# A new family of a posteriori error estimates for non-conforming finite element methods leading to stabilization-free error bounds

Théophile Chaumont-Frelet

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Théophile Chaumont-Frelet. A new family of a posteriori error estimates for non-conforming finite element methods leading to stabilization-free error bounds. IMA Journal of Numerical Analysis, In press. hal-05135065

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Submitted on 29 Jun 2025

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# A new family of a posteriori error estimates for non-conforming finite element methods leading to stabilization-free error bounds

T. CHAUMONT-FRELET*

**ABSTRACT.** We propose new a posteriori error estimators for non-conforming finite element discretizations of second-order elliptic PDE problems. These estimators are based on novel reformulations of the standard Prager-Synge identity, and enable to prove efficiency estimates without extra stabilization terms in the error measure for a large class of discretization schemes. We propose a residual-based estimator for which the efficiency constant scales optimally in polynomial degree, as well as two equilibrated estimators that are polynomial-degree-robust. One of the two estimators further leads to guaranteed error bounds.

**KEYWORDS.** a posteriori error estimate; discontinuous Galerkin method; equilibrated flux; high-order method; mixed finite element method; non-conforming finite element method; residual-based estimator

## 1. INTRODUCTION

In this work we are interested in finite element and discontinuous Galerkin discretizations of the following second-order elliptic PDE problem:

$$ (1.1) \qquad \begin{cases} -\nabla \cdot (\underline{\mathbf{A}}\nabla u) = f & \text{in } \Omega, \\ u = 0 & \text{on } \Gamma_D, \\ \underline{\mathbf{A}}\nabla u \cdot \mathbf{n} = 0 & \text{on } \Gamma_N. \end{cases} $$

This problem is set in a three dimensional domain $\Omega \subset \mathbb{R}^3$ with its boundary $\partial\Omega$ partitioned into two disjoint subsets $\Gamma_D$ and $\Gamma_N$. The right-hand side $f: \Omega \to \mathbb{R}$ and the diffusion coefficient $\underline{\mathbf{A}}: \Omega \to \mathbb{R}^{3 \times 3}$ are given, whereas $u: \Omega \to \mathbb{R}$ is the unknown. Here, we wish to numerically approximate the gradient $\mathcal{G} := \nabla u$ of $u$ by a piecewise polynomial ansatz $\mathcal{G}_h$ on a simplicial mesh $\mathcal{T}_h$ of $\Omega$. More specifically, we focus on a posteriori error estimation, where the goal is to provide a fully computable quantity $\eta$ which can be used to certify the discretization error $\mathcal{G} - \mathcal{G}_h$ in a suitable norm.

An alternative way to state (1.1) is that $\mathcal{G}$ is the unique vector field such that (i) $(-\underline{\mathbf{A}}\mathcal{G}) \cdot \mathbf{n} = 0$ on $\Gamma_N$ and $\nabla \cdot (-\underline{\mathbf{A}}\mathcal{G}) = f$ in $\Omega$, and (ii) there exists a scalar potential $v$ such that $v = 0$ on $\Gamma_D$ and $\mathcal{G} = \nabla v$. As a result, a possible starting point for the design of the estimator $\eta$ is the following Prager-Synge identity

$$ (1.2) \qquad \| \mathcal{G} - \mathcal{G}_h \|_{\underline{\mathbf{A}}, \Omega}^2 = \min_{\substack{\sigma \in H_{\Gamma_N}(\text{div}, \Omega) \\ \nabla \cdot \sigma = f}} \| \underline{\mathbf{A}}^{-1} \sigma + \mathcal{G}_h \|_{\underline{\mathbf{A}}, \Omega}^2 + \min_{v \in H_{\Gamma_D}^1(\Omega)} \| \nabla v - \mathcal{G}_h \|_{\underline{\mathbf{A}}, \Omega}^2 $$

which measures the deviation of the discrete approximation from (i) and (ii). The notation in (1.2) are standard, and rigorously introduced below. The seminal work from which the identity in (1.2) originates is [48]. This identity also appears in [35] under an extra assumption on $\mathcal{G}_h$, and it is rigorously established for general $L^2(\Omega)$ vector-field below.

<footnote>*Inria, Univ. Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé</footnote>
For conforming methods, one looks for an approximation $u_h$ of $u$ in a subspace of $H_{\Gamma_D}^1(\Omega)$.
As a result, setting $\mathfrak{G}_h := \nabla u_h$, the second term in (1.2) vanishes. In this case, residual-based
and flux-equilibrated estimator techniques are very well established, and control the first term
in the right-hand side of (1.2) provided $\mathfrak{G}_h$ satisfies suitable orthogonality conditions, which
in turn is guaranteed by the numerical method producing $u_h$. We refer the reader to [13, 47]
for an overview. In contrast, for mixed and non-conforming methods, the second term in (1.2)
does not vanish and must be controlled.

In cases where $\mathfrak{G}_h = \nabla_h u_h$ is element-wise given by the gradient of a known potential,
a standard procedure is to postprocess the discrete (non-conforming) potential $u_h$ into a
conforming version $\tilde{u}_h$, leading to the fully explicit estimation

$$ (1.3) \qquad \min_{v \in H_{\Gamma_D}^1(\Omega)} \| \nabla v - \mathfrak{G}_h \|_{\mathbf{A}, \Omega} \le \| \nabla \tilde{u}_h - \mathfrak{G}_h \|_{\mathbf{A}, \Omega} $$

of the second term in the right-hand side of (1.2). This approach is employed, e.g., in [4, 6, 35].
An important observation is that since $\tilde{u}_h$ is typically obtained from averaging the values
of $u_h$ across neighboring elements, $\nabla \tilde{u}_h$ is not (in general) solely defined in terms of $\mathfrak{G}_h$.
Alternatively, if one does not wish to explicitly compute $\tilde{u}_h$, one might simply invoke the
existence of such a conforming function, and use the estimate

$$ (1.4) \qquad \| \nabla \tilde{u}_h - \mathfrak{G}_h \|_{\mathbf{A}, \Omega}^2 \le C \sum_{K \in \mathcal{T}_h} \frac{p^2}{h_K} \| [u_h] \|_{\partial K \setminus \Gamma_N}^2 $$

which again provides the desired bound in (1.2). Here, $[u_h]$ denotes the jump of $u_h$, $p$ is the
polynomial degree, and $h_K$ is the diameter of the element $K \in \mathcal{T}_h$. We note that in case too,
the estimator cannot (in general) be determined only in terms of $\mathfrak{G}_h$. For simplicity, we will
respectively refer to the right-hand side of (1.3) and (1.4) as the standard equilibrated and
residual-based estimators in the remainder of this introduction.

The fact that the estimators in (1.3) and (1.4) are not solely defined in terms of $\mathfrak{G}_h$ can
lead to complications. First of all, if $\mathfrak{G}_h$ is not given as the gradient of a discrete potential,
the above techniques cannot readily apply. In addition, to the best of the author's knowledge,
general error lower bounds of the form

$$ \frac{p^2}{h_K} \| [u_h] \|_{\partial K \setminus \Gamma_N} \le C \| \nabla \tilde{u}_h - \mathfrak{G}_h \|_{\mathbf{A}, \omega_K} $$

where $\omega_K$ is a local region around $K$, have never been established in the literature. It means
that in general, estimators defined using a potential are not efficient in the natural error
measure.

In this work, we propose different characterizations of the term measuring the level of non-
conformity in (1.2). Specifically, we are going to show that for many numerical schemes of
interest, we have

$$ (1.5) \qquad \min_{v \in H_{\Gamma_D}^1(\Omega)} \| \nabla v - \mathfrak{G}_h \|_{\mathbf{A}, \Omega} = \min_{\phi \in H_{\Gamma_D}(\mathrm{curl}^0, \Omega)} \| \phi - \mathfrak{G}_h \|_{\mathbf{A}, \Omega} = \max_{\substack{\theta \in H_{\Gamma_N}(\mathrm{curl}, \Omega) \\ \| \nabla \times \theta \|_{\mathbf{A}^{-1}, \Omega} = 1}} (\mathfrak{G}_h, \nabla \times \theta)_\Omega $$

When the domain $\Omega$ and the boundary decomposition correspond to a topologically trivial
setting, the identities in (1.5) in fact hold true for any vector field $\mathfrak{G}_h \in L^2(\Omega)$. However, in
the general case, they are only true under specific orthogonality conditions, which turn out
to be automatically satisfied by the solution of several numerical schemes.
We respectively employ the first and second identities in (1.5) to derive new equilibrated and residual-based error estimators for non-conforming finite element methods, with special emphasis on Raviart–Thomas mixed discretization and the interior penalty discontinuous Galerkin (IPDG) scheme. For the equilibrated estimator, we propose a construction of a curl-free field $\phi_h$ built from localized Nédélec finite element problems to be used in place of the minimizer $\phi$ in (1.5), whereas the residual based estimator takes the form

$$ (1.6) \quad \min_{v \in H_{\Gamma_D}^1(\Omega)} \| \nabla v - \mathfrak{G}_h \|_{\underline{\mathbf{A}}, \Omega} \le C \sum_{K \in \mathcal{T}_h} \left\{ \frac{h_K^2}{p^2} \| \nabla \times \mathfrak{G}_h \|_K^2 + \frac{h_K}{p} \| [\mathfrak{G}_h] \|_{\partial K \setminus \Gamma_N}^2 \times \mathbf{n} \|_{\partial K \setminus \Gamma_N}^2 \right\}. $$

We additionally propose an alternative flux reconstruction technique hinging on a further reformulation of the curl-constrained minimization problem in (1.5).

In all cases, the proposed estimators are shown to be efficient in the natural energy norm, i.e., without adding any term linked to the potential to the local error measure $\|\mathfrak{G} - \mathfrak{G}_h\|_{\underline{\mathbf{A}}, \omega_K}$. In addition, the equilibrated error estimators are polynomial-degree-robust, whereas the efficiency constant of the residual-based estimator grows with the optimal known rate with the polynomial degree. We finally emphasize that $\mathfrak{G}_h$ does not need to be given element-wise as the gradient of a potential. Below, we highlight the advantage of the proposed estimators as compared to standard ones in (1.3) and (1.4). Specifically, we focus on (a) non-conforming methods of Crouzeix-Raviart type, (b) mixed methods, and (c) discontinuous Galerkin methods.

(a) We first discuss the case where $\mathfrak{G}_h = \nabla_h u_h$ and the average of the jump vanishes over each mesh faces, i.e. ($\|u_h\|$, $1)_F = 0$ for all mesh face $F$. This applies, e.g., to Crouzeix-Raviart [29] and primal hybrid [49] discretizations of (1.1). In this case, for the standard residual-based estimator, we can employ a face-wise Poincaré inequality (see e.g. [10]) leading to the estimate

$$ (1.7) \quad \sum_{K \in \mathcal{T}_h} \frac{p^2}{h_K} \| [u_h] \|_{\partial K \setminus \Gamma_N}^2 \le C \sum_{K \in \mathcal{T}_h} p^2 h_K \| [\nabla u_h] \|_{\partial K \setminus \Gamma_N}^2. $$

As a result, the standard residual-based estimator is indeed efficient since the right-hand side essentially correspond to the estimator proposed in this work. However, we can see that the right-hand side is multiplied by $p^3$ as compared to (1.6), indicating that the estimator may not behave optimally as $p$ increases. As a result, our residual-based estimator exhibit an improved $p$-dependence over the standard one. When a potential reconstruction is carefully constructed, the construction in [35, 36] is already polynomial-degree-robust. Therefore, for equilibrated estimators, although we expect that the present approach brings new insight, the present work does not lead to quantitative improvements in practice.

(b) For mixed finite-element methods, it is natural to consider $\mathfrak{G}_h = \underline{\mathbf{A}}^{-1} \sigma_h$, where $\sigma_h$ is the Raviart-Thomas approximation of the flux. Since $\mathfrak{G}_h$ is not a piecewise gradient, the standard procedure described above cannot readily apply. In [28] and [35], the standard residual-based and equilibrated estimators are therefore applied to a postprocessing $\nabla_h u_h^*$ of $\sigma_h$. In both cases, the estimator can be shown to be efficient, with an optimal scaling in $p$ for the efficiency constant. It is therefore true that our new estimators do not seem to present significant improvements over the standard ones as far as efficiency is concerned. However, these standard estimators are not suited for adaptivity purposes, and adaptive algorithms rather use residual-based estimators similar to the one we propose in (1.6) that were previously established in [7, Theorem 3.1] and [16, Lemma 5.1]. Unfortunately, both works are limited to two-dimensional problems, and trivial topologies. More specifically,
neither work consider mixed boundary conditions, and the domain is respectively assumed
to be simply-connected and convex in [7] and [16]. Besides, the estimates obtained in [7, 16]
are not hp-optimal. As previously mentioned, to the best of the author's knowledge, all
works dealing with (optimal) convergence of adaptive mixed finite element discretization either
explicitly requires a trivial topology [19, 26], or are based on [7, 16] where the assumption is
made [11, 17, 18]. The present results thus allow the above mentioned adaptive convergence
results to apply to general domains and boundary conditions.

(c) Finally, for discontinuous Galerkin discretizations, an estimation such as (1.7) does not hold. As a result, error estimators including the jumps of the potential are never shown to be efficient, unless said jumps are included in the error measure, see e.g. [4, 27, 35, 40]. This lack of efficiency in a natural error measure further causes problems in the analysis of adaptive schemes based on the estimator [38, 42, 43, 46]. To the best of our knowledge, the estimators proposed in this work are the first to be efficient for discontinuous Galerkin discretizations in a natural norm.

We finally mention two ideas related to the one developed here that recently appeared in the literature. (i) Closest to the present results are the previous work [22] of the author and [12]. In a nutshell, the key idea of both works is to employ extra orthogonality conditions satisfied by $\mathcal{G}_h$ to replace the gradient vector field by a curl-free one in (1.2). However, the setting in [22] is fairly different from here, since it considers magnetostatic problems discretized by mixed methods, and only covers flux-equilibrated estimators. On the other hand, whereas the present results are very closely related to [12], they represent a significant improvement. First, the identity in (1.5) is not explicitly formulated in [12], and the technique used there leads to unknown constants depending on the domain. This can be specifically seen in [12, Theorem 1]. Besides, the estimator proposed in [12, Equation (4)] is suited for lowest-order discretization, it is not hp-optimal. In contrast, the residual-based estimator proposed Lemma 4.1 below has an optimal hp-scaling. (ii) As already alluded to earlier, another displeasing aspect of the jump term in (1.4) is the asymmetry of the $p$ and $h_K$ powers in the multiplicative factor, which suggest an hp-suboptimality. This is a well-known problem that stems from the (apparently necessary) use of an inverse inequality. In [15, 31], ideas related to the one employed below and in [12] are used to bypass such an inverse inequality. Yet, the focus is different, since the final error estimators in [15, 31] still contains a jump term from the potential (albeit with an improved $p$-scaling), and is therefore not efficient in the natural energy norm.

The remainder of this work is organized as follows. In Section 2, we make the setting precise, introduce key notation and state useful results from the literature. Section 3 is dedicated to the abstract derivation of (1.5) under generic assumptions on $\mathcal{G}_h$. Finally, Sections 4, 5 and 6 are respectively dedicated to the three practical estimators we construct from (1.5), namely, a residual-based estimator, a fully-equilibrated estimator, and an alternative equilibration strategy.

## 2. SETTING

2.1. **Domain.** Throughout this manuscript, $\Omega \subset \mathbb{R}^3$ is an open domain with a Lipschitz polyhedral boundary $\partial\Omega$. The boundary is split into two Lipschitz relatively open polyhedral subdomains such that $\Gamma_D$ and $\Gamma_N$ in such a way that $\bar{\Gamma}_D \cup \bar{\Gamma}_N = \partial\Omega$. Notice that we do not assume that $\Omega$, nor $\Gamma_D$ or $\Gamma_N$ are simply connected. We denote by $\mathbf{n}$ the unit vector normal to $\partial\Omega$ and pointing outside $\Omega$.
2.2. **Diffusion coefficient.** The diffusion coefficient $\underline{\mathbf{A}}: \Omega \to \mathbb{R}^{3 \times 3}$ is a measurable symmetric matrix-valued function. For any measurable set $U \subset \Omega$, we set

$$ \alpha_{\min,U} := \essinf_{\mathbf{x} \in U} \min_{\substack{\mathbf{e} \in \mathbb{R}^d \\ |\mathbf{e}|=1}} \underline{\mathbf{A}}(\mathbf{x})\mathbf{e} \cdot \mathbf{e}, \qquad \alpha_{\max,U} := \esssup_{\mathbf{x} \in U} \max_{\substack{\mathbf{e} \in \mathbb{R}^d \\ |\mathbf{e}|=1}} \underline{\mathbf{A}}(\mathbf{x})\mathbf{e} \cdot \mathbf{e}, $$

and we assume that $\alpha_{\min,\Omega} > 0$ and $\alpha_{\max,\Omega} < +\infty$. Some results below will require that $\underline{\mathbf{A}}$ is piecewise polynomial, which we will specify case by case.

2.3. **Function spaces.** For any open set $U \subset \Omega$ with Lipschitz boundary, $L^2(U)$ is the Lebesgue space of square-integrable functions and $L^2(U) := [L^2(U)]^3$. The usual norm and inner product of both spaces are denoted by $\|\cdot\|_U$ and $(\cdot, \cdot)_U$. If $\underline{\mathbf{M}}: U \to \mathbb{R}^{3 \times 3}$ is a measurable symmetric matrix-valued functions, we employ the notation $\|\mathbf{v}\|_{\underline{\mathbf{M}},U}^2 := (\underline{\mathbf{M}}\mathbf{v}, \mathbf{v})_U$ for $\mathbf{v} \in L^2(U)$. If $\underline{\mathbf{M}}$ is uniformly bounded from above and below, then $\|\cdot\|_{\underline{\mathbf{M}},U}$ is a norm on $L^2(U)$ equivalent to standard one. We will also employ the same notation for Lebesgue spaces on two-dimensional relatively open affine subsets $V \subset \Omega$.

The standard Sobolev space of scalar functions $v \in L^2(U)$ with $\nabla v \in L^2(U)$ is denoted by $H^1(U)$, and we let $\underline{\mathbf{H}}^1(U) := [H^1(U)]^3$. If $\gamma \subset \partial U$ is a relatively open subset, then $H^1(U) \subset H^1(U)$ collects functions with vanishing trace on $\gamma$. For $\mathbf{v} \in \underline{\mathbf{H}}^1(U)$, we denote by $\underline{\nabla} \mathbf{v} = (\nabla \mathbf{v}_1, \dots, \nabla \mathbf{v}_d)$ its gradient, and the application

$$ \|\underline{\nabla} \mathbf{v}\|_{\underline{\mathbf{M}},\Omega}^2 := \sum_{k=1}^{d} \|\nabla \mathbf{v}_k\|_{\underline{\mathbf{M}},\Omega}^2, $$

is then equivalent to the standard semi-norm of $\underline{\mathbf{H}}^1(\Omega)$ if $\underline{\mathbf{M}}: U \to \mathbb{R}^{3 \times 3}$ is measurable and uniformly bounded from abvoe and below. We skip the $\underline{\mathbf{M}}$ subscript when $\underline{\mathbf{M}} = \underline{\mathbf{I}}$. Lebesgue and Sobolev spaces are extensively described, e.g. in [1].

If $v: U \to \mathbb{R}$ is measurable, we will employ the notation

$$ \|v\|_{L^{\infty}(U)} := \esssup_{\mathbf{x} \in U} |v(\mathbf{x})| $$

for its essential supremum. Similarly, if $v: U \to \mathbb{R}^3$, then $\|v\|_{L^\infty(U)} = \|v\|_{L^\infty(\Omega)}$.

We will also Sobolev spaces of vector-valued functions with well-defined curl and divergence. Namely, $\underline{\mathbf{H}}(\text{curl}, U)$ and $\underline{\mathbf{H}}(\text{div}, U)$ respectively contain functions $\mathbf{v} \in L^2(U)$ such that $\nabla \times \mathbf{v} \in L^2(U)$ and $\nabla \cdot \mathbf{v} \in L^2(U)$, where $\nabla \times$ and $\nabla \cdot$ are the curl and divergence operators defined in the sense of distributions. We further denote by $\underline{\mathbf{H}}_\gamma(\text{curl}, U)$ and $\underline{\mathbf{H}}_\gamma(\text{div}, U)$ the subspaces of functions with vanishing tangential and normal traces on $\gamma$. For more details, we refer the reader to [37], and in particular to [8] for the definition of traces on portions of the boundary.

