let $\mathrm{II}_{11}$ and $\mathrm{II}_{12}$ denote the two summations on the right-hand side. Using the Cauchy-Schwarz inequality and $\|\psi_a\|_{L^\infty(F)} = 1$, we infer that

$$ |\mathrm{II}_{11}| \le \left\{ \sum_{F \in \mathcal{F}_a} \left( \frac{h_F}{p} \right) \| \llbracket \nabla_h \times \mathbf{E}_h \rrbracket_F^d \|_F^2 \right\}^{\frac{1}{2}} \left\{ \sum_{F \in \mathcal{F}_a} \left( \frac{p}{h_F} \right) \| \xi - \mathcal{I}_a^g(\xi) \|_F^2 \right\}^{\frac{1}{2}}. $$

We now invoke the approximation result (2.8) on $\mathcal{I}_a^g$. For every $F \in \mathcal{F}_a$, we can pick a mesh cell $K \in \mathcal{T}_a$ of which $F$ is a face and obtain

$$ \left(\frac{p}{h_F}\right)^{\frac{1}{2}} \|\xi - \mathcal{I}_a^g(\xi)\|_F \lesssim \|\nabla\xi\|_K \le \|\nabla\xi\|_{D_a} \le \|\nabla_h \times \delta_a\|_{D_a}, $$

where we used (4.1b) in the last bound. This gives

$$ |\mathrm{II}_{11}| \lesssim \left\{ \sum_{F \in \mathcal{F}_a} \left( \frac{h_F}{p} \right) \| \llbracket \nabla_h \times \mathbf{E}_h \rrbracket_F^d \|_F^2 \right\}^{\frac{1}{2}} \| \nabla_h \times \delta_a \|_{D_a}. $$

Moreover, using the Cauchy-Schwarz inequality and $\|\nabla \psi_a\|_{L^\infty(F)} \lesssim h_F^{-1}$ gives

$$ |\mathrm{II}_{12}| \lesssim \left\{ \sum_{F \in \mathcal{F}_a} \left( \frac{h_F}{p} \right) h_F^{-2} \| \llbracket \mathbf{E}_h \rrbracket_F^c \|_F^2 \right\}^{\frac{1}{2}} \left\{ \sum_{F \in \mathcal{F}_a} \left( \frac{p}{h_F} \right) \| \xi - \mathcal{I}_a^g(\xi) \|_F^2 \right\}^{\frac{1}{2}}. $$
Invoking the same arguments as above to bound the factor involving $\xi$, we obtain

$$|\mathrm{II}_{12}| \lesssim p^{-\frac{3}{2}} \left\{ \sum_{F \in \mathcal{F}_a} \left( \frac{p^2}{h_F} \right) \| \llbracket E_h \rrbracket_F^c \|_F^2 \right\}^{\frac{1}{2}} \| \nabla_h \times \delta_a \|_{D_a}.$$

Putting everything together, we conclude that

$$|\mathrm{II}_1| \lesssim \left\{ \sum_{F \in \mathcal{F}_a} \left( \frac{h_F}{p} \right) \| \llbracket \nabla_h \times E_h \rrbracket_F^d \|_F^2 + p^{-3} \left( \frac{p^2}{h_F} \right) \| \llbracket E_h \rrbracket_F^c \|_F^2 \right\}^{\frac{1}{2}} \| \nabla_h \times \delta_a \|_{D_a}.$$

(2b) Bound on $\mathrm{II}_2$. This time we integrate by parts the curl operator in all the cells composing $\mathcal{T}_a$. Using similar arguments as above, we obtain

$$\mathrm{II}_2 = \sum_{K \in \mathcal{T}_a} (\psi_a E_h, n_K \times \nabla \mathcal{I}_a^g(\xi))_{\partial K} = \sum_{F \in \mathcal{F}_a} (\psi_a \llbracket E_h \rrbracket_F^c, \nabla \mathcal{I}_a^g(\xi))_F.$$

Invoking the Cauchy–Schwarz inequality, the discrete trace inequality (2.6), and $\|\psi_a\|_{L^\infty(F)} = 1$, we infer that

$$\begin{aligned} |\mathrm{II}_2| & \le \left\{ \sum_{F \in \mathcal{F}_a} \left( \frac{p^2}{h_K} \right) \| \llbracket E_h \rrbracket_F^c \|_F^2 \right\}^{\frac{1}{2}} \left\{ \sum_{F \in \mathcal{F}_a} \left( \frac{h_K}{p^2} \right) \| \nabla \mathcal{I}_a^g(\xi) \|_F^2 \right\}^{\frac{1}{2}} \\ & \lesssim \left\{ \sum_{F \in \mathcal{F}_a} \left( \frac{p^2}{h_K} \right) \| \llbracket E_h \rrbracket_F^c \|_F^2 \right\}^{\frac{1}{2}} \left\{ \sum_{K \in \mathcal{T}_a} \| \nabla \mathcal{I}_a^g(\xi) \|_K^2 \right\}^{\frac{1}{2}} \\ & \lesssim \left\{ \sum_{F \in \mathcal{F}_a} \left( \frac{p^2}{h_K} \right) \| \llbracket E_h \rrbracket_F^c \|_F^2 \right\}^{\frac{1}{2}} \| \nabla \xi \|_{D_a}, \end{aligned}$$

