<figure><img src="image_1.png" /></figure>

HAL
open science

# A stable local commuting projector with local-best approximation properties in H1

Théophile Chaumont-Frelet, Martin Vohralík

► To cite this version:

Théophile Chaumont-Frelet, Martin Vohralík. A stable local commuting projector with local-best approximation properties in H1. 2025. hal-05383571

HAL Id: hal-05383571

https://inria.hal.science/hal-05383571v1

Preprint submitted on 26 Nov 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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

Distributed under a Creative Commons Attribution 4.0 International License
# A STABLE LOCAL COMMUTING PROJECTOR WITH LOCAL-BEST APPROXIMATION PROPERTIES IN $H^1$

T. CHAUMONT-FRELET<fnref n="1" /> AND M. VOHRALÍK<fnref n="2" />

**ABSTRACT.** We introduce a quasi-interpolation operator that maps the infinite-dimensional Sobolev space $H^1$ into its finite-dimensional Lagrange finite element subspace formed by piecewise polynomials on a tetrahedral mesh. This operator enjoys the following key properties: 1) it is defined over the entire $H^1$ and includes essentially boundary conditions imposed on a part of the boundary; 2) it is defined locally in a neighborhood of each tetrahedron of the mesh; 3) it is based on simple piecewise polynomial projections; 4) it is stable in the $H^1$-seminorm; 5) it has optimal (local-best) approximation properties; 6) it commutes with its sibling operator on $H(\text{curl})$; 7) it is a projector.

## 1. INTRODUCTION

Quasi-interpolation operators $P_h^g$ from the infinite-dimensional Sobolev space $H^1(\Omega)$ into its finite-dimensional Lagrange subspace formed by piecewise polynomials on a tetrahedral mesh are crucial in analysis of finite element methods. Important examples are the operators by Clément [8], Scott-Zhang [15], or Gawlik, Holst, and Licht [13], see also the references therein, which can indeed act on the entire $H^1(\Omega)$ space; there is in particular no point evaluation as in the standard Lagrange interpolation operator (cf. Ciarlet [7] or Ern and Guermond [10]). Unfortunately, $P_h^g$ may not be a projection, i.e. $P_h^g u_h = u_h$ may not hold for all Lagrange piecewise polynomial $u_h$, and $P_h^g$ may not commute with the forthcoming operator in the de Rham sequence in that it is not true that $\nabla P_h^g(u) = P_h^c(\nabla u)$ for all $u \in H^1(\Omega)$, where $P_h^c$ is a quasi-interpolation operator on $H(\text{curl}, \Omega)$.

Stable local commuting projections, designed namely in the finite element exterior calculus framework, see Falk and Winther [11], Arnold and Guzmán [2], and the references therein, remove the two above deficiencies. Still, these operators typically do not satisfy another important requirement: their approximation property may not be that of elementwise orthogonal projection, leading to the so-called local-best-global-best equivalence as per Veeser [16].

Recently, quasi-interpolation operators $P_h^d$ over $H(\text{div}, \Omega)$ and $P_h^c$ over $H(\text{curl}, \Omega)$ have been designed in respectively Ern et al. [9] and Chaumont-Frelet and Vohralík [5]. These operators satisfy the following properties: 1) they are defined over the entire Sobolev spaces $H(\text{div}, \Omega)$ and $H(\text{curl}, \Omega)$, respectively, with essentially boundary conditions possibly imposed on a part of the boundary of $\Omega$ only; 2) they are defined locally, in a neighborhood of each tetrahedron of the mesh; 3) they are based on simple piecewise polynomial projections; 4) they are stable in the $L^2$-norm, up to data oscillation; 5) they have optimal approximation properties, that of unconstrained elementwise orthogonal projection, yielding local-best

<footnote n="1">¹Inria, Univ. Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé, 40 Av. Halley, 59650 Villeneuve-d'Ascq, France. theophile.chaumont@inria.fr</footnote>