We use the short-hand notation $L^2_\gamma(U) := \nabla \cdot \underline{\mathbf{H}}_\gamma(\text{div}, U)$. This space coincides with $L^2(U)$ unless $\gamma = \partial U$, in which case it is the subset of functions with vanishing mean-value. Finally, $\underline{\mathbf{H}}_\gamma(\text{div}^0, U)$ is the subspace of $\underline{\mathbf{H}}_\gamma(\text{div}, U)$ collecting divergence-free functions.

2.4. **Mesh.** We consider a simplicial mesh $\mathcal{T}_h$ of $\Omega$ consisting of (open) tetrahedral elements $K$. We assume that the mesh is conforming, i.e. $\cup_{K \in \mathcal{T}_h} \bar{K} = \bar{\Omega}$, and matching, meaning that if the intersection $\bar{K}_- \cap \bar{K}_+$ of two distinct elements $K_\pm \in \mathcal{T}_h$ is non-empty, then it is a subsimplex (i.e. a vertex, a full edge or a full face) of both elements. These assumptions are fully standard, see e.g. [34, Definition 8.11], but excludes hanging-nodes. We also classically assume that the mesh matches the boundary decomposition, meaning that if $F \subset \partial\Omega$, then either $F \subset \Gamma_D$ or $F \subset \Gamma_N$.
We respectively denote by $\mathcal{V}_h$, $\mathcal{E}_h$ and $\mathcal{F}_h$ the set of vertices, edges and faces of the mesh $\mathcal{T}_h$. For an element $K \in \mathcal{T}_h$, $\mathcal{V}(K) \subset \mathcal{V}_h$, $\mathcal{E}(K) \subset \mathcal{E}_h$, and $\mathcal{F}(K) \subset \mathcal{F}_h$ are the set of vertices, edges and faces of $K$. We also introduces the subsets $\mathcal{F}_h^1$, $\mathcal{F}_h^D$, $\mathcal{F}_h^N \subset \mathcal{F}_h$ of faces respectively lying inside $\Omega$, on $\Gamma_D$ and on $\Gamma_N$. We associate with each face $F \in \mathcal{F}_h$ a unit normal vector $\mathbf{n}_F$. When $F \in \mathcal{F}_h^D \cup \mathcal{F}_h^N$, we require that $\mathbf{n}_F$ points outside $\Omega$, whereas the orientation is arbitrary but fixed if $F \in \mathcal{F}_h^I$.

For all $K \in \mathcal{T}_h$, $h_K$ is the diameter of $K$ and $\rho_K$ is the diameter of the largest ball contained in $\bar{K}$. $\kappa_K := h_K/\rho_K \ge 1$ is then the shape-regularity parameter of $K$. For any submesh $\mathcal{T} \subset \mathcal{T}_h$, we let $\kappa_{\mathcal{T}} := \max_{K \in \mathcal{T}} \kappa_K$. Finally, if $U \subset \Omega$ is a subset covering the elements $K \in \mathcal{T}$, we will also alternatively employ the notation $\kappa_U := \kappa_{\mathcal{T}}$.

2.5. **Element patches.** For $a \in \mathcal{V}_h$, we denote by $\mathcal{T}_h^a \subset \mathcal{T}_h$ the edge patch of elements $K \in \mathcal{T}_h$ such that $a \in \mathcal{V}(K)$, and by $\mathcal{V}_h^a$ the corresponding set of vertices. We also employ the notation $\omega_a \subset \Omega$ for the associated open domain. We further employ the notation $\tilde{\mathcal{T}}_h^a = \cup_{b \in \mathcal{V}_h^a} \mathcal{T}_h^a$ for the extended vertex patch including one extra later of elements, and denote by $\tilde{\omega}_a \subset \Omega$ the corresponding open domain.

For $K \in \mathcal{T}_h$, the element patch $\omega_K \subset \Omega$ is the open domain covering all the lements $K' \in \mathcal{T}_h$ sharing a vertex with $K$. We also respecetively denote by $\tilde{\omega}_K$ and $\tilde{\omega}_K$ the element patches extended by one and two layers of elements.

2.6. **Jumps and averages.** Let $H^1(\mathcal{T}_h)$ stand for the sets of $v \in L^2(\Omega)$ such that $v|_K \in H^1(K)$ for all $K \in \mathcal{T}_h$. If $F \in \mathcal{F}_h^I$ is an internal face, we introduce the jump and average of a function $v \in H^1(\mathcal{T}_h)$ through $F$ as follows

$$ \llbracket v \rrbracket_F := v_+ \mathbf{n}_+ \cdot \mathbf{n}_F + v_- \mathbf{n}_- \cdot \mathbf{n}_F \qquad \llbracket v \rrbracket_F := \frac{1}{2}(v_+ + v_-), $$

where $K_{\pm} \in \mathcal{T}_h$ are the only two elements such that $F = \partial K_- \cap \partial K_+$, $v_{\pm} := v|_{K_{\pm}}|_F$, and $\mathbf{n}_{\pm}$ is unit normal pointing outside $K_{\pm}$. On the other hand, for a boundary face $F \in \mathcal{F}_h^D \cup \mathcal{F}_h^N$, we simply set

$$ \llbracket v \rrbracket_F := v|_F \qquad \llbracket v \rrbracket_F := v|_F. $$

We omit the subscript $F$ when it is clear from context, and also employ similar notation for vector-valued functions.

2.7. **Finite element spaces.** For $q \ge 0$ and $K \in \mathcal{T}_h$, $\mathcal{P}_q(K)$ is the space of polynomials of degree at most $q$ defined on $K$, $\mathcal{P}_q(K) := [\mathcal{P}_q(K)]^3$, and $\mathcal{P}_q(K) := [\mathcal{P}_q(K)]^{3 \times 3}$. For $q \ge 1$, we will also need the Raviart–Thomas and Nédélec polynomial spaces $RT_q(K) := \mathcal{P}_{q-1}(K) + x\mathcal{P}_{q-1}(K)$ and $N_q(K) := \mathcal{P}_{q-1}(K) + x \times \mathcal{P}_{q-1}(K)$. These spaces are extensively discussed in [34, Chapters 7, 14 and 15]. When $\mathcal{T} \subset \mathcal{T}_h$, $\mathcal{P}_q(\mathcal{T})$ stands for the set of functions whose restriction to each $K \in \mathcal{T}$ belongs to $\mathcal{P}_q(K)$, and we employ similar definitions for $\mathcal{P}_q(\mathcal{T})$, $\mathcal{P}_q(\mathcal{T})$, $RT_q(\mathcal{T})$ and $N_q(\mathcal{T})$. Notice that these spaces do not have build-in compatibility conditions.

For a face $F \in \mathcal{F}_h$ and $q \ge 0$, $\mathcal{P}_q(F)$ denotes the set of polynomials of degree at most $q$ defined on $F$, and if $v_q \in \mathcal{P}_q(F)$ and $\nabla_F v$ is its tangential gradient along $F$.

2.8. **Data-oscillation.** For $v \in L^2(\Omega)$ and $q \ge 1$, we employ the notation

$$ (2.1) \qquad \Pi_{hq} v := \arg \min_{v_h \in \mathcal{P}_{q-1}(\mathcal{T}_h)} \|v - \Pi_{hq} v\|_{\Omega}. $$
This projection is local, and we have

$$ (2.2) \qquad \|v - \Pi_{hq}v\|_{K} \le \frac{h_K}{\pi} \|\nabla v\|_{K} $$

as well as

$$ (2.3) \qquad \|v - \Pi_{hq}v\|_{K} \le C(\kappa_K) \frac{h_K}{q} \|\nabla v\|_{K} $$

for all $K \in \mathcal{T}_h$. These estimates are respectively established in [10] and [9].

2.9. **Cohomology and cuts.** There exists an integer $N \ge 0$ and, for $1 \le j \le N$, $N$ oriented surfaces $\Sigma^j$ with unit normal vector $n^j$ such that for all $\xi \in \mathbf{H}_{\Gamma_N}(\mathrm{div}^0, \Omega)$ there exist $\theta \in \mathbf{H}_{\Gamma_N}(\mathrm{curl}, \Omega)$ such that $\nabla \times \theta = \xi$ if and only if the periods

$$ \int_{\Sigma^j} \xi \cdot n^j $$

vanish for all $j \in \{1, \dots, N\}$, see e.g. [33] and the discussion in [22, Section 2.3].

In fact, we can assume without loss of generality that these surfaces are exactly covered by mesh faces. It is then easy to see that for $1 \le j \le N$ there exists a lower-order Raviart–Thomas function $\sigma_h^j \in \mathbf{RT}_1(\mathcal{T}_h) \cap \mathbf{H}_{\Gamma_N}(\mathrm{div}^0, \Omega)$ such that

$$ (2.4) \qquad \int_{\Sigma^k} \sigma_h^j \cdot n^k = \delta_{jk} $$

for $1 \le k \le N$.

These surfaces are only needed in the proofs below, but not in the practical implementation of the numerical schemes or error estimators considered in this work. In other words, the surfaces do not have to be computed.

2.10. **Model problem.** Throughout the manuscript we consider a fixed load term $f : \Omega \to \mathbb{R}$, and we look for $u : \Omega \to \mathbb{R}$ such that

$$ (2.5) \qquad \left\{ \begin{array}{ll} -\nabla \cdot (\underline{\mathbf{A}}\nabla u) & = f \quad \text{in } \Omega, \\ u & = 0 \quad \text{on } \Gamma_D, \\ \underline{\mathbf{A}}\nabla u \cdot \mathbf{n} & = 0 \quad \text{on } \Gamma_N. \end{array} \right. $$

More formally, we assume that $f \in L^2(\Omega)$, and $u$ is the unique element of $H_{\Gamma_D}^1(\Omega)$ such that

$$ (2.6) \qquad (\underline{\mathbf{A}}\nabla u, \nabla v)_{\Omega} = (f, v)_{\Omega} $$

for all $v \in H_{\Gamma_D}^1(\Omega)$.

2.11. **Numerical schemes.** We conclude this section with the numerical schemes that we will analyze.

2.11.1. *Discontinuous Galerkin scheme.* For $v_h \in \mathcal{P}_p(\mathcal{T}_h)$, we define its jump lifting $\mathcal{L}_h(v_h) \in \mathcal{P}_{p-1}(\mathcal{T}_h)$ by requiring that

$$ (\mathcal{L}_h(v_h), w_h)_{\Omega} = \sum_{F \in \mathcal{F}_h^1 \cup \mathcal{F}_h^D} (\llbracket v_h \rrbracket, \{w_h\} \cdot n_F)_F $$

for all $w_h \in \mathcal{P}_{p-1}(\mathcal{T}_h)$. We then introduce the discrete gradient of $v_h \in \mathcal{P}_p(\mathcal{T}_h)$ by setting

$$ \mathcal{G}_h(v_h) := \nabla_h v_h + \mathcal{L}_h(v_h). $$
One easily check (see also [20, Section 3]) that

$$ (2.7) \qquad (\mathfrak{G}_h(v_h), \xi_h)_{\Omega} = (v_h, \nabla \cdot v_h)_{\Omega} $$

for all $\xi_h \in \mathcal{P}_{p-1}(\mathcal{T}_h) \cap \mathcal{H}_{\Gamma_N}(\text{div}, \Omega)$.

Consider a symmetric bilinear stabilization form $s_h : \mathcal{P}_p(\mathcal{T}_h) \times \mathcal{P}_p(\mathcal{T}_h) \to \mathbb{R}$ such that $s_h(v_h, w_h) = 0$ for all $v_h \in \mathcal{P}_p(\mathcal{T}_h) \cap H_{\Gamma_D}^1(\Omega)$ and $w_h \in \mathcal{P}(\mathcal{T}_h)$, and, for all $z_h \in \mathcal{P}(\mathcal{T}_h)$, $s_h(z_h, z_h) > 0$ and $s_h(z_h, z_h) = 0$ implies that $z_h \in H_{\Gamma_D}^1(\Omega)$. Then, the IPDG schemes reads as follows: There exists a unique $u_h \in \mathcal{P}_p(\mathcal{T}_h)$ such that

$$ (2.8) \qquad (\underline{\mathcal{A}}\mathfrak{G}_h(u_h), \mathfrak{G}(v_h))_{\Omega} + s_h(u_h, v_h) = (f, v_h) $$

for all $v_h \in \mathcal{P}_p(\mathcal{T}_h)$.

For later references, we observe that

$$ (2.9) \qquad (\underline{\mathcal{A}}\mathfrak{G}_h(u_h), \nabla v_h)_{\Omega} = (f, v_h)_{\Omega} $$

for all $v_h \in \mathcal{P}_p(\mathcal{T}_h) \cap H_{\Gamma_D}^1(\Omega)$, since the stabilization vanishes and $\mathfrak{G}_h(v_h) = \nabla v_h$ for conforming functions.

**Remark 2.1 (Stability).** In practice, one needs to require that

$$ (2.10) \qquad s(v_h, v_h) \ge C(\kappa_{\mathcal{T}_h}) \sum_{F \in \mathcal{F}_h^1 \cap \mathcal{F}_h^D} \alpha_{\max, \omega_F} \frac{p^2}{h_F} \| [u_h] \|_F^2 $$

to ensure good stability properties of the bilinear form in the left-hand side of (2.8), and optimal convergence rates for the approximation $u_h$ to $u$. However, we do not need such an assumption here from the a posteriori point of view. In fact, our analysis applies to any function $u_h \in \mathcal{P}_p(\mathcal{T}_h)$ such that (2.9) holds true.

**Remark 2.2 (Standard formulation).** The IPDG scheme is typically not formulated (and implemented) as presented in (2.8), but rather, one looks for $u_h \in \mathcal{P}(\mathcal{T}_h)$ such that $a_h(u_h, v_h) = (f, v_h)_{\Omega}$ for all $v_h \in \mathcal{P}_p(\mathcal{T}_h)$ where

$$ a_h(u_h, v_h) := (\underline{\mathcal{A}}\nabla_h u_h, \nabla_h v_h)_{\Omega} \\ - \sum_{F \in \mathcal{F}_h^1 \cup \mathcal{F}_h^D} (\llbracket u_h \rrbracket, \llbracket \underline{\mathcal{A}}\nabla v_h \rrbracket \cdot \mathbf{n}_F)_F - \sum_{F \in \mathcal{F}_h^1 \cup \mathcal{F}_h^D} (\llbracket \underline{\mathcal{A}}\nabla u_h \rrbracket \cdot \mathbf{n}_F, v_h)_F \\ + \beta \sum_{F \in \mathcal{F}_h^1 \cup \mathcal{F}_h^D} \alpha_{\max, \omega_F} \frac{p^2}{h_F} (\llbracket u_h \rrbracket, \llbracket v_h \rrbracket)_F $$

with $\beta > 0$ large enough. When $\underline{\mathcal{A}} \in \mathcal{P}_0(\mathcal{T}_h)$, the definition of the lifting $\mathcal{L}_h$ shows that in fact

$$ a_h(u_h, v_h) = (\underline{\mathcal{A}}\mathfrak{G}_h(u_h), \mathfrak{G}(v_h))_{\Omega} + s_h(u_h, v_h) $$

where

$$ s_h(u_h, v_h) := \beta \sum_{F \in \mathcal{F}_h^1 \cup \mathcal{F}_h^D} \alpha_{\max, \omega_F} \frac{p^2}{h_F} (\llbracket u_h \rrbracket, \llbracket v_h \rrbracket)_F - (\underline{\mathcal{A}}\mathcal{L}_h(u_h), \mathcal{L}_h(v_h))_{\Omega}, $$

satisfies (2.10).
2.11.2. *Mixed finite element method.* The mixed finite element formulation consists in finding $(\sigma_h, p_h) \in \mathbf{R}\mathbf{T}_p(\mathcal{T}_h) \cap \mathbf{H}_{\Gamma_N}(\mathrm{div}, \Omega) \times \mathcal{P}_{p-1}(\mathcal{T}_h) \cap L^2_{\Gamma_N}(\Omega)$ such that

$$ (2.11) \qquad \left\{ \begin{array}{l} (\underline{\mathbf{A}}^{-1}\boldsymbol{\sigma}_h, \boldsymbol{v}_h)_\Omega + (r_h, \nabla \cdot \boldsymbol{v}_h)_\Omega = 0 \\ (\nabla \cdot \boldsymbol{\sigma}_h, q_h)_\Omega = (f, q_h)_\Omega \end{array} \right. $$

for all $\boldsymbol{v}_h \in \mathbf{R}\mathbf{T}_p(\mathcal{T}_h) \cap \mathbf{H}_{\Gamma_N}(\mathrm{div}, \Omega)$ and $q_h \in \mathcal{P}_{p-1}(\mathcal{T}_h) \cap L^2_{\Gamma_N}(\Omega)$. This saddle-point problem is well-posed due to standard inf-sup theory [14].

For later references, we record that the first and second equations in (2.11) respectively imply that

$$ (2.12) \qquad (\underline{\mathbf{A}}^{-1}\boldsymbol{\sigma}_h, \boldsymbol{\xi}_h) = 0 $$

for all $\boldsymbol{\xi}_h \in \mathbf{R}\mathbf{T}_p(\mathcal{T}_h) \cap \mathbf{H}_{\Gamma_N}(\mathrm{div}^0, \Omega)$ and

$$ (2.13) \qquad \nabla \cdot \boldsymbol{\sigma}_h = \Pi_{hp}f. $$

These two equations are the key properties needed in our a posteriori analysis. Crucially, our estimator does not use the Lagrange multiplier $p_h$, and can be solely computed from $\boldsymbol{\sigma}_h$.

2.11.3. *Other examples of covered schemes.* We finally mention that other types of scheme are covered by our abstract framework, but that we won't cover in detail. These include mixed conforming discretizations of the primal-hybrid formulation of (2.5) as described in [49] and non-conforming discretization of (2.6) by Crouzeix-Raviart elements [29, 41].

### 3. GENERALIZED PRAGER-SYNGE IDENTITIES

As motivated in the introduction, our core contribution is to derived a generalized Prager-Synge identity. The “standard” Prager-Synge identity is established in (3.1) below for completeness. It was first seminally established in [48] for conforming primal finite element discretizations, and later extended to non-conforming discretizations, as in [35]. Our goal here is to provide an alternative version, whereby the second term in the right-hand (which deals with the non-conforming nature of the discretization) is reformulated in a way that leads to new error estimators.

3.1. **The standard Prager-Synge identity.** For completeness, we start by recapping the standard Prager-Synge identity. Our proof largely follows [35] but is marginally modified, since the statement we give below is slightly more general than the one given in [35].