where the last bound follows from the $H^1$-stability of $\mathcal{I}_a^g$ (see (2.8)). Proceeding as above to bound $\|\nabla\xi\|_{D_a}$, we conclude that

$$|\mathrm{II}_2| \lesssim \left\{ \sum_{F \in \mathcal{F}_a} \left( \frac{p^2}{h_K} \right) \| \llbracket E_h \rrbracket_F^c \|_F^2 \right\}^{\frac{1}{2}} \| \nabla_h \times \delta_a \|_{D_a}.$$

(3) Putting the bounds on $\mathrm{II}_1$ and $\mathrm{II}_2$ together and since $p \ge 1$, we conclude the proof.

## 5.2 Proof of Lemma 3.4

We want to prove that

$$\| \delta_a \|_{D_a} \lesssim p \left\{ \sum_{F \in \mathcal{F}_a} \left( \frac{h_F}{p} \right)^3 \| \llbracket \nabla_h \times E_h \rrbracket_F^d \|_F^2 + h_F \| \llbracket E_h \rrbracket_F^c \|_F^2 \right\}^{\frac{1}{2}}.$$

We apply the local broken-curl, divergence-preserving Poincaré inequality to $v := \psi_a E_h \in P_{p+1}^b(\mathcal{T}_a) \subset H^1(\mathcal{T}_a)$ and select $v_a^c := E_a$, which indeed satisfies $v_a^c \in H_0(\mathrm{curl}; D_a)$ and $v_a^c - v = \psi_a E_h - U_a - \nabla\theta_a \in H(\mathrm{div} = 0; D_a)$ since $U_a \in H(\mathrm{div} = 0; D_a)$ by construction and $\psi_a E_h - \nabla\theta_a \in H(\mathrm{div} = 0; D_a)$ by definition of $\theta_a$. Observing that $\delta_a = E_a - \psi_a E_h = v_a^c - v$, we infer that

$$\begin{aligned} \| \delta_a \|_{D_a} & \lesssim h_a \left\{ \| \nabla_h \times \delta_a \|_{D_a} + \sum_{F \in \mathcal{F}_a} h_F^{-1} \| \llbracket \psi_a E_h \rrbracket_F^c \|_F^2 + \sum_{F \in \mathcal{F}_a^\theta} h_F^{-1} \| \gamma_{K(F),F}^c (\psi_a E_h) \|_F^2 \right\}^{\frac{1}{2}} \\ & \le h_a \left\{ \| \nabla_h \times \delta_a \|_{D_a} + \sum_{F \in \mathcal{F}_a} h_F^{-1} \| \llbracket E_h \rrbracket_F^c \|_F^2 \right\}^{\frac{1}{2}}, \end{aligned}$$
where we used that $\psi_a$ is a continuous function over $D_a$, vanishing on $\partial D_a$ and bounded by one. Invoking (3.10) to bound the first term on the right-hand side involving the broken curl and using the mesh shape-regularity gives

$$ \|\delta_a\|_{D_a} \lesssim \left\{ \sum_{F \in \mathcal{F}_a} \left( \frac{h_F^3}{p} \right) \|\nabla \times \mathbf{E}_h\|_F^d \|_F^2 + p^2 h_F \|\mathbf{E}_h\|_F^c \|_F^2 \right\}^{\frac{1}{2}}. $$

Factoring out $p^2$ completes the proof.

**Remark 5.1** (Definition of $\theta_a$). The above proof highlights the relevance of the local problem (3.5) defining the curl-free part of $\mathbf{E}_a$, i.e., $\nabla\theta_a$.

## 5.3 Proof of Lemma 3.6

We want to prove that, provided Assumption 3.5 holds true,

$$ \|\delta_a\|_{D_a} \lesssim \left\{ \sum_{F \in \mathcal{F}_a} h_F \|\mathbf{E}_h\|_F^c \|_F^2 \right\}^{\frac{1}{2}}. $$

