# An Arbitrary-Order Discrete de Rham Complex on Polyhedral Meshes: Exactness, Poincaré Inequalities, and Consistency

Daniele A Di Pietro, Jérôme Droniou

► To cite this version:

Daniele A Di Pietro, Jérôme Droniou. An Arbitrary-Order Discrete de Rham Complex on Polyhedral Meshes: Exactness, Poincaré Inequalities, and Consistency. Foundations of Computational Mathematics, 2021, 23 (1), pp.85-164. 10.1007/s10208-021-09542-8. hal-04536890

HAL Id: hal-04536890

https://hal.science/hal-04536890

Submitted on 8 Apr 2024

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_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# An Arbitrary-Order Discrete de Rham Complex on Polyhedral Meshes: Exactness, Poincaré Inequalities, and Consistency

Daniele A Di Pietro, Jérôme Droniou

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

## To cite this version:

Daniele A Di Pietro, Jérôme Droniou. An Arbitrary-Order Discrete de Rham Complex on Polyhedral Meshes: Exactness, Poincaré Inequalities, and Consistency. Foundations of Computational Mathematics, 2021, 23 (1), pp.85-164. 10.1007/s10208-021-09542-8. hal-04536890

HAL Id: hal-04536890

https://hal.science/hal-04536890v1

Submitted on 8 Apr 2024

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published 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
# An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness, Poincaré inequalities, and consistency

Daniele A. Di Pietro · Jérôme Droniou

Received: 28 January 2021 / Revised: 06 August 2021 / Accepted: 07 August 2021

**Abstract** In this paper we present a novel arbitrary-order discrete de Rham (DDR) complex on general polyhedral meshes based on the decomposition of polynomial spaces into ranges of vector calculus operators and complements linked to the spaces in the Koszul complex. The DDR complex is fully discrete, meaning that both the spaces and discrete calculus operators are replaced by discrete counterparts, and satisfies suitable exactness properties depending on the topology of the domain. In conjunction with bespoke discrete counterparts of $L^2$-products, it can be used to design schemes for partial differential equations that benefit from the exactness of the sequence but, unlike classical (e.g., Raviart-Thomas-Nédélec) finite elements, are nonconforming. We prove a complete panel of results for the analysis of such schemes: exactness properties, uniform Poincaré inequalities, as well as primal and adjoint consistency. We also show how this DDR complex enables the design of a numerical scheme for a magnetostatics problem, and use the aforementioned results to prove stability and optimal error estimates for this scheme.

**Keywords** Discrete de Rham complex · compatible discretisations · polyhedral methods · arbitrary order

<footnote>Communicated by Douglas Arnold.</footnote>

<footnote>The authors acknowledge the support of Agence Nationale de la Recherche through the grant NEMESIS (ANR-20-MRS2-0004-01). Daniele Di Pietro's work was also partially supported by the fast4hho grant (ANR-17-CE23-0019). Jérôme Droniou was partially supported by the Australian Government through the Australian Research Council's Discovery Projects funding scheme (grant number DP170100605). The authors also wish to thank Daniel Matthews for suggesting an explicit basis for $G^{c,l}(T)$, which was used in the implementation of DDR in versions 4.0+ of the HArDCore3D library.</footnote>

<footnote>D. A. Di Pietro</footnote>

<footnote>IMAG, Univ Montpellier, CNRS, Montpellier, France</footnote>

<footnote>E-mail: daniele.di-pietro@umontpellier.fr</footnote>

<footnote>J. Droniou</footnote>

<footnote>School of Mathematics, Monash University, Melbourne, Australia</footnote>

<footnote>E-mail: jerome.droniou@monash.edu</footnote>
# An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness, Poincaré inequalities, and consistency

Daniele A. Di Pietro · Jérôme Droniou

Received: 28 January 2021 / Revised: 06 August 2021 / Accepted: 07 August 2021

**Abstract** In this paper we present a novel arbitrary-order discrete de Rham (DDR) complex on general polyhedral meshes based on the decomposition of polynomial spaces into ranges of vector calculus operators and complements linked to the spaces in the Koszul complex. The DDR complex is fully discrete, meaning that both the spaces and discrete calculus operators are replaced by discrete counterparts, and satisfies suitable exactness properties depending on the topology of the domain. In conjunction with bespoke discrete counterparts of $L^2$-products, it can be used to design schemes for partial differential equations that benefit from the exactness of the sequence but, unlike classical (e.g., Raviart-Thomas-Nédélec) finite elements, are nonconforming. We prove a complete panel of results for the analysis of such schemes: exactness properties, uniform Poincaré inequalities, as well as primal and adjoint consistency. We also show how this DDR complex enables the design of a numerical scheme for a magnetostatics problem, and use the aforementioned results to prove stability and optimal error estimates for this scheme.

**Keywords** Discrete de Rham complex · compatible discretisations · polyhedral methods · arbitrary order

<footnote>Communicated by Douglas Arnold.</footnote>

<footnote>The authors acknowledge the support of Agence Nationale de la Recherche through the grant NEMESIS (ANR-20-MRS2-0004-01). Daniele Di Pietro's work was also partially supported by the fast4hho grant (ANR-17-CE23-0019). Jérôme Droniou was partially supported by the Australian Government through the Australian Research Council's Discovery Projects funding scheme (grant number DP170100605). The authors also wish to thank Daniel Matthews for suggesting an explicit basis for $G^{c,l}(T)$, which was used in the implementation of DDR in versions 4.0+ of the HArDCore3D library.</footnote>

<footnote>D. A. Di Pietro</footnote>

<footnote>IMAG, Univ Montpellier, CNRS, Montpellier, France</footnote>

<footnote>E-mail: daniele.di-pietro@umontpellier.fr</footnote>

<footnote>J. Droniou</footnote>

<footnote>School of Mathematics, Monash University, Melbourne, Australia</footnote>

<footnote>E-mail: jerome.droniou@monash.edu</footnote>
Mathematics Subject Classification (2020) 65N30 · 65N99 · 78A30

Contents

<table><tbody><tr><td>1</td><td>Introduction</td><td>3</td></tr><tr><td>2</td><td>Setting</td><td>7</td></tr><tr><td>2.1</td><td>Domain and mesh</td><td>7</td></tr><tr><td>2.2</td><td>Orientation of mesh entities and vector calculus operators on faces</td><td>8</td></tr><tr><td>2.3</td><td>Lebesgue and Sobolev spaces</td><td>8</td></tr><tr><td>2.4</td><td>Polynomial spaces and decompositions</td><td>9</td></tr><tr><td>2.5</td><td>Recovery operator</td><td>10</td></tr><tr><td>3</td><td>Discrete de Rham complex</td><td>15</td></tr><tr><td>3.1</td><td>Discrete spaces</td><td>15</td></tr><tr><td>3.2</td><td>Interpolators</td><td>17</td></tr><tr><td>3.3</td><td>Discrete vector calculus operators</td><td>18</td></tr><tr><td>3.3.1</td><td>Gradient</td><td>18</td></tr><tr><td>3.3.2</td><td>Curl</td><td>21</td></tr><tr><td>3.3.3</td><td>Divergence</td><td>25</td></tr><tr><td>3.4</td><td>Discrete sequence</td><td>26</td></tr><tr><td>3.5</td><td>Commutation properties</td><td>26</td></tr><tr><td>3.6</td><td>Complex and exactness properties</td><td>28</td></tr><tr><td>4</td><td>Potential reconstructions and L<sup>2</sup>-products on discrete spaces</td><td>32</td></tr><tr><td>4.1</td><td>Scalar potential on X<sub>grad,T</sub><sup>k</sup></td><td>33</td></tr><tr><td>4.2</td><td>Vector potential on X<sub>curl,T</sub><sup>k</sup></td><td>33</td></tr><tr><td>4.3</td><td>Vector potential on X<sub>div,T</sub><sup>k</sup></td><td>34</td></tr><tr><td>4.4</td><td>Discrete L<sup>2</sup>-products</td><td>35</td></tr><tr><td>4.5</td><td>Component L<sup>2</sup>-norms, bounds, and equivalence properties</td><td>36</td></tr><tr><td>4.6</td><td>Links between discrete vector potentials and vector calculus operators</td><td>39</td></tr><tr><td>5</td><td>Poincaré inequalities</td><td>41</td></tr><tr><td>5.1</td><td>Discrete Poincaré inequalities</td><td>41</td></tr><tr><td>5.2</td><td>Proof of the discrete Poincaré inequality for the gradient</td><td>42</td></tr><tr><td>6</td><td>Consistency results</td><td>47</td></tr><tr><td>6.1</td><td>Primal consistency</td><td>47</td></tr><tr><td>6.2</td><td>Adjoint consistency</td><td>48</td></tr><tr><td>6.3</td><td>Proof of the primal consistency</td><td>50</td></tr><tr><td>6.4</td><td>Proof of the adjoint consistency for the gradient</td><td>52</td></tr><tr><td>6.5</td><td>Proof of the adjoint consistency for the curl</td><td>53</td></tr><tr><td>6.6</td><td>Proof of the adjoint consistency for the divergence</td><td>56</td></tr><tr><td>7</td><td>Convergence analysis for a DDR discretisation of magnetostatics</td><td>57</td></tr><tr><td>7.1</td><td>Scheme</td><td>57</td></tr><tr><td>7.2</td><td>Error estimate</td><td>58</td></tr><tr><td>7.3</td><td>Numerical tests</td><td>60</td></tr><tr><td>A</td><td>Results on local spaces</td><td>64</td></tr><tr><td>B</td><td>Curl lifting</td><td>67</td></tr><tr><td>B.1</td><td>Face lifting R<sub>curl,F</sub></td><td>67</td></tr><tr><td>B.1.1</td><td>Existence of φ<sub>ν_F</sub></td><td>67</td></tr><tr><td>B.1.2</td><td>Existence of ψ<sub>ν_F</sub></td><td>68</td></tr><tr><td>B.1.3</td><td>Orthogonality property of R<sub>curl,F</sub></td><td>69</td></tr><tr><td>B.2</td><td>Element lifting R<sub>curl,T</sub></td><td>69</td></tr><tr><td>B.2.1</td><td>Existence of δ<sub>Tν_T</sub></td><td>69</td></tr><tr><td>B.2.2</td><td>Existence of R<sub>curl,Tν_T</sub></td><td>70</td></tr><tr><td>B.2.3</td><td>Bound on R<sub>curl,T</sub></td><td>71</td></tr><tr><td>C</td><td>Notations</td><td>74</td></tr></tbody></table>
Mathematics Subject Classification (2020) 65N30 · 65N99 · 78A30

Contents

<table><tbody><tr><td>1</td><td>Introduction</td><td>3</td></tr><tr><td>2</td><td>Setting</td><td>7</td></tr><tr><td>2.1</td><td>Domain and mesh</td><td>7</td></tr><tr><td>2.2</td><td>Orientation of mesh entities and vector calculus operators on faces</td><td>8</td></tr><tr><td>2.3</td><td>Lebesgue and Sobolev spaces</td><td>8</td></tr><tr><td>2.4</td><td>Polynomial spaces and decompositions</td><td>9</td></tr><tr><td>2.5</td><td>Recovery operator</td><td>10</td></tr><tr><td>3</td><td>Discrete de Rham complex</td><td>15</td></tr><tr><td>3.1</td><td>Discrete spaces</td><td>15</td></tr><tr><td>3.2</td><td>Interpolators</td><td>17</td></tr><tr><td>3.3</td><td>Discrete vector calculus operators</td><td>18</td></tr><tr><td>3.3.1</td><td>Gradient</td><td>18</td></tr><tr><td>3.3.2</td><td>Curl</td><td>21</td></tr><tr><td>3.3.3</td><td>Divergence</td><td>25</td></tr><tr><td>3.4</td><td>Discrete sequence</td><td>26</td></tr><tr><td>3.5</td><td>Commutation properties</td><td>26</td></tr><tr><td>3.6</td><td>Complex and exactness properties</td><td>28</td></tr><tr><td>4</td><td>Potential reconstructions and L<sup>2</sup>-products on discrete spaces</td><td>32</td></tr><tr><td>4.1</td><td>Scalar potential on X<sub>grad,T</sub><sup>k</sup></td><td>33</td></tr><tr><td>4.2</td><td>Vector potential on X<sub>curl,T</sub><sup>k</sup></td><td>33</td></tr><tr><td>4.3</td><td>Vector potential on X<sub>div,T</sub><sup>k</sup></td><td>34</td></tr><tr><td>4.4</td><td>Discrete L<sup>2</sup>-products</td><td>35</td></tr><tr><td>4.5</td><td>Component L<sup>2</sup>-norms, bounds, and equivalence properties</td><td>36</td></tr><tr><td>4.6</td><td>Links between discrete vector potentials and vector calculus operators</td><td>39</td></tr><tr><td>5</td><td>Poincaré inequalities</td><td>41</td></tr><tr><td>5.1</td><td>Discrete Poincaré inequalities</td><td>41</td></tr><tr><td>5.2</td><td>Proof of the discrete Poincaré inequality for the gradient</td><td>42</td></tr><tr><td>6</td><td>Consistency results</td><td>47</td></tr><tr><td>6.1</td><td>Primal consistency</td><td>47</td></tr><tr><td>6.2</td><td>Adjoint consistency</td><td>48</td></tr><tr><td>6.3</td><td>Proof of the primal consistency</td><td>50</td></tr><tr><td>6.4</td><td>Proof of the adjoint consistency for the gradient</td><td>52</td></tr><tr><td>6.5</td><td>Proof of the adjoint consistency for the curl</td><td>53</td></tr><tr><td>6.6</td><td>Proof of the adjoint consistency for the divergence</td><td>56</td></tr><tr><td>7</td><td>Convergence analysis for a DDR discretisation of magnetostatics</td><td>57</td></tr><tr><td>7.1</td><td>Scheme</td><td>57</td></tr><tr><td>7.2</td><td>Error estimate</td><td>58</td></tr><tr><td>7.3</td><td>Numerical tests</td><td>60</td></tr><tr><td>A</td><td>Results on local spaces</td><td>64</td></tr><tr><td>B</td><td>Curl lifting</td><td>67</td></tr><tr><td>B.1</td><td>Face lifting R<sub>curl,F</sub></td><td>67</td></tr><tr><td>B.1.1</td><td>Existence of φ<sub>ν_F</sub></td><td>67</td></tr><tr><td>B.1.2</td><td>Existence of ψ<sub>ν_F</sub></td><td>68</td></tr><tr><td>B.1.3</td><td>Orthogonality property of R<sub>curl,F</sub></td><td>69</td></tr><tr><td>B.2</td><td>Element lifting R<sub>curl,T</sub></td><td>69</td></tr><tr><td>B.2.1</td><td>Existence of δ<sub>Tν_T</sub></td><td>69</td></tr><tr><td>B.2.2</td><td>Existence of R<sub>curl,Tν_T</sub></td><td>70</td></tr><tr><td>B.2.3</td><td>Bound on R<sub>curl,T</sub></td><td>71</td></tr><tr><td>C</td><td>Notations</td><td>74</td></tr></tbody></table>
# 1 Introduction