**Theorem 3.1 (Prager-Synge identities).** For all $\mathfrak{G} \in L^2(\Omega)$, we have

$$ (3.1) \qquad \| \nabla u - \mathfrak{G} \|_{\underline{\mathbf{A}}, \Omega}^2 = \min_{\substack{\boldsymbol{\sigma} \in \mathbf{H}_{\Gamma_N}(\mathrm{div}, \Omega) \\ \nabla \cdot \boldsymbol{\sigma} = f}} \| \mathfrak{G} + \underline{\mathbf{A}}^{-1} \boldsymbol{\sigma} \|_{\underline{\mathbf{A}}, \Omega}^2 + \min_{s \in \mathbf{H}_{\Gamma_D}^1(\Omega)} \| \mathfrak{G} - \nabla s \|_{\underline{\mathbf{A}}, \Omega}^2. $$

*Proof.* We first introduce $s^*$ as the unique element of $\mathbf{H}_{\Gamma_D}^1(\Omega)$ such that

$$ (3.2) \qquad (\underline{\mathbf{A}}\nabla s^*, \nabla v)_\Omega = (\underline{\mathbf{A}}\mathfrak{G}, \nabla v)_\Omega $$

so that

$$ \begin{aligned} \| \nabla u - \mathfrak{G} \|_{\underline{\mathbf{A}}, \Omega}^2 &= \| \nabla u - \nabla s^* \|_{\underline{\mathbf{A}}, \Omega}^2 + \| \mathfrak{G} - \nabla s^* \|_{\underline{\mathbf{A}}, \Omega}^2 \\ &= \| \nabla (u - s^*) \|_{\underline{\mathbf{A}}, \Omega}^2 + \min_{s \in \mathbf{H}_{\Gamma_D}^1(\Omega)} \| \mathfrak{G} - \nabla s \|_{\underline{\mathbf{A}}, \Omega}^2. \end{aligned} $$
For all $\boldsymbol{\sigma} \in \mathbf{H}_{\Gamma_N}(\Omega)$ and $v \in H_{\Gamma_D}^1(\Omega)$, we have

$$ (\underline{\mathbf{A}}\nabla u, \nabla v)_{\Omega} = (f, v)_{\Omega} = (\nabla \cdot \boldsymbol{\sigma}, v)_{\Omega} = -(\boldsymbol{\sigma}, \nabla v)_{\Omega} $$

so that

$$ (\underline{\mathbf{A}}\nabla(u - s^*), \nabla v)_{\Omega} = -(\boldsymbol{\sigma} + \underline{\mathbf{A}}\mathfrak{G}, \nabla v)_{\Omega} $$

and therefore

$$ \|\nabla(u - s^*)\|_{\underline{\mathbf{A}}, \Omega} \le \|\underline{\mathbf{A}}^{-1}\boldsymbol{\sigma} + \mathfrak{G}\|_{\underline{\mathbf{A}}, \Omega}, $$

by picking $v = u - s^*$. It follows that

$$ (3.3) \qquad \|\nabla(u - s^*)\|_{\underline{\mathbf{A}}, \Omega} \le \min_{\substack{\boldsymbol{\sigma} \in \mathbf{H}_{\Gamma_N}(\mathrm{div}, \Omega) \\ \nabla \cdot \boldsymbol{\sigma} = f}} \|\underline{\mathbf{A}}^{-1}\boldsymbol{\sigma} + \mathfrak{G}\|_{\underline{\mathbf{A}}, \Omega} $$

since $\boldsymbol{\sigma}$ was arbitrary. This proves the upper bound in (3.1). For the converse bound, we need to analyze the minimizer $\boldsymbol{\sigma}^*$ in the right-hand side of (3.3). The Euler-Lagrange equations associated with the minimization problem are to find $\boldsymbol{\sigma}^* \in \mathbf{H}_{\Gamma_N}(\mathrm{div}, \Omega)$ and $q \in L_{\Gamma_N}^2(\Omega)$ such that

$$ \left\{ \begin{array}{l} (\underline{\mathbf{A}}^{-1}\boldsymbol{\sigma}^*, v)_{\Omega} + (q, \nabla \cdot v)_{\Omega} = -(\mathfrak{G}, v)_{\Omega} \\ (\nabla \cdot \boldsymbol{\sigma}^*, r)_{\Omega} = (f, r)_{\Omega} \end{array} \right. $$

for all $v \in \mathbf{H}_{\Gamma_N}(\mathrm{div}, \Omega)$ and $r \in L_{\Gamma_N}^2(\Omega)$. We read from the first equation that $q \in H_{\Gamma_D}^1(\Omega)$ with

$$ \nabla q = \underline{\mathbf{A}}^{-1}\boldsymbol{\sigma}^* + \mathfrak{G}, $$

and it remains to show that

$$ \|\nabla q\|_{\underline{\mathbf{A}}, \Omega} \le \|\nabla(u - s^*)\|_{\underline{\mathbf{A}}, \Omega}. $$

We have

$$ \|\nabla q\|_{\underline{\mathbf{A}}, \Omega}^2 = (\underline{\mathbf{A}}\nabla q, \nabla q) = (\boldsymbol{\sigma}^*, \nabla q) + (\underline{\mathbf{A}}\mathfrak{G}, \nabla q). $$

On the one hand,

$$ (\boldsymbol{\sigma}^*, \nabla q) = -(\nabla \cdot \boldsymbol{\sigma}^*, q)_{\Omega} = -(f, q)_{\Omega} = - (\underline{\mathbf{A}}\nabla u, \nabla q)_{\Omega} $$

whereas on the other hand

$$ (\underline{\mathbf{A}}\mathfrak{G}, \nabla q) = (\underline{\mathbf{A}}\nabla s^*, \nabla q), $$

leading to

$$ \|\nabla q\|_{\underline{\mathbf{A}}, \Omega}^2 = (\underline{\mathbf{A}}\nabla(s^* - u), \nabla q) \le \|\nabla(s^* - u)\|_{\underline{\mathbf{A}}, \Omega} \|\nabla q\|_{\underline{\mathbf{A}}, \Omega}, $$

which concludes the proof. $\square$

**3.2. Generalization.** We now aim at rewriting the second term in the right-hand side of (3.1) in alternative ways. Our first result applies without any assumption on the domain, nor on the field $\mathfrak{G}$.

**Lemma 3.2 (Divergence-free maximimization).** For all $\mathfrak{G} \in L^2(\Omega)$, we have

$$ (3.4) \qquad \min_{s \in H_{\Gamma_D}^1(\Omega)} \|\mathfrak{G} - \nabla s\|_{\underline{\mathbf{A}}, \Omega} = \max_{\substack{\boldsymbol{\xi} \in \mathbf{H}_{\Gamma_N}(\mathrm{div}^0 \Omega) \\ \|\boldsymbol{\xi}\|_{\underline{\mathbf{A}}^{-1}, \Omega} = 1}} (\mathfrak{G}, \boldsymbol{\xi})_{\Omega}. $$
Proof. For the sake of simplicity, we introduce

$$s^* := \arg \min_{s \in H_{\Gamma_D}^1(\Omega)} \|\mathfrak{G} - \nabla s\|_{\mathbf{A}, \Omega}.$$

Classically, $s^*$ is the only element of $H_{\Gamma_D}^1(\Omega)$ such that

$$(3.5) \qquad (\underline{\mathbf{A}}\nabla s^*, \nabla v)_\Omega = (\underline{\mathbf{A}}\mathfrak{G}, \nabla v)_\Omega$$

for all $v \in H_{\Gamma_D}^1(\Omega)$. Let us set $\boldsymbol{\sigma}^* = \underline{\mathbf{A}}(\nabla s^* - \mathfrak{G})$. Then, it follows from (3.5) that $\boldsymbol{\sigma}^* \in \mathbf{H}_{\Gamma_N}(\mathrm{div}^0, \Omega)$.

On the one hand, by definition of $\boldsymbol{\sigma}^*$, we have

$$\|\boldsymbol{\sigma}^*\|_{\underline{\mathbf{A}}^{-1}, \Omega}^2 = (\underline{\mathbf{A}}^{-1}\boldsymbol{\sigma}^*, \boldsymbol{\sigma}^*)_\Omega = (\nabla s^* - \mathfrak{G}, \underline{\mathbf{A}}(\nabla s^* - \mathfrak{G}))_\Omega = \|\nabla s^* - \mathfrak{G}\|_{\underline{\mathbf{A}}, \Omega}^2.$$

On the other hand, we have

$$\|\boldsymbol{\sigma}^*\|_{\underline{\mathbf{A}}^{-1}, \Omega} = \sup_{\substack{\boldsymbol{\xi} \in \mathbf{H}(\mathrm{div}^0, \Omega) \\ \|\boldsymbol{\xi}\|_{\underline{\mathbf{A}}^{-1}, \Omega} = 1}} (\underline{\mathbf{A}}^{-1}\boldsymbol{\sigma}^*, \boldsymbol{\xi})_\Omega = \sup_{\substack{\boldsymbol{\xi} \in \mathbf{H}(\mathrm{div}^0, \Omega) \\ \|\boldsymbol{\xi}\|_{\underline{\mathbf{A}}^{-1}, \Omega} = 1}} (\nabla s^* - \mathfrak{G}, \boldsymbol{\xi})_\Omega = \sup_{\substack{\boldsymbol{\xi} \in \mathbf{H}(\mathrm{div}^0, \Omega) \\ \|\boldsymbol{\xi}\|_{\underline{\mathbf{A}}^{-1}, \Omega} = 1}} (\mathfrak{G}, -\boldsymbol{\xi})_\Omega,$$

since $(\nabla s^*, \boldsymbol{\xi})_\Omega = (s^*, \nabla \cdot \boldsymbol{\xi})_\Omega = 0$. This shows (3.4) with a “sup” instead of a “max”. To show that the supremum is indeed achieved, we simply observe that the maximizer is $\boldsymbol{\xi} = \boldsymbol{\sigma}^*/\|\boldsymbol{\sigma}^*\|_{\underline{\mathbf{A}}^{-1}, \Omega}$. $\square$

The formulation in (3.4) above is still not satisfactory, since working with divergence-free functions is not convenient. We would rather work with the curl of vector potentials. However, these two notions are only equivalent is the topology of the domain is trivial. Here, we show that we can still resort to vector potentials for general domains if we assume some orthogonality conditions for the field $\mathfrak{G}$. Similar ideas have been used in [12, Lemma 2].

**Theorem 3.3 (Alternative reformulations).** Assume that $\mathfrak{G} \in L^2(\Omega)$ satisfies $(\mathfrak{G}, \boldsymbol{\xi}_h)_\Omega = 0$ for all $\boldsymbol{\xi}_h \in \mathbf{R}\mathbf{T}_1(\mathcal{T}_h) \cap \mathbf{H}_{\Gamma_N}(\mathrm{div}^0, \Omega)$. Then,

$$(3.6) \quad \min_{s \in H_{\Gamma_D}^1(\Omega)} \|\mathfrak{G} - \nabla s\|_{\underline{\mathbf{A}}, \Omega} = \max_{\substack{\boldsymbol{\theta} \in \mathbf{H}_{\Gamma_N}(\mathrm{curl}, \Omega) \\ \|\boldsymbol{\nabla} \times \boldsymbol{\theta}\|_{\underline{\mathbf{A}}^{-1}, \Omega} = 1}} (\mathfrak{G}, \nabla \times \boldsymbol{\theta})_\Omega = \min_{\boldsymbol{\phi} \in \mathbf{H}_{\Gamma_D}(\mathrm{curl}^0, \Omega)} \|\mathfrak{G} - \boldsymbol{\phi}\|_{\underline{\mathbf{A}}, \Omega}.$$

**Remark 3.4 (Trivial topology).** In a trivial topology, (3.6) simply follows from the identities $\nabla H_{\Gamma_D}^1(\Omega) = \mathbf{H}_{\Gamma_D}(\mathrm{curl}^0, \Omega)$ and $\nabla \times \mathbf{H}_{\Gamma_N}(\mathrm{curl}, \Omega) = \mathbf{H}_{\Gamma_N}(\mathrm{div}^0, \Omega)$ combined with (3.4). In this case, (3.6) even holds true without the orthogonality assumption on $\mathfrak{G}$. As we show below, the orthogonality assumption enables to extend the validity of (3.6) to general topologies.

Proof. Consider an arbitrary $\boldsymbol{\xi} \in \mathbf{H}_{\Gamma_N}(\mathrm{div}^0, \Omega)$, and, for $1 \le j \le N$, let

$$\tau^j := \int_{\Sigma_j} \boldsymbol{\xi} \cdot \mathbf{n}^j$$

denote the period of $\boldsymbol{\xi}$. We then set

$$\tilde{\boldsymbol{\xi}} = \boldsymbol{\xi} - \sum_{j=1}^{N} \tau^j \boldsymbol{\sigma}_h^j \in \mathbf{H}_{\Gamma_N}(\mathrm{div}^0, \Omega),$$

with the $\boldsymbol{\sigma}_h^j \in \mathbf{R}\mathbf{T}_0(\mathcal{T}_h) \cap \mathbf{H}_{\Gamma_N}(\mathrm{div}^0, \Omega)$ from (2.4). By definition, all the periods of $\tilde{\boldsymbol{\xi}}$ vanish, so that there exists $\boldsymbol{\theta} \in \mathbf{H}_{\Gamma_N}(\mathrm{curl}, \Omega)$ such that $\tilde{\boldsymbol{\xi}} = \nabla \times \boldsymbol{\theta}$.
Recalling our assumption on $\mathfrak{G}$, we have

$$ (\mathfrak{G}, \xi)_{\Omega} = (\mathfrak{G}, \tilde{\xi})_{\Omega} = (\mathfrak{G}, \nabla \times \theta)_{\Omega}, $$

and since $\xi$ was arbitrary, it follows that

$$ \begin{array}{l} \max_{\substack{\xi \in H_{\Gamma_N}(\mathrm{div}^0 \Omega) \\ \|\sigma\|_{\underline{\Lambda}^{-1}, \Omega}=1}} (\mathfrak{G}, \xi)_{\Omega} \le \max_{\substack{\theta \in H_{\Gamma_N}(\mathrm{curl}, \Omega) \\ \|\nabla \times \theta\|_{\underline{\Lambda}^{-1}, \Omega}=1}} (\mathfrak{G}, \nabla \times \theta)_{\Omega}. \\ \max_{\substack{\theta \in H_{\Gamma_N}(\mathrm{curl}, \Omega) \\ \|\nabla \times \theta\|_{\underline{\Lambda}^{-1}, \Omega}=1}} (\mathfrak{G}, \nabla \times \theta^*)_{\Omega} \le \max_{\substack{\theta \in H_{\Gamma_N}(\mathrm{curl}, \Omega) \\ \|\nabla \times \theta\|_{\underline{\Lambda}^{-1}, \Omega}=1}} (\mathfrak{G}, \nabla \times \theta^*)_{\Omega} = (\mathfrak{G} - \phi, \nabla \times \theta)_{\Omega}. \end{array} $$

The converse inequality is trivial since $\nabla \times H_{\Gamma_N}(\mathrm{curl}, \Omega) \subset H_{\Gamma_N}(\mathrm{div}^0, \Omega)$, and therefore, the first identity in (3.6) is established.

The second identity follows from a usual duality principle. Namely, we write that the any maximizer $\theta^* \in H_{\Gamma_N}(\mathrm{curl}, \Omega)$ satisfies

$$ (\underline{\Lambda}^{-1} \nabla \times \theta^*, \nabla \times v)_{\Omega} = (\underline{\Lambda}^{-1}(\underline{\Lambda} \mathfrak{G}), \nabla \times v)_{\Omega} = (\mathfrak{G} - \phi, \nabla \times v)_{\Omega}. $$

for all $\phi \in H_{\Gamma_D}(\mathrm{curl}^0, \Omega)$ and $v \in H_{\Gamma_N}(\mathrm{curl}, \Omega)$. It follows in particular that

$$ \max_{\substack{\theta \in H_{\Gamma_N}(\mathrm{curl}, \Omega) \\ \|\nabla \times \theta\|_{\underline{\Lambda}^{-1}, \Omega}=1}} (\mathfrak{G}, \nabla \times \theta)_{\Omega} = \|\nabla \times \theta^*\|_{\underline{\Lambda}^{-1}, \Omega} \le \|\underline{\Lambda}(\mathfrak{G} - \phi)\|_{\underline{\Lambda}^{-1}, \Omega} = \|\mathfrak{G} - \phi\|_{\underline{\Lambda}, \Omega} $$

for all $\phi \in H_{\Gamma_D}(\mathrm{curl}^0, \Omega)$. Hence,

$$ \max_{\substack{\theta \in H_{\Gamma_N}(\mathrm{curl}, \Omega) \\ \|\nabla \times \theta\|_{\underline{\Lambda}^{-1}, \Omega}=1}} (\mathfrak{G}, \nabla \times \theta)_{\Omega} \le \min_{\substack{\phi \in H_{\Gamma_D}(\mathrm{curl}^0, \Omega) \\ \|\nabla \times \theta\|_{\underline{\Lambda}^{-1}, \Omega}=1}} \|\mathfrak{G} - \phi\|_{\underline{\Lambda}, \Omega}. $$

The converse inequality is trivial since $\nabla H_{\Gamma_D}^1(\Omega) \subset H_{\Gamma_D}(\mathrm{curl}^0, \Omega)$. This proves the second identity. $\square$

As a direct consequence of the above reformulations, we can recast the Prager-Synge identity in (3.1) into two alternative formulations that are respectively suited for residual-based and equilibrated error estimators.

**Corollary 3.5 (Generalized Prager-Synge identity).** For all $\mathfrak{G} \in L^2(\Omega)$ such that $(\mathfrak{G}, \xi_h)_{\Omega} = 0$ for all $\xi_h \in \mathcal{R}T_1(\mathcal{T}_h) \cap H_{\Gamma_N}(\mathrm{div}^0, \Omega)$, we have

$$ (3.7a) \quad \|\nabla u - \mathfrak{G}\|_{\underline{\Lambda}, \Omega}^2 = \max_{\substack{v \in H_{\Gamma_D}^1(\Omega) \\ \|\nabla v\|_{\underline{\Lambda}, \Omega}=1}} |(f, v)_{\Omega} - (\underline{\Lambda}\mathfrak{G}, \nabla v)_{\Omega}|^2 + \max_{\substack{\theta \in H_{\Gamma_N}(\mathrm{curl}, \Omega) \\ \|\nabla \times \theta\|_{\underline{\Lambda}^{-1}, \Omega}=1}} |(\mathfrak{G}, \nabla \times \theta)_{\Omega}|^2 $$

and

$$ (3.7b) \quad \|\nabla u - \mathfrak{G}\|_{\underline{\Lambda}, \Omega}^2 = \min_{\substack{\sigma \in H_{\Gamma_N}(\mathrm{div}, \Omega) \\ \nabla \cdot \sigma = f}} \|\mathfrak{G} + \underline{\Lambda}^{-1}\sigma\|_{\underline{\Lambda}, \Omega}^2 + \min_{\substack{\phi \in H_{\Gamma_D}(\mathrm{curl}^0, \Omega) \\ \|\nabla \times \phi\|_{\underline{\Lambda}, \Omega}=1}} \|\mathfrak{G} - \phi\|_{\underline{\Lambda}, \Omega}^2. $$

# 4. RESIDUAL-BASED ESTIMATORS

In this section, we first leverage the generalized Prager-Synge identity established in (3.7a) to design residual-based error estimators.

## 4.1. Tools.
We first review a set of standard tools from the literature that are instrumental for the analysis of residual-based estimators.
4.1.1. Face patches. The following notation for patches of elements will be useful. If $F = \partial K_{-} \cap \partial K_{+} \in \mathcal{F}_h^1$, then $\omega_F$ is the open set such that $\overline{\omega_F} = \overline{K_{-}} \cup \overline{K_{+}}$, and similarly, we let $\omega_F := K$ for an exterior face $F \in \mathcal{F}_h$ with $F \subset \partial K \cap \partial \Omega$. For $K \in \mathcal{T}_h$, we then employ the notation $\hat{\omega}_K := \cup_{F \in \mathcal{F}(K)} \omega_F$.

4.1.2. Bubble functions. We start with the “bubble functions” used in the efficiency proofs. This is a standard technique [5, 52], and here, we use the polynomial-degree explicit bounds from [47]. The result in [47] are explicitly established for two-dimensional problems, but the results extend without change for three-dimensional configurations, see e.g. [32, Theorem 4.1.2].

For all $K \in \mathcal{T}_h$ there exists a “bubble” function $b_K : K \to \mathbb{R}$ such that for all $q \ge 1$ and $v_q \in \mathcal{P}_q(K)$, we have

$$ (4.1a) \qquad \|v_q\|_K \le C(\kappa_K)q \|b_K^{1/2}v_q\|_K $$

and

$$ (4.1b) \qquad \|\nabla(b_K v_q)\|_K \le C(\kappa_K) \frac{q}{h_K} \|b_K^{1/2} v_q\|_K, $$

see [47, Theorem 2.5]. Similarly, for all $F \in \mathcal{F}_h$, there exists a function $b_F : F \to \mathbb{R}$ such that for all $q \ge 1$ and $w_q \in \mathcal{P}_q(F)$

$$ (4.2) \qquad \|w_q\|_F \le C(\kappa_{\omega_F})q \|b_F^{1/2}w_q\|_F. $$

In addition, there exists an extension operator $\mathcal{E}_F : \mathcal{P}_q(F) \to H^1(\omega_F)$ such that $\mathcal{E}_F(w_q) = b_F w_q$ on $F$, $\mathcal{E}_F(w_q) = 0$ on $\partial \omega_F \setminus F$, and

$$ (4.3) \qquad qh_F^{-1/2} \|\mathcal{E}_F(w_q)\|_{\omega_F} + q^{-1}h_F^{1/2} \|\nabla \mathcal{E}_F(w_q)\|_{\omega_F} \le C(\kappa_{\omega_F}) \|b_F^{1/2} w_q\|_F $$

for all $w_q \in \mathcal{P}_q(F)$. These results follow from applying and [47, Lemma 2.6] with $\alpha = 1$ and $\varepsilon = q^{-2}$.