We use the Aubin-Nitsche duality argument to bound $\delta_a$ in the $L^2$-norm on the vertex patch $D_a$. Specifically, we consider the following dual problem: Find $Z_a \in X_0^c(D_a)$ such that

$$ \nabla \times (\nabla_0 \times Z_a) = \delta_a \quad \text{in } D_a, \qquad (5.3a) $$

$$ Z_a \times n_{D_a} = 0 \quad \text{on } \partial D_a. \qquad (5.3b) $$

This problem is well-posed since $D_a$ has trivial topology and $\delta_a$ is divergence-free. Indeed, $\nabla \cdot \delta_a = \nabla \cdot (\mathbf{E}_a - (\psi_a \mathbf{E}_h)) = \nabla \cdot U_a + \nabla \cdot (\nabla \theta_a - \psi_a \mathbf{E}_h) = 0$ since $U_a$ is divergence-free by construction and the definition (3.5) of $\theta_a$ implies that $\nabla \theta_a - \psi_a \mathbf{E}_h$ is divergence-free as well. Recalling the regularity pickup from [16, Section 5.2], and [23, Lemma 44.2], we infer that there is $\delta_a \in (0, \frac{1}{2}]$ (here, $\delta_a$ can depend on $a \in V_h$) so that $\nabla_0 \times Z_a \in H^{\frac{1}{2}+\delta}(D_a)$. This implies in particular that $\nabla_0 \times Z_a$ has meaningful (tangential) traces in $L^2(\partial K)$ for all $K \in \mathcal{T}_a$. Taking the $L^2(D_a)$-inner product with $\delta_a$ in (5.3a) and using the definition of $\delta_a$, we obtain

$$ \begin{aligned} \|\delta_a\|_{D_a}^2 &= (\nabla \times (\nabla_0 \times Z_a), \delta_a)_{D_a} \\ &= (\nabla \times (\nabla_0 \times Z_a), U_a)_{D_a} + (\nabla \times (\nabla_0 \times Z_a), \nabla \theta_a)_{D_a} - (\nabla \times (\nabla_0 \times Z_a), \psi_a \mathbf{E}_h)_{D_a}. \end{aligned} $$

Integrating by parts the curl operator, we infer that

$$ \begin{aligned} \|\delta_a\|_{D_a}^2 &= (\nabla_0 \times Z_a, \nabla_0 \times U_a)_{D_a} - (\nabla_0 \times Z_a, \nabla_h \times (\psi_a \mathbf{E}_h))_{D_a} \\ &\quad - \sum_{K \in \mathcal{T}_a} (\nabla_0 \times Z_a, (\psi_a \mathbf{E}_h) \times n_K)_{\partial K}. \end{aligned} $$

Re-arranging the last term on the right-hand side leads to

$$ \|\delta_a\|_{D_a}^2 = (\nabla_0 \times Z_a, \nabla_h \times (U_a - (\psi_a \mathbf{E}_h)))_{D_a} - \sum_{F \in \mathcal{F}_a} (\nabla_0 \times Z_a, \psi_a [\mathbf{E}_h]_F^c)_F. $$

Next, invoking the relation (3.4) with $W_a := Z_a \in X_0^c(D_a)$ and the definition (2.9) of lifting operator, we infer that

$$ \begin{aligned} \|\delta_a\|_{D_a}^2 &= (\nabla_0 \times Z_a, L_h^p(\psi_a \mathbf{E}_h))_{D_a} - \sum_{F \in \mathcal{F}_a} (\nabla_0 \times Z_a, \psi_a [\mathbf{E}_h]_F^c)_F \\ &= (\Pi_a^b(\nabla_0 \times Z_a), L_h^p(\psi_a \mathbf{E}_h))_{D_a} - \sum_{F \in \mathcal{F}_a} (\nabla_0 \times Z_a, \psi_a [\mathbf{E}_h]_F^c)_F \qquad (5.4) \\ &= - \sum_{F \in \mathcal{F}_a} (\{\nabla_0 \times Z_a - \Pi_a^b(\nabla_0 \times Z_a)\}^g_F, \psi_a [\mathbf{E}_h]_F^c)_F, \end{aligned} $$
where $\Pi_a^b$ denotes the $L^2$-orthogonal projection onto $P_p^b(T_a)$. Using the Cauchy-Schwarz inequality, $\|\psi_a\|_{L^\infty(F)} = 1$, bounding the mean-values by the traces in the (one or) two cell(s) sharing $F \in \mathcal{F}_a$, and invoking the mesh shape-regularity, we infer that

$$ \begin{aligned} \|\delta_a\|_{D_a}^2 &\le \left\{ \sum_{F \in \mathcal{F}_a} \left(\frac{p}{h_F}\right)^{2\delta} \|\{\nabla_0 \times \mathbf{Z}_a - \Pi_a^b(\nabla_0 \times \mathbf{Z}_a)\}^g_F \|_F^2 \right\}^{\frac{1}{2}} \left\{ \sum_{F \in \mathcal{F}_a} \left(\frac{h_F}{p}\right)^{2\delta} \|\{[E_h]_F^c\}_F^2 \|_F^2 \right\}^{\frac{1}{2}} \\ &\lesssim \left\{ \sum_{K \in \mathcal{F}_a} \left(\frac{p}{h_K}\right)^{2\delta} \|\nabla_0 \times \mathbf{Z}_a - \Pi_K^b(\nabla_0 \times \mathbf{Z}_a)\|_{\partial K}^2 \right\}^{\frac{1}{2}} \left\{ \sum_{F \in \mathcal{F}_a} \left(\frac{h_F}{p}\right)^{2\delta} \|\{[E_h]_F^c\}_F^2 \|_F^2 \right\}^{\frac{1}{2}}, \end{aligned} $$