The design of stable and convergent schemes for the numerical approximation of certain classes of partial differential equations (PDEs) requires to reproduce, at the discrete level, the underlying geometric, topological, and algebraic structures. This leads to the notion of compatibility, which can be achieved either in a conforming or non-conforming setting. Relevant examples include PDEs that relate to the de Rham complex. For an open connected polyhedral domain $\Omega \subset \mathbb{R}^3$, this complex reads

$$ \mathbb{R} \xrightarrow{i_{\Omega}} H^1(\Omega) \xrightarrow{\text{grad}} \mathbf{H}(\text{curl}; \Omega) \xrightarrow{\text{curl}} \mathbf{H}(\text{div}; \Omega) \xrightarrow{\text{div}} L^2(\Omega) \xrightarrow{0} \{0\}, \quad (1.1) $$

where $i_{\Omega}$ denotes the operator that maps a real value to a constant function over $\Omega$, $H^1(\Omega)$ the space of scalar-valued functions over $\Omega$ that are square integrable along with their gradient, $\mathbf{H}(\text{curl}; \Omega)$ (resp. $\mathbf{H}(\text{div}; \Omega)$) the space of vector-valued functions over $\Omega$ that are square integrable along with their curl (resp. divergence). In order to serve as a basis for the numerical approximation of PDEs, discrete counterparts of this sequence of spaces and operators should enjoy the following key properties:

(P1) **Complex and exactness properties.** For the sequence to form a complex, the image of each discrete vector calculus operator should be contained in the kernel of the next one. Moreover, the following exactness properties should be reproduced at the discrete level: Im $i_{\Omega} = \text{Ker grad}$ (since $\Omega$ is connected); Im $\text{grad} = \text{Ker curl}$ if the first Betti number of $\Omega$ is zero; Im $\text{curl} = \text{Ker div}$ if the second Betti number of $\Omega$ is zero; Im $\text{div} = L^2(\Omega)$ (since we are in dimension three).

(P2) **Uniform Poincaré inequalities.** Whenever a function from a space in the sequence lies in some orthogonal complement of the kernel of the vector calculus operator defined on this space, its (discrete) $L^2$-norm should be controlled by the (discrete) $L^2$-norm of the operator up to a multiplicative constant independent of the mesh size.

(P3) **Primal and adjoint consistency.** The discrete vector calculus operators should satisfy appropriate commutation properties with the interpolators and their continuous counterparts. Additionally, these operators along with the corresponding (scalar or vector) potentials should approximate smooth fields with sufficient accuracy. Finally, whenever a formal integration by parts is used in the weak formulation of the problem at hand, the vector calculus operators should also enjoy suitable adjoint consistency properties. The notion of adjoint consistency accounts for the failure, in non-conforming settings, to verify global integration by parts formulas exactly.

In the context of Finite Element (FE) approximations, discrete counterparts of the de Rham complex are obtained replacing each space in the sequence with a finite-dimensional subspace. These subspaces are built upon a
# 1 Introduction

The design of stable and convergent schemes for the numerical approximation of certain classes of partial differential equations (PDEs) requires to reproduce, at the discrete level, the underlying geometric, topological, and algebraic structures. This leads to the notion of compatibility, which can be achieved either in a conforming or non-conforming setting. Relevant examples include PDEs that relate to the de Rham complex. For an open connected polyhedral domain $\Omega \subset \mathbb{R}^3$, this complex reads

$$ \mathbb{R} \xrightarrow{i_{\Omega}} H^1(\Omega) \xrightarrow{\text{grad}} \mathbf{H}(\text{curl}; \Omega) \xrightarrow{\text{curl}} \mathbf{H}(\text{div}; \Omega) \xrightarrow{\text{div}} L^2(\Omega) \xrightarrow{0} \{0\}, \quad (1.1) $$

where $i_{\Omega}$ denotes the operator that maps a real value to a constant function over $\Omega$, $H^1(\Omega)$ the space of scalar-valued functions over $\Omega$ that are square integrable along with their gradient, $\mathbf{H}(\text{curl}; \Omega)$ (resp. $\mathbf{H}(\text{div}; \Omega)$) the space of vector-valued functions over $\Omega$ that are square integrable along with their curl (resp. divergence). In order to serve as a basis for the numerical approximation of PDEs, discrete counterparts of this sequence of spaces and operators should enjoy the following key properties:

(P1) **Complex and exactness properties.** For the sequence to form a complex, the image of each discrete vector calculus operator should be contained in the kernel of the next one. Moreover, the following exactness properties should be reproduced at the discrete level: Im $i_{\Omega} = \text{Ker grad}$ (since $\Omega$ is connected); Im $\text{grad} = \text{Ker curl}$ if the first Betti number of $\Omega$ is zero; Im $\text{curl} = \text{Ker div}$ if the second Betti number of $\Omega$ is zero; Im $\text{div} = L^2(\Omega)$ (since we are in dimension three).

(P2) **Uniform Poincaré inequalities.** Whenever a function from a space in the sequence lies in some orthogonal complement of the kernel of the vector calculus operator defined on this space, its (discrete) $L^2$-norm should be controlled by the (discrete) $L^2$-norm of the operator up to a multiplicative constant independent of the mesh size.

(P3) **Primal and adjoint consistency.** The discrete vector calculus operators should satisfy appropriate commutation properties with the interpolators and their continuous counterparts. Additionally, these operators along with the corresponding (scalar or vector) potentials should approximate smooth fields with sufficient accuracy. Finally, whenever a formal integration by parts is used in the weak formulation of the problem at hand, the vector calculus operators should also enjoy suitable adjoint consistency properties. The notion of adjoint consistency accounts for the failure, in non-conforming settings, to verify global integration by parts formulas exactly.

In the context of Finite Element (FE) approximations, discrete counterparts of the de Rham complex are obtained replacing each space in the sequence with a finite-dimensional subspace. These subspaces are built upon a
conforming mesh of the domain, whose elements are restricted to a small num-
ber of shapes and, in practice, are most often tetrahedra; see [2] for a complete
and extremely general exposition including an exhaustive bibliography, and
also [21] on the link between Raviart–Thomas–Nédélec differential forms and
FE systems. The restriction to conforming meshes made of standard elements
can be a major shortcoming in advanced applications, limiting, for example,
the capacity for local refinement or mesh agglomeration; see, e.g., the preface
of [28]. The extension of the FE approach to more general meshes including,
e.g., polyhedral elements and non-matching interfaces, is not straightforward.
Recent efforts in this direction have been made in [39,19] (see also references
therein), focusing mainly on the lowest-order case and with some limitations
on the element shapes in three dimensions. The extension to specific element
shapes has also been considered in [37,26]. A recent generalisation of FE meth-
ods is provided by the Isogeometric Analysis, which is designed to facilitate
exchanges with Computer Assisted Design software. In this framework, spline
spaces and projection operators that verify a de Rham diagram have been
developed in [17]; see also [18].

General polytopal meshes can be handled by several lowest-order methods grounded, to a different extent, in the seminal work of Whitney on geometric integration [44]. These methods share the common feature that discrete de Rham complexes are obtained by replacing both the spaces and operators with discrete counterparts. Specifically, the spaces consist of vectors of real numbers attached to mesh entities of dimension equal to the index of the space in the sequence (vertices for $H^1(\Omega)$, edges for $H(\mathrm{curl};\Omega)$, faces for $H(\mathrm{div};\Omega)$, and elements for $L^2(\Omega)$). In Mimetic Finite Differences, discrete vector calculus operators and $L^2$-products are obtained by mimicking the Stokes theorem; see [8] for a complete exposition. Their extension to polytopal meshes has first been carried out in [41,42], then analysed in [16,15]; see also [36] for a link with the Mixed Hybrid Finite Volume methods of [34,38] and [33, Section 2.5] along with [32, Section 3.5] and [1] for links with Hybrid High-Order methods. In the Discrete Geometric Approach, originally introduced in [22] and extended to polyhedral meshes in [23,24], as well as in Compatible Discrete Operators [12,11], the key notions are topological vector calculus operators (expressed in terms of incidence matrices) along with the Hodge operator. The role of the latter is to establish a link, through the introduction of physical parameters, between quantities defined on primal and dual mesh entities. All of the above schemes are limited to the lowest-order, and their analysis often relies on an interplay of functional and topological arguments that is not required in our approach.

Discretisation methods that provide arbitrary-order approximations on
general polyhedral meshes have only recently appeared in the literature. A
first example is provided by the Virtual Element Method, which can be de-
scribed as a FE method where explicit expressions for the basis functions are
not available at each point. A de Rham complex of virtual spaces on polyhedra
has been recently proposed in [7]; important evolutions of this original virtual
complex are contained in [5,4], which also include applications to the Kikuchi
conforming mesh of the domain, whose elements are restricted to a small num-
ber of shapes and, in practice, are most often tetrahedra; see [2] for a complete
and extremely general exposition including an exhaustive bibliography, and
also [21] on the link between Raviart–Thomas–Nédélec differential forms and
FE systems. The restriction to conforming meshes made of standard elements
can be a major shortcoming in advanced applications, limiting, for example,
the capacity for local refinement or mesh agglomeration; see, e.g., the preface
of [28]. The extension of the FE approach to more general meshes including,
e.g., polyhedral elements and non-matching interfaces, is not straightforward.
Recent efforts in this direction have been made in [39,19] (see also references
therein), focusing mainly on the lowest-order case and with some limitations
on the element shapes in three dimensions. The extension to specific element
shapes has also been considered in [37,26]. A recent generalisation of FE meth-
ods is provided by the Isogeometric Analysis, which is designed to facilitate
exchanges with Computer Assisted Design software. In this framework, spline
spaces and projection operators that verify a de Rham diagram have been
developed in [17]; see also [18].

General polytopal meshes can be handled by several lowest-order methods grounded, to a different extent, in the seminal work of Whitney on geometric integration [44]. These methods share the common feature that discrete de Rham complexes are obtained by replacing both the spaces and operators with discrete counterparts. Specifically, the spaces consist of vectors of real numbers attached to mesh entities of dimension equal to the index of the space in the sequence (vertices for $H^1(\Omega)$, edges for $H(\text{curl};\Omega)$, faces for $H(\text{div};\Omega)$, and elements for $L^2(\Omega)$). In Mimetic Finite Differences, discrete vector calculus operators and $L^2$-products are obtained by mimicking the Stokes theorem; see [8] for a complete exposition. Their extension to polytopal meshes has first been carried out in [41,42], then analysed in [16,15]; see also [36] for a link with the Mixed Hybrid Finite Volume methods of [34,38] and [33, Section 2.5] along with [32, Section 3.5] and [1] for links with Hybrid High-Order methods. In the Discrete Geometric Approach, originally introduced in [22] and extended to polyhedral meshes in [23,24], as well as in Compatible Discrete Operators [12,11], the key notions are topological vector calculus operators (expressed in terms of incidence matrices) along with the Hodge operator. The role of the latter is to establish a link, through the introduction of physical parameters, between quantities defined on primal and dual mesh entities. All of the above schemes are limited to the lowest-order, and their analysis often relies on an interplay of functional and topological arguments that is not required in our approach.

Discretisation methods that provide arbitrary-order approximations on
general polyhedral meshes have only recently appeared in the literature. A
first example is provided by the Virtual Element Method, which can be de-
scribed as a FE method where explicit expressions for the basis functions are
not available at each point. A de Rham complex of virtual spaces on polyhedra
has been recently proposed in [7]; important evolutions of this original virtual
complex are contained in [5,4], which also include applications to the Kikuchi
formulation of magnetostatics, and in [9], which contains a detailed study of the interpolation and stability properties of the low-order VEM spaces. In order to derive an actual discretisation scheme starting from the sequence of virtual spaces, a variational crime involving projections is required. A different approach is pursued in [31,29], where a discrete de Rham (DDR) complex is presented, based on decompositions of full polynomial spaces into the range of vector calculus operators and their L²-orthogonal complements. This complex involves discrete spaces and operators that appear, through discrete L²-products, in the formulation of discretisation methods. The analysis in [31,29] focuses on a subset of properties (P1)–(P2) involved in the stability analysis of numerical schemes: local exactness ([31, Theorems 4.1 and 5.1]), global complex property, discrete counterparts of Im grad = Ker curl for domains that do not enclose voids and Im div = L²(Ω) ([29, Theorem 3]), as well as Poincaré inequalities for the divergence and the curl ([29, Theorems 18 and 20, respectively]). This approach completely avoids, both in the construction and in the analysis, the use of (virtual or piecewise polynomial) functions with global regularity, and is closer in spirit to Mimetic Finite Differences and Mixed Hybrid Finite Volume methods.

Regarding consistency properties (P3) for polytopal methods, and starting
from low-order methods, results for Compatible Discrete Operator approxima-
tions of the Poisson problem based on nodal unknowns can be found in [12];
see in particular the proof of Theorem 3.3 therein, which contains an adjoint
consistency result for a gradient reconstructed from vertex values. In the same
framework, an adjoint consistency estimate for a discrete curl constructed from
edge values can be found in [13, Lemma 2.3]. A rather complete set of con-
sistency results for Mimetic Finite Difference operators can be found in [8],
where they appear as intermediate steps in the error analyses of Chapters 5–7.
A notable exception is provided by the adjoint consistency of the curl operator,
which is not needed in the error estimate of [8, Theorem 7.3] since the authors
consider an approximation of the current density based on the knowledge of a
vector potential.