For vector-valued functions, we can apply the extension operator and the above inequalities component-wise. As a result, for all $v_q \in \mathcal{P}_q(K)$, we have

$$ (4.4a) \qquad \|v_q\|_K \le C(\kappa_K)q \|b_K^{1/2}v_q\|_K $$

and

$$ (4.4b) \qquad \|\nabla \times (b_K v_q)\|_K \le C(\kappa_K) \frac{q}{h_K} \|b_K^{1/2} v_q\|_K, $$

and similarly, for all $w_q \in \mathcal{P}_q(F)$,

$$ (4.5) \qquad \|w_q\|_F \le C(\kappa_{\omega_F})q \|b_F^{1/2}w_q\|_F $$

and $\mathcal{E}_F(w_q) \times n_F = b_F w_q \times n_F$ on $F$, $\mathcal{E}_F(w_q) \times n_{\omega_F} = o$ on $\partial \omega_F \setminus F$, with

$$ (4.6) \qquad qh_F^{-1/2} \|\mathcal{E}_F(w_q)\|_{\omega_F} + q^{-1}h_F^{1/2} \|\nabla \times \mathcal{E}_F(w_q)\|_{\omega_F} \le C(\kappa_{\omega_F}) \|b_F^{1/2} w_q\|_F $$

for all $w_q \in \mathcal{P}_q(F)$.
4.1.3. Quasi-interpolation operator. Next, we will need suitable interpolation operators for the reliability estimate. Following [44, Theorem 3.3], for all $q \ge 1$, there exists a quasi-interpolation operator $\mathcal{I}_{hq} : H_{\Gamma_D}^1(\Omega) \to \mathcal{P}_q(\mathcal{T}_h) \cap H_{\Gamma_D}^1(\Omega)$ such that

$$ (4.7) \qquad \frac{q}{h_K} \|v - \mathcal{I}_{hq} v\|_K + \sqrt{\frac{q}{h_K}} \|v - \mathcal{I}_{hq} v\|_{\partial K} \le C(\kappa_{\omega_K}) (\ell_{\Omega}^{-1} \|v\|_{\omega_K} + \|\nabla v\|_{\omega_K}) $$

for all $v \in H_{\Gamma_D}^1(\Omega)$, where $\ell_{\Omega}$ denotes the diameter of $\Omega$. Similarly, for vector-valued functions, we can construct $\mathcal{J}_{hq} : H_{\Gamma_N}^1(\Omega) \to \mathcal{N}_q(\mathcal{T}_h) \cap H_{\Gamma_N}(\text{curl}, \Omega)$ such that

$$ (4.8) \qquad \frac{q}{h_K} \|\psi - \mathcal{J}_{hq} \psi\|_K + \sqrt{\frac{q}{h_K}} \|(\psi - \mathcal{J}_{hq} \psi) \times \mathbf{n}_K\|_{\partial K} \le C(\kappa_{\omega_K}) (\ell_{\Omega}^{-1} \|\psi\|_{\omega_K} + \|\underline{\nabla} \psi\|_{\omega_K}) $$

for all $\psi \in H_{\Gamma_N}^1(\Omega)$. Indeed, if $q \ge 2$, we can define $\mathcal{J}_{hq}$ by using $\mathcal{I}_{h(q-1)}$ component-wise (with boundary conditions imposed on $\Gamma_N$ rather than on $\Gamma_D$), and if $q = 1$, a suitable operator is designed in [39].

4.1.4. Regular decomposition. In order to employ the quasi-interpolation operator $\mathcal{J}_{hq}$ introduced above, it will be crucial to replace a general function $\theta \in H_{\Gamma_N}(\text{curl}, \Omega)$ by a function $\psi \in H_{\Gamma_N}^1(\Omega)$ with the same curl. Specifically, we will employ [50, Theorem 1], stating that there exists an operator $S_h : H_{\Gamma_N}(\text{curl}, \Omega) \to \mathcal{N}_1(\mathcal{T}_h) \cap H_{\Gamma_N}(\text{curl}, \Omega)$ such that for all $\theta \in H_{\Gamma_N}(\text{curl}, \Omega)$,

$$ (4.9a) \qquad \theta - S_h \theta = \nabla q + \psi $$

with $q \in H_{\Gamma_N}^1(\Omega)$, $\psi \in H_{\Gamma_N}^1(\Omega)$ and

$$ (4.9b) \qquad h_K^{-1} \|\psi\|_K + \|\underline{\nabla} \psi\|_K \le C(\kappa_{\omega_K}) \|\nabla \times \theta\|_{\omega_K}. $$

4.1.5. Inverse inequalities. When dealing with the IPDG scheme, we will need the following results to show that our new error estimator is smaller than the standard one. For $K \in \mathcal{T}_h$ and $F \in \mathcal{F}_h$, the estimates

$$ (4.10a) \qquad \|\nabla v_q\|_K \le C(\kappa_K) \frac{q^2}{h_K} \|v_q\|_K \quad \forall v_q \in \mathcal{P}_q(K) $$

and

$$ (4.10b) \qquad \|\nabla_F w_q\|_F \le C(\kappa_{\omega_F}) \frac{q^2}{h_F} \|w_q\|_F \quad \forall w_q \in \mathcal{P}_q(F) $$

are established for all $q \ge 0$ in [51, Theorem 4.76]. By combining (4.10a) with a standard multiplicative trace inequality, for all $K \in \mathcal{T}_h$ and $F \subset \mathcal{F}(K)$, we also have

$$ (4.10c) \qquad \|v_q\|_F^2 \le C(\kappa_K) \frac{q^2}{h_K} \|v_q\|_K^2 \quad \forall v_q \in \mathcal{P}_q(K) $$

for all $q \ge 1$, see again [51, Theorem 4.76].
4.2. **The new error estimator.** The first term in the right-hand side of the Prager-Synge identity in (3.1) is usually controlled in the literature by the construction of an equilibrated flux, or with residual terms of the form

$$ \frac{h_K}{p} \| \nabla \cdot (\underline{\mathcal{A}}\mathfrak{G}) + f \|_K + \sqrt{\frac{h_K}{p}} \| [[\underline{\mathcal{A}}\mathfrak{G}]] \cdot \mathbf{n}_K \|_{\partial K \setminus \Gamma_D} $$

in each element $K \in \mathcal{T}_h$, where $p$ is the polynomial degree associated with the numerical method. This term is fine, and leads to satisfactory efficiency estimate.

Here, we rather focus on the second term in the right-hand side of (3.1). In the literature, this term is either estimated by constructing a potential [2, 3, 35, 45], or by exploiting the fact that $\mathfrak{G}$ is a broken gradient of scalar potential and measuring the jump of the potential across mesh faces [15, 40, 42]. The key issue is that neither approach leads to efficiency estimates that solely involve $\nabla u - \mathfrak{G}$. Here, we instead employ a residual term of the form residual terms of the form

$$ \frac{h_K}{p} \| \nabla \times \mathfrak{G} \|_K + \sqrt{\frac{h_K}{p}} \| [[\mathfrak{G}]] \times \mathbf{n}_K \|_{\partial K \setminus \Gamma_N} $$

that is still reliable, but leads to sharper lower bounds.

4.3. **Abstract reliability.** We start with a reliability result under abstract assumptions on $\mathfrak{G}$. We first focus on the second term in the Prager-Synge identity measuring the non-conformity of the discrete gradient, which is the new term in our estimator.

**Lemma 4.1** (Abstract reliability). Let $\mathfrak{G} \in L^2(\Omega)$ and $q \ge 1$. If $(\mathfrak{G}, \xi_h)_{\Omega} = 0$ for all $\xi_h \in RT_q(\mathcal{T}_h) \cap H_{\Gamma_N}(\mathrm{div}^0, \Omega)$, then

$$ (4.11) \quad \max_{\substack{\theta \in H_{\Gamma_N}(\mathrm{curl}, \Omega) \\ \|\nabla \times \theta\|_{\underline{\mathcal{A}}^{-1}, \Omega} = 1}} |(\mathfrak{G}, \nabla \times \theta)_{\Omega}|^2 \le C(\kappa_{\mathcal{T}_h})^2 \sum_{K \in \mathcal{T}_h} \alpha_{\max, \tilde{\omega}_K} \left\{ \left(\frac{h_K}{q}\right)^2 \|\nabla \times \mathfrak{G}\|_K^2 + \frac{h_K}{q} \|[[\mathfrak{G}]] \times \mathbf{n}_F\|_{\partial K \setminus \Gamma_N}^2 \right\} $$

*Proof.* Consider an arbitrary $\theta \in H_{\Gamma_N}(\mathrm{curl}, \Omega)$. Recalling the discussion in Section 4.1.4 and our assumption on $\mathfrak{G}$, since $\nabla \times (S_h \theta) \in RT_1(\mathcal{T}_h) \cap H_{\Gamma_N}(\mathrm{div}^0, \Omega)$, we have

$$ (\mathfrak{G}, \nabla \times \theta)_{\Omega} = (\mathfrak{G}, \nabla \times (\theta - S_h \theta))_{\Omega} = (\mathfrak{G}, \nabla \times \psi)_{\Omega} $$

for $\psi \in H_{\Gamma_N}^1(\Omega)$ satisfying (4.9b). Similarly, since $\mathcal{J}_{hq}\psi \in N_q(\mathcal{T}_h) \cap H_{\Gamma_N}(\mathrm{curl}, \Omega)$, we have $\nabla \times (\mathcal{J}_{hq}\psi) \in RT_q(\mathcal{T}_h) \cap H_{\Gamma_N}(\mathrm{div}^0, \Omega)$, and due to our assumption of $\mathfrak{G}$, we can further write that

$$ (\mathfrak{G}, \nabla \times \theta)_{\Omega} = (\mathfrak{G}, \nabla \times \tilde{\psi})_{\Omega} $$

with $\tilde{\psi} = \psi - \mathcal{J}_{hq}\psi$.
We perform standard integration by parts to show that

$$
\begin{align*}
(\mathfrak{G}, \nabla \times \boldsymbol{\theta})_{\Omega} &= \sum_{K \in \mathcal{T}_h} \left\{ (\nabla \times \mathfrak{G}, \tilde{\psi})_K + (\mathfrak{G} \times \mathbf{n}_K, \tilde{\psi})_{\partial K} \right\} \\
&= \sum_{K \in \mathcal{T}_h} (\nabla \times \mathfrak{G}, \tilde{\psi})_K + \sum_{F \in \mathcal{F}_h} (\llbracket \mathfrak{G} \rrbracket \times \mathbf{n}_F, \tilde{\psi})_{\partial K}, \\
&\le \sum_{K \in \mathcal{T}_h} \left\{ \|\nabla \times \mathfrak{G}\|_K \|\tilde{\psi}\|_K + \|\llbracket \mathfrak{G} \rrbracket \times \mathbf{n}\|_{\partial K \setminus \Gamma_N} \|\tilde{\psi} \times \mathbf{n}\|_{\partial K} \right\}
\end{align*}
$$

and continue with Cauchy-Schwarz inequalitie as follows

$$
\begin{align*}
(\mathfrak{G}, \nabla \times \boldsymbol{\theta})_{\Omega} &\le \sum_{K \in \mathcal{T}_h} \alpha_{\max, \tilde{\omega}_K}^{1/2} \left\{ \frac{h_K}{q} \|\nabla \times \mathfrak{G}\|_K + \sqrt{\frac{h_K}{q}} \|\llbracket \mathfrak{G} \rrbracket\| \times \mathbf{n}\|_{\partial K \setminus \Gamma_N} \right\}, \\
&\quad \times \alpha_{\max, \tilde{\omega}_K}^{-1/2} \left\{ \frac{q}{h_K} \|\tilde{\psi}\|_K + \sqrt{\frac{q}{h_K}} \|\tilde{\psi} \times \mathbf{n}\|_{\partial K} \right\} \\
&\le \sqrt{2} \left( \sum_{K \in \mathcal{T}_h} \alpha_{\max, \tilde{\omega}_K} \left\{ \left(\frac{h_K}{q}\right)^2 \|\nabla \times \mathfrak{G}\|_K^2 + \frac{h_K}{q} \|\llbracket \mathfrak{G} \rrbracket\| \times \mathbf{n}\|_{\partial K \setminus \Gamma_N}^2 \right\} \right)^{1/2} \\
&\quad \times \left( \sum_{K \in \mathcal{T}_h} \alpha_{\max, \tilde{\omega}_K} \left\{ \frac{q}{h_K} \|\tilde{\psi}\|_K + \sqrt{\frac{q}{h_K}} \|\tilde{\psi} \times \mathbf{n}\|_{\partial K} \right\}^2 \right)^{1/2}.
\end{align*}
$$

At that point, it remains to estimate the second sum. To do so, we consider a fixed element $K \in \mathcal{T}_h$. Using (4.8) and (4.9b), we have

$$
\begin{align*}
\frac{q}{h_K} \|\tilde{\psi}\|_K + \sqrt{\frac{q}{h_K}} \|\tilde{\psi} \times \mathbf{n}\|_{\partial K} &\le C(\kappa_{\mathcal{T}_h}) (\ell_{\Omega}^{-1} \|\psi\|_{\omega_K} + \|\nabla \psi\|_{\omega_K}) \\
&\le C(\kappa_{\mathcal{T}_h}) \sum_{\substack{K' \in \mathcal{T}_h \\ K \subset \omega_K}} (h_K^{-1} \|\psi\|_{K'} + \|\nabla \psi\|_{K'}) \\
&\le C(\kappa_{\mathcal{T}_h}) \sum_{\substack{K' \in \mathcal{T}_h \\ K \subset \omega_K}} \|\nabla \times \boldsymbol{\theta}\|_{\tilde{\omega}_K},
\end{align*}
$$

leading to

$$
\begin{align*}
\alpha_{\max, \tilde{\omega}_K}^{-1/2} \left\{ \frac{q}{h_K} \|\tilde{\psi}\|_K + \sqrt{\frac{q}{h_K}} \|\tilde{\psi} \times \mathbf{n}\|_{\partial K} \right\} &\le C(\kappa_{\mathcal{T}_h}) \alpha_{\max, \tilde{\omega}_K}^{-1/2} \sum_{\substack{K' \in \mathcal{T}_h \\ K \subset \omega_K}} \|\nabla \times \boldsymbol{\theta}\|_{\tilde{\omega}_K} \\
&\le C(\kappa_{\mathcal{T}_h}) \|\nabla \times \boldsymbol{\theta}\|_{\underline{\mathbf{A}}^{-1, \tilde{\omega}_K}},
\end{align*}
$$

and, due to a bounded number of overlaps, (4.11) follows by summation. $\square$

Although the two first terms of the estimator are standard, for completeness, we still analyze them here. For conforming discretizations, the treatment of these terms can be bound, e.g., in [5, 47, 52].

**Lemma 4.2** (Abstract reliability for the conforming part of the estimator). Let $\mathfrak{G} \in L^2(\Omega)$ such that $(\underline{\mathbf{A}}\mathfrak{G}, \nabla v_h)_\Omega = (f, v_h)_\Omega$ for all $v_h \in P_q(\mathcal{T}_h) \cap H_{\Gamma_D}^1(\Omega)$ for some $q \ge 1$. Then, we
have

$$
(4.12) \quad \sup_{\substack{v \in H_{\Gamma_D}^1(\Omega) \\ \|\nabla v\|_{\mathcal{A}, \Omega} = 1}} |(f, v)_{\Omega} - (\underline{\mathcal{A}}\mathfrak{G}, \nabla v)_{\Omega}|^2 \le C(\kappa_{\mathcal{T}_h})^2 \\
\sum_{K \in \mathcal{T}_h} \alpha_{\min, \tilde{\omega}_K}^{-1} \left\{ \left( \frac{h_K}{q} \right)^2 \|\nabla \cdot (\underline{\mathcal{A}}\mathfrak{G}) + f\|_K^2 + \frac{h_K}{q} \|\llbracket \underline{\mathcal{A}}\mathfrak{G} \rrbracket \cdot \mathbf{n}_K \|_{\partial K \setminus \Gamma_D}^2 \right\}.
$$

Proof. We fix $v \in H_{\Gamma_D}^1(\Omega)$ and observe that by assumption, we have

$$
(f, v)_{\Omega} - (\underline{\mathcal{A}}\mathfrak{G}, \nabla v)_{\Omega} = (f, \tilde{v})_{\Omega} - (\underline{\mathcal{A}}\mathfrak{G}, \nabla \tilde{v})_{\Omega}
$$

with $\tilde{v} = v - \mathcal{I}_{hq}v$, where $\mathcal{I}_{hq}$ is the quasi-interpolation operator from (4.7). Integration by parts then gives that

$$
\begin{align*}
(f, v)_{\Omega} - (\underline{\mathcal{A}}\mathfrak{G}, \nabla v)_{\Omega} &= \sum_{K \in \mathcal{T}_h} \{ (\nabla \cdot (\underline{\mathcal{A}}\mathfrak{G}) + f, \tilde{v})_K - (\underline{\mathcal{A}}\mathfrak{G} \cdot \mathbf{n}_K, \tilde{v})_{\partial K} \} \\
&= \sum_{K \in \mathcal{T}_h} (\nabla \cdot (\underline{\mathcal{A}}\mathfrak{G}) + f, \tilde{v})_K - \sum_{F \in \mathcal{F}_h} (\llbracket \underline{\mathcal{A}}\mathfrak{G}(u_h) \rrbracket \cdot \mathbf{n}_F, \tilde{v})_F \\
&\le \sum_{K \in \mathcal{T}_h} \{ \|\nabla \cdot (\underline{\mathcal{A}}\mathfrak{G}) + f\|_K \|\tilde{v}\|_K + \|\llbracket \underline{\mathcal{A}}\mathfrak{G} \rrbracket \cdot \mathbf{n}_K \|_{\partial K \setminus \Gamma_D} \|\tilde{v}\|_{\partial K} \},
\end{align*}
$$

and by further employ Cauchy-Schwarz inequalities, we have

$$
\begin{align*}
(f, v)_{\Omega} - (\underline{\mathcal{A}}\mathfrak{G}, \nabla v)_{\Omega} &\le \sum_{K \in \mathcal{T}_h} \alpha_{\min, \omega_K}^{-1/2} \left( \frac{h_K}{q} \|\nabla \cdot (\underline{\mathcal{A}}\mathfrak{G}) + f\|_K + \sqrt{\frac{h_K}{q}} \|\llbracket \underline{\mathcal{A}}\mathfrak{G} \rrbracket \cdot \mathbf{n}_K \|_{\partial K \setminus \Gamma_D} \right) \\
&\qquad \times \alpha_{\min, \omega_K}^{1/2} \left( \frac{q}{h_K} \|\tilde{v}\|_K + \sqrt{\frac{q}{h_K}} \|\tilde{v}\|_{\partial K} \right)^2 \\
&\le \sqrt{\sum_{K \in \mathcal{T}_h} \alpha_{\min, \omega_K}^{-1} \left( \frac{h_K}{q} \|\nabla \cdot (\underline{\mathcal{A}}\mathfrak{G}) + f\|_K + \sqrt{\frac{h_K}{q}} \|\llbracket \underline{\mathcal{A}}\mathfrak{G} \rrbracket \cdot \mathbf{n}_K \|_{\partial K \setminus \Gamma_D} \right)^2} \\
&\qquad \times \sqrt{\sum_{K \in \mathcal{T}_h} \alpha_{\min, \omega_K} \left( \frac{q}{h_K} \|\tilde{v}\|_K + \sqrt{\frac{q}{h_K}} \|\tilde{v}\|_{\partial K} \right)^2},
\end{align*}
$$

and (4.12) follows provided we show that

$$
(4.13) \quad \sum_{K \in \mathcal{T}_h} \alpha_{\min, \omega_K} \left( \frac{q}{h_K} \|\tilde{v}\|_K + \sqrt{\frac{q}{h_K}} \|\tilde{v}\|_{\partial K} \right)^2 \le C(\kappa_{\mathcal{T}_h}) \|\nabla v\|^2_{\underline{\mathcal{A}}, \Omega}.
$$

To that end, we employ (4.7) leading to

$$
\alpha_{\min, \omega_K}^{1/2} \left( \frac{q}{h_K} \|\tilde{v}\|_K + \sqrt{\frac{q}{h_K}} \|\tilde{v}\|_{\partial K} \right) \le C(\kappa_{\omega_K}) \alpha_{\min, \omega_K}^{1/2} \|\nabla v\|_{\omega_K} \le C(\kappa_{\omega_K}) \|\nabla v\|_{\underline{\mathcal{A}}, \omega_K},
$$

and (4.13) follows by summation due to the finite number of overlap between patches. $\square$

<figure><img src="image_2.png" /></figure>
4.4. Abstract efficiency. The following efficiency results are proved using the standard bubble function techniques, as introduced in Section 4.1.2. We first establish a result for the new terms in the error estimator, and then revisit the usual proofs for the standard terms.