where $\delta \in (0, \frac{1}{2}]$ is given by Assumption 3.5. Using the approximation result (2.7) on the $L^2$-projection followed by Lemma 4.7 (notice that indeed $\nabla_0 \times \mathbf{Z}_a \in \mathbf{X}^c(D_a)$), we infer that, for all $K \in \mathcal{F}_a$,

$$ \left(\frac{p}{h_K}\right)^{\delta} \|\nabla_0 \times \mathbf{Z}_a - \Pi_K^b(\nabla_0 \times \mathbf{Z}_a)\|_{\partial K} \lesssim |\nabla_0 \times \mathbf{Z}_a|_{H^{\frac{1}{2}+\delta}(K)} \lesssim h_a^{\frac{1}{2}-\delta} \|\delta_a\|_{D_a}. $$

Combining the above two bounds and since $h_a \lesssim h_F$ for all $F \in \mathcal{F}_a$ and $p \ge 1$, we infer that the assertion holds true.

# 6 Application to hp-a posteriori error analysis

In this section, we show how the $H_0(\text{curl})$-reconstruction operator constructed above can be used in residual-based hp-a posteriori error analysis. We focus on the symmetric interior penalty dG approximation of some simplified forms of Maxwell's equations involving the curl-curl operator and a zero-order term. Other nonconforming approximation methods can be considered as well. For simplicity, we assume that Assumption 3.5 holds true.

## 6.1 Model problem

The model problem consists in finding $E \in H_0(\text{curl}; D)$ such that

$$ \omega^2 \epsilon E + \nabla \times (\nu \nabla_0 \times E) = J \quad \text{in } D, \qquad (6.1) $$

where $J \in H(\text{div}; D)$ is the source term. Both material properties $\epsilon$ and $\nu$ can vary in $D$ and are assumed to take positive-definite, symmetric, piecewise constant matrix values with eigenvalues uniformly bounded from above and below away from zero. Notice that, for simplicity, we are enforcing homogeneous Dirichlet boundary conditions (also called perfect electric conductor boundary conditions). The parameter $\omega$ scales as a frequency (reciprocal of a time scale) and can be either a real number (leading to the so-called definite Maxwell problem) or a pure imaginary umber (leading to the so-called indefinite problem). Whenever $\omega = 0$, the additional property $\nabla \cdot J = 0$ is prescribed. For the time being, we assume that $\omega$ is a nonzero real number and refer the reader to Remarks 6.2 and 6.10 for further insight into the other cases.

For a material property $\alpha \in \{\epsilon, \nu\}$, we introduce the inner product and associated norm weighted by $\alpha$, leading to the notation $(\cdot, \cdot)_\alpha$ and $\|\cdot\|_\alpha$. The weak formulation of (6.1) classically amounts to finding $E \in H_0(\text{curl}; D)$ such that

$$ b(E, w) = (J, w) \quad \forall w \in H_0(\text{curl}; D), \qquad (6.2) $$

with the bilinear form defined on $H_0(\text{curl}; D) \times H_0(\text{curl}; D)$ such that

$$ b(v, w) := \omega^2(v, w)_\epsilon + (\nabla_0 \times v, \nabla_0 \times w)_\nu. \qquad (6.3) $$
## 6.2 Symmetric interior penalty dG discretization

We employ the symmetric interior penalty dG method to discretize the weak formulation (6.2) (see [36, 29]). We assume that the mesh $\mathcal{T}_h$ is compatible with the domain partition on which $\epsilon$ and $\nu$ are piecewise constant. For $\alpha \in \{\epsilon, \nu\}$, we set $\alpha_K := \alpha|_K$ for all $K \in \mathcal{T}_h$, and denote by $\alpha_K^b$ and $\alpha_K^\sharp$ the smallest and largest eigenvalue of $\alpha_K$, respectively. It is convenient to set

$$ \alpha_F^\sharp := \max(\alpha_{K_l}^\sharp, \alpha_{K_r}^\sharp) \quad \forall F := \partial K_l \cap \partial K_r \in \mathcal{F}_h^{\text{int}}, \quad \alpha_F^\sharp := \alpha_{K_l}^\sharp \quad \forall F := \partial K_l \cap \partial \Omega \in \mathcal{F}_h^{\text{bd}}. \quad (6.4) $$

For all $K \in \mathcal{T}_h$, we also define $\alpha_{\partial K}^\sharp$ so that $\alpha_{\partial K}^\sharp|_F := \alpha_F^\sharp$ for all $F \in \mathcal{F}_K$. To avoid distracting technicalities, we assume that the material properties are only mildly heterogeneous and anisotropic. Thus, we hide anisotropy ratios such as $\alpha_K^\sharp/\alpha_K^b$ in the generic constants used in the error analysis. We do the same for contrast factors such as $\alpha_K^\sharp/\alpha_{K'}^\sharp$, where $K'$ lies in some neighborhood of $K$. We refer the reader to Remarks 6.1 and 6.4 for some further insight.