Moving to consistency properties for arbitrary-order polytopal methods, error estimates that involve the adjoint consistency of a gradient and the consistency of the corresponding potential have been recently derived in [14] in the framework of the H¹-conforming Virtual Element method. The same method is considered in [27, Section 3.2], where a different analysis is proposed based on the third Strang lemma. The estimate of the consistency error in [27, Theorem 19] involves, in particular, the adjoint consistency of a discrete gradient reconstructed as the gradient of a scalar polynomial rather than a vector-valued polynomial. We note, in passing, that the concept of adjoint consistency for (discrete) gradients is directly related to the notion of limit-conformity in the Gradient Discretisation Method [35], a generic framework which encompasses several polytopal methods. Primal and dual consistency estimates for a discrete divergence and the corresponding vector potential similar (but not identical) to the ones considered here have been established in [32] in the framework of Mixed High-Order methods. Note that these methods (the
formulation of magnetostatics, and in [9], which contains a detailed study of the interpolation and stability properties of the low-order VEM spaces. In order to derive an actual discretisation scheme starting from the sequence of virtual spaces, a variational crime involving projections is required. A different approach is pursued in [31,29], where a discrete de Rham (DDR) complex is presented, based on decompositions of full polynomial spaces into the range of vector calculus operators and their L²-orthogonal complements. This complex involves discrete spaces and operators that appear, through discrete L²-products, in the formulation of discretisation methods. The analysis in [31,29] focuses on a subset of properties (P1)–(P2) involved in the stability analysis of numerical schemes: local exactness ([31, Theorems 4.1 and 5.1]), global complex property, discrete counterparts of Im **grad** = Ker **curl** for domains that do not enclose voids and Im div = L²(**Ω**) ([29, Theorem 3]), as well as Poincaré inequalities for the divergence and the curl ([29, Theorems 18 and 20, respectively]). This approach completely avoids, both in the construction and in the analysis, the use of (virtual or piecewise polynomial) functions with global regularity, and is closer in spirit to Mimetic Finite Differences and Mixed Hybrid Finite Volume methods.

Regarding consistency properties (P3) for polytopal methods, and starting
from low-order methods, results for Compatible Discrete Operator approxima-
tions of the Poisson problem based on nodal unknowns can be found in [12];
see in particular the proof of Theorem 3.3 therein, which contains an adjoint
consistency result for a gradient reconstructed from vertex values. In the same
framework, an adjoint consistency estimate for a discrete curl constructed from
edge values can be found in [13, Lemma 2.3]. A rather complete set of con-
sistency results for Mimetic Finite Difference operators can be found in [8],
where they appear as intermediate steps in the error analyses of Chapters 5–7.
A notable exception is provided by the adjoint consistency of the curl operator,
which is not needed in the error estimate of [8, Theorem 7.3] since the authors
consider an approximation of the current density based on the knowledge of a
vector potential.

Moving to consistency properties for arbitrary-order polytopal methods, error estimates that involve the adjoint consistency of a gradient and the consistency of the corresponding potential have been recently derived in [14] in the framework of the H¹-conforming Virtual Element method. The same method is considered in [27, Section 3.2], where a different analysis is proposed based on the third Strang lemma. The estimate of the consistency error in [27, Theorem 19] involves, in particular, the adjoint consistency of a discrete gradient reconstructed as the gradient of a scalar polynomial rather than a vector-valued polynomial. We note, in passing, that the concept of adjoint consistency for (discrete) gradients is directly related to the notion of limit-conformity in the Gradient Discretisation Method [35], a generic framework which encompasses several polytopal methods. Primal and dual consistency estimates for a discrete divergence and the corresponding vector potential similar (but not identical) to the ones considered here have been established in [32] in the framework of Mixed High-Order methods. Note that these methods (the
H¹-conforming Virtual Element method and the Mixed High-Order method) do not lead to a discrete de Rham complex. In the framework of arbitrary-order compatible discretisations, on the other hand, primal consistency results for the curl appear as intermediate results in [4], where an error analysis for a Virtual Element approximation of magnetostatics is carried out assuming interpolation estimates for three-dimensional vector valued virtual spaces; see Remark 4.4 therein. However, [4] does not establish any adjoint consistency property of the discrete curl (the formulation of magnetostatics considered in this reference does not require it).

*Content of the paper.* We present a new DDR sequence based, contrary to [31, 29], on explicit complements of the ranges of vector calculus operators inspired by the ones used in [4]; these complements are easier to implement, and enable a complete proof of the full set of properties (P1)–(P3). To the best of our knowledge, this is the first time that such a complete panel of results is available for an arbitrary-order polyhedral method compatible with the de Rham complex. The complements considered here are linked to the spaces appearing in the Koszul complex (see, e.g., [2, Chapter 7]) and enjoy two key properties on general polyhedral meshes: they are hierarchical (see Remark 1 below) and their traces on polyhedral faces or edges lie in appropriate polynomial spaces (cf. Proposition 8). These properties make it possible to prove discrete integration by parts formulas for the discrete potentials (see Remarks 8, 17, 18, and 19 below) which, in turn, are essential to the proof of the adjoint consistency properties.

The key ingredients to establish primal consistency are the polynomial consistency of discrete vector calculus operators along with the corresponding potentials, and their boundedness when applied to the interpolates of smooth functions. The proofs of adjoint consistency, on the other hand, rely on operator-specific techniques and are all grounded in the above-mentioned discrete integration by parts formulas for the corresponding potential reconstructions. Specifically, the key point for the adjoint consistency of the gradient are estimates for local H¹-like seminorms of the scalar potentials. The adjoint consistency of curl requires, on the other hand, the construction of liftings of the discrete face potentials that satisfy an orthogonality and a boundedness condition. These reconstructions are inspired by the minimal reconstruction operators of [8, Chapter 3], with a key novelty provided by a curl correction which ensures the well-posedness of the reconstruction inside mesh elements and relies on fine results from [25,3].

In order to showcase the theoretical results derived here, we carry out a full convergence analysis for a DDR approximation of magnetostatics. This is, to the best of our knowledge, the first full theoretical result of this kind for arbitrary-order polytopal methods.

The key innovation of the DDR complex presented here, compared to the one in [31,29], precisely lies in the fact that it enables all mathematical results required to prove error estimates for schemes built from this sequence. The
H¹-conforming Virtual Element method and the Mixed High-Order method) do not lead to a discrete de Rham complex. In the framework of arbitrary-order compatible discretisations, on the other hand, primal consistency results for the curl appear as intermediate results in [4], where an error analysis for a Virtual Element approximation of magnetostatics is carried out assuming interpolation estimates for three-dimensional vector valued virtual spaces; see Remark 4.4 therein. However, [4] does not establish any adjoint consistency property of the discrete curl (the formulation of magnetostatics considered in this reference does not require it).

*Content of the paper.* We present a new DDR sequence based, contrary to [31, 29], on explicit complements of the ranges of vector calculus operators inspired by the ones used in [4]; these complements are easier to implement, and enable a complete proof of the full set of properties (P1)–(P3). To the best of our knowledge, this is the first time that such a complete panel of results is available for an arbitrary-order polyhedral method compatible with the de Rham complex. The complements considered here are linked to the spaces appearing in the Koszul complex (see, e.g., [2, Chapter 7]) and enjoy two key properties on general polyhedral meshes: they are hierarchical (see Remark 1 below) and their traces on polyhedral faces or edges lie in appropriate polynomial spaces (cf. Proposition 8). These properties make it possible to prove discrete integration by parts formulas for the discrete potentials (see Remarks 8, 17, 18, and 19 below) which, in turn, are essential to the proof of the adjoint consistency properties.

The key ingredients to establish primal consistency are the polynomial consistency of discrete vector calculus operators along with the corresponding potentials, and their boundedness when applied to the interpolates of smooth functions. The proofs of adjoint consistency, on the other hand, rely on operator-specific techniques and are all grounded in the above-mentioned discrete integration by parts formulas for the corresponding potential reconstructions. Specifically, the key point for the adjoint consistency of the gradient are estimates for local H¹-like seminorms of the scalar potentials. The adjoint consistency of curl requires, on the other hand, the construction of liftings of the discrete face potentials that satisfy an orthogonality and a boundedness condition. These reconstructions are inspired by the minimal reconstruction operators of [8, Chapter 3], with a key novelty provided by a curl correction which ensures the well-posedness of the reconstruction inside mesh elements and relies on fine results from [25,3].

In order to showcase the theoretical results derived here, we carry out a full convergence analysis for a DDR approximation of magnetostatics. This is, to the best of our knowledge, the first full theoretical result of this kind for arbitrary-order polytopal methods.

The key innovation of the DDR complex presented here, compared to the one in [31,29], precisely lies in the fact that it enables all mathematical results required to prove error estimates for schemes built from this sequence. The
only analytical results available in [31,29] are Poincaré inequalities for the curl and the divergence and, as a matter of fact, it seems that the sequence in these references does not satisfy the critical discrete integration by parts formulas mentioned above, and is therefore not amenable to an adjoint consistency analysis.

The rest of the paper is organised as follows. In Section 2 we establish the general setting. Section 3 contains the definition of the DDR sequence along with key intermediate results for the discrete vector calculus operators (including the commutation property in (P3)) and the proof of (P1). In Section 4, we introduce tools for the design and analysis of schemes based on the DDR sequence: polynomial potential reconstructions and $L^2$-products on the discrete spaces. Discrete Poincaré inequalities corresponding to (P2) are covered in Section 5. Section 6 contains the statement and proofs of the primal and adjoint consistency results corresponding to (P3). The application of the theoretical tools to the error analysis of a DDR approximation of magnetostatics is considered in Section 7, where numerical evidence supporting the error estimates is also provided. The paper is completed by three appendices. Appendix A contains results on local polynomial spaces including those on the traces of the trimmed spaces constructed from the Koszul complements. Appendix B contains an in-depth and novel study of the div-curl problems defining the curl liftings on polytopal elements: well-posedness, orthogonality and boundedness properties. Finally, Appendix C details the conventions of notation adopted throughout the paper, and lists the main spaces and operators of the DDR complex.

# 2 Setting

## 2.1 Domain and mesh

For any (measurable) set $Y \subset \mathbb{R}^3$, we denote by $h_Y := \sup\{|\mathbf{x} - \mathbf{y}| : \mathbf{x}, \mathbf{y} \in Y\}$ its diameter and by $|Y|$ its Hausdorff measure. We consider meshes $\mathcal{M}_h := \mathcal{T}_h \cup \mathcal{F}_h \cup \mathcal{E}_h \cup \mathcal{V}_h$, where: $\mathcal{T}_h$ is a finite collection of open disjoint polyhedral elements such that $\bar{\Omega} = \cup_{T \in \mathcal{T}_h} \bar{T}$ and $h = \max_{T \in \mathcal{T}_h} h_T > 0$; $\mathcal{F}_h$ is a finite collection of open planar faces; $\mathcal{E}_h$ is the set collecting the open polygonal edges (line segments) of the faces; $\mathcal{V}_h$ is the set collecting the edge endpoints. It is assumed, in what follows, that $(\mathcal{T}_h, \mathcal{F}_h)$ matches the conditions in [28, Definition 1.4], so that the faces form a partition of the mesh skeleton $\cup_{T \in \mathcal{T}_h} \partial T$. We additionally assume that the polytopes in $\mathcal{T}_h \cup \mathcal{F}_h$ are simply connected and have connected Lipschitz-continuous boundaries. This notion of mesh is related to that of cellular (or CW) complex from algebraic topology; see, e.g., [43, Chapter 7].

The set collecting the mesh faces that lie on the boundary of a mesh element $T \in \mathcal{T}_h$ is denoted by $\mathcal{F}_T$. For any mesh element or face $Y \in \mathcal{T}_h \cup \mathcal{F}_h$, we denote, respectively, by $\mathcal{E}_Y$ and $\mathcal{V}_Y$ the set of edges and vertices of $Y$.

Throughout the paper, unless otherwise specified, we write $a \lesssim b$ in place of $a \le Cb$ with $C$ depending only on $\Omega$, the mesh regularity parameter $\rho$
only analytical results available in [31,29] are Poincaré inequalities for the curl and the divergence and, as a matter of fact, it seems that the sequence in these references does not satisfy the critical discrete integration by parts formulas mentioned above, and is therefore not amenable to an adjoint consistency analysis.

The rest of the paper is organised as follows. In Section 2 we establish the general setting. Section 3 contains the definition of the DDR sequence along with key intermediate results for the discrete vector calculus operators (including the commutation property in (P3)) and the proof of (P1). In Section 4, we introduce tools for the design and analysis of schemes based on the DDR sequence: polynomial potential reconstructions and $L^2$-products on the discrete spaces. Discrete Poincaré inequalities corresponding to (P2) are covered in Section 5. Section 6 contains the statement and proofs of the primal and adjoint consistency results corresponding to (P3). The application of the theoretical tools to the error analysis of a DDR approximation of magnetostatics is considered in Section 7, where numerical evidence supporting the error estimates is also provided. The paper is completed by three appendices. Appendix A contains results on local polynomial spaces including those on the traces of the trimmed spaces constructed from the Koszul complements. Appendix B contains an in-depth and novel study of the div-curl problems defining the curl liftings on polytopal elements: well-posedness, orthogonality and boundedness properties. Finally, Appendix C details the conventions of notation adopted throughout the paper, and lists the main spaces and operators of the DDR complex.

## 2 Setting

### 2.1 Domain and mesh

For any (measurable) set $Y \subset \mathbb{R}^3$, we denote by $h_Y := \sup\{|\mathbf{x} - \mathbf{y}| : \mathbf{x}, \mathbf{y} \in Y\}$ its diameter and by $|Y|$ its Hausdorff measure. We consider meshes $\mathcal{M}_h := \mathcal{T}_h \cup \mathcal{F}_h \cup \mathcal{E}_h \cup \mathcal{V}_h$, where: $\mathcal{T}_h$ is a finite collection of open disjoint polyhedral elements such that $\bar{\Omega} = \cup_{T \in \mathcal{T}_h} \bar{T}$ and $h = \max_{T \in \mathcal{T}_h} h_T > 0$; $\mathcal{F}_h$ is a finite collection of open planar faces; $\mathcal{E}_h$ is the set collecting the open polygonal edges (line segments) of the faces; $\mathcal{V}_h$ is the set collecting the edge endpoints. It is assumed, in what follows, that $(\mathcal{T}_h, \mathcal{F}_h)$ matches the conditions in [28, Definition 1.4], so that the faces form a partition of the mesh skeleton $\cup_{T \in \mathcal{T}_h} \partial T$. We additionally assume that the polytopes in $\mathcal{T}_h \cup \mathcal{F}_h$ are simply connected and have connected Lipschitz-continuous boundaries. This notion of mesh is related to that of cellular (or CW) complex from algebraic topology; see, e.g., [43, Chapter 7].

The set collecting the mesh faces that lie on the boundary of a mesh element $T \in \mathcal{T}_h$ is denoted by $\mathcal{F}_T$. For any mesh element or face $Y \in \mathcal{T}_h \cup \mathcal{F}_h$, we denote, respectively, by $\mathcal{E}_Y$ and $\mathcal{V}_Y$ the set of edges and vertices of $Y$.