**Lemma 4.3 (Abstract efficiency).** Let $q \ge 1$ and assume that $\mathfrak{G} \in \mathcal{P}_q(\mathcal{T}_h)$. Then,

$$ (4.14) \qquad \frac{h_K}{q} \|\nabla \times \mathfrak{G}\|_K \le C(\kappa_K)q \|\nabla u - \mathfrak{G}\|_K $$

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

$$ (4.15) \qquad \left(\frac{h_K}{q}\right)^{1/2} \|[[\mathfrak{G}]\] \times \mathbf{n}_F\|_F \le C(\kappa_{\omega_F}) q^{3/2} \|\nabla u - \mathfrak{G}\|_{\omega_F} $$

for all $F \in \mathcal{F}_h^1 \cup \mathcal{F}_h^D$.

*Proof.* Let $K \in \mathcal{T}_h$. Since $\nabla \times \mathfrak{G} \in \mathcal{P}_{q-1}(\mathcal{T}_h) \subset \mathcal{P}_q(\mathcal{T}_h)$, we can apply (4.4) with $v_q = \nabla \times \mathfrak{G}$. Using the second estimate in (4.4), we have

$$ \begin{aligned} \|b_K^{1/2} \nabla \times \mathfrak{G}\|_K^2 &= (b_K \nabla \times \mathfrak{G}, \nabla \times (\mathfrak{G} - \nabla u))_K = (\nabla \times (b_K \nabla \times \mathfrak{G}), \mathfrak{G} - \nabla u)_K \\ &\le \|\nabla \times (b_K \nabla \times \mathfrak{G})\|_K \|\mathfrak{G} - \nabla u\|_K \le C(\kappa_K) \frac{q}{h_K} \|b_K^{1/2} \nabla \times \mathfrak{G}\|_K^2 \|\mathfrak{G} - \nabla u\|_K \end{aligned} $$

leading to

$$ \frac{h_K^2}{q^2} (b_K \nabla \times \mathfrak{G}, \nabla \times \mathfrak{G})_K \le C(\kappa_K) \|\mathfrak{G} - \nabla u\|_K^2. $$

We then show (4.14) with the first estimate in (4.4) as follows:

$$ \frac{h_K^2}{q^2} \|\nabla \times \mathfrak{G}\|_K^2 \le C(\kappa_K) \frac{h_K^2}{q^2} q^2 (b_K \nabla \times \mathfrak{G}, \nabla \times \mathfrak{G})_K \le C(\kappa_K) q^2 \|\mathfrak{G} - \nabla u\|_K^2. $$

On the other hand, for $F \in \mathcal{F}_h^1 \cap \mathcal{F}_h^D$, we have $[[\mathfrak{G}]\] \times \mathbf{n}_F \in \mathcal{P}_q(F)$, and we can write
$$ \begin{aligned} (b_F[[\mathfrak{G}]\] \times \mathbf{n}_F, [[\mathfrak{G}]\] \times \mathbf{n}_F)_F &= (\mathcal{E}_F([[\mathfrak{G}]\] \times \mathbf{n}_F), [[\mathfrak{G}]\] \times \mathbf{n}_F)_F \\ &= (\nabla \times \mathcal{E}_F([[\mathfrak{G}]\] \times \mathbf{n}_F), \mathfrak{G})_{\omega_F} - (\mathcal{E}_F([[\mathfrak{G}]\] \times \mathbf{n}_F), \nabla_h \times \mathfrak{G})_{\omega_F} \\ &= (\nabla \times \mathcal{E}_F([[\mathfrak{G}]\] \times \mathbf{n}_F), \mathfrak{G} - \nabla u)_{\omega_F} - (\mathcal{E}_F([[\mathfrak{G}]\] \times \mathbf{n}_F), \nabla_h \times \mathfrak{G})_{\omega_F}, \end{aligned} $$

where we are allowed to plug in $\nabla u$ since $u \in H^1(\omega_F)$ with $u = 0$ on $F$ when $F \in \mathcal{F}_h^D$. For the first term, we use (4.6) and write that

$$ \begin{aligned} |(\nabla \times \mathcal{E}_F([[\mathfrak{G}]\] \times \mathbf{n}_F), \mathfrak{G} - \nabla u)_{\omega_F}| &\le \|\nabla \times \mathcal{E}_F([[\mathfrak{G}]\] \times \mathbf{n}_F)\|_{\omega_F} \|\mathfrak{G} - \nabla u\|_{\omega_F} \\ &\le \|\nabla \times \mathcal{E}_F([[\mathfrak{G}]\] \times \mathbf{n}_F)\|_{\omega_F} \|\mathfrak{G} - \nabla u\|_{\omega_F} \\ &\le C(\kappa_{\omega_F}) \frac{q}{h_F^{1/2}} \|b_F^{1/2}[[\mathfrak{G}]\] \times \mathbf{n}_F\|_F \|\mathfrak{G} - \nabla u\|_{\omega_F}. \end{aligned} $$

For the second term, we use (4.14) and (4.6), which gives

$$ \begin{aligned} |(\mathcal{E}_F([[\mathfrak{G}]\] \times \mathbf{n}_F), \nabla_h \times \mathfrak{G})_{\omega_F}| &\le \|\mathcal{E}_F([[\mathfrak{G}]\] \times \mathbf{n}_F)\|_{\omega_F} \|\nabla_h \times \mathfrak{G}\|_{\omega_F} \\ &\le C(\kappa_{\omega_F}) \frac{h_F^{1/2}}{q} \|b_F^{1/2}[[\mathfrak{G}]\] \times \mathbf{n}_F\|_F \frac{q^2}{h_F} \|\mathfrak{G} - \nabla u\|_{\omega_F} \\ &\le C(\kappa_{\omega_F}) \frac{q}{h_F^{1/2}} \|b_F^{1/2}[[\mathfrak{G}]\] \times \mathbf{n}_F\|_F \|\mathfrak{G} - \nabla u\|_{\omega_F}. \end{aligned} $$
Combining the last two bounds gives that

$$ \|b_F^{1/2}[[\mathfrak{G}]] \times \mathbf{n}_F\|_F \le C(\kappa_{\omega_F}) \frac{q}{h_F^{1/2}} \|\mathfrak{G} - \nabla u\|_{\omega_F}, $$

and therefore, with (4.5), we have

$$ \sqrt{\frac{h_F}{q}} \|[[\mathfrak{G}]] \times \mathbf{n}_F\|_F \le C(\kappa_{\omega_F}) \sqrt{qh_F} \|b_F^{1/2}[[\mathfrak{G}]] \times \mathbf{n}_F\|_F \le C(\kappa_{\omega_F}) q^{3/2} \|\mathfrak{G} - \nabla u\|_{\omega_F}, $$

which concludes the proof of (4.15).

□

**Lemma 4.4** (Abstract efficiency of the conforming part of the estimator). Let $q \ge 1$ and assume that $\mathfrak{G} \in \mathcal{P}_q(\mathcal{T}_h)$ and $\underline{\mathbf{A}} \in \underline{\mathcal{P}}_q(\mathcal{T}_h)$. We have

$$ (4.16) \quad \alpha_{\min,K}^{-1/2} \frac{h_K}{q} \|\nabla \cdot (\underline{\mathbf{A}}\mathfrak{G}) + f\|_K \le C(\kappa_K) \qquad q \left( \sqrt{\frac{\alpha_{\max,K}}{\alpha_{\min,K}}} \|\nabla u - \mathfrak{G}\|_{\underline{\mathbf{A}},K} + \alpha_{\min,K}^{-1/2} \frac{h_K}{q} \|f - \Pi_{hq}f\|_K \right) $$

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

$$ (4.17) \quad \alpha_{\min,\omega_F}^{-1/2} \sqrt{\frac{h_F}{q}} \|[[\mathfrak{G}]] \cdot \mathbf{n}_F\|_F \le C(\kappa_{\omega_F}) \qquad q^{3/2} \left( \sqrt{\frac{\alpha_{\max,\omega_F}}{\alpha_{\min,\omega_F}}} \|\nabla u - \mathfrak{G}\|_{\underline{\mathbf{A}},\omega_F} + \alpha_{\min,\omega_F}^{-1/2} \frac{h_K}{q} \|f - \Pi_{hq}f\|_{\omega_F} \right) $$

for all $F \in \mathcal{F}_h^I \cup \mathcal{F}_h^N$

*Proof.* We first write that

$$ (4.18) \quad \|\nabla \cdot (\underline{\mathbf{A}}\mathfrak{G}) + f\|_K = \|\nabla \cdot (\underline{\mathbf{A}}\mathfrak{G}) + \Pi_{hq}f\|_K + \|f - \Pi_{hq}f\|_K $$

where $\Pi_{hq}f \in \mathcal{P}_{q-1}(\mathcal{T}_h)$ is the projection of $f$ introduced in (2.1). Now, we can work with the polynomial data $r_K := \nabla \cdot (\underline{\mathbf{A}}\mathfrak{G}) + \Pi_{hq}f \in \mathcal{P}_{2q-1}(K)$. We have

$$ \|b_K^{1/2}r_K\|_K^2 = (b_K r_K, r_K)_K = (\Pi_{hq}f, b_K r_K)_K + (\nabla \cdot (\underline{\mathbf{A}}\mathfrak{G}), b_K r_K)_K \\ = (\Pi_{hq}f - f, b_K r_K)_K + (\underline{\mathbf{A}}(\nabla u - \mathfrak{G}), \nabla(b_K r_K))_K $$

since $b_K r_K \in H_0^1(K)$. For the first term, we write

$$ (\Pi_{hq}f - f, b_K r_K)_K \le \|f - \Pi_{hq}f\|_K \|b_K r_K\|_K \le C(\kappa_K) \|f - \Pi_{hq}f\|_K \|b_K^{1/2}r_K\|_K $$

As for the second term, we have

$$ (\underline{\mathbf{A}}(\nabla u - \mathfrak{G}), \nabla(b_K r_K))_K \le \|\nabla u - \mathfrak{G}\|_{\underline{\mathbf{A}},K} \alpha_{\max,K}^{1/2} \|\nabla(b_K r_K)\|_K \\ \le C(\kappa_K) \frac{q}{h_K} \|\nabla u - \mathfrak{G}\|_{\underline{\mathbf{A}},K} \alpha_{\max,K}^{1/2} \|b_K^{1/2}r_K\|_K, $$

leading to

$$ \frac{h_K}{q} \|r_K\|_K \le C(\kappa_K) q \left( \alpha_{\max,K}^{1/2} \|\nabla u - \mathfrak{G}\|_{\underline{\mathbf{A}},K} + \frac{h_K}{q} \|f - \Pi_{hq}f\|_K \right), $$

due to (4.1). Then, (4.16) simply follows from (4.18).
For (4.17), we fix $F \in \mathcal{F}_h^I \cup \mathcal{F}_h^N$ and let $r_F := [[\underline{\mathcal{A}}\mathfrak{G}]\cdot \mathbf{n}_F \in \mathcal{P}_q(F)]$. Then, we write that

$$
\begin{align*}
\|b_F^{1/2} r_F\|_F^2 &= (r_F, b_F r_F)_F = (r_F, \mathcal{E}_F(r_F))_F \\
&= (\underline{\mathcal{A}}\mathfrak{G}, \nabla \mathcal{E}_F(r_F))_{\omega_F} + (\nabla_h \cdot (\underline{\mathcal{A}}\mathfrak{G}), \mathcal{E}_F(r_F))_{\omega_F} \\
&= (\underline{\mathcal{A}}(\mathfrak{G} - \nabla u), \nabla \mathcal{E}_F(r_F))_{\omega_F} + (\nabla_h \cdot (\underline{\mathcal{A}}\mathfrak{G}) + f, \mathcal{E}_F(r_F))_{\omega_F} \\
&\le \|\mathfrak{G} - \nabla u\|_{\underline{\mathcal{A}}, \omega_F} \alpha_{\max, \omega_F}^{1/2} \|\nabla \mathcal{E}_F(r_F)\|_{\omega_F} + \|\nabla_h \cdot (\underline{\mathcal{A}}(\mathfrak{G}) + f)\|_{\omega_F} \mathcal{E}(r_F)\|_{\omega_F}
\end{align*}
$$

and it follows from (4.3) that

$$
\|b_F^{1/2} r_F\|_F \le C(\kappa_{\omega_F}) \left( \alpha_{\max, \omega_F}^{1/2} q h_F^{-1/2} \|\mathfrak{G} - \nabla u\|_{\underline{\mathcal{A}}, \omega_F} + q^{-1} h_F^{1/2} \|\nabla_h \cdot (\underline{\mathcal{A}}\mathfrak{G}) + f\|_{\omega_F} \right)
$$

and

$$
\sqrt{\frac{h_F}{p}} \|r_F\|_F \le C(\kappa_{\omega_F}) \left( \alpha_{\max, \omega_F}^{1/2} q^{3/2} \|\mathfrak{G} - \nabla u\|_{\underline{\mathcal{A}}, \omega_F} + q^{1/2} \frac{h_F}{q} \|\nabla_h \cdot (\underline{\mathcal{A}}\mathfrak{G}_h) + f\|_{\omega_F} \right)
$$

due to (4.2). Then, (4.17) follows from (4.16). $\square$

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

**Remark 4.5** (Piecewise polynomial coefficient). It is interesting to remark that we do not require any piecewise polynomial assumption on $\underline{\mathcal{A}}$ in Lemma 4.3, whereas such an assumption is necessary in Lemma 4.4. There, we have assumed for simplicity that $\underline{\mathcal{A}} \in \underline{\mathcal{P}}_q(\mathcal{T}_h)$, but we could more generally assume that $\underline{\mathcal{A}} \in \underline{\mathcal{P}}_p(\mathcal{T}_h)$ for any $p \ge 1$ if we replace the first occurences of $q$ by $p+q$ in the right-hand sides of (4.16) and (4.17).

4.5. **An alternative error estimator for the IPDG scheme.** We are now ready to propose a new error estimate for the IPDG scheme introduced in Section 2.11.1. Specifically, for each $K \in \mathcal{T}_h$, we introduce the estimator

$$
\eta_K^2 := \frac{1}{\alpha_{\min, \tilde{\omega}_K}} \left(\frac{h_K}{p}\right)^2 \|\nabla \cdot (\underline{\mathcal{A}}\mathfrak{G}_h(u_h)) + f\|_K^2 + \frac{1}{\alpha_{\min, \tilde{\omega}_K}} \frac{h_K}{p} \|[[\underline{\mathcal{A}}\mathfrak{G}_h(u_h)]]\| \cdot \mathbf{n}\|_{\partial K \setminus \Gamma_D}^2 \\
+ \alpha_{\max, \tilde{\omega}_K} \left(\frac{h_K}{p}\right)^2 \|\nabla \times \mathfrak{G}_h(u_h)\|_K^2 + \alpha_{\max, \tilde{\omega}_K} \frac{h_K}{p} \|[[\mathfrak{G}_h(u_h)]]\| \cdot \mathbf{n}\|_{\partial K \setminus \Gamma_N}^2.
$$

We have the following estimates:

**Theorem 4.6** (Residual-based estimator for IPDG). We have

$$
(4.19) \qquad \|\mathfrak{G}_h(u - u_h)\|_{\underline{\mathcal{A}}, \Omega}^2 \le C(\kappa_{\mathcal{T}_h}) \sum_{K \in \mathcal{T}_h} \eta_K^2.
$$

Furthermore, if $\underline{\mathcal{A}} \in \underline{\mathcal{P}}_p(\mathcal{T}_h)$, then

$$
(4.20) \quad \eta_K \le C(\kappa_{\hat{\omega}_K}) p^{3/2} \left( \sqrt{\frac{\alpha_{\max, \tilde{\omega}_K}}{\alpha_{\min, \tilde{\omega}_K}}} \|\mathfrak{G}_h(u-u_h)\|_{\underline{\mathcal{A}}, \hat{\omega}_K} + \alpha_{\min, \hat{\omega}_K}^{-1/2} \frac{h_K}{q} \|f - \Pi_{hp} f\|_{\hat{\omega}_K} \right)
$$

for all $K \in \mathcal{T}_h$.

*Proof.* For the upper bound in (4.19), we invoke the generalized Prager-Synge inequality in (3.7a), which is feasible since

$$
(4.21) \qquad (\mathfrak{G}_h(u_h), \xi_h)_\Omega = (u_h, \nabla \cdot \xi_h)_\Omega = 0
$$
for all $\xi_h \in \mathbf{RT}_p(\mathcal{T}_h) \cap \mathbf{H}_{\Gamma_N}(\mathrm{div}^0, \Omega)$ due to (2.7). We then bound the two terms using Lemmas 4.1 and 4.2 with $\mathfrak{G} = \mathfrak{G}_h(u_h)$ and $q = p$. The assumptions of Lemma 4.1 are satisfied due to (4.21) and the ones of Lemma 4.2 due to (2.9).

As for (4.20), we simply apply Lemmas 4.3 and 4.4, again with $\mathfrak{G} = \mathfrak{G}_h(u_h)$ and $q = p$. $\square$

The first two terms in the definition of the estimator $\eta_K$ are standard: They are essentially already present for conforming finite element discretizations, see e.g. [5, 47, 52]. The novelty of the proposed estimator is the way the non-conformity of the scheme is handled.

The standard error estimator for the IPDG scheme typically reads

$$ (4.22) \quad \begin{aligned} \tilde{\eta}_K^2 := \frac{1}{\alpha_{\min, \tilde{\omega}_K}} \left(\frac{h_K}{p}\right)^2 \|\nabla \cdot (\underline{\mathbf{A}}\nabla u_h) + f\|_K^2 &+ \frac{1}{\alpha_{\min, \tilde{\omega}_K}} \frac{h_K}{p} \|\llbracket \underline{\mathbf{A}}\nabla u_h \rrbracket \cdot \mathbf{n}\|_{\partial K \setminus \Gamma_D}^2 \\ &+ \alpha_{\max, \tilde{\omega}_K} \frac{p^2}{h_K} \|\llbracket u_h \rrbracket\|_{\partial K \setminus \Gamma_N}^2 \end{aligned} $$

for all $K \in \mathcal{T}_h$. The next result shows that the proposed estimator is sharper than the standard one, in the sense that it is smaller while remaining efficient (in a norm with less terms).

It follows from the discrete trace inequality in (4.10c) that

$$ (4.23) \quad \|\mathcal{L}_h(u_h)\|_K^2 \le C(\kappa_K) \frac{p^2}{h_K} \|\llbracket u_h \rrbracket\|_{\partial K \setminus \Gamma_N}^2. $$

As a result, we can first note that

$$ \|\nabla u_h - \mathfrak{G}_h(u_h)\|_{\underline{\mathbf{A}}, K}^2 = \|\mathcal{L}_h(u_h)\|_{\underline{\mathbf{A}}, K}^2 \le C(\kappa_K) \alpha_{\max, K} \frac{p^2}{h_K} \|\llbracket u_h \rrbracket\|_{\partial K \setminus \Gamma_N}^2 $$

and that therefore using the broken gradient $\nabla_h u_h$ or the discrete gradient $\mathfrak{G}_h(u_h)$ in (4.22) is not important as soon as the jumps of $u_h$ are included in the estimator and the error measure.

**Theorem 4.7 (Comparison with the standard estimator).** For $K \in \mathcal{T}_h$, we have

$$ (4.24a) \quad \frac{h_K^2}{p^2} \|\nabla \times \mathfrak{G}_h(u_h)\|_K^2 \le C(\kappa_K) p^2 \left( \frac{p^2}{h_K} \|\llbracket u_h \rrbracket\|_{\partial K \setminus \Gamma_N}^2 \right) $$

and

$$ (4.24b) \quad \frac{h_K}{p} \|\llbracket \mathfrak{G}_h(u_h) \rrbracket\|_{\partial K}^2 \le C(\kappa_{\hat{\omega}_K}) p^2 \left( \sum_{\substack{K' \in \mathcal{T}_h \\ K' \subset \hat{\omega}_K}} \frac{p^2}{h_{K'}} \|\llbracket u_h \rrbracket\|_{\partial K' \setminus \Gamma_N}^2 \right). $$

In particular, we have

$$ (4.25) \quad \eta_K^2 \le C(\kappa_{\hat{\omega}_K}) p^2 \sum_{\substack{K' \in \mathcal{T}_h \\ K' \subset \hat{\omega}_K}} \tilde{\eta}_{K'}^2 $$

for all $K \in \mathcal{T}_h$.