<footnote n="2">²Inria, 48 rue Barrault, 75647 Paris, France & CERMICS, Ecole nationale des ponts et chaussées, IP Paris, 77455 Marne-la-Vallée, France. martin.vohralik@inria.fr</footnote>
global-best equivalence; 6) they commute with the forthcoming operators in the de Rham sequence, namely $\nabla \cdot P_h^d(v) = \Pi_h(\nabla \cdot v)$ for all $v \in H(\text{div}, \Omega)$, where $\Pi_h$ is the $L^2$-orthogonal projection onto (discontinuous) piecewise polynomials, and $\nabla \times P_h^c(v) = P_h^d(\nabla \times v)$ for all $v \in H(\text{curl}, \Omega)$; 7) are projections.

The goal of the present contribution is to complement the operators $P_h^d$ and $P_h^c$ from [9, 5] by an operator $P_h^g$ satisfying all the 7 optimal properties above. Namely, in our main result, Theorem 3.1 below, we achieve the requested commuting property (3.5) and the optimal local-best approximation property (3.3). An optimal $L^2$-approximation property is achieved as well, see (3.4).

## 2. NOTATION

**2.1. Domain and boundary partition.** We consider an open weakly Lipschitz polyhedral domain $\Omega \subset \mathbb{R}^3$. The boundary of $\Omega$ is partitioned as $\partial\Omega = \overline{\Gamma_D} \cup \overline{\Gamma_N}$, where $\Gamma_D$ and $\Gamma_N$ are two relatively open polygonal subsets of $\partial\Omega$, both with Lipschitz boundaries (this excludes checkerboard patterns).

**2.2. Function spaces.** For an open weakly Lipschitz subset $D \subset \Omega$, $L^2(D)$ denotes the usual Lebesgue space of square-integrable functions, and $L^2(D) := [L^2(D)]^3$ collects square-integrable vector-valued fields. The usual norm and inner product of both spaces are respectively denoted by $\|\cdot\|_D$ and $(\cdot, \cdot)_D$. We write $H^1(D)$, $H(\text{curl}, D)$, and $H(\text{div}, D)$ for the Sobolev spaces respectively containing functions $v \in L^2(D)$ with $\nabla v \in L^2(D)$, $z \in L^2(D)$ with $\nabla \times z \in L^2(D)$, and $w \in L^2(D)$ with $\nabla \cdot w \in L^2(D)$, where $\nabla$, $\nabla \times$, and $\nabla \cdot$ respectively stand for the weak gradient, curl, and divergence. For a general presentation of these spaces, we refer the reader to [1, 14]. If $\gamma \subset \partial D$ is a relatively open subset of the boundary, we respectively denote by $H_\gamma^1(D)$, $H_\gamma(\text{curl}, D)$, and $H_\gamma(\text{div}, D)$ the subspaces containing functions with vanishing trace, tangential trace, and normal trace on $\gamma$. For spaces of vector-valued fields, this notion is precisely defined in [12].

**2.3. Tetrahedral mesh.** In the remainder of this work, we fix a matching simplicial mesh $\mathcal{T}_h$ of $\Omega$ consisting of a finite number of (open) simplices $K$. By matching, we mean that if the intersection $\overline{K}_- \cap \overline{K}_+$ of two elements $K_\pm \in \mathcal{T}_h$ is non-empty, then it is a full subsimplex (i.e. a vertex, an edge, or a face) of both simplices. We also demand that the union of the (closure of the) elements is (the closure of) $\Omega$, and that each boundary face either belongs to $\Gamma_D$ or to $\Gamma_N$.