Throughout the paper, unless otherwise specified, we write $a \lesssim b$ in place of $a \le Cb$ with $C$ depending only on $\Omega$, the mesh regularity parameter $\rho$
of [28, Definition 1.9], and the considered polynomial degree. We note that this mesh regularity parameter $\rho \in (0, 1)$ is bounded away from 0 when a shape-regular matching simplicial submesh of $\mathcal{T}_h$ exists such that each $T \in \mathcal{T}_h$ is partitioned into simplices of size uniformly comparable to $T$. We also use $a \approx b$ as a shorthand for “$a \le b$ and $b \le a$”.

## 2.2 Orientation of mesh entities and vector calculus operators on faces

For any face $F \in \mathcal{T}_h$, an orientation is set by prescribing a unit normal vector $n_F$ and, for any mesh element $T \in \mathcal{T}_h$ sharing $F$, we denote by $\omega_{TF} \in \{-1, 1\}$ the orientation of $F$ relative to $T$, that is, $\omega_{TF} = 1$ if $n_F$ points out of $T$, $-1$ otherwise. With this choice, $\omega_{TF}n_F$ is the unit vector normal to $F$ that points out of $T$. For any edge $E \in \mathcal{E}_h$, an orientation is set by prescribing the unit tangent vector $t_E$. Denoting by $F \in \mathcal{T}_h$ a face such that $E \in \mathcal{E}_F$, its boundary $\partial F$ is oriented counter-clockwise with respect to $n_F$, and we denote by $\omega_{FE} \in \{-1, 1\}$ the (opposite of the) orientation of $E$ relative to that $\partial F$: $\omega_{FE} = 1$ if $t_E$ points on $E$ in the opposite orientation to $\partial F$, $\omega_{FE} = -1$ otherwise. We also denote by $n_{FE}$ the unit vector normal to $E$ lying in the plane of $F$ such that $(t_E, n_{FE})$ forms a system of right-handed coordinates in the plane of $F$, so that the system of coordinates $(t_E, n_{FE}, n_F)$ is right-handed in $\mathbb{R}^3$. It can be checked that $\omega_{FE}n_{FE}$ is the normal to $E$, in the plane where $F$ lies, pointing out of $F$.

For any mesh face $F \in \mathcal{T}_h$, we denote by $\text{grad}_F$ and $\text{div}_F$ the tangent gradient and divergence operators acting on smooth enough functions. Moreover, for any $r: F \to \mathbb{R}$ and $z: F \to \mathbb{R}^2$ smooth enough, we define the two-dimensional vector and scalar curl operators such that

$$ \text{rot}_F r := \varrho_{-\pi/2}(\text{grad}_F r) \quad \text{and} \quad \text{rot}_F z = \text{div}_F(\varrho_{-\pi/2}z), \quad (2.1) $$

where $\varrho_{-\pi/2}$ is the rotation of angle $-\pi/2$ in the oriented tangent space to $F$.

## 2.3 Lebesgue and Sobolev spaces

Let $Y$ be a measurable subset of $\mathbb{R}^3$. We denote by $L^2(Y)$ the Lebesgue space spanned by functions that are square-integrable over $Y$. When $Y$ is a subset of an $n$-dimensional variety, we will use the boldface notation $L^2(Y) := L^2(Y)^n$ for the space of vector-valued fields over $Y$ with square-integrable components. Given an integer $l$ and $Y \in \{\Omega\} \cup \mathcal{T}_h \cup \mathcal{T}_h$, $H^l(Y)$ will denote the Sobolev space spanned by square-integrable functions whose partial derivatives of order up to $l$ are also square-integrable. Denoting again by $n$ the dimension of $Y$, we let $H^l(Y) := H^l(Y)^n$ and $C^l(Y) := C^l(Y)^n$. For all $F \in \mathcal{T}_h$, we let $H(\text{rot}; F) := \{v \in L^2(F) : \text{rot}_F v \in L^2(F)\}$. Similarly, for all $Y \in \{\Omega\} \cup \mathcal{T}_h$, $H(\text{curl}; Y) := \{v \in L^2(Y) : \text{curl} v \in L^2(Y)\}$ and $H(\text{div}; Y) := \{w \in L^2(Y) : \text{div} w \in L^2(Y)\}$.
of [28, Definition 1.9], and the considered polynomial degree. We note that this mesh regularity parameter $\rho \in (0, 1)$ is bounded away from 0 when a shape-regular matching simplicial submesh of $\mathcal{T}_h$ exists such that each $T \in \mathcal{T}_h$ is partitioned into simplices of size uniformly comparable to $T$. We also use $a \approx b$ as a shorthand for “$a \le b$ and $b \le a$”.

## 2.2 Orientation of mesh entities and vector calculus operators on faces

For any face $F \in \mathcal{T}_h$, an orientation is set by prescribing a unit normal vector $n_F$ and, for any mesh element $T \in \mathcal{T}_h$ sharing $F$, we denote by $\omega_{TF} \in \{-1, 1\}$ the orientation of $F$ relative to $T$, that is, $\omega_{TF} = 1$ if $n_F$ points out of $T$, $-1$ otherwise. With this choice, $\omega_{TF}n_F$ is the unit vector normal to $F$ that points out of $T$. For any edge $E \in \mathcal{E}_h$, an orientation is set by prescribing the unit tangent vector $t_E$. Denoting by $F \in \mathcal{T}_h$ a face such that $E \in \mathcal{E}_F$, its boundary $\partial F$ is oriented counter-clockwise with respect to $n_F$, and we denote by $\omega_{FE} \in \{-1, 1\}$ the (opposite of the) orientation of $E$ relative to that $\partial F$: $\omega_{FE} = 1$ if $t_E$ points on $E$ in the opposite orientation to $\partial F$, $\omega_{FE} = -1$ otherwise. We also denote by $n_{FE}$ the unit vector normal to $E$ lying in the plane of $F$ such that $(t_E, n_{FE})$ forms a system of right-handed coordinates in the plane of $F$, so that the system of coordinates $(t_E, n_{FE}, n_F)$ is right-handed in $\mathbb{R}^3$. It can be checked that $\omega_{FE}n_{FE}$ is the normal to $E$, in the plane where $F$ lies, pointing out of $F$.

For any mesh face $F \in \mathcal{T}_h$, we denote by $\text{grad}_F$ and $\text{div}_F$ the tangent gradient and divergence operators acting on smooth enough functions. Moreover, for any $r: F \to \mathbb{R}$ and $z: F \to \mathbb{R}^2$ smooth enough, we define the two-dimensional vector and scalar curl operators such that

$$ \text{rot}_F r := \varrho_{-\pi/2}(\text{grad}_F r) \quad \text{and} \quad \text{rot}_F z = \text{div}_F(\varrho_{-\pi/2}z), \quad (2.1) $$

where $\varrho_{-\pi/2}$ is the rotation of angle $-\pi/2$ in the oriented tangent space to $F$.

## 2.3 Lebesgue and Sobolev spaces

Let $Y$ be a measurable subset of $\mathbb{R}^3$. We denote by $L^2(Y)$ the Lebesgue space spanned by functions that are square-integrable over $Y$. When $Y$ is a subset of an $n$-dimensional variety, we will use the boldface notation $L^2(Y) := L^2(Y)^n$ for the space of vector-valued fields over $Y$ with square-integrable components. Given an integer $l$ and $Y \in \{\Omega\} \cup \mathcal{T}_h \cup \mathcal{T}_h$, $H^l(Y)$ will denote the Sobolev space spanned by square-integrable functions whose partial derivatives of order up to $l$ are also square-integrable. Denoting again by $n$ the dimension of $Y$, we let $\mathbf{H}^l(Y) := \mathbf{H}^l(Y)^n$ and $\mathbf{C}^l(Y) := \mathbf{C}^l(Y)^n$. For all $F \in \mathcal{T}_h$, we let $\mathbf{H}(\text{rot}; F) := \{v \in L^2(F) : \text{rot}_F v \in L^2(F)\}$. Similarly, for all $Y \in \{\Omega\} \cup \mathcal{T}_h$, $\mathbf{H}(\text{curl}; Y) := \{v \in L^2(Y) : \text{curl} v \in L^2(Y)\}$ and $\mathbf{H}(\text{div}; Y) := \{w \in L^2(Y) : \text{div} w \in L^2(Y)\}$.
## 2.4 Polynomial spaces and decompositions

For a given integer $\ell \ge 0$, $\mathbb{P}_h^\ell$ denotes the space of $n$-variate polynomials of total degree $\le \ell$, with the convention that $\mathbb{P}_0^\ell = \mathbb{R}$ for any $\ell$ and that $\mathbb{P}_n^{-1} := \{0\}$ for any $n$. For any $Y \in \mathcal{T}_h \cup \mathcal{F}_h \cup \mathcal{E}_h$, we denote by $\mathcal{P}^\ell(Y)$ the space spanned by the restriction to $Y$ of the functions in $\mathbb{P}_3^\ell$. Denoting by $1 \le n \le 3$ the dimension of $Y$, $\mathcal{P}^\ell(Y)$ is isomorphic to $\mathbb{P}_h^\ell$ (see [28, Proposition 1.23]). In what follows, with a little abuse of notation, both spaces are denoted by $\mathcal{P}^\ell(Y)$. We additionally denote by $\pi_{\mathcal{P},Y}^\ell$ the corresponding $\mathbb{L}^2$-orthogonal projector and let $\mathcal{P}^{0,\ell}(Y)$ denote the subspace of $\mathcal{P}^\ell(Y)$ made of polynomials with zero average over $Y$. For the sake of brevity, we also introduce the boldface notations $\mathcal{P}^\ell(T) := \mathcal{P}^\ell(T)^3$ for all $T \in \mathcal{T}_h$ and $\mathcal{P}^\ell(F) := \mathcal{P}^\ell(F)^2$ for all $F \in \mathcal{F}_h$.

Let again an integer $\ell \ge 1$ be given, and denote by $\mathfrak{G} \subset \mathcal{E}_h$ a collection of edges such that $S_{\mathfrak{G}} := \bigcup_{E \in \mathfrak{G}} \bar{E}$ forms a connected set. We denote by $\mathcal{P}_c^\ell(\mathfrak{G}) := \{q_{\mathfrak{G}} \in \mathbb{C}^0(S_{\mathfrak{G}}) : (q_{\mathfrak{G}})_{|E} \in \mathcal{P}^\ell(E) \text{ for all } E \in \mathfrak{G}\}$ the space of functions over $S_{\mathfrak{G}}$ whose restriction to each edge $E \in \mathfrak{G}$ is a polynomial of total degree $\le \ell$ and that are continuous at the edges endpoints; these endpoints are collected in the set $\mathcal{V}_{\mathfrak{G}} \subset \mathcal{V}_h$. Denoting by $\mathbf{x}_V$ the coordinates vector of a vertex $V \in \mathcal{V}_h$, it can be easily checked that the following mapping is an isomorphism:

$$ \mathcal{P}_c^\ell(\mathfrak{G}) \ni q_{\mathfrak{G}} \mapsto ((\pi_{\mathcal{P},E}^{\ell-2}(q_{\mathfrak{G}})|_E)_{E \in \mathfrak{G}}, (q_{\mathfrak{G}}(\mathbf{x}_V))_{V \in \mathcal{V}_{\mathfrak{G}}}) \in \left( \bigtimes_{E \in \mathfrak{G}} \mathcal{P}^{\ell-2}(E) \right) \times \mathbb{R}^{\mathcal{V}_{\mathfrak{G}}}. \quad (2.2) $$

For all $Y \in \mathcal{T}_h \cup \mathcal{F}_h$, denote by $\mathbf{x}_Y$ a point inside $Y$ such that $Y$ contains a ball centered at $\mathbf{x}_Y$ of radius $\rho h_Y$, where $\rho$ is the mesh regularity parameter in [28, Definition 1.9]. For any mesh face $F \in \mathcal{F}_h$ and any integer $\ell \ge 0$, we define the following relevant subspaces of $\mathcal{P}^\ell(F)$:

$$ \mathcal{G}^\ell(F) := \operatorname{grad}_F \mathcal{P}^{\ell+1}(F), \qquad \mathcal{G}^{c,\ell}(F) := (\mathbf{x} - \mathbf{x}_F)^\perp \mathcal{P}^{\ell-1}(F), \quad (2.3a) $$

$$ \mathcal{R}^\ell(F) := \operatorname{rot}_F \mathcal{P}^{\ell+1}(F), \qquad \mathcal{R}^{c,\ell}(F) := (\mathbf{x} - \mathbf{x}_F)\mathcal{P}^{\ell-1}(F), \quad (2.3b) $$

(where $\mathbf{y}^\perp$ is a shorthand for the rotated vector $\mathcal{Q}_{-\pi/2}\mathbf{y}$) so that

$$ \mathcal{P}^{\ell}(F) = \mathcal{G}^{\ell}(F) \oplus \mathcal{G}^{c,\ell}(F) = \mathcal{R}^{\ell}(F) \oplus \mathcal{R}^{c,\ell}(F). \quad (2.4) $$

These decompositions of $\mathcal{P}^\ell(F)$ (as well as those of $\mathcal{P}^\ell(T)$ in (2.6) below) result from [2, Corollary 7.4]. Notice that the direct sums in the above expression are not $\mathbb{L}^2$-orthogonal in general. The $\mathbb{L}^2$-orthogonal projectors on the spaces (2.3) are, with obvious notation, $\pi_{\mathcal{G},F}^\ell$, $\pi_{\mathcal{G},F}^{c,\ell}$, $\pi_{\mathcal{R},F}^\ell$, and $\pi_{\mathcal{R},F}^{c,\ell}$. Similarly, for any mesh element $T \in \mathcal{T}_h$ and any integer $\ell \ge 0$ we introduce the following subspaces of $\mathcal{P}^\ell(T)$:

$$ \mathcal{G}^\ell(T) := \operatorname{grad} \mathcal{P}^{\ell+1}(T), \qquad \mathcal{G}^{c,\ell}(T) := (\mathbf{x} - \mathbf{x}_T) \times \mathcal{P}^{\ell-1}(T), \quad (2.5a) $$

$$ \mathcal{R}^\ell(T) := \operatorname{curl} \mathcal{P}^{\ell+1}(T), \qquad \mathcal{R}^{c,\ell}(T) := (\mathbf{x} - \mathbf{x}_T)\mathcal{P}^{\ell-1}(T), \quad (2.5b) $$

so that

$$ \mathcal{P}^{\ell}(T) = \mathcal{G}^{\ell}(T) \oplus \mathcal{G}^{c,\ell}(T) = \mathcal{R}^{\ell}(T) \oplus \mathcal{R}^{c,\ell}(T). \quad (2.6) $$
## 2.4 Polynomial spaces and decompositions

For a given integer $\ell \ge 0$, $\mathbb{P}_h^\ell$ denotes the space of $n$-variate polynomials of total degree $\le \ell$, with the convention that $\mathbb{P}_0^\ell = \mathbb{R}$ for any $\ell$ and that $\mathbb{P}_n^{-1} := \{0\}$ for any $n$. For any $Y \in \mathcal{T}_h \cup \mathcal{F}_h \cup \mathcal{E}_h$, we denote by $\mathcal{P}^\ell(Y)$ the space spanned by the restriction to $Y$ of the functions in $\mathbb{P}_3^\ell$. Denoting by $1 \le n \le 3$ the dimension of $Y$, $\mathcal{P}^\ell(Y)$ is isomorphic to $\mathbb{P}_h^\ell$ (see [28, Proposition 1.23]). In what follows, with a little abuse of notation, both spaces are denoted by $\mathcal{P}^\ell(Y)$. We additionally denote by $\pi_{\mathcal{P},Y}^\ell$ the corresponding $\mathbb{L}^2$-orthogonal projector and let $\mathcal{P}^{0,\ell}(Y)$ denote the subspace of $\mathcal{P}^\ell(Y)$ made of polynomials with zero average over $Y$. For the sake of brevity, we also introduce the boldface notations $\mathcal{P}^\ell(T) := \mathcal{P}^\ell(T)^3$ for all $T \in \mathcal{T}_h$ and $\mathcal{P}^\ell(F) := \mathcal{P}^\ell(F)^2$ for all $F \in \mathcal{F}_h$.