The discrete problem reads as follows: Find $E_h \in \mathbf{P}_p^b(\mathcal{T}_h)$ (recall that $p \ge 1$) such that

$$ b_h(E_h, w_h) = (J, w_h) \quad \forall w_h \in \mathbf{P}_p^b(\mathcal{T}_h), \quad (6.5) $$

with the discrete bilinear form such that, for all $v_h, w_h \in \mathbf{P}_p^b(\mathcal{T}_h)$,

$$ b_h(v_h, w_h) := \omega^2(v_h, w_h)_\epsilon + (\nabla_h \times v_h, \nabla_h \times w_h)_\nu + \eta_* s_h(v_h, w_h) + \sum_{F \in \mathcal{F}_h} \{ (\llbracket \nu \nabla_h \times v_h \rrbracket_F^g)_F, (\llbracket w_h \rrbracket_F^c)_F + (\llbracket v_h \rrbracket_F^c, \llbracket \nu \nabla_h \times w_h \rrbracket_F^g)_F \}. \quad (6.6) $$

The stabilization bilinear form $s_h$ is defined as

$$ s_h(v_h, w_h) := \sum_{F \in \mathcal{F}_h} \nu_F^\sharp \left( \frac{p^2}{h_F} \right) (\llbracket v_h \rrbracket_F^c, \llbracket w_h \rrbracket_F^c)_F, \quad (6.7) $$

and the user-dependent parameter $\eta_* > 0$ is taken large enough (see below). Using the definition (2.9) of the lifting operator $\mathbf{L}_h^p$, we infer that

$$ b_h(v_h, w_h) := \omega^2(v_h, w_h)_\epsilon + (\nabla_h \times v_h, \nabla_h \times w_h)_\nu + \eta_* s_h(v_h, w_h) + (\mathbf{L}_h^p(w_h), \nabla_h \times v_h)_\nu + (\mathbf{L}_h^p(v_h), \nabla_h \times w_h)_\nu. \quad (6.8) $$

We equip the discrete space $\mathbf{P}_p^b(\mathcal{T}_h)$ with the norm

$$ \|v_h\|_{\text{dG}}^2 := \omega^2 \|v_h\|_{\epsilon}^2 + \|\nabla_h \times v_h\|_{\nu}^2 + s_h(v_h, v_h) \quad \forall v_h \in \mathbf{P}_p^b(\mathcal{T}_h). \quad (6.9) $$

Recalling (2.10), we infer that

$$ \| \mathbf{L}_h^p(v_h) \|_{\nu}^2 \le C(\kappa_{\mathcal{T}_h}) s_h(v_h, v_h) \quad \forall v_h \in \mathbf{P}_p^b(\mathcal{T}_h), \quad (6.10) $$

whence we infer, provided we take $\eta_* \ge \frac{1}{2} + 2C(\kappa_{\mathcal{T}_h})$ with $C(\kappa_{\mathcal{T}_h})$ from (6.10), that the following coercivity property holds:

$$ b_h(v_h, v_h) \ge \frac{1}{2} \|v_h\|_{\text{dG}}^2 \quad \forall v_h \in \mathbf{P}_p^b(\mathcal{T}_h). \quad (6.11) $$

**Remark 6.1 (Weighted averages).** Whenever the jumps of $\nu$ across the mesh interfaces are large, one can consider weighted ($\nu$-dependent) averages to evaluate the last two terms on the right-hand side of (6.6). In the case of strong anisotropy, the weights depend on the normal-normal component of $\nu$ on both sides of the interface. We refer the reader, e.g., to [24] for an example in the context of a scalar diffusion problems. Accordingly, the definition (2.9) of the lifting operator $\mathbf{L}_h^p$ can be modified by using a weighted average on the right-hand side.