**2.4. Shape-regularity parameter.** We denote by $h_K$ the diameter of an element $K \in \mathcal{T}_h$ and by $\rho_K$ the diameter of the largest ball contained in $K$. The shape regularity parameter of the tetrahedron $K$ is then defined as $\kappa_K := h_K/\rho_K \ge 1$. If $\mathcal{T} \subset \mathcal{T}_h$, we also employ the notation

$$ (2.1) \qquad \kappa_{\mathcal{T}} := \max_{K \in \mathcal{T}} \kappa_K. $$

**2.5. Patches.** For $\ell = 0$, we let $\mathcal{T}_K^{(0)} := \{K\}$. Then, by induction, we let

$$ \mathcal{T}_K^{(\ell)} := \{ K'' \in \mathcal{T}_h \mid \exists K' \in \mathcal{T}_K^{(\ell-1)}; \overline{K''} \cap \overline{K'} \neq \emptyset \} $$

for $\ell \ge 1$. For $\ell = 1$, $\mathcal{T}_K^{(1)}$ is the set of mesh elements sharing a vertex with $K$ and any further increase of $\ell$ brings an additional layer of vertex neighbors.
We denote by $\mathcal{V}_h$ the set of vertices of the mesh. If $\mathbf{a} \in \mathcal{V}_h$, we denote by

$$ \mathcal{T}_{\mathbf{a}} := \{K \in \mathcal{T}_h \mid \mathbf{a} \in \overline{K}\} $$

the so-called “vertex patch” of $\mathbf{a}$. The corresponding open subdomain of $\Omega$ is denoted by $\omega_{\mathbf{a}}$ and its diameter by $h_{\mathbf{a}}$. We also generalize this notion by setting $\mathcal{T}_{\mathbf{a}}^{(0)} := \mathcal{T}_{\mathbf{a}}$ and then,

$$ \mathcal{T}_{\mathbf{a}}^{(\ell)} := \{K'' \in \mathcal{T}_h \mid \exists K' \in \mathcal{T}_{\mathbf{a}}^{(\ell-1)}; \overline{K''} \cap \overline{K'} \neq \emptyset\} $$

for $\ell \ge 1$. As above, increasing $\ell$ adds layers of vertex neighbors. We denote by $\tilde{\omega}_{\mathbf{a}} \subset \Omega$ the open domain covered by the elements in $\mathcal{T}_{\mathbf{a}}^{(2)}$.

**2.6. Mesh fineness assumption.** In the remainder of this work, we ask that the mesh is fine enough so that $\tilde{\omega}_{\mathbf{a}}$ is simply connected, and that $\partial\tilde{\omega}_{\mathbf{a}} \setminus \Gamma_N$ is non-empty and connected. In this case, we let

$$ H_0^1(\tilde{\omega}_{\mathbf{a}}) := H_{\partial\tilde{\omega}_{\mathbf{a}} \setminus \Gamma_N}^1(\tilde{\omega}_{\mathbf{a}}) $$

and

$$ H_0(\mathrm{curl}, \tilde{\omega}_{\mathbf{a}}) := H_{\partial\tilde{\omega}_{\mathbf{a}} \setminus \Gamma_N}(\tilde{\omega}_{\mathbf{a}}). $$

In this setting, the Poincaré inequality

$$ (2.2) \qquad \|v\|_{\tilde{\omega}_{\mathbf{a}}} \lesssim h_{\mathbf{a}} \|\nabla v\|_{\tilde{\omega}_{\mathbf{a}}} $$

holds true for all $v \in H_0^1(\tilde{\omega}_{\mathbf{a}})$ with a hidden constant solely depending on $\kappa_{\mathcal{T}_h}$, see e.g. [17]. Besides, we have

$$ (2.3) \qquad H_0(\mathrm{curl}^0, \tilde{\omega}_{\mathbf{a}}) = \nabla H_0^1(\tilde{\omega}_{\mathbf{a}}), $$

where $H_0(\mathrm{curl}^0, \tilde{\omega}_{\mathbf{a}})$ is the subset of $H_0(\mathrm{curl}, \tilde{\omega}_{\mathbf{a}})$ consisting of curl-free vector fields.