Let again an integer $\ell \ge 1$ be given, and denote by $\mathfrak{G} \subset \mathcal{E}_h$ a collection of edges such that $\mathcal{S}_{\mathfrak{G}} := \bigcup_{E \in \mathfrak{G}} \bar{E}$ forms a connected set. We denote by $\mathcal{P}_c^\ell(\mathfrak{G}) := \{q_{\mathfrak{G}} \in \mathbb{C}^0(\mathcal{S}_{\mathfrak{G}}) : (q_{\mathfrak{G}})_{|E} \in \mathcal{P}^\ell(E) \text{ for all } E \in \mathfrak{G}\}$ the space of functions over $\mathcal{S}_{\mathfrak{G}}$ whose restriction to each edge $E \in \mathfrak{G}$ is a polynomial of total degree $\le \ell$ and that are continuous at the edges endpoints; these endpoints are collected in the set $\mathcal{V}_{\mathfrak{G}} \subset \mathcal{V}_h$. Denoting by $\mathbf{x}_V$ the coordinates vector of a vertex $V \in \mathcal{V}_h$, it can be easily checked that the following mapping is an isomorphism:

$$ \mathcal{P}_c^\ell(\mathfrak{G}) \ni q_{\mathfrak{G}} \mapsto ((\pi_{\mathcal{P},E}^{\ell-2}(q_{\mathfrak{G}})|_E)_{E \in \mathfrak{G}}, (q_{\mathfrak{G}}(\mathbf{x}_V))_{V \in \mathcal{V}_{\mathfrak{G}}}) \in \left( \bigtimes_{E \in \mathfrak{G}} \mathcal{P}^{\ell-2}(E) \right) \times \mathbb{R}^{\mathcal{V}_{\mathfrak{G}}}. \quad (2.2) $$

For all $Y \in \mathcal{T}_h \cup \mathcal{F}_h$, denote by $\mathbf{x}_Y$ a point inside $Y$ such that $Y$ contains a ball centered at $\mathbf{x}_Y$ of radius $\rho h_Y$, where $\rho$ is the mesh regularity parameter in [28, Definition 1.9]. For any mesh face $F \in \mathcal{F}_h$ and any integer $\ell \ge 0$, we define the following relevant subspaces of $\mathcal{P}^\ell(F)$:

$$ \mathcal{G}^\ell(F) := \operatorname{grad}_F \mathcal{P}^{\ell+1}(F), \qquad \mathcal{G}^{c,\ell}(F) := (\mathbf{x} - \mathbf{x}_F)^\perp \mathcal{P}^{\ell-1}(F), \quad (2.3\text{a}) $$

$$ \mathcal{R}^\ell(F) := \operatorname{rot}_F \mathcal{P}^{\ell+1}(F), \qquad \mathcal{R}^{c,\ell}(F) := (\mathbf{x} - \mathbf{x}_F)\mathcal{P}^{\ell-1}(F), \quad (2.3\text{b}) $$

(where $\mathbf{y}^\perp$ is a shorthand for the rotated vector $\mathcal{Q}_{-\pi/2}\mathbf{y}$) so that

$$ \mathcal{P}^{\ell}(F) = \mathcal{G}^{\ell}(F) \oplus \mathcal{G}^{c,\ell}(F) = \mathcal{R}^{\ell}(F) \oplus \mathcal{R}^{c,\ell}(F). \quad (2.4) $$

These decompositions of $\mathcal{P}^\ell(F)$ (as well as those of $\mathcal{P}^\ell(T)$ in (2.6) below) result from [2, Corollary 7.4]. Notice that the direct sums in the above expression are not $\mathbb{L}^2$-orthogonal in general. The $\mathbb{L}^2$-orthogonal projectors on the spaces (2.3) are, with obvious notation, $\pi_{\mathcal{G},F}^\ell$, $\pi_{\mathcal{G},F}^{c,\ell}$, $\pi_{\mathcal{R},F}^\ell$, and $\pi_{\mathcal{R},F}^{c,\ell}$. Similarly, for any mesh element $T \in \mathcal{T}_h$ and any integer $\ell \ge 0$ we introduce the following subspaces of $\mathcal{P}^\ell(T)$:

$$ \mathcal{G}^\ell(T) := \operatorname{grad} \mathcal{P}^{\ell+1}(T), \qquad \mathcal{G}^{c,\ell}(T) := (\mathbf{x} - \mathbf{x}_T) \times \mathcal{P}^{\ell-1}(T), \quad (2.5\text{a}) $$

$$ \mathcal{R}^\ell(T) := \operatorname{curl} \mathcal{P}^{\ell+1}(T), \qquad \mathcal{R}^{c,\ell}(T) := (\mathbf{x} - \mathbf{x}_T)\mathcal{P}^{\ell-1}(T), \quad (2.5\text{b}) $$

so that

$$ \mathcal{P}^{\ell}(T) = \mathcal{G}^{\ell}(T) \oplus \mathcal{G}^{c,\ell}(T) = \mathcal{R}^{\ell}(T) \oplus \mathcal{R}^{c,\ell}(T). \quad (2.6) $$
Also in this case, the direct sums above are not L²-orthogonal in general. The L²-orthogonal projectors on the spaces (2.5) are $\pi_{G,T}^l$, $\pi_{G,T}^{c,l}$, $\pi_{R,T}^l$, and $\pi_{R,T}^{c,l}$.

**Remark 1 (Hierarchical complements)** Unlike the L²-orthogonal complements considered in [31], the Koszul complements in (2.4) and (2.6) satisfy, for all $Y \in \mathcal{T}_h \cup \mathcal{F}_h$ and all $l \ge 1$,

$$ \mathcal{G}^{c,l-1}(Y) \subset \mathcal{G}^{c,l}(Y) \quad \text{and} \quad \mathcal{R}^{c,l-1}(Y) \subset \mathcal{R}^{c,l}(Y). \qquad (2.7) $$

**Remark 2 (Vector calculus isomorphisms on local polynomial spaces)** For any polygon F, polyhedron T, and polynomial degree $l \ge 0$, a consequence of the polynomial exactness [2, Corollary 7.3] is that the following mappings are isomorphisms:

$$ \mathrm{rot}_F : \mathcal{P}^{0,l}(F) \stackrel{\sim}{\to} \mathcal{R}^{l-1}(F) \qquad (2.8) $$

$$ \mathrm{div}_F : \mathcal{R}^{c,l}(F) \stackrel{\sim}{\to} \mathcal{P}^{l-1}(F), \quad \mathrm{div} : \mathcal{R}^{c,l}(T) \stackrel{\sim}{\to} \mathcal{P}^{l-1}(T), \qquad (2.9) $$

$$ \mathrm{curl} : \mathcal{G}^{c,l}(T) \stackrel{\sim}{\to} \mathcal{R}^{l-1}(T). \qquad (2.10) $$

An estimate of the norms of the inverses of these differential isomorphisms is provided in Lemma 9 in Appendix A.

**Remark 3 (Composition of L²-orthogonal projectors)** Let $\mathcal{X} \in \{\mathcal{G}, \mathcal{R}\}$, $l \ge -1$, and $Y \in \mathcal{T}_h \cup \mathcal{F}_h$. Using the definition of the L²-orthogonal projectors, and denoting by $\pi_{\mathcal{P},Y}^l$ the L²-orthogonal projector on $\mathcal{P}^l(Y)$, it holds

$$ \pi_{\mathcal{X},Y}^{l} = \pi_{\mathcal{X},Y}^{l} \circ \pi_{\mathcal{P},Y}^{l} \quad \text{and} \quad \pi_{\mathcal{X},Y}^{c,l} = \pi_{\mathcal{X},Y}^{c,l} \circ \pi_{\mathcal{P},Y}^{l}. \qquad (2.11) $$

In what follows, we will need the local Nédélec and Raviart-Thomas spaces: For $Y \in \mathcal{T}_h \cup \mathcal{F}_h$,

$$ \mathcal{N}^l(Y) := \mathcal{G}^{l-1}(Y) \oplus \mathcal{G}^{c,l}(Y), \quad \mathcal{R}\mathcal{T}^l(Y) := \mathcal{R}^{l-1}(Y) \oplus \mathcal{R}^{c,l}(Y). \qquad (2.12) $$

These spaces sit between $\mathcal{P}^{l-1}(Y)$ and $\mathcal{P}^l(Y)$ and are therefore referred to as trimmed in the FE literature. Notice that we have selected the index in (2.12) so as to reflect the maximum polynomial degrees of functions in each space and, as a result, it is shifted by +1 with respect to [31,29].

## 2.5 Recovery operator

As mentioned above, the direct sums in (2.4) and (2.6) are not L²-orthogonal. The following lemma however shows that, for any of these decompositions, a given polynomial can be recovered from its orthogonal projections on each space in the sum.
Also in this case, the direct sums above are not L²-orthogonal in general. The L²-orthogonal projectors on the spaces (2.5) are $\pi_{G,T}^l$, $\pi_{G,T}^{c,l}$, $\pi_{R,T}^l$, and $\pi_{R,T}^{c,l}$.

**Remark 1 (Hierarchical complements)** Unlike the L²-orthogonal complements considered in [31], the Koszul complements in (2.4) and (2.6) satisfy, for all $Y \in \mathcal{T}_h \cup \mathcal{F}_h$ and all $l \ge 1$,

$$ \mathcal{G}^{c,l-1}(Y) \subset \mathcal{G}^{c,l}(Y) \quad \text{and} \quad \mathcal{R}^{c,l-1}(Y) \subset \mathcal{R}^{c,l}(Y). \qquad (2.7) $$

**Remark 2 (Vector calculus isomorphisms on local polynomial spaces)** For any polygon F, polyhedron T, and polynomial degree $l \ge 0$, a consequence of the polynomial exactness [2, Corollary 7.3] is that the following mappings are isomorphisms:

$$ \mathrm{rot}_F : \mathcal{P}^{0,l}(F) \stackrel{\sim}{\to} \mathcal{R}^{l-1}(F) \qquad (2.8) $$

$$ \mathrm{div}_F : \mathcal{R}^{c,l}(F) \stackrel{\sim}{\to} \mathcal{P}^{l-1}(F), \quad \mathrm{div} : \mathcal{R}^{c,l}(T) \stackrel{\sim}{\to} \mathcal{P}^{l-1}(T), \qquad (2.9) $$

$$ \mathrm{curl} : \mathcal{G}^{c,l}(T) \stackrel{\sim}{\to} \mathcal{R}^{l-1}(T). \qquad (2.10) $$

An estimate of the norms of the inverses of these differential isomorphisms is provided in Lemma 9 in Appendix A.

**Remark 3 (Composition of L²-orthogonal projectors)** Let $\mathcal{X} \in \{\mathcal{G}, \mathcal{R}\}$, $l \ge -1$, and $Y \in \mathcal{T}_h \cup \mathcal{F}_h$. Using the definition of the L²-orthogonal projectors, and denoting by $\pi_{\mathcal{P},Y}^l$ the L²-orthogonal projector on $\mathcal{P}^l(Y)$, it holds

$$ \pi_{\mathcal{X},Y}^{l} = \pi_{\mathcal{X},Y}^{l} \circ \pi_{\mathcal{P},Y}^{l} \quad \text{and} \quad \pi_{\mathcal{X},Y}^{c,l} = \pi_{\mathcal{X},Y}^{c,l} \circ \pi_{\mathcal{P},Y}^{l}. \qquad (2.11) $$

In what follows, we will need the local Nédélec and Raviart-Thomas spaces: For $Y \in \mathcal{T}_h \cup \mathcal{F}_h$,

$$ \mathcal{N}^l(Y) := \mathcal{G}^{l-1}(Y) \oplus \mathcal{G}^{c,l}(Y), \quad \mathcal{R}\mathcal{T}^l(Y) := \mathcal{R}^{l-1}(Y) \oplus \mathcal{R}^{c,l}(Y). \qquad (2.12) $$

These spaces sit between $\mathcal{P}^{l-1}(Y)$ and $\mathcal{P}^l(Y)$ and are therefore referred to as trimmed in the FE literature. Notice that we have selected the index in (2.12) so as to reflect the maximum polynomial degrees of functions in each space and, as a result, it is shifted by +1 with respect to [31, 29].

## 2.5 Recovery operator

As mentioned above, the direct sums in (2.4) and (2.6) are not L²-orthogonal. The following lemma however shows that, for any of these decompositions, a given polynomial can be recovered from its orthogonal projections on each space in the sum.
**Lemma 1 (Recovery operator)** Let E be a Euclidean space, S be a subspace of E, and S<sup>c</sup> be a complement (not necessarily orthogonal) of S in E. Let π<sub>S</sub> and π<sub>S</sub><sup>c</sup> be, respectively, the orthogonal projections on S and S<sup>c</sup>. Then, the mappings Id − π<sub>S</sub>π<sub>S</sub><sup>c</sup> : E → E and Id − π<sub>S</sub><sup>c</sup>π<sub>S</sub> : E → E are isomorphisms.