**Remark 6.2 (Indefinite case).** In the indefinite case, the coercivity property (6.11) is replaced by a Gårding inequality. Assuming that $\omega$ is not a resonant frequency and invoking a duality argument à la Schatz then leads to a discrete inf-sup condition if the mesh size is small enough. We refer the reader to [12, Sec. 5.4] for more details.
## 6.3 Estimator and error measure

The a posteriori error estimator is written as the sum over the mesh cells of local error indicators $\eta_K$ for all $K \in \mathcal{T}_h$. The local error indicator consists of three pieces. The first two respectively measure the residuals of the divergence constraint and of Maxwell's equations:

$$ \eta_{K, \text{div}}^2 := \frac{1}{\epsilon_K^\sharp} \left\{ \frac{1}{\omega^2} \left(\frac{h_K}{p}\right)^2 \|\nabla \cdot (\mathbf{J} - \omega^2 \epsilon \mathbf{E}_h)\|_K^2 + \omega^2 \left(\frac{h_K}{p}\right) \|\llbracket \epsilon \mathbf{E}_h \rrbracket_{\partial K}^\text{d} \|_{\partial K \setminus \partial \Omega}^2 \right\}, \quad (6.12a) $$

and

$$ \eta_{K, \text{curl}}^2 := \frac{1}{\nu_K^\sharp} \left\{ \left(\frac{h_K}{p}\right)^2 \|\mathbf{J} - \omega^2 \epsilon \mathbf{E}_h - \nabla \times (\nu \nabla \times \mathbf{E}_h)\|_K^2 \right. \\ \left. + \left(\frac{h_K}{p}\right) \|\llbracket \nu \nabla_h \times \mathbf{E}_h \rrbracket_{\partial K}^\text{c} \|_{\partial K \setminus \partial \Omega}^2 + \nu_{\partial K}^\sharp \left(\frac{p^2}{h_K}\right) \|\llbracket \mathbf{E}_h \rrbracket_{\partial K}^\text{c} \|_{\partial K}^2 \right\}. \quad (6.12b) $$

The last part of the estimator is related to the nonconformity of the discrete field $\mathbf{E}_h$ and reads

$$ \eta_{K, \text{nc}}^2 := \nu_{\partial K}^\sharp \left(\frac{h_K}{p}\right) \|\llbracket \nabla_h \times \mathbf{E}_h \rrbracket_{\partial K}^\text{d} \|_{\partial K}^2 + \left\{ \omega^2 \epsilon_{\partial K}^\sharp h_K + \nu_{\partial K}^\sharp \left(\frac{p^2}{h_K}\right) \right\} \|\llbracket \mathbf{E}_h \rrbracket_{\partial K}^\text{c} \|_{\partial K}^2. \quad (6.12c) $$

Notice that this latter estimator is suitable under Assumption 3.5. It is convenient to introduce the following shorthand notation:

$$ \eta_K^2 := \eta_{K, \text{div}}^2 + \eta_{K, \text{curl}}^2 + \eta_{K, \text{nc}}^2, \quad \eta_{\bullet}^2 := \sum_{K \in \mathcal{T}_h} \eta_{\bullet, K}^2, \quad \eta^2 := \sum_{K \in \mathcal{T}_h} \eta_K^2, \quad (6.13) $$

with $\bullet \in \{\text{div}, \text{curl}, \text{nc}\}$.

The approximation error is defined as

$$ e := E - E_h \in \mathcal{V}_\sharp := H_0(\text{curl}; D) + P_p^b(\mathcal{T}_h). \quad (6.14) $$

Although the sum defining $\mathcal{V}_\sharp$ is not direct, any field $v_h \in H_0(\text{curl}; D) \cap P_p^b(\mathcal{T}_h)$ satisfies $\|v_h\|_F^c = 0$ for all $F \in \mathcal{F}_h$. It is therefore legitimate to extend the jump operator to $\mathcal{V}_\sharp$ by setting $\|v\|_F^c := \|v_h\|_F^c$ for all $v = \tilde{v} + v_h \in \mathcal{V}_\sharp$. Similarly, the lifting operator $L_h^p$ can be extended to $\mathcal{V}_\sharp$ by setting $L_h^p(v) := L_h^p(v_h)$. The discrete bilinear form $b_h$ is then extended to $\mathcal{V}_\sharp \times \mathcal{V}_\sharp$ by using the expression (6.8), i.e., we set, for all $v, w \in \mathcal{V}_\sharp$,

$$ b_{\sharp}(v, w) := \omega^2(v, w)_{\epsilon} + (\nabla_h \times v, \nabla_h \times w)_{\nu} + \eta_* s_h(v, w) \\ + (L_h^p(w), \nabla_h \times v)_{\nu} + (L_h^p(v), \nabla_h \times w)_{\nu}, \quad (6.15) $$

keeping in mind that $\nabla_h \times v := \nabla_0 \times v$ whenever $v \in H_0(\text{curl}; D)$.

For any subset $T \subset \mathcal{T}_h$, we define the error measure

$$ \|e\|_{\sharp, T}^2 := \sum_{K \in T} \left\{ \omega^2 \|e\|_{\epsilon, K}^2 + \|\nabla \times e\|_{\nu, K}^2 + \nu_{\partial K}^\sharp \left(\frac{p^2}{h_K}\right) \|\llbracket e \rrbracket_{\partial K}^\text{c} \|_{\partial K}^2 \right\}, \quad (6.16) $$

and omit the subscript $T$ whenever $T = \mathcal{T}_h$. When restricted to $P_p^b(\mathcal{T}_h)$, $\|\cdot\|_{\sharp}$ is equivalent to the discrete dG norm $\|\cdot\|_{\text{dG}}$ defined in (6.9) up to a rewriting of the jump contribution as a sum over the mesh cell boundaries rather than over the mesh faces.