**2.7. Finite element spaces.** Throughout this work, for a tetrahedron $K \in \mathcal{T}_h$ and a polynomial degree $q \ge 0$, $\mathcal{P}_q(K)$ is the space of polynomials of degree at most $q$ defined on $K$, and we set $\mathcal{P}_q(K) := [\mathcal{P}_q(K)]^3$. We further respectively denote by $\mathcal{N}_q(K) := \mathcal{P}_q(K) + \mathbf{x} \times \mathcal{P}_q(K)$ and $\mathcal{R}\mathcal{T}_q(K) := \mathcal{P}_q(K) + \mathbf{x}\mathcal{P}_q(K)$ the spaces of Nédélec and Raviart-Thomas polynomials, cf. [3, 10]. If $\mathcal{T} \subset \mathcal{T}_h$ is a collection of elements, then $\mathcal{P}_q(\mathcal{T})$, $\mathcal{N}_q(\mathcal{T})$, and $\mathcal{R}\mathcal{T}_q(\mathcal{T})$ respectively consist of functions whose elementwise restriction to each $K \in \mathcal{T}$ belongs to $\mathcal{P}_q(K)$, $\mathcal{N}_q(K)$, and $\mathcal{R}\mathcal{T}_q(K)$, without any requirement of interelement connection.

**2.8. Partition of unity.** We will need the standard partition of unity formed by the “hat” functions. Specifically, for each vertex $\mathbf{a} \in \mathcal{V}_h$, there exists a unique $\psi_{\mathbf{a}} \in H^1(\Omega) \cap \mathcal{P}_1(\mathcal{T}_h)$ such that $\psi_{\mathbf{a}}(\mathbf{b}) = \delta_{\mathbf{a},\mathbf{b}}$ for all $\mathbf{b} \in \mathcal{V}_h$, where $\delta$ is the Kronecker symbol. For each $\mathbf{a} \in \mathcal{V}_h$, we have

$$ (2.4) \qquad \operatorname{supp} \psi_{\mathbf{a}} = \bigcup_{K \in \mathcal{T}_{\mathbf{a}}^{(0)}} \overline{K} = \overline{\omega_{\mathbf{a}}} $$

and

$$ (2.5) \qquad \|\psi_{\mathbf{a}}\|_{L^{\infty}(\Omega)} + h_{\mathbf{a}} \|\nabla \psi_{\mathbf{a}}\|_{L^{\infty}(\Omega)} \lesssim 1, $$

with a hidden constant only depending on the local shape-regularity parameter $\kappa_{\mathcal{T}_{\mathbf{a}}}$ from (2.1). Besides, the partition of unity property

$$ (2.6) \qquad \sum_{\mathbf{a} \in \mathcal{V}_h} \psi_{\mathbf{a}} = 1 $$

holds true pointwise in $\Omega$.
2.9. **Commuting projectors.** In the remainder of this work, we respectively denote by $P_h^c: H_{\Gamma_D}(\text{curl}, \Omega) \to \mathcal{N}_p(\mathcal{T}_h) \cap H_{\Gamma_D}(\text{curl}, \Omega)$ and $P_h^d: H_{\Gamma_D}(\text{div}, \Omega) \to \mathcal{RT}_p(\mathcal{T}_h) \cap H_{\Gamma_D}(\text{div}, \Omega)$ the projectors introduced in [5] and [9]. These projectors have, among others, the property that

$$ (2.7) \qquad \nabla \times P_h^c(v) = P_h^d(\nabla \times v) $$

for all $v \in H_{\Gamma_D}(\text{curl}, \Omega)$.

### 3. MAIN RESULT

**Theorem 3.1** (A stable local commuting projector with local-best approximation in $H^1$). Let $p \ge 0$ be a fixed polynomial degree. There exists an operator

$$ (3.1) \qquad P_h^g: H_{\Gamma_D}^1(\Omega) \to \mathcal{P}_{p+1}(\mathcal{T}_h) \cap H_{\Gamma_D}^1(\Omega) $$

with the following properties. 1) It is defined over the entire $H_{\Gamma_D}^1(\Omega)$. 2) It is defined locally in a neighborhood of each tetrahedron $K \in \mathcal{T}_h$. 3) It is based on simple piecewise polynomial projections of $\nabla(\psi_a u)$. 4) It is stable in the $H^1$-seminorm norm: for all $u \in H_{\Gamma_D}^1(\Omega)$ and all $K \in \mathcal{T}_h$, there holds

$$ (3.2) \qquad \|\nabla(P_h^g(u))\|_K \lesssim \sum_{K' \in \mathcal{T}_K^{(2)}} \|\nabla u\|_{K'} $$