We can therefore define the recovery operator ℜ<sub>S,S<sup>c</sup></sub>(·,·) : S×S<sup>c</sup> → E such that

$$
\begin{equation}
\begin{split}
\Re_{S,S^c}(\mathbf{b}, \mathbf{c}) := (\mathrm{Id} - \pi_S \pi_S^c)^{-1}(\mathbf{b} - \pi_S \mathbf{c}) &+ (\mathrm{Id} - \pi_S^c \pi_S)^{-1}(\mathbf{c} - \pi_S^c \mathbf{b}) \\
& \quad \forall (\mathbf{b}, \mathbf{c}) \in S \times S^c.
\end{split}
\tag{2.13}
\end{equation}
$$

This operator satisfies the following properties:

$$
\pi_S(\mathfrak{R}_{S,S^c}(\mathbf{b}, \mathbf{c})) = \mathbf{b} \quad \text{and} \quad \pi_S^c(\mathfrak{R}_{S,S^c}(\mathbf{b}, \mathbf{c})) = \mathbf{c} \quad \forall (\mathbf{b}, \mathbf{c}) \in S \times S^c, \quad (2.14)
$$

$$
\mathbf{a} = \mathfrak{R}_{S,S^c}(\pi_S \mathbf{a}, \pi_S^c \mathbf{a}) \quad \forall \mathbf{a} \in E. \tag{2.15}
$$

*Proof* Let us denote by $\|\cdot\|$ the norm in $E$. To prove that $\text{Id}-\pi_S\pi_S^c$ is invertible, we show that the mapping $\pi_S\pi_S^c$ has a norm $< 1$, which implies

$$
(\mathrm{Id} - \pi_S \pi_S^c)^{-1} = \sum_{n \ge 0} (\pi_S \pi_S^c)^n. \qquad (2.16)
$$

The space E being finite dimensional, it suffices to see that, for any **x** ∈ E
with ||**x**|| = 1, we have ||π_S(π_S^c**x**)|| < 1. Since π_S is an orthogonal projector,
by Pythagoras' theorem we have ||π_S(π_S^c**x**)|| ≤ ||π_S^c**x**||, with equality only if
π_S^c**x** ∈ S, that is, only if π_S^c**x** = **0** since π_S^c**x** ∈ S<sup>c</sup>. In this case, ||π_S(π_S^c**x**)|| = 0 < 1.
Otherwise, ||π_S(π_S^c**x**)|| < ||π_S^c**x**|| ≤ ||**x**|| = 1, where the second inequality is a
consequence of the fact that π_S^c is an orthogonal projection. This concludes
the proof that Id − π_Sπ_S^c is an isomorphism. The invertibility of Id − π_S^cπ_S is
obtained similarly, exchanging the roles of S and S<sup>c</sup>.

Let us prove the first relation in (2.14). The second follows using the same arguments. We expand $(\text{Id} - \pi_S \pi_S^c)^{-1}$ in (2.13) using the series (2.16) (and similarly for $(\text{Id} - \pi_S^c \pi_S)^{-1}$) to write

$$
\begin{align*}
\pi_S(\mathfrak{R}_{S,S^c}(\mathbf{b}, \mathbf{c})) &= \pi_S \sum_{n \ge 0} (\pi_S \pi_S^c)^n (\mathbf{b} - \pi_S \mathbf{c}) + \pi_S \sum_{n \ge 0} (\pi_S^c \pi_S)^n (\mathbf{c} - \pi_S^c \mathbf{b}) \\
&= \left[ \pi_S \sum_{n \ge 0} (\pi_S \pi_S^c)^n - \pi_S \sum_{n \ge 0} (\pi_S^c \pi_S)^n \pi_S^c \right] \mathbf{b} \\
&\quad + \left[ \pi_S \sum_{n \ge 0} (\pi_S^c \pi_S)^n - \pi_S \sum_{n \ge 0} (\pi_S \pi_S^c)^n \pi_S \right] \mathbf{c}.
\end{align*}
$$

We have $\pi_S \sum_{n \ge 0} (\pi_S^c \pi_S)^n \pi_S^c = \pi_S \sum_{n \ge 1} (\pi_S \pi_S^c)^n$ (we have used $\pi_S \pi_S = \pi_S$ to introduce the pre-factor $\pi_S$) and the operator acting on **b** above therefore reduces to $\pi_S$, and returns **b** since $\mathbf{b} \in S$. As for the operator acting on **c**, using again $\pi_S \pi_S = \pi_S$ shows that it is equal to 0. This concludes the proof of the first relation in (2.14).
**Lemma 1 (Recovery operator)** Let E be a Euclidean space, S be a subspace of E, and S<sup>c</sup> be a complement (not necessarily orthogonal) of S in E. Let π<sub>S</sub> and π<sub>S</sub><sup>c</sup> be, respectively, the orthogonal projections on S and S<sup>c</sup>. Then, the mappings Id − π<sub>S</sub>π<sub>S</sub><sup>c</sup> : E → E and Id − π<sub>S</sub><sup>c</sup>π<sub>S</sub> : E → E are isomorphisms.

We can therefore define the recovery operator ℜ<sub>S,S<sup>c</sup></sub>(·,·) : S×S<sup>c</sup> → E such that

$$
\mathfrak{R}_{S,S^c}(\mathbf{b}, \mathbf{c}) := (\mathrm{Id} - \pi_S \pi_S^c)^{-1}(\mathbf{b} - \pi_S \mathbf{c}) + (\mathrm{Id} - \pi_S^c \pi_S)^{-1}(\mathbf{c} - \pi_S^c \mathbf{b}) \\
\qquad \forall (\mathbf{b}, \mathbf{c}) \in S \times S^c. \quad (2.13)
$$

This operator satisfies the following properties:

$$
\pi_S(\mathfrak{R}_{S,S^c}(\mathbf{b}, \mathbf{c})) = \mathbf{b} \quad \text{and} \quad \pi_S^c(\mathfrak{R}_{S,S^c}(\mathbf{b}, \mathbf{c})) = \mathbf{c} \quad \forall (\mathbf{b}, \mathbf{c}) \in S \times S^c, \quad (2.14)
$$

$$
a = \mathfrak{R}_{S,S^c}(\pi_S a, \pi_S^c a) \quad \forall a \in E. \qquad (2.15)
$$

*Proof* Let us denote by $\|\cdot\|$ the norm in $E$. To prove that $\mathrm{Id}-\pi_S\pi_S^c$ is invertible, we show that the mapping $\pi_S\pi_S^c$ has a norm $< 1$, which implies

$$
(\mathrm{Id} - \pi_S \pi_S^c)^{-1} = \sum_{n \ge 0} (\pi_S \pi_S^c)^n. \qquad (2.16)
$$

The space E being finite dimensional, it suffices to see that, for any **x** ∈ E
with ||**x**|| = 1, we have ||π_S(π_S^c**x**)|| < 1. Since π_S is an orthogonal projector,
by Pythagoras' theorem we have ||π_S(π_S^c**x**)|| ≤ ||π_S^c**x**||, with equality only if
π_S^c**x** ∈ S, that is, only if π_S^c**x** = **0** since π_S^c**x** ∈ S<sup>c</sup>. In this case, ||π_S(π_S^c**x**)|| = 0 < 1.
Otherwise, ||π_S(π_S^c**x**)|| < ||π_S^c**x**|| ≤ ||**x**|| = 1, where the second inequality is a
consequence of the fact that π_S^c is an orthogonal projection. This concludes
the proof that Id − π_Sπ_S^c is an isomorphism. The invertibility of Id − π_S^cπ_S is
obtained similarly, exchanging the roles of S and S<sup>c</sup>.

Let us prove the first relation in (2.14). The second follows using the same arguments. We expand $(\mathrm{Id} - \pi_S \pi_S^c)^{-1}$ in (2.13) using the series (2.16) (and similarly for $(\mathrm{Id} - \pi_S^c \pi_S)^{-1}$) to write

$$
\begin{align*}
\pi_S(\mathfrak{R}_{S,S^c}(\mathbf{b}, \mathbf{c})) &= \pi_S \sum_{n \ge 0} (\pi_S \pi_S^c)^n (\mathbf{b} - \pi_S \mathbf{c}) + \pi_S \sum_{n \ge 0} (\pi_S^c \pi_S)^n (\mathbf{c} - \pi_S^c \mathbf{b}) \\
&= \left[ \pi_S \sum_{n \ge 0} (\pi_S \pi_S^c)^n - \pi_S \sum_{n \ge 0} (\pi_S^c \pi_S)^n \pi_S^c \right] \mathbf{b} \\
&\quad + \left[ \pi_S \sum_{n \ge 0} (\pi_S^c \pi_S)^n - \pi_S \sum_{n \ge 0} (\pi_S \pi_S^c)^n \pi_S \right] \mathbf{c}.
\end{align*}
$$

We have $\pi_S \sum_{n \ge 0} (\pi_S^c \pi_S)^n \pi_S^c = \pi_S \sum_{n \ge 1} (\pi_S \pi_S^c)^n$ (we have used $\pi_S \pi_S = \pi_S$ to introduce the pre-factor $\pi_S$) and the operator acting on $\mathbf{b}$ above therefore reduces to $\pi_S$, and returns $\mathbf{b}$ since $\mathbf{b} \in S$. As for the operator acting on $\mathbf{c}$, using again $\pi_S \pi_S = \pi_S$ shows that it is equal to 0. This concludes the proof of the first relation in (2.14).
Fix now $\mathbf{a} \in E$ and set $\mathbf{z} := \mathbf{a} - \mathfrak{R}_{S,S^c}(\pi_S \mathbf{a}, \pi_S^c \mathbf{a})$. Applying (2.14) to $\mathbf{b} = \pi_S \mathbf{a}$ and $\mathbf{c} = \pi_S^c \mathbf{a}$ shows that $\pi_S \mathbf{z} = \pi_S^c \mathbf{z} = \mathbf{0}$. Since $E = S \oplus S^c$, we can write $\mathbf{z} = \mathbf{z}_S + \mathbf{z}_S^c$ with $\mathbf{z}_S \in S$ and $\mathbf{z}_S^c \in S^c$, and the definition of the orthogonal projectors on $S$ and $S^c$ therefore yields, with $(\cdot, \cdot)_E$ the scalar product on $E$,

$$ \|\mathbf{z}\|^2 = (\mathbf{z}, \mathbf{z})_E = (\mathbf{z}, \mathbf{z}_S)_E + (\mathbf{z}, \mathbf{z}_S^c)_E = (\pi_S \mathbf{z}, \mathbf{z}_S)_E + (\pi_S^c \mathbf{z}, \mathbf{z}_S^c)_E = 0. $$

Hence, $\mathbf{z} = \mathbf{0}$ and (2.15) is established. $\square$

The following lemma shows that the norm of the recovery operator for the decompositions (2.4) and (2.6) is equivalent to the sum of the norms of its arguments, uniformly in $h$. In other words, it states that the decompositions are not just algebraic but also topological (uniformly in $h$). Since the recovery operator will mostly be of interest to us for these pairs of spaces, to alleviate the notations from here on we will write

$$ \mathfrak{R}_{\mathcal{X},Y}^{\ell}(\cdot, \cdot) := \mathfrak{R}_{\mathcal{X}^{\ell}(Y), \mathcal{X}^{c,\ell}(Y)}(\cdot, \cdot) \quad \forall \mathcal{X} \in \{\mathcal{R}, \mathcal{G}\}, \forall Y \in \mathcal{T}_h \cup \mathcal{F}_h. \quad (2.17) $$

**Lemma 2 (Estimate on the norm of the recovery operator)** For all $\ell \ge 0$, there exists $\alpha < 1$ depending only on the mesh regularity parameter in [28, Definition 1.9] such that, for all $\mathcal{X} \in \{\mathcal{R}, \mathcal{G}\}$ and all $Y \in \mathcal{T}_h \cup \mathcal{F}_h$,

$$ \|\pi_{\mathcal{X},Y}^{\ell} \pi_{\mathcal{X},Y}^{c,\ell}\|_{Y} \le \alpha \quad \text{and} \quad \|\pi_{\mathcal{X},Y}^{c,\ell} \pi_{\mathcal{X},Y}^{\ell}\|_{Y} \le \alpha, \quad (2.18) $$

where $\|\cdot\|_Y$ denotes the norm induced by $\|\cdot\|_{L^2(Y)}$ on the space of endomorphisms of $\mathcal{P}^\ell(Y)$. As a result,

$$ \|\mathfrak{R}_{\mathcal{X},Y}^{\ell}(\mathbf{v}, \mathbf{w})\|_{L^2(Y)} \approx \|\mathbf{v}\|_{L^2(Y)} + \|\mathbf{w}\|_{L^2(Y)} \quad \forall (\mathbf{v}, \mathbf{w}) \in \mathcal{X}^{\ell}(Y) \times \mathcal{X}^{c,\ell}(Y). \quad (2.19) $$

**Remark 4 (Recovery operator and $L^2$-orthogonal complements)** When working with $L^2$-orthogonal complements to $\mathcal{G}^\ell(Y)$ and $\mathcal{R}^\ell(Y)$, instead of the Koszul complements in (2.3) and (2.5), the recovery operator is trivial since it consists in the sum of its two arguments (its topological property (2.19) is also obvious). As mentioned in the introduction, however, the Koszul complements enable proofs of commutation and consistency properties that do not seem straightforward with orthogonal complements; the trade-off lies in having to deal with a less trivial recovery operator (although it remains a purely theoretical tool, see Remark 11), whose topological properties are more complex to establish.