*Proof.* Let $K \in \mathcal{T}_h$. To establish (4.24a), we first invoke the inverse inequality in (4.10a), leading to

$$ \frac{h_K^2}{p^2} \|\nabla \times \mathfrak{G}_h(u_h)\|_K^2 = \frac{h_K^2}{p^2} \|\nabla \times \mathcal{L}_h(\llbracket u_h \rrbracket)\|_K^2 \le C(\kappa_K) p^2 \|\mathcal{L}_h(\llbracket u_h \rrbracket)\|_K^2, $$
and the result follows from (4.23). To prove (4.24b), we first write that

$$ \begin{aligned} \|[[\mathfrak{G}_h(u_h)]] \times \mathbf{n}_F\|_F &\le \|[[\nabla_h u_h]] \times \mathbf{n}_F\|_F + \|[[\mathcal{L}(u_h)]] \times \mathbf{n}_F\|_F \\ &= \|\nabla_F[[u_h]]\|_F + \|[[\mathcal{L}(u_h)]] \times \mathbf{n}_F\|_F. \end{aligned} $$

For the first term, we simply employ the inverse inequality in (4.10b). For the second term, if $F = \partial K_+ \cap \partial K_+ \in \mathcal{F}_h^1$ is an interior face, we write

$$ \|[[\mathcal{L}(u_h)]] \times \mathbf{n}_F\|_F \le \|\mathcal{L}(u_h)|_{K_-} \times \mathbf{n}_F\|_F + \|\mathcal{L}(u_h)|_{K_+} \times \mathbf{n}_F\|_F $$

and simply

$$ \|[[\mathcal{L}(u_h)]] \times \mathbf{n}_F\|_F \le \|\mathcal{L}(u_h)|_K \times \mathbf{n}_F\|_F $$

$F \in \mathcal{F}_h^D$, where $K \in \mathcal{T}_h$ is the only element such that $F \subset \partial K$. We will only detail the second case. Using the trace inequality in (4.10c), we have

$$ \|\mathcal{L}(u_h) \times \mathbf{n}_F\|_F \le \|\mathcal{L}(u_h)\|_F \le C(\kappa_K) \frac{p^2}{\sqrt{h_K}} \|\mathcal{L}(u_h)\|_K \le C(\kappa_K) \frac{p^2}{h_K} \|[[u_h]]\|_{\partial K \setminus \Gamma_N}, $$

where we employ (4.23) in the last inequality. $\square$

4.6. **An alternative error estimator for mixed FEM discretizations.** We next consider the mixed FEM discretization presented in Section 2.11.2. Here, our error estimator reads

$$ \eta_K^2 := \alpha_{\max, \tilde{\omega}_K} \left\{ \left( \frac{h_K}{p} \right)^2 \| \nabla \times (\underline{\mathbf{A}}^{-1} \boldsymbol{\sigma}_h) \|_K^2 + \frac{h_K}{p} \| [[\underline{\mathbf{A}}^{-1} \boldsymbol{\sigma}_h]] \times \mathbf{n} \|_{\partial K \setminus \Gamma_N}^2 \right\} $$

for all $K \in \mathcal{T}_h$.

**Theorem 4.8 (Residual-based estimator for mixed FEM).** We have

$$ (4.26) \qquad \|\underline{\mathbf{A}}\nabla u + \boldsymbol{\sigma}_h\|_{\underline{\mathbf{A}}^{-1}, \Omega}^2 \le C(\kappa_{\mathcal{T}_h}) \sum_{K \in \mathcal{T}_h} \left\{ \eta_K^2 + \left(\frac{h_K}{p}\right)^2 \|f - \Pi_{hp}f\|_K^2 \right\} $$

and, if $\underline{\mathbf{A}} \in \mathcal{P}_0(\mathcal{T}_h)$, then

$$ (4.27) \qquad \eta_K \le C(\kappa_{\tilde{\omega}_K}) \sqrt{\frac{\alpha_{\max, \tilde{\omega}_K}}{\alpha_{\min, \tilde{\omega}_K}}} p^{3/2} \|\underline{\mathbf{A}}\nabla u + \boldsymbol{\sigma}_h\|_{\underline{\mathbf{A}}^{-1}, \tilde{\omega}_K} $$

for all $K \in \mathcal{T}_h$

*Proof.* We first show (4.26). Here, we utilize the generalized Prager-Synge identity (3.7a) with $\mathfrak{G} = -\underline{\mathbf{A}}^{-1}\boldsymbol{\sigma}$, and we estimate the second term using Lemma 4.1. Notice that the assumptions of both results are satisfied due to (2.12). For the first term, for $v \in H_{\Gamma_D}^1(\Omega)$, we write

$$ (f, v)_{\Omega} - (\underline{\mathbf{A}}\mathfrak{G}, \nabla v)_{\Omega} = (f, v)_{\Omega} - (\nabla \cdot \boldsymbol{\sigma}_h, v)_{\Omega} = (f - \Pi_{hp}f, v)_{\Omega} = (f - \Pi_{hp}f, v - \Pi_{hp}v)_{\Omega}, $$

and the conclusion follows from (2.3), since

$$ \begin{aligned} (f - \Pi_{hp}f, v - \Pi_{hp}v)_{\Omega} &\le C(\kappa_{\mathcal{T}_h}) \sum_{K \in \mathcal{T}_h} \frac{h_K}{p} \|f - \Pi_{hp}f\|_K \|\nabla v\|_K \\ &\le C(\kappa_{\mathcal{T}_h}) \sum_{K \in \mathcal{T}_h} \alpha_{\min,K}^{-1/2} \frac{h_K}{p} \|f - \Pi_{hp}f\|_K \|\nabla v\|_{\underline{\mathbf{A}},K}. \end{aligned} $$
For (4.27), since $\underline{\mathbf{A}} \in \mathcal{P}_0(\mathcal{T}_h)$ by assumption here, we can apply Lemma 4.3 with $\mathfrak{G} = \underline{\mathbf{A}}^{-1}\boldsymbol{\sigma}_h$, leading to result

$$
\begin{align*}
\eta_K &\le C(\kappa_{\hat{\omega}_K}) \alpha_{\max, \tilde{\omega}_K}^{1/2} p^{3/2} \|\nabla u - \underline{\mathbf{A}}^{-1} \boldsymbol{\sigma}_h\|_{\hat{\omega}_K}^2 \\
&\le C(\kappa_{\hat{\omega}_K}) \sqrt{\frac{\alpha_{\max, \tilde{\omega}_K}}{\alpha_{\min, \tilde{\omega}_K}} p^{3/2}} \|\nabla u - \underline{\mathbf{A}}^{-1} \boldsymbol{\sigma}_h\|_{\underline{\mathbf{A}}, \hat{\omega}_K}^2 \\
&\le C(\kappa_{\hat{\omega}_K}) \sqrt{\frac{\alpha_{\max, \tilde{\omega}_K}}{\alpha_{\min, \tilde{\omega}_K}} p^{3/2}} \|\underline{\mathbf{A}} \nabla u - \boldsymbol{\sigma}_h\|_{\underline{\mathbf{A}}^{-1}, \hat{\omega}_K}^2.
\end{align*}
$$

□

## 5. FULLY-EQUILIBRATED ESTIMATORS

We now propose to exploit the alternative Prager-Synge estimate in (3.7b) to derive a flux-equilibrated error estimator. Crucially, in contrast to the standard theory in [2, 3, 35, 45], we will reconstruct a curl-free approximation of the flux variable, rather than a gradient field.

**5.1. Tools.** We will need some tools developed for curl-constrained equilibration developed in the context of the discretization of curl-curl problems in [24].

**5.1.1. Vertex partition of unity.** For all vertices $\boldsymbol{a} \in \mathcal{T}_h$, there exists a unique “hat” function $\psi_{\boldsymbol{a}} \in \mathcal{P}_1(\mathcal{T}_h) \cap H^1(\Omega)$ such that $\psi_{\boldsymbol{a}}(\boldsymbol{b}) = \delta_{\boldsymbol{a}, \boldsymbol{b}}$ for all vertices $\boldsymbol{b} \in \mathcal{T}_h$. We have $\psi_{\boldsymbol{a}} \in H^1(\Omega)$ and

$$ (5.1) \qquad \|\psi_{\boldsymbol{a}}\|_{L^{\infty}(\Omega)} \le 1, \qquad \|\nabla \psi_{\boldsymbol{a}}\|_{L^{\infty}(\Omega)} \le C(\kappa_{\mathcal{T}_h^a}) h_{\omega_{\boldsymbol{a}}}. $$

Each hat function $\psi_{\boldsymbol{a}}$ is supported in $\omega_{\boldsymbol{a}}$, and they form a partition of unity, i.e.

$$ (5.2) \qquad \sum_{\boldsymbol{a} \in \mathcal{V}_h} \psi_{\boldsymbol{a}} = 1. $$

**5.1.2. Local function spaces and inequalities.** Consider a vertex $\boldsymbol{a} \in \mathcal{V}_h$. If $\boldsymbol{a} \notin \overline{\Gamma_N}$, we let $\gamma_N := \emptyset$ and $\gamma_N^c := \partial\omega_{\boldsymbol{a}}$. On the other, when $\boldsymbol{a} \in \overline{\Gamma_N}$, we denote by $\gamma_N$ the set of faces sharing the vertex $\boldsymbol{a}$ and lying in $\overline{\Gamma_N}$. We then let $H_0(\text{curl}, \omega_{\boldsymbol{a}}) := H_{\gamma_N^c}(\text{curl}, \omega_{\boldsymbol{a}})$ and $H_*(\text{curl}, \omega_{\boldsymbol{a}}) := H_{\gamma_N}(\text{curl}, \omega_{\boldsymbol{a}})$, as well as $H_0(\text{div}^0, \omega_{\boldsymbol{a}}) := H_{\gamma_N^c}(\text{div}^0, \omega_{\boldsymbol{a}})$. Finally, we define $H_*(^1(\omega_{\boldsymbol{a}}))$ by setting $H_*(^1(\omega_{\boldsymbol{a}})) := H_{\gamma_N}^1(\omega_{\boldsymbol{a}})$ when $\gamma_N \ne \emptyset$, and as the subspace of $H^1(\omega_{\boldsymbol{a}})$ of functions with vanishing mean-value otherwise.

We have the following Poincaré-like inequality

$$ (5.3) \qquad \|\boldsymbol{v}\|_{\omega_{\boldsymbol{a}}} \le C(\kappa_{\mathcal{T}_h^a}) h_{\omega_{\boldsymbol{a}}} \|\nabla \times \boldsymbol{v}\|_{\omega_{\boldsymbol{a}}}. $$

for all $\boldsymbol{v} \in H_*(\text{curl}, \omega_{\boldsymbol{a}}) \cap H_0(\text{div}^0, \omega_{\boldsymbol{a}})$. The fact that the constant only depends on $\kappa_{\mathcal{T}_h^a}$ can be shown following [23, Appendix A].

**5.2. Curl-free field reconstruction.** In this section, given $\mathfrak{G} \in L^2(\Omega)$ with $(\mathfrak{G}, \xi_h)_\Omega = 0$ for all $\xi_h \in \mathcal{R}T_1(\mathcal{T}_h) \cap H_{\Gamma_N}(\text{div}^0, \Omega)$, we follow the equilibration philosophy and propose a strategy to build a field $\phi_h \in \mathcal{N}_p(\mathcal{T}_h) \cap H_{\Gamma_D}(\text{curl}^0, \Omega)$ in order to bound the error as follows:

$$ \min_{s \in H_{\Gamma_D}^1(\Omega)} \|\mathfrak{G} - \nabla s\|_{\underline{\mathbf{A}}, \Omega} = \min_{\phi \in H_{\Gamma_D}(\text{curl}^0, \Omega)} \|\mathfrak{G} - \phi\|_{\underline{\mathbf{A}}, \Omega} \le \|\mathfrak{G} - \phi_h\|_{\underline{\mathbf{A}}, \Omega}. $$
5.2.1. Main ideas. We follow the construction introduced in [24], where the idea is to construct $\phi_h$ from local contributions $\phi_h^a \in H_0(\text{curl}, \omega_a)$ meant to mimic $\psi_a \mathfrak{G}$. The product rule

$$ \nabla \times (\psi_a \mathfrak{G}) = \nabla \psi_a \times \mathfrak{G} + \psi_a \nabla \times \mathfrak{G} $$

motivates that each $\phi_h^a$ should be constructed in such way that $\nabla \times \phi_h^a = \nabla \psi_a \times \mathfrak{G}$. Unfortunately, this is not possible since in general $\nabla \psi_a \times \mathfrak{G} \notin H_0(\text{div}^0, \omega_a)$ for the methods considered in this work. We therefore introduce a suitable local curl-constrained $\theta_h^a \sim \nabla \psi_a \times \mathfrak{G}$ such that $\theta_h^a \in H_0(\text{div}, \omega_a)$. Following [24], this construction is done locally in two steps, whereby $\theta^{a_h} = \hat{\theta}_h^a - \tilde{\theta}_h^a$.

5.2.2. Reconstruction. Our reconstruction is rigorously constructed as follows:

**Theorem 5.1 (Curl-free reconstruction).** Assume that $\mathfrak{G} \in \mathcal{P}_q(\mathcal{T}_h)$ satisfies $(\mathfrak{G}, \xi_h)_{\Omega} = 0$ for all $\xi_h \in \mathcal{R}T_1(\mathcal{T}_h) \cap \mathcal{H}_{\Gamma_N}(\text{div}^0, \Omega)$. Then, the following holds true.

First (i), for all $a \in \mathcal{V}_h$, the over-constrained local problems

$$ (5.4) \qquad \hat{\theta}_h^a := \arg \min_{\substack{w_h \in \mathcal{R}T_q(\mathcal{T}_h^a) \cap \mathcal{H}_0(\text{div}, \omega_a) \\ (w_h, r_0)_{\omega_a} = (\nabla \psi_a \times \mathfrak{G}, r_0)_{\omega_a} \quad \forall r_0 \in \mathcal{P}_0(\mathcal{T}_h^a) \\ \nabla \cdot w_h = 0}} \| \nabla \psi_a \times \mathfrak{G} - w_h \|_{\mathcal{A}, \omega_a} $$

are well-posed, and we set

$$ (5.5) \qquad \hat{\theta}_h := \sum_{a \in \mathcal{T}_h} \hat{\theta}_h^a. $$

Besides (ii), for all $a \in \mathcal{V}_h$ we introduce $\tilde{\theta}_h^a$ by solving the well-posed problem

$$ (5.6) \qquad \tilde{\theta}_h^a|_{K} := \arg \min_{\substack{v_h \in \mathcal{R}T_{q+1}(K) \\ \nabla \cdot v_h = 0 \\ v_h \cdot n_K = \psi_a \hat{\theta}_h \cdot n_K}} \|v_h - \psi_a \hat{\theta}_h\|_K $$

for all $K \in \mathcal{T}_h^a$, and we set

$$ (5.7) \qquad \theta_h^a := \hat{\theta}_h^a - \tilde{\theta}_h^a. $$

Then (iii), for all $a \in \mathcal{V}_h$, the local problem

$$ (5.8) \qquad \phi_h^a := \arg \min_{\substack{v_h \in \mathcal{N}_{q+1}(\mathcal{T}_h) \cap \mathcal{H}_0(\text{curl}, \omega_a) \\ \nabla \times v_h = \theta_h^a}} \| \psi_a \mathfrak{G} - v_h \|_{\mathcal{A}, \omega_a} $$

is well-posed, and we have

$$ (5.9) \qquad \phi_h := \sum_{a \in \mathcal{V}_h} \phi_h^a \in \mathcal{N}_{q+1}(\mathcal{T}_h) \cap \mathcal{H}_{\Gamma_D}(\text{curl}^0, \Omega). $$

**Proof.** We start with (5.4), which is treated in [24, Appendix A] with $\tau_h^a = \nabla \psi_a \times \mathfrak{G}$ and $g^a = 0$. We first need to check [24, Assumption A.1], and the only non-trivial point is that

$$ (\tau^a, \nabla q_h)_{\omega_a} + (g^a, q_h)_{\omega_a} = (\nabla \psi_a \times \mathfrak{G}, \nabla q_h)_{\omega_a} = (\mathfrak{G}, \nabla q_h \times \nabla \psi_a)_{\Omega} = 0 $$

for all $q_h \in \mathcal{P}_1(\mathcal{T}_h^a) \cap H_*(\omega_a)$, due to our assumptions since $\nabla p_h \times \nabla \psi_a \in \mathcal{R}T_1(\mathcal{T}_h) \cap \mathcal{H}_{\Gamma_N}(\text{div}^0, \Omega)$. Then, the well-posedness of (5.4) follows from [24, Theorem A.2].

We then turn to (5.6), which is analyzed in [24, Appendix B]. The well-posedness of (5.6) follows from the observation that for all $K \in \mathcal{T}_h$

$$ (\hat{\theta}_h, 1)_{\partial K} = \sum_{a \in \mathcal{V}(K)} (\hat{\theta}_h^a, 1)_{\partial K} = \sum_{a \in \mathcal{V}(K)} (\nabla \cdot \hat{\theta}_h^a, 1)_K = \sum_{a \in \mathcal{V}(K)} (\nabla \psi_a \times \mathfrak{G}, 1)_K = (\nabla(1) \times \mathfrak{G}, 1)_K = 0 $$
due to (5.2).

Since $\theta_h^a \in \mathbf{RT}_{p+1}(\mathcal{T}_h^a) \cap \mathbf{H}_0(\mathrm{div}, \omega_a)$ by construction, it is clear that the local problem defining $\phi_h^a$ in (5.8) is well-posed for each $a \in \mathcal{V}_h$, and the fact that $\phi \in \mathbf{H}_{\Gamma_D}(\mathrm{curl}^0, \Omega)$ follows from [24, Theorem B.1]. $\square$

**5.3. A posteriori error estimates.** We are now ready to show that the proposed reconstruction may be used to produce sharp upper and lower bounds on the error. The upper bounds immediately follows from Theorems 3.3 and 5.1

**Corollary 5.2 (Abstract reliability).** Let $\mathfrak{G} \in \mathcal{P}_q(\mathcal{T}_h)$ with $(\mathfrak{G}, \xi_h)_{\Omega} = 0$ for all $\xi_h \in \mathbf{RT}_1(\mathcal{T}_h) \cap \mathbf{H}_{\Gamma_N}(\mathrm{div}^0, \Omega)$ and let $\phi_h \in \mathbf{H}_{\Gamma_D}(\mathrm{curl}^0, \Omega)$ be constructed from (5.9) in Theorem 5.1. Then, we have

$$ \min_{s \in H_{\Gamma_D}^1(\Omega)} \|\mathfrak{G} - \nabla s\|_{\mathbf{A}, \Omega} \le \|\mathfrak{G} - \phi_h\|_{\mathbf{A}, \Omega}. $$

We next establish local lower bounds. Crucially, there is no dependence on the polynomial degree $q$ in those bounds.

**Theorem 5.3 (Abstract efficiency).** Assume that $\mathfrak{G} \in \mathcal{P}_q(\mathcal{T}_h)$ is such that $(\mathfrak{G}, \xi_h)_{\Omega} = 0$ for all $\xi_h \in \mathbf{RT}_1(\mathcal{T}_h) \cap \mathbf{H}_{\Gamma_N}(\mathrm{div}^0, \Omega)$. For $a \in \mathcal{V}_h$, let $\phi_h^a$ be defined from (5.8), and $\phi_h$ be constructed with (5.9). Then, we have

$$ (5.10) \qquad \|\psi_a \mathfrak{G} - \phi_h^a\|_{\mathbf{A}, \omega_a} \le C(\kappa_{\mathcal{T}_h^a}) \left( \frac{\alpha_{\max, \tilde{\omega}_a}}{\alpha_{\min, \tilde{\omega}_a}} \right)^{3/2} \|\nabla u - \mathfrak{G}\|_{\mathbf{A}, \tilde{\omega}_a} $$

for all $a \in \mathcal{T}_h$. In addition, for all $K \in \mathcal{T}_h$, the estimate

$$ (5.11) \qquad \|\mathfrak{G} - \phi_h\|_{\mathbf{A}, K} \le C(\kappa_{\tilde{\omega}_K}) \left( \frac{\alpha_{\max, \tilde{\omega}_K}}{\alpha_{\min, \tilde{\omega}_K}} \right)^{3/2} \|\nabla u - \mathfrak{G}\|_{\mathbf{A}, \tilde{\omega}_K} $$

holds true.