**Remark 6.3** (Stabilization weight). We observe that the error measure defined in (6.16) is independent of the stabilization parameter $\eta_*$ introduced in the dG scheme.

**Remark 6.4** (Heterogeneous materials). Whenever one wishes to account for the heterogeneity of the material properties, some weights in the error indicators need to be adapted. For the error
indicators defined in (6.12), one needs to consider a contrast factor involving three layers of vertex-based neighbors around K, as detailed in [12]. For the nonconformity indicator defined in (6.12c), one layer of neighbors is sufficient. We also notice that it is possible to modify Definition 3.1 so as to reconstruct on each vertex patch an $H_0(\text{curl})$-conforming field by using weighted $L^2$-inner products. However, the estimates satisfied by this modified reconstruction in the broken curl and $L^2$-norms will depend anyway on the contrast factors associated with the underlying weights.

## 6.4 hp-a posteriori error analysis

We decompose the approximation error into two components as follows:

$$e = (E - E_c) + (E_c - E_h) := e_c + e_{nc}, \quad E_c := R^c(E_h) \in H_0(\text{curl}; D). \quad (6.17)$$

We call $e_c$ the conforming error and $e_{nc}$ the nonconforming error. Notice that we use the $H_0(\text{curl})$-reconstruction operator from Section 3 to define the two error components. Actually, the precise definition of $E_c$ is irrelevant for bounding $e_c$ (we only use that $e_c \in H_0(\text{curl}; D)$), and is only relevant for bounding $e_{nc}$.

**Lemma 6.5 (Residual).** Let $\eta_{dc}^2 := \eta_{\text{div}}^2 + \eta_{\text{curl}}^2$. The following holds:

$$|b_{\#}(e, e_c)| \le C(\kappa_{T_h})\eta_{dc}\|e_c\|_{\#}. \quad (6.18)$$

*Proof.* The proof essentially follows the arguments from the proof of [12, Lem. 6.2], up to minor adaptations (the proof therein tracks the dependency of the constants on the heterogeneity of the material properties, and deals with the indefinite Maxwell problem). □

**Lemma 6.6 (Bound on $e_c$).** The following holds:

$$\|e_c\|_{\#} \le C(\kappa_{T_h})(\eta_{dc} + \|e_{nc}\|_{\#}). \quad (6.19)$$

*Proof.* Since $e_c \in H_0(\text{curl}; D)$, we have $s_h(e_c, e_c) = 0$ and $L_h^p(e_c) = 0$, so that

$$\begin{aligned} \|e_c\|_{\#}^2 &= \omega^2 \|e_c\|_{\epsilon}^2 + \|\nabla_0 \times e_c\|_{\nu}^2 \\ &= b_{\#}(e_c, e_c) = b_{\#}(e, e_c) - b_{\#}(e_{nc}, e_c) =: T_1 + T_2. \end{aligned}$$

Applying the bound (6.18) from Lemma 6.5, we get

$$|T_1| \lesssim \eta_{dc} \|e_c\|_{\#}.$$

Moreover, we observe that

$$T_2 = b_{\#}(e_{nc}, e_c) = \omega^2(e_{nc}, e_c)_{\epsilon} + (\nabla_h \times e_{nc}, \nabla_0 \times e_c)_{\nu} - (L_h^p(E_h), \nabla_0 \times e_c)_{\nu}.$$

Invoking the Cauchy–Schwarz inequality and the bound (2.10) on the discrete lifting operator, we infer that

$$|T_2| \le 2 \{ \omega^2 \|e_{nc}\|_{\epsilon}^2 + \|\nabla_h \times e_{nc}\|_{\nu}^2 + \|L_h^p(E_h)\|_{\nu}^2 \}^{\frac{1}{2}} \|e_c\|_{\#} \lesssim \|e_{nc}\|_{\#} \|e_c\|_{\#}.$$

Combining the above bounds completes the proof. □

**Lemma 6.7 (Bound on $e_{nc}$).** Under Assumption 3.5, the following holds:

$$\|e_{nc}\|_{\#} \le C(\kappa_{T_h})\eta_{nc}. \quad (6.20)$$
Proof. Recall that, for all $K \in \mathcal{T}_h$, $e_{nc}|_K = \sum_{a \in \mathcal{V}_K} \delta_a|_K$ with $\delta_a := E_a - \psi_a E_h$ (see (3.9)). Reasoning as in the proof of Theorem 3.7, this gives

$$ \|\nabla \times e_{nc}\|_{\nu, K}^2 \le \nu_K \sum_{a \in \mathcal{V}_K} \|\nabla_h \times \delta_a\|_{D_a}^2. $$