**Proof 1.** *Proof of (2.18).* We estimate $\|\pi_{\mathcal{G},T}^{\ell} \pi_{\mathcal{G},T}^{c,\ell}\|_T$ for an element $T \in \mathcal{T}_h$, the other cases being identical. The linear mapping $\mathbb{R}^3 \ni \mathbf{x} \mapsto h_T^{-1}(\mathbf{x} - \mathbf{x}_T) \in \mathbb{R}^3$ maps $T$ onto a polyhedron $\tilde{T}$ of diameter 1, transports the spaces $\mathcal{P}^\ell(T)$, $\mathcal{G}^\ell(T)$ and $\mathcal{G}^{c,\ell}(T)$ on their equivalent over $\tilde{T}$, and simply scales the $L^2$-norm of functions. As a consequence, $\|\pi_{\mathcal{G},T}^{\ell} \pi_{\mathcal{G},T}^{c,\ell}\|_T = \|\pi_{\mathcal{G},\tilde{T}}^{\ell} \pi_{\mathcal{G},\tilde{T}}^{c,\ell}\|_{\tilde{T}}$, and we only have to estimate the latter quantity.
Fix now $\mathbf{a} \in E$ and set $\mathbf{z} := \mathbf{a} - \mathfrak{R}_{S,S^c}(\pi_S \mathbf{a}, \pi_S^c \mathbf{a})$. Applying (2.14) to $\mathbf{b} = \pi_S \mathbf{a}$ and $\mathbf{c} = \pi_S^c \mathbf{a}$ shows that $\pi_S \mathbf{z} = \pi_S^c \mathbf{z} = \mathbf{0}$. Since $E = S \oplus S^c$, we can write $\mathbf{z} = \mathbf{z}_S + \mathbf{z}_S^c$ with $\mathbf{z}_S \in S$ and $\mathbf{z}_S^c \in S^c$, and the definition of the orthogonal projectors on $S$ and $S^c$ therefore yields, with $(\cdot, \cdot)_E$ the scalar product on $E$,

$$ \|\mathbf{z}\|^2 = (\mathbf{z}, \mathbf{z})_E = (\mathbf{z}, \mathbf{z}_S)_E + (\mathbf{z}, \mathbf{z}_S^c)_E = (\pi_S \mathbf{z}, \mathbf{z}_S)_E + (\pi_S^c \mathbf{z}, \mathbf{z}_S^c)_E = 0. $$

Hence, $\mathbf{z} = \mathbf{0}$ and (2.15) is established. $\square$

The following lemma shows that the norm of the recovery operator for the decompositions (2.4) and (2.6) is equivalent to the sum of the norms of its arguments, uniformly in $h$. In other words, it states that the decompositions are not just algebraic but also topological (uniformly in $h$). Since the recovery operator will mostly be of interest to us for these pairs of spaces, to alleviate the notations from here on we will write

$$ \mathfrak{R}_{\mathcal{X},Y}^{\ell}(\cdot, \cdot) := \mathfrak{R}_{\mathcal{X}^{\ell}(Y), \mathcal{X}^{c,\ell}(Y)}(\cdot, \cdot) \quad \forall \mathcal{X} \in \{\mathcal{R}, \mathcal{G}\}, \forall Y \in \mathcal{T}_h \cup \mathcal{F}_h. \quad (2.17) $$

**Lemma 2 (Estimate on the norm of the recovery operator)** For all $\ell \ge 0$, there exists $\alpha < 1$ depending only on the mesh regularity parameter in [28, Definition 1.9] such that, for all $\mathcal{X} \in \{\mathcal{R}, \mathcal{G}\}$ and all $Y \in \mathcal{T}_h \cup \mathcal{F}_h$,

$$ \|\pi_{\mathcal{X},Y}^{\ell} \pi_{\mathcal{X},Y}^{c,\ell}\|_{Y} \le \alpha \quad \text{and} \quad \|\pi_{\mathcal{X},Y}^{c,\ell} \pi_{\mathcal{X},Y}^{\ell}\|_{Y} \le \alpha, \quad (2.18) $$

where $\|\cdot\|_Y$ denotes the norm induced by $\|\cdot\|_{L^2(Y)}$ on the space of endomorphisms of $\mathcal{P}^\ell(Y)$. As a result,

$$ \|\mathfrak{R}_{\mathcal{X},Y}^{\ell}(\mathbf{v}, \mathbf{w})\|_{L^2(Y)} \approx \|\mathbf{v}\|_{L^2(Y)} + \|\mathbf{w}\|_{L^2(Y)} \quad \forall (\mathbf{v}, \mathbf{w}) \in \mathcal{X}^{\ell}(Y) \times \mathcal{X}^{c,\ell}(Y). \quad (2.19) $$

**Remark 4 (Recovery operator and $L^2$-orthogonal complements)** When working with $L^2$-orthogonal complements to $\mathcal{G}^\ell(Y)$ and $\mathcal{R}^\ell(Y)$, instead of the Koszul complements in (2.3) and (2.5), the recovery operator is trivial since it consists in the sum of its two arguments (its topological property (2.19) is also obvious). As mentioned in the introduction, however, the Koszul complements enable proofs of commutation and consistency properties that do not seem straightforward with orthogonal complements; the trade-off lies in having to deal with a less trivial recovery operator (although it remains a purely theoretical tool, see Remark 11), whose topological properties are more complex to establish.

**Proof 1.** *Proof of (2.18).* We estimate $\|\pi_{\mathcal{G},T}^{\ell} \pi_{\mathcal{G},T}^{c,\ell}\|_T$ for an element $T \in \mathcal{T}_h$, the other cases being identical. The linear mapping $\mathbb{R}^3 \ni \mathbf{x} \mapsto h_T^{-1}(\mathbf{x} - \mathbf{x}_T) \in \mathbb{R}^3$ maps $T$ onto a polyhedron $\tilde{T}$ of diameter 1, transports the spaces $\mathcal{P}^\ell(T)$, $\mathcal{G}^\ell(T)$ and $\mathcal{G}^{c,\ell}(T)$ on their equivalent over $\tilde{T}$, and simply scales the $L^2$-norm of functions. As a consequence, $\|\pi_{\mathcal{G},T}^{\ell} \pi_{\mathcal{G},T}^{c,\ell}\|_T = \|\pi_{\mathcal{G},\tilde{T}}^{\ell} \pi_{\mathcal{G},\tilde{T}}^{c,\ell}\|_{\tilde{T}}$, and we only have to estimate the latter quantity.
Assume that we establish the existence of $\alpha < 1$, depending only on the mesh regularity parameter, such that

$$ \int_{\hat{T}} \nu \cdot \pi_{\hat{G}, \hat{T}}^{\ell} \nu \le \alpha^2 \|\nu\|_{L^2(\hat{T})} \|\pi_{\hat{G}, \hat{T}}^{\ell} \nu\|_{L^2(\hat{T})} \quad \forall \nu \in \mathcal{G}^{c, \ell}(\hat{T}) = \mathbf{x} \times \mathcal{P}^{\ell-1}(\hat{T}). \quad (2.20) $$

Notice that, with the selected mapping, $\mathbf{x}_T$ is mapped onto $\mathbf{0} \in \hat{T}$. Then, for all $\mathbf{w} \in \mathcal{P}^{\ell}(T)$,

$$ \begin{aligned} \int_{\hat{T}} \pi_{\hat{G}, \hat{T}}^{\ell}(\pi_{\hat{G}, \hat{T}}^{c, \ell} \mathbf{w}) \cdot \pi_{\hat{G}, \hat{T}}^{\ell}(\pi_{\hat{G}, \hat{T}}^{c, \ell} \mathbf{w}) &= \int_{\hat{T}} \pi_{\hat{G}, \hat{T}}^{c, \ell} \mathbf{w} \cdot \pi_{\hat{G}, \hat{T}}^{\ell}(\pi_{\hat{G}, \hat{T}}^{c, \ell} \mathbf{w}) \\ &\le \alpha^2 \|\pi_{\hat{G}, \hat{T}}^{c, \ell} \mathbf{w}\|_{L^2(\hat{T})} \|\pi_{\hat{G}, \hat{T}}^{\ell} \pi_{\hat{G}, \hat{T}}^{c, \ell} \mathbf{w}\|_{L^2(\hat{T})} \\ &\le \alpha^2 \|\mathbf{w}\|_{L^2(\hat{T})}^2, \end{aligned} \quad (2.21) $$

where the equality comes from the definition of $\pi_{\hat{G}, \hat{T}}^{\ell}$, the first inequality is obtained applying (2.20) to $\nu = \pi_{\hat{G}, \hat{T}}^{c, \ell} \mathbf{w}$, and the conclusion is obtained using the fact that $\pi_{\hat{G}, \hat{T}}^{\ell}$ and $\pi_{\hat{G}, \hat{T}}^{c, \ell}$ are both $L^2(\hat{T})$-orthogonal projectors and have thus norm 1. The bound (2.21) shows that $\|\pi_{\hat{G}, \hat{T}}^{\ell} \pi_{\hat{G}, \hat{T}}^{c, \ell}\|_{\hat{T}} \le \alpha$ and concludes the proof.

We therefore only have to establish (2.20). Note that, in the rest of the proof, polynomials are indifferently considered over $\mathbb{R}^3$ or some of its open subsets. We also remark that, by choice of $\mathbf{x}_T$ in $T$ and of the mapping $T \mapsto \hat{T}$, we have $B(\rho) \subset \hat{T} \subset B(1)$, where $B(r)$ is the ball in $\mathbb{R}^d$ centered at $\mathbf{0}$ and of radius $r$. The proof of (2.20) is done by contradiction: if this relation does not hold, there exists a sequence $(\hat{T}_n)_{n \in \mathbb{N}}$ of open sets between $B(\rho)$ and $B(1)$, a sequence $(\alpha_n)_{n \in \mathbb{N}}$ converging to 1, and a sequence $(\nu_n)_{n \in \mathbb{N}}$ in $\mathbf{x} \times \mathcal{P}^{\ell-1}(\mathbb{R}^3)$ such that

$$ \int_{\hat{T}_n} \nu_n \cdot \pi_{\hat{G}, \hat{T}_n}^{\ell} \nu_n > \alpha_n^2 \|\nu_n\|_{L^2(\hat{T}_n)} \|\pi_{\hat{G}, \hat{T}_n}^{\ell} \nu_n\|_{L^2(\hat{T}_n)}. \quad (2.22) $$

Upon replacing $\nu_n$ by $\nu_n/\|\nu_n\|_{L^2(\hat{T}_n)}$, we can assume that $\|\nu_n\|_{L^2(\hat{T}_n)} = 1$. Since $B(\rho) \subset \hat{T}_n$, we infer that $\|\nu_n\|_{L^2(B(\rho))} \le \|\nu_n\|_{L^2(\hat{T}_n)} = 1$; hence, $(\nu_n)_{n \in \mathbb{N}}$ is bounded for the $L^2(B(\rho))$-norm in the finite-dimensional space $\mathbf{x} \times \mathcal{P}^{\ell-1}(\mathbb{R}^3)$, and converges up to a subsequence to some $\nu \in \mathbf{x} \times \mathcal{P}^{\ell-1}(\mathbb{R}^3)$. Likewise, we can assume that $\pi_{\hat{G}, \hat{T}_n}^{\ell} \nu_n \to \mathbf{w}$ in $\mathcal{G}^{\ell}(\mathbb{R}^3)$. The characteristic function $1_{\hat{T}_n}$ satisfies $1_{B(\rho)} \le 1_{\hat{T}_n} \le 1_{B(1)}$ and converges therefore, up to a subsequence, in $L^\infty(B(1))$ weak-★ towards some function $\theta$ satisfying $1_{B(\rho)} \le \theta \le 1_{B(1)}$. Noting that

$$ \begin{aligned} & \int_{\hat{T}_n} \nu_n \cdot \pi_{\hat{G}, \hat{T}_n}^{\ell} \nu_n = \int_{B(1)} 1_{\hat{T}_n} \nu_n \cdot \pi_{\hat{G}, \hat{T}_n}^{\ell} \nu_n, \\ & \|\nu_n\|_{L^2(\hat{T}_n)}^2 = \int_{B(1)} 1_{\hat{T}_n} |\nu_n|^2, \quad \text{and} \\ & \|\pi_{\hat{G}, \hat{T}_n}^{\ell} \nu_n\|_{L^2(\hat{T}_n)}^2 = \int_{B(1)} 1_{\hat{T}_n} |\pi_{\hat{G}, \hat{T}_n}^{\ell} \nu_n|^2, \end{aligned} $$
Assume that we establish the existence of $\alpha < 1$, depending only on the mesh regularity parameter, such that

$$ \int_{\hat{T}} \nu \cdot \pi_{\hat{G}, \hat{T}}^{\ell} \nu \le \alpha^2 \|\nu\|_{L^2(\hat{T})} \|\pi_{\hat{G}, \hat{T}}^{\ell} \nu\|_{L^2(\hat{T})} \quad \forall \nu \in \mathcal{G}^{c, \ell}(\hat{T}) = \mathbf{x} \times \mathcal{P}^{\ell-1}(\hat{T}). \quad (2.20) $$

Notice that, with the selected mapping, $\mathbf{x}_T$ is mapped onto $\mathbf{0} \in \hat{T}$. Then, for all $\mathbf{w} \in \mathcal{P}^{\ell}(T)$,

$$ \begin{aligned} \int_{\hat{T}} \pi_{\hat{G}, \hat{T}}^{\ell} (\pi_{\hat{G}, \hat{T}}^{c, \ell} \mathbf{w}) \cdot \pi_{\hat{G}, \hat{T}}^{\ell} (\pi_{\hat{G}, \hat{T}}^{c, \ell} \mathbf{w}) &= \int_{\hat{T}} \pi_{\hat{G}, \hat{T}}^{c, \ell} \mathbf{w} \cdot \pi_{\hat{G}, \hat{T}}^{\ell} (\pi_{\hat{G}, \hat{T}}^{c, \ell} \mathbf{w}) \\ &\le \alpha^2 \|\pi_{\hat{G}, \hat{T}}^{c, \ell} \mathbf{w}\|_{L^2(\hat{T})} \|\pi_{\hat{G}, \hat{T}}^{\ell} \pi_{\hat{G}, \hat{T}}^{c, \ell} \mathbf{w}\|_{L^2(\hat{T})} \\ &\le \alpha^2 \|\mathbf{w}\|_{L^2(\hat{T})}^2, \end{aligned} \quad (2.21) $$

where the equality comes from the definition of $\pi_{\hat{G}, \hat{T}}^{\ell}$, the first inequality is obtained applying (2.20) to $\nu = \pi_{\hat{G}, \hat{T}}^{c, \ell} \mathbf{w}$, and the conclusion is obtained using the fact that $\pi_{\hat{G}, \hat{T}}^{\ell}$ and $\pi_{\hat{G}, \hat{T}}^{c, \ell}$ are both $L^2(\hat{T})$-orthogonal projectors and have thus norm 1. The bound (2.21) shows that $\|\pi_{\hat{G}, \hat{T}}^{\ell} \pi_{\hat{G}, \hat{T}}^{c, \ell}\|_{\hat{T}} \le \alpha$ and concludes the proof.