**Proof.** We first invoke [24, Theorem A.2], which states that

$$ \|\psi_a \mathfrak{G} - \phi_h^a\|_{\mathbf{A}, \omega_a} \le C(\kappa_{\mathcal{T}_h^a}) \sqrt{\frac{\alpha_{\max, \omega_a}}{\alpha_{\min, \omega_a}}} \min_{\substack{\phi^a \in \mathbf{H}_0(\mathrm{curl}, \omega_a) \\ \nabla \times \phi^a = \theta_h^a}} \|\psi_a \mathfrak{G} - \phi^a\|_{\mathbf{A}, \omega_a}. $$

The result in [24] is given in unweighted norms, but the estimate above from norm equivalence. The Euler-Lagrange equation for the minimizer in the right-hand side consists in finding $\phi^a \in \mathbf{H}_0(\mathrm{curl}, \omega_a)$ and $\rho^a \in \mathbf{H}_0(\mathrm{div}^0, \omega_a)$ such that

$$ \left\{ \begin{array}{l} (\mathbf{A}\phi^a, \mathbf{v})_{\omega_a} + (\rho^a, \nabla \times \mathbf{v})_{\omega_a} = (\mathbf{A}\psi_a \mathfrak{G}, \mathbf{v})_{\omega_a} \\ (\nabla \times \phi^a, \mathbf{w})_{\omega_a} = (\theta_h^a, \mathbf{v})_{\omega_a} \end{array} \right. $$
for all $v \in H_0(\text{curl}, \omega_a)$ and $\rho^a \in H_0(\text{div}^0, \omega_a)$. The first equation guarantees that $\rho^a \in H_*(\text{curl}, \omega_a)$ with $\nabla \times \rho^a = \underline{A}(\phi^a - \psi_a \mathfrak{G})$. We then write that

$$
\begin{align*}
\| \nabla \times \rho^a \|_{\underline{A}^{-1}, \omega_a}^2 &= (\phi^a - \psi_a \mathfrak{G}, \nabla \times \rho^a)_{\omega_a} \\
&= -(\psi_a(\nabla u - \mathfrak{G}), \nabla \times \rho^a)_{\omega_a} + (\phi^a - \psi_a \nabla u, \nabla \times \rho^a)_{\omega_a} \\
&= -(\psi_a(\nabla u - \mathfrak{G}), \nabla \times \rho^a)_{\omega_a} + (\nabla \times (\phi^a - \psi_a \nabla u), \rho^a)_{\omega_a} \\
&= -(\underline{A}^{-1} \underline{A} \psi_a (\nabla u - \mathfrak{G}), \nabla \times \rho^a)_{\omega_a} + (\underline{A}^{-1} \underline{A} (\theta_h^a - \nabla \psi_a \times \nabla u), \rho^a)_{\omega_a} \\
&\le \|\psi_a(\nabla u - \mathfrak{G})\|_{\underline{A}, \omega_a} \|\nabla \times \rho^a\|_{\underline{A}^{-1}, \omega_a} + \|\theta_h^a - \nabla \psi_a \times \nabla u\|_{\underline{A}, \omega_a} \|\rho^a\|_{\underline{A}^{-1}, \omega_a} \\
&\le \left\{ \|\nabla u - \mathfrak{G}\|_{\underline{A}, \omega_a} + \sqrt{\frac{\alpha_{\max, \omega_a}}{\alpha_{\min, \omega_a}}} h_{\omega_a} \|\theta_h^a - \nabla \psi_a \times \nabla u\|_{\underline{A}, \omega_a} \right\} \|\nabla \times \rho^a\|_{\underline{A}^{-1}, \omega_a},
\end{align*}
$$

where we employed (5.3). This leads to

$$
\|\phi^a - \psi_a \mathfrak{G}\|_{\underline{A}, \omega_a} \le \|\nabla u - \mathfrak{G}\|_{\underline{A}, \omega_a} + C(\kappa \tau_h^a) \sqrt{\frac{\alpha_{\max, \omega_a}}{\alpha_{\min, \omega_a}}} h_{\omega_a} \|\theta_h^a - \nabla \psi_a \times \nabla u\|_{\underline{A}, \omega_a}.
$$

and

$$
\begin{align*}
h_{\omega_a} \|\theta_h^a - \nabla \psi_a \times \nabla u\|_{\underline{A}, \omega_a} &\le h_{\omega_a} \|\nabla \psi_a \times (\nabla u - \mathfrak{G})\|_{\underline{A}, \omega_a} + h_{\omega_a} \|\theta_h^a - \nabla \psi_a \times \mathfrak{G}\|_{\underline{A}, \omega_a} \\
&\le C(\kappa \tau_h^a) \|\nabla u - \mathfrak{G}\|_{\underline{A}, \omega_a} + h_{\omega_a} \|\theta_h^a - \nabla \psi_a \times \mathfrak{G}\|_{\underline{A}, \omega_a}, \\
&\le C(\kappa \tau_h^a) \|\nabla u - \mathfrak{G}\|_{\underline{A}, \omega_a} + h_{\omega_a} \|\tilde{\theta}_h^a - \nabla \psi_a \times \mathfrak{G}\|_{\underline{A}, \omega_a} + h_{\omega_a} \|\tilde{\theta}_h^a\|_{\underline{A}, \omega_a}
\end{align*}
$$

due to (5.1) and the definition of $\theta_h^a$. To prove (5.10), we estimate the two terms in the
right-hand side.

We estimate the first term with

$$
\begin{align*}
(5.12) \quad h_{\omega_a} \|\hat{\theta}_h^a - \nabla \psi_a \times \mathfrak{G}\|_{\underline{A}, \omega_a} &\le C(\kappa \tau_h^a) \sqrt{\frac{\alpha_{\max, \omega_a}}{\alpha_{\min, \omega_a}}} h_{\omega_a} \min_{\substack{v \in H_0(\text{div}, \omega_a) \\ \nabla \cdot v = 0}} \|v - \nabla \psi_a \times \mathfrak{G}\|_{\underline{A}, \omega_a} \\
&\le C(\kappa \tau_h^a) \sqrt{\frac{\alpha_{\max, \omega_a}}{\alpha_{\min, \omega_a}}} h_{\omega_a} \|\nabla \psi_a \times (\nabla u - \mathfrak{G})\|_{\underline{A}, \omega_a} \\
&\le C(\kappa \tau_h^a) \sqrt{\frac{\alpha_{\max, \omega_a}}{\alpha_{\min, \omega_a}}} \|\nabla u - \mathfrak{G}\|_{\underline{A}, \omega_a},
\end{align*}
$$

where we used the fact that $\nabla \psi_a \times \nabla u \in H_0(\text{div}^0, \omega_a)$ and (5.1). For the second term, we first employ [24, Theorem B.1], which gives

$$
h_{\omega_a} \| \tilde{\theta}_h^a \|_{\underline{A}, \omega_a} = C(\kappa \tau_h^a) \sqrt{\frac{\alpha_{\max, \omega_a}}{\alpha_{\min, \omega_a}}} h_{\omega_a} \| \hat{\theta}_h \|_{\underline{A}, \omega_a}.
$$

We then write

$$
\hat{\theta}_h = \sum_{a \in V_h} (\nabla \psi_a \times \nabla u - \hat{\theta}_h^a),
$$

so that

$$
h_{\omega_a} \| \tilde{\theta}_h^a \|_{\underline{A}, \omega_a} \le C(\kappa \tilde{\tau}_h^a) \frac{\alpha_{\max, \tilde{\omega}_a}}{\alpha_{\min, \tilde{\omega}_a}} \| \nabla u - \mathfrak{G} \|_{\underline{A}, \tilde{\omega}_a},
$$

due to to small support of each $\psi_a$ and (5.12). This concludes the proof of (5.10).

For (5.11), we simply use that due to (5.2) and the definition of $\phi_h$ in (5.9), we have

$$
\| \mathfrak{G} - \phi_h \|_{\underline{A}, K} = \| \sum_{a \in V(K)} (\psi_a \mathfrak{G} - \phi_h^a) \|_{\underline{A}, K} \le \sum_{a \in V(K)} \| \psi_a \mathfrak{G} - \phi_h^a \|_{\underline{A}, \tilde{\omega}_a},
$$
and the result follows from (5.10).

5.4. **An alternative error estimator for the IPDG scheme.** Here, the curl-free reconstruction $\phi_h$ is used to control the non-conformity of the solution, i.e. the second term in the right-hand of the side of the standard Prager-Synge identity in (3.1). For the other term, we follow the standard flux-equilibration approach from [13, 30, 35].

**Remark 5.4** (Lower-order lifting). Given the IPDG solution $u_h \in \mathcal{P}_p(\mathcal{T}_h)$ as been computed (typically through the formulation described in Remark 2.2), the construction detailed below utilizes the generalized gradient $\mathfrak{G}_h u_h \in \mathcal{P}_{p-1}(\mathcal{T}_h)$. However, as already proposed in [35], the results presented in this section hold unchanged if the jumps of are lifting in a smaller polynomial space, i.e. if we employ the space $\mathcal{P}_0(\mathcal{T}_h)$ in the definition of the lifting $\mathcal{L}_h(u_h)$, irrespectively of the polynomial degree $p$ of the discretization.

5.4.1. *Flux reconstruction.* For all $a \in \mathcal{V}_h$, the local problem

$$ \sigma_h^a := \min_{\substack{w_h \in \mathcal{R}\mathcal{T}_{p+1}(\mathcal{T}_h^a) \cap \mathcal{H}_0(\mathrm{div}, \omega_a) \\ \nabla \cdot w_h = \Pi_{hp}(\psi_a f - \underline{\mathcal{A}}\nabla \psi_a \cdot \mathfrak{G}_h(u_h))}} \|\underline{\mathcal{A}}^{-1}w_h + \psi_a \mathfrak{G}_h(u_h)\|_{\underline{\mathcal{A}}, \omega_a} $$

is well-posed since Stokes' compatibility condition is satisfied due to the Galerkin orthogonality property in (2.9), and whenever $\underline{\mathcal{A}} \in \mathcal{P}_0(\mathcal{T}_h)$, we have

$$ (5.13) \qquad \nabla \cdot \sigma_h = \Pi_{hp} f $$

for the reconstructed flux

$$ \sigma_h := \sum_{a \in \mathcal{V}_h} \sigma_h^a $$

since the $\psi_a$ form a partition of unity as per (5.2).

5.4.2. *Error estimates.* Once the divergence-conforming and curl-free reconstructions $\sigma_h$ and $\phi_h$ have been computed, we can easily introduce our error estimator. For all $K \in \mathcal{T}_h$, we set

$$ \eta_K^2 := \|\underline{\mathcal{A}}^{-1}\sigma_h + \mathfrak{G}_h(u)\|_{\underline{\mathcal{A}}, K}^2 + \|\mathfrak{G}_h(u) - \phi_h\|_{\underline{\mathcal{A}}, K}^2. $$

**Theorem 5.5** (Equilibrated estimator for IPDG). Assume that $\underline{\mathcal{A}} \in \mathcal{P}_0(\mathcal{T}_h)$. Then, we have

$$ (5.14) \qquad \|\mathfrak{G}_h(u - u_h)\|_{\underline{\mathcal{A}}, \Omega}^2 \le \sum_{K \in \mathcal{T}_h} \left\{ \eta_K^2 + \left(\frac{h_K}{\pi}\right)^2 \|f - \Pi_{hp} f\|_K^2 \right\}. $$

Besides, the following estimate

$$ (5.15) \qquad \eta_K \le C(\kappa_{\tilde{\omega}_K}) \left( \frac{\alpha_{\max, \tilde{\omega}_K}}{\alpha_{\min, \tilde{\omega}_K}} \right)^{3/2} \left\{ \|\mathfrak{G}_h(u - u_h)\|_{\underline{\mathcal{A}}, \tilde{\omega}_K}^2 + \left(\frac{h_K}{p}\right)^2 \|f - \Pi_{hp} f\|_{\tilde{\omega}_K}^2 \right\} $$

holds true for all $K \in \mathcal{T}_h$.

*Proof.* The proof of (5.14) simply lies on the standard Prager-Synge identity in (3.1). For the term in the right-hand is bounded by the second term in the estimator as per Corollary 5.2, whereas the first term is bounded by the first term in the estimator plus the data oscillation term, which reflects the fact that we do not exactly have $\nabla \cdot \sigma_h = f$, but only $\nabla \cdot \sigma = \Pi_{hp} f$, as per (5.13). The proof that can be found in [35, Theorem 3.3], or by following the lines of the proof of Theorem 4.8, but using the Poincaré inequality in (2.2) rather than the one in (2.3).
On the other hand, the lower bounds in (5.15) are obtained by combining Theorem 5.3 for the second term in the estimator and the state-of-the art theory of equilibrated estimator for conforming methods as per [13, 25, 35, 36] for the first term. □

5.5. **An alternative error estimator for mixed FEM discretizations.** Let $\phi_h$ be constructed following Theorem 5.1 with $\mathfrak{G} = \underline{\mathbf{A}}^{-1}\sigma_h$. Notice that this is possible whenever $\underline{\mathbf{A}} \in \underline{\mathcal{P}}_0(\mathcal{T}_h)$ since $\mathfrak{G} \in \mathcal{P}_p(\mathcal{T}_h)$ with $(\mathfrak{G}, \xi_h)_{\Omega} = 0$ for all $\xi_h \in \mathcal{R}\mathcal{T}_p(\mathcal{T}_h) \cap \mathbf{H}_{\Gamma_N}(\mathrm{div}^0, \Omega)$ due to (2.12). Then, we let

$$ \eta_K := \|\sigma_h + \underline{\mathbf{A}}\phi_h\|_{\underline{\mathbf{A}}^{-1,K}} $$

for all $K \in \mathcal{T}_h$, and the following result from Corollay 5.2 and Theorem 5.3, whereby the oscillation term appears exactly as in the proof of Theorems 4.8 and 5.5.

**Theorem 5.6** (Equilibrated estimator for mixed FEM). Assume that $\underline{\mathbf{A}} \in \underline{\mathcal{P}}_0(\mathcal{T}_h)$. Then, we have

$$ \|\underline{\mathbf{A}}\nabla u + \sigma_h\|_{\underline{\mathbf{A}}^{-1}, \Omega}^2 \le \sum_{K \in \mathcal{T}_h} \left\{ \eta_K^2 + \left(\frac{h_K}{\pi}\right)^2 \|f - \Pi_{hp}f\|_K^2 \right\}. $$

as well as

$$ \eta_K \le C(\kappa_{\tilde{\omega}_K}) \left( \frac{\alpha_{\max, \tilde{\omega}_K}}{\alpha_{\min, \tilde{\omega}_K}} \right)^{3/2} \left\{ \| \underline{\mathbf{A}} \nabla u + \sigma_h \|_{\underline{\mathbf{A}}^{-1, \tilde{\omega}_K}}^2 + \left( \frac{h_K}{p} \right)^2 \| f - \Pi_{hp} f \|_{\tilde{\omega}_K}^2 \right\} $$

for all $K \in \mathcal{T}_h$.

5.6. **More general coefficients.** For the construction of the estimator above, we need that $\underline{\mathbf{A}} \in \underline{\mathcal{P}}_0(\mathcal{T}_h)$ if we want to reconstruct $\sigma_h \in \mathcal{R}\mathcal{T}_{p+1}(\mathcal{T}_h)$ for the IPDG scheme and $\phi_h \in \mathcal{N}_{p+1}(\mathcal{T}_h)$ for the mixed FEM. This assumption can be lifted, and we can assume that $\underline{\mathbf{A}} \in \underline{\mathcal{P}}_q(\mathcal{T}_h)$ if we are ready to using higher-order spaces for the reconstructions in such a way that $\sigma_h \in \mathcal{R}\mathcal{T}_{p+q+1}(\mathcal{T}_h)$ for the IPDG scheme and $\phi_h \in \mathcal{N}_{p+q+1}(\mathcal{T}_h)$ for the mixed FEM.

## 6. ALTERNATIVE EQUILIBRATION TECHNIQUES

In this section, we explore an alternative to the flux equilibration technique described above. This alternative is less expensive and easier to implement. It preserves the polynomial-dgeree-robust, but introduces an a priori unknown constant in the upper bound.

6.1. **Tools.** We start by listing some key technical results useful throughout this section.

6.1.1. **Edge patches.** For $\ell \in \mathcal{E}_h$, $\mathcal{T}_h^\ell$ is the edge patch collecting elements $K \in \mathcal{T}_h$ such that $\ell \in \mathcal{E}(K)$, and we denote by $\omega_\ell \subset \Omega$ the associated open domain. For an element $K \in \mathcal{T}_h$ we then also employ the notation $\tilde{\omega}_K = \cup_{\ell \in \mathcal{E}(K)} \omega_\ell$.

6.1.2. **Edge partition of unity.** For each edge $\ell \in \mathcal{E}_h$, we denote by $\psi_\ell$ the “edge” function associated with $\ell$, i.e. the only element of $\mathcal{N}_1(\mathcal{T}_h) \cap \mathbf{H}(\mathrm{curl}, \Omega)$ such that $(\psi_\ell, 1)_\ell = |\ell|\delta_{\ell,\ell'}$ for all $\ell' \in \mathcal{E}_h$, where $|\ell|$ is the length of edge. The suppot of $\psi_\ell$ coincides with $\omega_\ell$, and we have

$$ (6.1) \qquad \|\psi_\ell\|_{\mathcal{L}^\infty(\Omega)} \le C(\kappa_{\omega_\ell}), \qquad \|\nabla \times \psi_\ell\|_{\mathcal{L}^\infty(\Omega)} \le C(\kappa_{\omega_\ell})h_{\omega_\ell}^{-1}. $$

In addition, if $w \in \mathcal{L}^2(\Omega)$, then

$$ (6.2) \qquad w = \sum_{\ell \in \mathcal{E}_h} (w \cdot \tau_\ell)\psi_\ell. $$
6.1.3. Local function spaces and inequalities. For each $\ell \in \mathcal{E}_h$, if $\ell \subset \bar{\Gamma}_N$, we let $\gamma_\ell := \partial\omega_\ell$ and $\gamma_\ell^c := \emptyset$. On the other hand, if $\ell \subset \bar{\Gamma}_N$, then we let $\gamma_\ell := \partial\omega_\ell \cap \gamma_\ell^c$, where $\gamma_\ell^c$ corresponds to the faces $F \in \mathcal{F}_h$ having $\ell$ as an edge and lying in $\bar{\Gamma}_N$. Notice that except when the mesh is very coarse, $\gamma_\ell^c = \partial\omega_\ell \cap \Gamma_N$. We then set $H_0(\text{div}, \omega_\ell) := H_{\gamma_\ell}(\text{div}, \omega_\ell)$, $H_\star^1(\omega_\ell) := H_{\gamma_\ell^c}^1(\omega_\ell)$, and $L_0^2(\omega_\ell) := \nabla \cdot H_0(\text{div}, \omega_\ell)$. $L_0^2(\omega_\ell) = L^2(\omega_\ell)$ when $\gamma_\ell^c \neq \emptyset$, and it contains functions with vanishing mean-value otherwise.

We observe that whenever $r \in H_\star^1(\omega_\ell)$, we have $r\psi_\ell \subset H_{\gamma_\ell, \mathrm{D}}(\mathrm{curl}, \omega_\ell)$, where $\gamma_{\ell, \mathrm{D}}$ collects the faces of $F \in \mathcal{F}_h$ having $\ell$ as an edge and lying in $\bar{\Gamma}_\mathrm{D}$. Besides, the estimate

$$ (6.3) \qquad \| \nabla \times (r\psi_{\ell}) \|_{\omega_{\ell}} \le C(\kappa_{\mathcal{T}_h^{\ell}}) \| \nabla r \|_{\omega_{\ell}} \quad \forall r \in H_{\star}^{1}(\omega_{\ell}) \cap L_{0}^{2}(\omega_{\ell}) $$

may be obtained by combining the product rule, the bounds in (6.1) and a Poincaré inequality in the patch $\omega_\ell$. The resulting constant only depends on the shape-regularity of the mesh, see [21, Theorem 3.2]. We emphasize that a Poincaré inequality is always available in $H_\star^1(\omega_\ell) \cap L_0^2(\omega_\ell)$ since either $\gamma_\ell^c \neq \emptyset$ and the functions have vanishing traces in a subset of $\partial\omega_\ell$, or they have vanishing mean-value.