5) It has the optimal (local-best) approximation property: for all $u \in H_{\Gamma_D}^1(\Omega)$ and all $K \in \mathcal{T}_h$, there holds

$$ (3.3) \qquad \|\nabla(u - P_h^g(u))\|_K \lesssim \sum_{K' \in \mathcal{T}_K^{(2)}} \min_{v_h \in \mathcal{N}_p(K')} \|\nabla u - v_h\|_{K'} $$

and

$$ (3.4) \qquad \|u - P_h^g(u)\|_K \lesssim \sum_{K' \in \mathcal{T}_K^{(4)}} h_{K'} \min_{v_h \in \mathcal{N}_p(K')} \|\nabla u - v_h\|_{K'} $$

The constants hidden in (3.2)-(3.4) only depend on the local shape-regularities $\kappa_{\mathcal{T}_K^{(2)}}$ and $\kappa_{\mathcal{T}_K^{(4)}}$, respectively, as well as on the polynomial degree p. 6) It commutes with $P_h^c$,

$$ (3.5) \qquad \nabla P_h^g(u) = P_h^c(\nabla u) \quad \forall u \in H_{\Gamma_D}^1(\Omega). $$

7) It is a projector,

$$ (3.6) \qquad P_h^g(u_h) = u_h \quad \forall u_h \in \mathcal{P}_{p+1}(\mathcal{T}_h) \cap H_{\Gamma_D}^1(\Omega). $$

### 4. PROOF OF THE MAIN RESULT

In this section, we provide the definition of $P_h^g$ from Theorem 3.1 and establish its main properties. We first note that since we want the commuting property (3.5), the operator $P_h^g$ is essentially already defined since (3.5) states that

$$ \nabla P_h^g(u) = P_h^c(\nabla u) $$

for all $u \in H_{\Gamma_D}^1(\Omega)$. This equation indeed defines $P_h^g$ if $\Gamma_D \neq \emptyset$ (otherwise up to constant). However, this definition is not a local one, and is not suited to obtain error estimates in the $L^2(\Omega)$ norm (3.4). We thus need to come up with an equivalent local definition, which is as follows:
**Definition 4.1** (Local definition of $P_h^g$). Let $u \in H_{\Gamma_D}^1(\Omega)$. We define

$$ (4.1) \qquad P_h^g(u) := \sum_{a \in \mathcal{V}_h} u_h^a, $$

where, for each mesh vertex $a \in \mathcal{V}_h$, the local contributions $u_h^a \in \mathcal{P}_{p+1}(\mathcal{T}_a^{(2)}) \cap H_0^1(\tilde{\omega}_a)$ (extended by zero) are given by

$$ (4.2) \qquad \nabla u_h^a := P_h^c(\nabla(\psi_a u)). $$

**Lemma 4.2** (Well-posedness of the local definitions (4.2)). For all $u \in H_{\Gamma_D}^1(\Omega)$ and all $a \in \mathcal{V}_h$, there exists a unique $u_h^a \in \mathcal{P}_{p+1}(\mathcal{T}_a^{(2)}) \cap H_0^1(\tilde{\omega}_a)$ satisfying (4.2).