Invoking the bound on $\|\nabla_h \times \delta_a\|_{D_a}$ from Lemma 3.3, and hiding the contrast factor on $\nu$ in the generic constants, we infer that

$$ \|\nabla \times e_{nc}\|_{\nu, K}^2 \lesssim \sum_{a \in \mathcal{V}_K} \sum_{F \in \mathcal{F}_a} \left\{ \nu_F^{\sharp} \left( \frac{h_F}{p} \right) \|\|\nabla_h \times E_h\|_F^d \|_F^2 + \nu_F^{\sharp} \left( \frac{p^2}{h_F} \right) \|\|\mathbf{E}_h\|_F^c \|_F^2 \right\}. $$

Reasoning similarly to bound $\omega^2 \|e_{nc}\|_{\epsilon,K}^2$, but this time invoking the bound from Lemma 3.6, we infer that

$$ \omega^2 \|e_{nc}\|_{\epsilon,K}^2 \lesssim \sum_{a \in \mathcal{V}_K} \sum_{F \in \mathcal{F}_a} \omega^2 \epsilon_F^{\sharp} h_F \|\|\mathbf{E}_h\|_F^c \|_F^2. $$

Combining the above two bounds together, re-arranging the summations over the mesh cells, invoking again the mild heterogeneity of the material properties, and observing that $\|\mathbf{e}_{nc}\|_{\partial K}^c = -\|\mathbf{E}_h\|_{\partial K}^c$ for all $K \in \mathcal{T}_h$, leads to the expected bound on $\|\mathbf{e}_{nc}\|_{\sharp}$. $\square$

**Theorem 6.8** (Global upper bound (reliability)). Under Assumption 3.5, the following holds:

$$ \|\mathbf{e}\|_{\sharp} \le C(\kappa_{\mathcal{T}_h})\eta. \qquad (6.21) $$

Proof. Invoke the triangle inequality together with Lemmas 6.6 and 6.7, and recall the definition of $\eta$ from (6.13). $\square$

Finally, we derive local efficiency estimates.

**Proposition 6.9** (Local lower bound (local efficiency)). For all $K \in \mathcal{T}_h$, we have

$$ \eta_K \le C(\kappa_{\mathcal{T}_h}) p^{\frac{3}{2}} \left\{ \left( 1 + \left( \frac{\omega^2 \epsilon_K^{\sharp}}{\nu_K^{\sharp}} \right)^{\frac{1}{2}} \left( \frac{h_K}{p} \right) \right) \|\mathbf{E} - \mathbf{E}_h\|_{\sharp, K^f} + \mathrm{osc}_{K^f} \right\}, \quad (6.22) $$

with the data oscillation term

$$ \mathrm{osc}_{K^f}^2 := \sum_{K' \in K^f} \min_{J_h \in \mathcal{P}_p^b(\mathcal{T}_h)} \left\{ \frac{1}{\nu_{K'}^{\sharp}} \left( \frac{h_{K'}}{p} \right)^2 \|\mathbf{J} - \mathbf{J}_h\|_{K'}^2 + \frac{1}{\omega^2 \epsilon_{K'}^{\sharp}} \left( \frac{h_{K'}}{p} \right)^2 \|\nabla \cdot (\mathbf{J} - \mathbf{J}_h)\|_{K'}^2 \right\}, \quad (6.23) $$

and $K^f$ is the collection of the mesh cells sharing a face with $K$.

Proof. Local bounds on all terms composing the error estimator $\eta_K$ can be found from [12, Theorem 6.5] except for the term involving $\|\nabla_h \times E_h\|_F^d$. Invoking the $H^1$ to $L^2$ norm inverse inequality from [37, Theorem 4.76], we infer that

$$ \nu_{\partial K}^{\sharp} \left( \frac{h_K}{p} \right) \|\|\nabla_h \times E_h\|_{\partial K}^d \|_{\partial K}^2 \lesssim \nu_{\partial K}^{\sharp} \left( \frac{p^3}{h_K} \right) \|\|\mathbf{E}_h\|_{\partial K}^c \|_{\partial K}^2 \lesssim p \|\mathbf{E} - \mathbf{E}_h\|_{\sharp, K^f}. $$

This completes the proof. $\square$

**Remark 6.10** (Indefinite case). Whenever $\omega = 0$, the above hp-a posteriori error analysis can be applied with the additional simplification of discarding the $L^2$-norm estimate, so that Assumption 3.5 is no longer relevant. (Notice also that the last term on the right-hand side of (6.23) is no longer relevant since $J$ is divergence-free). In the time-harmonic regime with a nonzero, pure imaginary $\omega$, the techniques in [12] can be combined with the present $H_0(\mathrm{curl})$-conforming reconstruction.
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