6.1.4. Regular decomposition. Similar to the discussion in Section 4.1.4 for the residual-based estimator, we are going to use a regular decomposition of a vector field in $H_{\Gamma_N}(\mathrm{curl}, \Omega)$ with suitable orthogonality constraint into a $H_{\Gamma_N}^1(\Omega)$ component and a gradient. Here, however, the resulting stability constant is the only unknown quantity in the reliability bound, so that we take great care to identify it sharply. We refer the reader to [21, Remark 5.2] for more details.

We introduce

$$ C_L := \sup_{g \in \mathbf{X}_{hq}^{\perp}(\Omega)} \sup_{\phi \in H_{\Gamma_N}(\mathrm{curl}, \Omega)} \min_{\theta \in H_{\Gamma_N}^1(\Omega)} \| \nabla \theta \|_{\underline{\mathbf{A}}^{-1, \Omega}} \\ \| \nabla \times \phi \|_{\underline{\mathbf{A}}^{-1, \Omega}} = 1 (g, \nabla \times \theta)_{\Omega} = (g, \nabla \times \phi)_{\Omega} $$

where, for $q \ge 1$,

$$ \mathbf{X}_{hq}^{\perp}(\Omega) := \{g \in L^2(\Omega) \mid (g, \xi_h)_{\Omega} = 0 \quad \forall \xi_h \in \mathbf{R}T_q(\mathcal{T}_h) \cap H_{\Gamma_N}(\text{div}, \Omega)\}. $$

As we show below, this constant is (at least) controlled uniformly by the shape-regularity parameter and the contrast in the diffusion coefficient.

**Lemma 6.1** (Regular decomposition constant). We have

$$ C_L \le C(\kappa_{\mathcal{T}_h}) \max_{K \in \mathcal{T}_h} \sqrt{\frac{\alpha_{\max, \tilde{\omega}_K}}{\alpha_{\min, \tilde{\omega}_K}}} $$

*Proof.* Let $g \in \mathbf{X}_{hq}^{\perp}(\Omega)$ and $\phi \in H_{\Gamma_N}(\mathrm{curl}, \Omega)$ with $\|\nabla \times \phi\|_{\underline{\mathbf{A}}^{-1, \Omega}} = 1$. According to [50, Theorem 1], we can find $\phi_h \in N_1(\mathcal{T}_h) \cap H_{\Gamma_N}(\mathrm{curl}, \Omega)$ and $\theta \in H_{\Gamma_N}^1(\Omega)$ such that

$$ \nabla \times \theta = \nabla \times (\phi - \phi_h), $$

and

$$ \| \underline{\nabla} \theta \|_{K} \le C(\kappa_{\mathcal{T}_h}) \| \nabla \times \phi \|_{\tilde{\omega}_K}. $$

We now observe that

$$ (g, \nabla \times \theta)_{\Omega} = (g, \nabla \times (\phi - \phi_h))_{\Omega} = (g, \nabla \times \phi)_{\Omega}, $$
since $\nabla \times \phi_h \in RT_1(\mathcal{T}_h) \cap H_{\Gamma_N}(\text{div}, \Omega)$. Hence, $\theta$ is in the minimization set, and it remains to estimate $\|\underline{\nabla}\theta\|_{\underline{\mathbf{A}}^{-1}, \Omega}$. We have

$$ \begin{aligned} \|\underline{\nabla}\theta\|^2_{\underline{\mathbf{A}}^{-1}, \Omega} &\le \sum_{K \in \mathcal{T}_h} \|\underline{\nabla}\theta\|^2_{\underline{\mathbf{A}}^{-1}, K} \le \sum_{K \in \mathcal{T}_h} \alpha_{\min, \tilde{\omega}_K}^{-1} \|\underline{\nabla}\theta\|^2_K \\ &\le C(\kappa_{\mathcal{T}_h}) \sum_{K \in \mathcal{T}_h} \alpha_{\min, \tilde{\omega}_K}^{-1} \|\nabla \times \phi\|^2_{\tilde{\omega}_K} \le C(\kappa_{\mathcal{T}_h}) \sum_{K \in \mathcal{T}_h} \frac{\alpha_{\max, \tilde{\omega}_K}}{\alpha_{\min, \tilde{\omega}_K}} \|\nabla \times \phi\|^2_{\underline{\mathbf{A}}^{-1}, \tilde{\omega}_K} \\ &\le C(\kappa_{\mathcal{T}_h}) \max_{K \in \mathcal{T}_h} \frac{\alpha_{\max, \tilde{\omega}_K}}{\alpha_{\min, \tilde{\omega}_K}} \|\nabla \times \phi\|^2_{\underline{\mathbf{A}}^{-1}, \Omega}, \end{aligned} $$

where we used the fact that the number of overlaps is controlled by the shape-regularity parameter $\kappa_{\mathcal{T}_h}$. $\square$

**6.2. Construction of the estimator.** We are now ready to propose the last error estimator studied in this work. To motivate the construction, we start by establishing an alternative equilibration principle.

**Theorem 6.2 (Alternative equilibration).** Let $\mathfrak{G} \in L^2(\Omega)$ such that $(\mathfrak{G}, \xi_h)_{\Omega} = 0$ for all $\xi_h \in RT_1(\mathcal{T}_h) \cap H_{\Gamma_N}(\text{div}^0, \Omega)$. Then

$$ \min_{s \in H_{\Gamma_D}^1(\Omega)} \|\mathfrak{G} - \nabla s\|^2_{\underline{\mathbf{A}}, \Omega} \le C_L^2 \sum_{k=1}^3 \min_{\sigma^k \in H_{\Gamma_D}(\text{div}^0, \Omega)} \|\mathfrak{G} \times e^k - \sigma^k\|^2_{\underline{\mathbf{A}}, \Omega} $$

*Proof.* Let $\theta \in H_{\Gamma_N}^1(\Omega)$. We have

$$ (\mathfrak{G}, \nabla \times \theta)_{\Omega} = \sum_{k=1}^{3} (\mathfrak{G}, \nabla \times (\theta_k e^k))_{\Omega} = \sum_{k=1}^{3} (\mathfrak{G}, \nabla \theta_k \times e^k)_{\Omega} = - \sum_{k=1}^{3} (\mathfrak{G} \times e^k, \nabla \theta_k)_{\Omega}. $$

If $\sigma^k \in H_{\Gamma_D}(\text{div}^0, \Omega)$, then

$$ (\mathfrak{G} \times e^k, \nabla \theta_k)_{\Omega} = (\mathfrak{G} \times e^k - \sigma^k, \nabla \theta_k)_{\Omega}, $$

so that

$$ \begin{aligned} |(\mathfrak{G} \times e^k, \nabla \theta_k)_{\Omega}| &\le \min_{\sigma^k \in H_{\Gamma_D}(\text{div}^0, \Omega)} \|\underline{\mathbf{A}}(\mathfrak{G} \times e^k - \sigma^k)\|_{\underline{\mathbf{A}}^{-1}, \Omega} \|\nabla \theta_k\|_{\underline{\mathbf{A}}^{-1}, \Omega} \\ &= \min_{\sigma^k \in H_{\Gamma_D}(\text{div}^0, \Omega)} \|\mathfrak{G} \times e^k - \sigma^k\|_{\underline{\mathbf{A}}, \Omega} \|\nabla \theta_k\|_{\underline{\mathbf{A}}^{-1}, \Omega}. \end{aligned} $$

Therefore,

$$ (\mathfrak{G}, \nabla \times \theta)_{\Omega} \le \left( \sum_{k=1}^{3} \min_{\sigma^k \in H_{\Gamma_D}(\text{div}^0, \Omega)} \|\mathfrak{G} \times e^k - \sigma^k\|^2_{\underline{\mathbf{A}}, \Omega} \right)^{1/2} \left( \sum_{k=1}^{3} \|\nabla \theta_k\|^2_{\underline{\mathbf{A}}^{-1}, \Omega} \right)^{1/2}. \square $$

Following, Theorem 6.2, we propose the following construction. Assuming that $\mathfrak{G} \in \mathcal{P}_q(\mathcal{T}_h)$ for some $q \ge 0$, for each $\ell \in \mathcal{E}_h$, we introduce

$$ (6.4a) \qquad \sigma_{h,\ell} := \arg \min_{v_h \in H_0(\text{div}, \omega_\ell) \cap RT_{q+2}(\mathcal{T}_h^\ell)} \|\mathfrak{G} \times \psi_\ell - v_h\|_{\underline{\mathbf{A}}, \omega_\ell}. $$
Then, for $1 \le k \le 3$, we combined these local contributions into global objects

$$ (6.4\text{b}) \qquad \sigma_h^k := \sum_{\ell \in \mathcal{E}_h} (\tau_\ell \cdot e^k) \sigma_{h,\ell}. $$

**Theorem 6.3 (Construction of the estimator).** Assume that $\mathfrak{G} \in \mathcal{P}_q(\mathcal{T}_h)$ for some $q \ge 0$ with $(\mathfrak{G}, \xi_h)_{\Omega} = 0$ for all $\xi_h \in \mathcal{R}\mathcal{T}_1(\mathcal{T}_h) \cap \mathcal{H}_{\Gamma_N}(\text{div}^0, \Omega)$. For each $\ell \in \mathcal{E}_h$, the local problem in (6.4a) is well-posed and uniquely defines $\sigma_{h,\ell}$. Besides, for $1 \le k \le 3$, we have $\sigma_h^k \in \mathcal{H}_{\Gamma_D}(\text{div}^0, \Omega)$.

*Proof.* As far as the well-posedness of the local problems in (6.4a) are well-posed, we simply need to check that

$$ (\mathfrak{G} \cdot \nabla \times \psi_\ell, 1)_{\omega_\ell} = (\mathfrak{G}, \nabla \times \psi_\ell)_{\Omega} = 0 $$

whenever $\ell \notin \bar{\Gamma}_N$. This, however, immediately follows by assumption, since $\nabla \times \psi_\ell \in \mathcal{R}\mathcal{T}_1(\mathcal{T}_h) \cap \mathcal{H}_{\Gamma_N}(\text{div}^0 \Omega)$ in this case.

Let $1 \le k \le 3$. The fact that $\sigma_h^k \in \mathcal{H}_{\Gamma_D}(\text{div}, \Omega)$ readily follows from the fact that each $\sigma_{h,\ell} \in \mathcal{H}_0(\text{div}, \omega_\ell)$. Besides, the constraint $\nabla \cdot \sigma_{h,\ell} = 0$ may be obtained directly from the partition of unity property of the edge functions stated in (6.2). Indeed, we have

$$ \nabla \cdot \sigma_h^k = \mathfrak{G} \cdot \nabla \times \sum_{\ell \in \mathcal{E}_h} (\tau_\ell \cdot e^k) \psi_\ell = \mathfrak{G} \cdot \nabla \times e^k = 0. $$

□

**Remark 6.4 (Reconstruction in smaller discrete spaces).** It is also possible to reconstruct $\sigma_{h,\ell}$ in $\mathcal{P}_{q+1}(\mathcal{T}_h^\ell)$ rather than $\mathcal{R}\mathcal{T}_{q+2}(\mathcal{T}_h^\ell)$, resulting in a more cost effective reconstruction. However, while it is still possible to show that the resulting estimator is efficient, its polynomial-degree-robustness cannot be directly infere from the literature in this case.

**6.3. Reliability.** As a direct consequence of Theorems 6.2 and 6.3, we have the following reliability result.

**Theorem 6.5 (Reliability).** Let $\mathfrak{G} \in \mathcal{P}_q(\mathcal{T}_h)$ for some $q \ge 0$ such that $(\mathfrak{G}, \xi_h)_{\Omega} = 0$ for all $\xi_h \in \mathcal{R}\mathcal{T}_1(\mathcal{T}_h) \cap \mathcal{H}_{\Gamma_N}(\text{div}^0, \Omega)$, and assume that for $1 \le k \le 3$, $\sigma_h^k \in \mathcal{R}\mathcal{T}_{q+1}(\mathcal{T}_h)$ has been constructed following (6.4). Then, we have

$$ \min_{s \in H_{\Gamma_D}^1(\Omega)} \|\mathfrak{G} - \nabla s\|_{\underline{\mathcal{A}}, \Omega}^2 \le C_L^2 \sum_{k=1}^3 \|\mathfrak{G} \times e^k - \sigma_h^k\|_{\underline{\mathcal{A}}, \Omega}^2. $$

**6.4. Efficiency.** We now establish an efficiency result. We start with a preliminary result concerning the local contributions $\sigma_{h,\ell}$.

**Lemma 6.6 (Efficiency of contributions).** Assume that $\mathfrak{G} \in \mathcal{P}_q(\mathcal{T}_h)$ for some $q \ge 0$. Then for all $\ell \in \mathcal{E}_h$, the local lower bound

$$ (6.5) \qquad \|\mathfrak{G} \times \psi_\ell - \sigma_{h,\ell}\|_{\underline{\mathcal{A}},\omega_\ell} \le C(\kappa_{\mathcal{T}_h^\ell}) \sqrt{\frac{\alpha_{\max,\omega_\ell}}{\alpha_{\min,\omega_\ell}}} \min_{s \in H_{\ell,\mathrm{D}}^1(\omega_\ell)} \|\mathfrak{G} - \nabla s\|_{\underline{\mathcal{A}},\omega_\ell} $$

holds true.

*Proof.* We start by considering the “continuous” version of the minimization problem in (6.4a), namely:

$$ \sigma_{\ell} := \arg \min_{v \in H_0^1(\text{div}, \omega_{\ell})} \|\mathfrak{G} \times \psi_{\ell} - v\|_{\underline{\mathcal{A}}, \omega_{\ell}}. $$
Following [13, 21, 36, 25], a discrete stable minimization result shows that

$$ (6.6) \qquad \|\mathfrak{G} \times \psi_{\ell} - \sigma_{h,\ell}\|_{\underline{\mathcal{A}},\omega_{\ell}} \le C(\kappa_{\mathcal{T}_h^\ell}) \sqrt{\frac{\alpha_{\max,\omega_{\ell}}}{\alpha_{\min,\omega_{\ell}}}} \|\mathfrak{G} \times \psi_{\ell} - \sigma_{\ell}\|_{\underline{\mathcal{A}},\omega_{\ell}}, $$

and it remains to estimate the right-hand side.

The associated Euler-Lagrange equations are to find $\sigma_{\ell} \in H_0(\text{div}, \omega_{\ell})$ and $q_{\ell} \in L_0^2(\omega_{\ell})$ such that

$$ (6.7) \qquad \left\{ \begin{array}{l} (\underline{\mathcal{A}}\sigma_{\ell}, \mathbf{v})_{\omega_{\ell}} + (q_{\ell}, \nabla \cdot \mathbf{v})_{\omega_{\ell}} = (\underline{\mathcal{A}}\mathfrak{G} \times \psi_{\ell}, \mathbf{v})_{\omega_{\ell}} \\ (\nabla \cdot \sigma_{\ell}, r)_{\omega_{\ell}} = (\mathfrak{G} \cdot \nabla \times \psi_{\ell}, r)_{\omega_{\ell}} \end{array} \right. $$

for all $\mathbf{v} \in H_0(\text{div}, \omega_{\ell})$ and $r \in L_0^2(\omega_{\ell})$.

From the first equation in (6.7), we read that $q_{\ell} \in H_{\star}^1(\omega_{\ell})$ with $\underline{\mathcal{A}}^{-1}\nabla q_{\ell} = \sigma_{\ell} - \mathfrak{G} \times \psi_{\ell}$. In particular, we have

$$ (6.8) \qquad \|\mathfrak{G} \times \psi_{\ell} - \sigma_{\ell}\|_{\underline{\mathcal{A}},\omega_{\ell}} = \|\nabla q_{\ell}\|_{\underline{\mathcal{A}}^{-1},\Omega}. $$

On the other hand, taking arguments in $r \in H_{\star}^1(\omega_{\ell}) \cap L_0^2(\omega_{\ell})$ in the second equation shows that

$$ -(\sigma_{\ell}, \nabla r)_{\omega_{\ell}} = (\mathfrak{G} \cdot \nabla \times \psi_{\ell}, r)_{\omega_{\ell}} $$

and it follows that

$$ -(\underline{\mathcal{A}}^{-1}\nabla q_{\ell} + \mathfrak{G} \times \psi_{\ell}, \nabla r)_{\omega_{\ell}} = (\mathfrak{G} \cdot \nabla \times \psi_{\ell}, r)_{\omega_{\ell}}, $$

which we rewrite as

$$ \begin{aligned} (\underline{\mathcal{A}}^{-1}\nabla q_{\ell}, \nabla r)_{\omega_{\ell}} &= -(\mathfrak{G} \cdot \nabla \times \psi_{\ell}, r)_{\omega_{\ell}} - (\mathfrak{G} \times \psi_{\ell}, \nabla r)_{\omega_{\ell}} \\ &= -(\mathfrak{G}, r\nabla \times \psi_{\ell} + \nabla r \times \psi_{\ell})_{\omega_{\ell}} = -(\mathfrak{G}, \nabla \times (r\psi_{\ell}))_{\omega_{\ell}} = -(\mathfrak{G} - \nabla s, \nabla \times (r\psi_{\ell}))_{\omega_{\ell}} \end{aligned} $$

for all $s \in H_{\gamma_{\ell,D}}^1(\omega_{\ell})$. Recalling that $q_{\ell} \in H_{\star}^1(\omega_{\ell}) \cap L_0^2(\omega_{\ell})$ and taking $r = q_{\ell}$, it follows that

$$ (6.9) \qquad \|\nabla q_{\ell}\|_{\underline{\mathcal{A}}^{-1},\omega_{\ell}} \le C(\kappa_{\mathcal{T}_h^\ell}) \min_{s \in H_{\gamma_{\ell,D}}^1(\omega_{\ell})} \|\mathfrak{G} - \nabla s\|_{\underline{\mathcal{A}},\omega_{\ell}}, $$

where we employed (6.3). The estimate in (6.5) now follows from (6.6), (6.8) and (6.9). $\square$

<figure><img src="image_4.png" /></figure>

We are now ready to give the main efficiency estimate of this section.

**Theorem 6.7 (Efficiency).** Let $\mathfrak{G} \in \mathcal{P}_q(\mathcal{T}_h)$ for some $q \ge 0$. For all $K \in \mathcal{T}_h$, the local lower bound

$$ (6.10) \qquad \sum_{k=1}^{3} \|\mathfrak{G} \times e^k - \sigma_h^k\|_{\underline{\mathcal{A}},K}^2 \le C(\kappa_{\mathcal{T}_h^\ell}) \frac{\alpha_{\max,\omega_K}}{\alpha_{\min,\omega_K}} \|\mathfrak{G} - \nabla u\|_{\underline{\mathcal{A}},\tilde{\omega}_K}^2 $$

holds true.

*Proof.* Fix $K \in \mathcal{T}_h$ and $1 \le k \le 3$. We use the partition of unity property (6.2) of the edge functions to write that

$$ \mathfrak{G} \times e^k|_K = \sum_{\ell \in \mathcal{E}(K)} (\tau_\ell \cdot e^k) \mathfrak{G} \times \tau_\ell, $$

and it follows from the definition of $\sigma_h^k$ that

$$ \|\mathfrak{G} \times e^k - \sigma_h^k\|_{\underline{\mathcal{A}},K} = \| \sum_{\ell \in \mathcal{E}(K)} (\tau_\ell \cdot e^k)(\mathfrak{G} \times \psi_\ell - \sigma_{h,\ell}) \|_{\underline{\mathcal{A}},K} \le C(\kappa_{\mathcal{T}_h^\ell}) \sum_{\ell \in \mathcal{E}(K)} \|\mathfrak{G} \times \psi_\ell - \sigma_{h,\ell}\|_{\underline{\mathcal{A}},\omega_\ell}, $$

and (6.10) follows from (6.5), with $s = u$. $\square$

<figure><img src="image_5.png" /></figure>
6.5. **Alternative error estimators for IPDG and mixed FEM.** The application of the alternatively equilibrated fluxes to the IPDG scheme and the mixed FEM discretization follows closely the procedure described in Sections 5.4 and 5.5 and is not detailed here for shortness.

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