*Proof.* Let $u \in H_{\Gamma_D}^1(\Omega)$ and $a \in \mathcal{V}_h$ be fixed. There holds $\nabla(\psi_a u) \in H_{\Gamma_D}(\text{curl}, \Omega)$, so that we can apply the projector $P_h^c : H_{\Gamma_D}(\text{curl}, \Omega) \to \mathcal{N}_p(\mathcal{T}_h) \cap H_{\Gamma_D}(\text{curl}, \Omega)$ from [5] to $\nabla(\psi_a u)$. Thanks to the local construction and local stability of $P_h^c$, namely using property (3.18) established in [5, Theorem 1], we know that $P_h^c(\nabla(\psi_a u))|_K = 0$ if $\mathcal{T}_K^{(0)} \cap \mathcal{T}_a^{(2)} = \emptyset$, since $\psi_a$ and thus $\nabla(\psi_a u)$ are locally supported on $\mathcal{T}_a^{(0)}$ as per (2.4). Since $P_h^c(\nabla(\psi_a u)) \in H_{\Gamma_D}(\text{curl}, \Omega)$ by construction, we conclude that $P_h^c(\nabla(\psi_a u)) \in \mathcal{H}_0(\text{curl}, \tilde{\omega}_a)$. Due to the commuting property in (2.7), since $\nabla \times P_h^c(\nabla(\psi_a u)) = P_h^d(\nabla \times \nabla(\psi_a u)) = P_h^d(0) = 0$, we conclude that $P_h^c(\nabla(\psi_a u)) \in \mathcal{H}_0(\text{curl}^0, \tilde{\omega}_a)$. Since the patch $\tilde{\omega}_a$ equipped with the associated boundary conditions is topologically trivial by assumption, the existence of a suitable $u_h^a \in H_0^1(\tilde{\omega}_a)$ is ensured by (2.3). Uniqueness then readily follows from the fact that the Dirichlet boundary condition on $\tilde{\omega}_a$ is non-empty by assumption. Finally, since $P_h^c$ maps into $\mathcal{N}_p(\mathcal{T}_h)$ and as curl-free Nédélec functions are $\mathcal{P}_p(\mathcal{T}_h)$ polynomials, see [3, Lemma 2.3.4 and Corollary 2.3.1], we must have $u_h^a \in \mathcal{P}_{p+1}(\mathcal{T}_h)$. $\square$

We next establish the following $L^2$ stability estimate:

**Lemma 4.3** ($L^2$ stability estimate). For all $u \in H_{\Gamma_D}^1(\Omega)$ and all $K \in \mathcal{T}_h$, we have

$$ (4.3) \qquad \|P_h^g(u)\|_K \lesssim \sum_{K' \in \mathcal{T}_K^{(3)}} (\|u\|_{K'} + h_{K'} \|\nabla u\|_{K'}), $$

where the hidden constant only depends on the local shape-regularity $\kappa_{\mathcal{T}_K^{(3)}}$.

*Proof.* Setting

$$ \mathcal{V}_K := \{a \in \mathcal{V}_h \mid K \in \mathcal{T}_a^{(2)}\}, $$

we exploit the local definition of $P_h^g(u)$ to write that

$$ \|P_h^g(u)\|_K = \left\| \sum_{a \in \mathcal{V}_K} u_h^a \right\|_K \le \sum_{a \in \mathcal{V}_K} \|u_h^a\|_{\tilde{\omega}_a}. $$

Since each $u_h^a \in H_0^1(\tilde{\omega}_a)$, the Poincaré inequality in (2.2) gives that

$$ \|P_h^g(u)\|_K \lesssim \sum_{a \in \mathcal{V}_K} h_a \|\nabla u_h^a\|_{\tilde{\omega}_a} = \sum_{a \in \mathcal{V}_K} h_a \|P_h^c(\nabla(\psi_a u))\|_{\tilde{\omega}_a}. $$

We then employ the stability estimate (3.18) in [5, Theorem 1]. This gives

$$ \|P_h^g(u)\|_K \lesssim \sum_{a \in \mathcal{V}_K} h_a \|\nabla(\psi_a u)\|_{\omega_a}, $$
since $\nabla \times \nabla(\psi_a u) = 0$. We further employ the product rule together with (2.5)

$$ \|\nabla(\psi_a u)\|_{\omega_a} \le \|\nabla \psi_a u\|_{\omega_a} + \|\psi_a \nabla u\|_{\omega_a} \stackrel{(2.5)}{\lesssim} \|\nabla u\|_{\omega_a} + h_a^{-1} \|u\|_{\omega_a}, $$

which leads us to

$$ \|P_h^g(u)\|_K \lesssim \sum_{a \in \mathcal{V}_K} \{\|u\|_{\omega_a} + h_a \|\nabla u\|_{\omega_a}\}. $$

The conclusion now follows from observing that all elements involved in the sum belong to $\mathcal{T}_K^{(3)}$, and using the fact that for such elements $K'$, we have $h_{K'} \sim h_a$ for one of the involved vertices up to constants depending on $\kappa_{\mathcal{T}_K^{(3)}}$. □

*Proof of Theorem 3.1.* Properties 1)–3) Properties 1)–3) follow immediately from the construction of Definition 4.1 and from the definition of the projector $P_h^c : H_{\Gamma_D}(\text{curl}, \Omega) \to \mathcal{N}_p(\mathcal{T}_h) \cap H_{\Gamma_D}^1(\text{curl}, \Omega)$ in [5].

Property 6) We observe that by linearity, since the hat functions form a partition of unity as per (2.6), we do have

$$ \nabla P_h^g(u) = \sum_{a \in \mathcal{V}_h} \nabla u_h^a = \sum_{a \in \mathcal{V}_h} P_h^c(\nabla(\psi_a u)) = P_h^c(\nabla u) $$

for all $u \in H_{\Gamma_D}^1(\Omega)$, which is the commuting property in (3.5).

Property 7) The fact that $P_h^g$ is a projector is a direct consequence of (3.3) and (4.2).

Property 5) We notice that (3.3) is a direct consequence of property (3.17) in [5, Theorem 1]. Indeed, since $\nabla \times \nabla u = 0$, we have

$$ \|\nabla(u - P_h^g(u))\|_K = \|(\nabla u) - P_h^c(\nabla u)\|_K \lesssim \sum_{K' \in \mathcal{T}_K^{(2)}} \min_{v_h \in \mathcal{N}_p(K')} \|\nabla u - v_h\|_{K'}. $$

To prove (3.4), we will use the quasi-interpolation operator $J_h^g : H_{\Gamma_D}^1(\Omega) \to \mathcal{P}_{p+1}(\mathcal{T}_h) \cap H_{\Gamma_D}^1(\Omega)$, from [6] (recall that we start from $p=0$ here). Specifically, employing Theorems 4.2 and 4.3 of this reference, we infer

$$ \|u - J_h^g(u)\|_K + h_K \|\nabla(u - J_h^g(u))\|_K \lesssim \sum_{K' \in \mathcal{T}_K^{(1)}} h_{K'} \min_{v_h \in \mathcal{N}_p(K')} \|\nabla u - v_h\|_{K'}. $$

Since $\nabla \mathcal{P}_{p+1}(K') = \{v_h \in \mathcal{N}_p(K') | \nabla \times v_h = 0\}$, cf. [3, equation (2.3.56)], this for the moment only gives the approximation power of curl-free Nédélec polynomials on $K'$. However, employing further [4, Lemma A.1], this actually yields the approximation power of all Nédélec polynomials on $K'$,

$$ (4.4) \qquad \|u - J_h^g(u)\|_K + h_K \|\nabla(u - J_h^g(u))\|_K \lesssim \sum_{K' \in \mathcal{T}_K^{(1)}} h_{K'} \min_{v_h \in \mathcal{N}_p(K')} \|\nabla u - v_h\|_{K'}. $$

Adding and subtracting $J_h^g(u)$ and using the triangle inequality, we now write that

$$ \|u - P_h^g(u)\|_K \le \|u - J_h^g(u)\|_K + \|P_h^g(u - J_h^g(u))\|_K; $$

we have also crucially used that we have already established that $P_h^g$ is a projector as per (3.6). It now follows from (4.3) that

$$ \|u - P_h^g(u)\|_K \lesssim \sum_{K' \in \mathcal{T}_K^{(3)}} (\|u - J_h^g(u)\|_{K'} + h_{K'} \|\nabla(u - J_h^g(u))\|_{K'}), $$
so that (3.4) follows from (4.4).

Property 4) The stability (3.2) is a direct consequence of (3.3). Indeed, by the triangle inequality and (3.3), we have

$$ \begin{aligned} \|\nabla(P_h^g(u))\|_K &= \|\nabla(u - P_h^g(u)) + \nabla u\|_K \\ &\lesssim \sum_{K' \in \mathcal{T}_K^{(2)}} \min_{v_h \in \mathcal{N}_p(K')} \|\nabla u - v_h\|_{K'} + \|\nabla u\|_K \le 2 \sum_{K' \in \mathcal{T}_K^{(2)}} \|\nabla u\|_{K'}. \end{aligned} $$

□

## REFERENCES

[1] Adams, R. A., and Fournier, J. J. F. Sobolev spaces, second ed., vol. 140 of Pure and Applied Mathematics (Amsterdam). Elsevier/Academic Press, Amsterdam, 2003.

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

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

[4] Chaumont-Frelet, T., and Vohralík, M. Equivalence of local-best and global-best approximations in $H(\text{curl})$. Calcolo 58 (2021), 53. https://doi.org/10.1007/s10092-021-00430-9.

[5] Chaumont-Frelet, T., and Vohralík, M. A stable local commuting projector and optimal hp approximation estimates in $H(\text{curl})$. Numer. Math. 156 (2024), 2293–2342. https://doi.org/10.1007/s00211-024-01431-w.

[6] Chaumont-Frelet, T., and Vohralík, M. A quasi-interpolation operator yielding fully computable error bounds. ESAIM Math. Model. Numer. Anal. (2025). DOI 10.1051/m2an/2025091, https://doi.org/10.1051/m2an/2025091.

[7] Ciarlet, P. G. The finite element method for elliptic problems, vol. 40 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. https://doi.org/10.1137/1.9780898719208.

[8] Clément, P. Approximation by finite element functions using local regularization. RAIRO Anal. Numer. 9 (1975), 77–84.

[9] Ern, A., Gudi, T., Smears, I., and Vohralík, M. Equivalence of local- and global-best approximations, a simple stable local commuting projector, and optimal hp approximation estimates in $H(\text{div})$. IMA J. Numer. Anal. 42 (2022), 1023–1049. https://doi.org/10.1093/imanum/draa103.

[10] Ern, A., and Guermond, J.-L. Finite Elements I. Approximation and Interpolation, vol. 72 of Texts in Applied Mathematics. Springer International Publishing, Springer Nature Switzerland AG, 2021. https://doi.org/10.1007/978-3-030-56341-7.

[11] Falk, R. S., and Winther, R. Local bounded cochain projections. Math. Comp. 83 (2014), 2631–2656. https://doi.org/10.1090/S0025-5718-2014-02827-5.

[12] Fernandes, P., and Gilardi, G. Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. Math. Models Methods Appl. Sci. 7 (1997), 957–991. https://doi.org/10.1142/S0218202597000487.

[13] Gawlik, E., Holst, M. J., and Licht, M. W. Local finite element approximation of Sobolev differential forms. ESAIM Math. Model. Numer. Anal. 55 (2021), 2075–2099. https://doi.org/10.1051/m2an/2021034.

[14] Girault, V., and Raviart, P.-A. Finite element methods for Navier-Stokes equations, vol. 5 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 1986. https://doi.org/10.1007/978-3-642-61623-5.

[15] Scott, L. R., and Zhang, S. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990), 483–493. https://doi.org/10.2307/2008497.

[16] Veeser, A. Approximating gradients with continuous piecewise polynomial functions. Found. Comput. Math. 16 (2016), 723–750. https://doi.org/10.1007/s10208-015-9262-z.

[17] Veeser, A., and Verfürth, R. Poincaré constants for finite element stars. IMA J. Numer. Anal. 32 (2012), 30–47. https://doi.org/10.1093/imanum/drr011.