We therefore only have to establish (2.20). Note that, in the rest of the proof, polynomials are indifferently considered over $\mathbb{R}^3$ or some of its open subsets. We also remark that, by choice of $\mathbf{x}_T$ in $T$ and of the mapping $T \mapsto \hat{T}$, we have $B(\rho) \subset \hat{T} \subset B(1)$, where $B(r)$ is the ball in $\mathbb{R}^d$ centered at $\mathbf{0}$ and of radius $r$. The proof of (2.20) is done by contradiction: if this relation does not hold, there exists a sequence $(\hat{T}_n)_{n \in \mathbb{N}}$ of open sets between $B(\rho)$ and $B(1)$, a sequence $(\alpha_n)_{n \in \mathbb{N}}$ converging to 1, and a sequence $(\nu_n)_{n \in \mathbb{N}}$ in $\mathbf{x} \times \mathcal{P}^{\ell-1}(\mathbb{R}^3)$ such that

$$ \int_{\hat{T}_n} \nu_n \cdot \pi_{\hat{G}, \hat{T}_n}^{\ell} \nu_n > \alpha_n^2 \|\nu_n\|_{L^2(\hat{T}_n)} \|\pi_{\hat{G}, \hat{T}_n}^{\ell} \nu_n\|_{L^2(\hat{T}_n)}. \quad (2.22) $$

Upon replacing $\nu_n$ by $\nu_n/\|\nu_n\|_{L^2(\hat{T}_n)}$, we can assume that $\|\nu_n\|_{L^2(\hat{T}_n)} = 1$. Since $B(\rho) \subset \hat{T}_n$, we infer that $\|\nu_n\|_{L^2(B(\rho))} \le \|\nu_n\|_{L^2(\hat{T}_n)} = 1$; hence, $(\nu_n)_{n \in \mathbb{N}}$ is bounded for the $L^2(B(\rho))$-norm in the finite-dimensional space $\mathbf{x} \times \mathcal{P}^{\ell-1}(\mathbb{R}^3)$, and converges up to a subsequence to some $\nu \in \mathbf{x} \times \mathcal{P}^{\ell-1}(\mathbb{R}^3)$. Likewise, we can assume that $\pi_{\hat{G}, \hat{T}_n}^{\ell} \nu_n \to \mathbf{w}$ in $\mathcal{G}^{\ell}(\mathbb{R}^3)$. The characteristic function $1_{\hat{T}_n}$ satisfies $1_{B(\rho)} \le 1_{\hat{T}_n} \le 1_{B(1)}$ and converges therefore, up to a subsequence, in $L^\infty(B(1))$ weak-★ towards some function $\theta$ satisfying $1_{B(\rho)} \le \theta \le 1_{B(1)}$. Noting that

$$ \begin{aligned} \int_{\hat{T}_n} \nu_n \cdot \pi_{\hat{G}, \hat{T}_n}^{\ell} \nu_n &= \int_{B(1)} 1_{\hat{T}_n} \nu_n \cdot \pi_{\hat{G}, \hat{T}_n}^{\ell} \nu_n, \\ \|\nu_n\|_{L^2(\hat{T}_n)}^2 &= \int_{B(1)} 1_{\hat{T}_n} |\nu_n|^2, \quad \text{and} \\ \|\pi_{\hat{G}, \hat{T}_n}^{\ell} \nu_n\|_{L^2(\hat{T}_n)}^2 &= \int_{B(1)} 1_{\hat{T}_n} |\pi_{\hat{G}, \hat{T}_n}^{\ell} \nu_n|^2, \end{aligned} $$
the aforementioned convergences enable us to take the limit $n \to \infty$ of (2.22) and find

$$ \int_{B(1)} \theta \mathbf{v} \cdot \mathbf{w} \geq \|\sqrt{\theta \mathbf{v}}\|_{L^2(B(1))} \|\sqrt{\theta \mathbf{w}}\|_{L^2(B(1))}. \quad (2.23) $$

The Cauchy-Schwarz inequality, on the other hand, gives

$$ \int_{B(1)} \theta \mathbf{v} \cdot \mathbf{w} = \int_{B(1)} \sqrt{\theta \mathbf{v}} \cdot \sqrt{\theta \mathbf{w}} \leq \|\sqrt{\theta \mathbf{v}}\|_{L^2(B(1))} \|\sqrt{\theta \mathbf{w}}\|_{L^2(B(1))}, $$

which, combined with (2.23), shows that,

$$ \int_{B(1)} \theta \mathbf{v} \cdot \mathbf{w} = \|\sqrt{\theta \mathbf{v}}\|_{L^2(B(1))} \|\sqrt{\theta \mathbf{w}}\|_{L^2(B(1))}. $$

Hence, $\sqrt{\theta \mathbf{v}}$ and $\sqrt{\theta \mathbf{w}}$ are co-linear. Restricted to $B(\rho)$, over which $\theta = 1$, this proves that $\mathbf{v}$ and $\mathbf{w}$ are co-linear. Since $\mathbf{v} \in \mathcal{G}^{c,\ell}(B(\rho))$ and $\mathbf{w} \in \mathcal{G}^{\ell}(B(\rho))$, we infer that $\mathbf{v} = \mathbf{w} = \mathbf{0}$ on $B(\rho)$, and thus on $\mathbb{R}^3$. This leads to $0 = \|\sqrt{\theta \mathbf{v}}\|_{L^2(B(1))} = \lim_{n \to \infty} \|\mathbf{v}_n\|_{L^2(\hat{T}_n)} = 1$, which yields the sought contradiction.

2. *Proof of (2.19).* By (2.13), recalling the abridged notation (2.17), we have

$$ \|\mathfrak{R}_{\chi,Y}^{\ell}(\mathbf{v}, \mathbf{w})\|_{L^2(Y)} \le \|(\mathrm{Id} - \pi_{\chi,Y}^{\ell} \pi_{\chi,Y}^{c,\ell})^{-1}\|_{Y} \left( \|\mathbf{v}\|_{L^2(Y)} + \|\pi_{\chi,Y}^{\ell} \mathbf{w}\|_{L^2(Y)} \right) \\ + \|(\mathrm{Id} - \pi_{\chi,Y}^{c,\ell} \pi_{\chi,Y}^{\ell})^{-1}\|_{Y} \left( \|\pi_{\chi,Y}^{c,\ell} \mathbf{v}\|_{L^2(Y)} + \|\mathbf{w}\|_{L^2(Y)} \right). $$

The expansion (2.16) and the estimates (2.18) show that

$$ \|(\mathrm{Id} - \pi_{\chi,Y}^{\ell} \pi_{\chi,Y}^{c,\ell})^{-1}\|_{Y} \le \sum_{n \ge 0} \|\pi_{\chi,Y}^{\ell} \pi_{\chi,Y}^{c,\ell}\|_{Y}^{n} \le \sum_{n \ge 0} \alpha^{n} = \frac{1}{1-\alpha} $$

and, similarly, $\|(\mathrm{Id} - \pi_{\chi,Y}^{c,\ell} \pi_{\chi,Y}^{\ell})^{-1}\|_{Y} \le \frac{1}{1-\alpha}$. Since $\|\pi_{\chi,Y}^{\ell} \mathbf{w}\|_{L^2(Y)} \le \|\mathbf{w}\|_{L^2(Y)}$ and $\|\pi_{\chi,Y}^{c,\ell} \mathbf{v}\|_{L^2(Y)} \le \|\mathbf{v}\|_{L^2(Y)}$ as both $\pi_{\chi,Y}^{\ell}$ and $\pi_{\chi,Y}^{c,\ell}$ are $L^2$-orthogonal projectors, we conclude that

$$ \|\mathfrak{R}_{\chi,Y}^{\ell}(\mathbf{v}, \mathbf{w})\|_{L^2(Y)} \le \frac{2}{1-\alpha} \left( \|\mathbf{v}\|_{L^2(Y)} + \|\mathbf{w}\|_{L^2(Y)} \right). $$

To prove the converse inequality, we use (2.14) along with the $L^2$-boundedness of $\pi_{\chi,Y}^{\ell}$ and $\pi_{\chi,Y}^{c,\ell}$ to write

$$ \|\mathbf{v}\|_{L^2(Y)} + \|\mathbf{w}\|_{L^2(Y)} = \|\pi_{\chi,Y}^{\ell} \mathfrak{R}_{\chi,Y}^{\ell}(\mathbf{v}, \mathbf{w})\|_{L^2(Y)} + \|\pi_{\chi,Y}^{c,\ell} \mathfrak{R}_{\chi,Y}^{\ell}(\mathbf{v}, \mathbf{w})\|_{L^2(Y)} \\ \le 2 \|\mathfrak{R}_{\chi,Y}^{\ell}(\mathbf{v}, \mathbf{w})\|_{L^2(Y)}. $$

This concludes the proof of the norm equivalence (2.19). $\square$
the aforementioned convergences enable us to take the limit $n \to \infty$ of (2.22) and find

$$ \int_{B(1)} \theta \mathbf{v} \cdot \mathbf{w} \geq \|\sqrt{\theta \mathbf{v}}\|_{L^2(B(1))} \|\sqrt{\theta \mathbf{w}}\|_{L^2(B(1))}. \quad (2.23) $$

The Cauchy–Schwarz inequality, on the other hand, gives

$$ \int_{B(1)} \theta \mathbf{v} \cdot \mathbf{w} = \int_{B(1)} \sqrt{\theta \mathbf{v}} \cdot \sqrt{\theta \mathbf{w}} \leq \|\sqrt{\theta \mathbf{v}}\|_{L^2(B(1))} \|\sqrt{\theta \mathbf{w}}\|_{L^2(B(1))}, $$

which, combined with (2.23), shows that,

$$ \int_{B(1)} \theta \mathbf{v} \cdot \mathbf{w} = \|\sqrt{\theta \mathbf{v}}\|_{L^2(B(1))} \|\sqrt{\theta \mathbf{w}}\|_{L^2(B(1))}. $$

Hence, $\sqrt{\theta \mathbf{v}}$ and $\sqrt{\theta \mathbf{w}}$ are co-linear. Restricted to $B(\rho)$, over which $\theta = 1$, this proves that $\mathbf{v}$ and $\mathbf{w}$ are co-linear. Since $\mathbf{v} \in \mathcal{G}^{c,l}(B(\rho))$ and $\mathbf{w} \in \mathcal{G}^{l}(B(\rho))$, we infer that $\mathbf{v} = \mathbf{w} = \mathbf{0}$ on $B(\rho)$, and thus on $\mathbb{R}^3$. This leads to $0 = \|\sqrt{\theta \mathbf{v}}\|_{L^2(B(1))} = \lim_{n \to \infty} \|\mathbf{v}_n\|_{L^2(\hat{T}_n)} = 1$, which yields the sought contradiction.

2. *Proof of (2.19).* By (2.13), recalling the abridged notation (2.17), we have

$$ \|\mathfrak{R}_{\chi,Y}^{l}(\mathbf{v}, \mathbf{w})\|_{L^2(Y)} \le \|(\mathrm{Id} - \pi_{\chi,Y}^{l} \pi_{\chi,Y}^{c,l})^{-1}\|_{Y} \left( \|\mathbf{v}\|_{L^2(Y)} + \|\pi_{\chi,Y}^{l} \mathbf{w}\|_{L^2(Y)} \right) \\ + \|(\mathrm{Id} - \pi_{\chi,Y}^{c,l} \pi_{\chi,Y}^{l})^{-1}\|_{Y} \left( \|\pi_{\chi,Y}^{c,l} \mathbf{v}\|_{L^2(Y)} + \|\mathbf{w}\|_{L^2(Y)} \right). $$

The expansion (2.16) and the estimates (2.18) show that

$$ \|(\mathrm{Id} - \pi_{\chi,Y}^{l} \pi_{\chi,Y}^{c,l})^{-1}\|_{Y} \le \sum_{n \ge 0} \|\pi_{\chi,Y}^{l} \pi_{\chi,Y}^{c,l}\|_{Y}^{n} \le \sum_{n \ge 0} \alpha^{n} = \frac{1}{1-\alpha} $$

and, similarly, $\|(\mathrm{Id} - \pi_{\chi,Y}^{c,l} \pi_{\chi,Y}^{l})^{-1}\|_{Y} \le \frac{1}{1-\alpha}$. Since $\|\pi_{\chi,Y}^{l} \mathbf{w}\|_{L^2(Y)} \le \|\mathbf{w}\|_{L^2(Y)}$ and $\|\pi_{\chi,Y}^{c,l} \mathbf{v}\|_{L^2(Y)} \le \|\mathbf{v}\|_{L^2(Y)}$ as both $\pi_{\chi,Y}^{l}$ and $\pi_{\chi,Y}^{c,l}$ are $L^2$-orthogonal projectors, we conclude that

$$ \|\mathfrak{R}_{\chi,Y}^{l}(\mathbf{v}, \mathbf{w})\|_{L^2(Y)} \le \frac{2}{1-\alpha} \left( \|\mathbf{v}\|_{L^2(Y)} + \|\mathbf{w}\|_{L^2(Y)} \right). $$

To prove the converse inequality, we use (2.14) along with the $L^2$-boundedness of $\pi_{\chi,Y}^{l}$ and $\pi_{\chi,Y}^{c,l}$ to write

$$ \|\mathbf{v}\|_{L^2(Y)} + \|\mathbf{w}\|_{L^2(Y)} = \|\pi_{\chi,Y}^{l} \mathfrak{R}_{\chi,Y}^{l}(\mathbf{v}, \mathbf{w})\|_{L^2(Y)} + \|\pi_{\chi,Y}^{c,l} \mathfrak{R}_{\chi,Y}^{l}(\mathbf{v}, \mathbf{w})\|_{L^2(Y)} \\ \le 2 \|\mathfrak{R}_{\chi,Y}^{l}(\mathbf{v}, \mathbf{w})\|_{L^2(Y)}. $$

This concludes the proof of the norm equivalence (2.19). □
### 3 Discrete de Rham complex

We define a discrete counterpart of the de Rham complex (1.1). Throughout the rest of this section, we fix an integer $k \ge 0$ corresponding to the polynomial degree of the discrete sequence. The rules used in the notations are detailed in Appendix C, and the main DDR-related notations are summarised in Table 4.

#### 3.1 Discrete spaces

The DDR spaces are spanned by vectors of polynomials whose components, each attached to a mesh entity, are selected in order to:

1) enable the reconstruction of consistent local discrete vector calculus operators and (scalar or vector) potentials in full polynomial spaces of total degree $\le k$ (or $\le k + 1$ for the potentials associated with the gradient);

2) give rise to exact local sequences on mesh elements and faces.

Specifically, the discrete counterparts of $H^1(\Omega)$, $H(\text{curl};\Omega)$, $H(\text{div};\Omega)$ and $L^2(\Omega)$ are respectively defined as follows:

$$ \underline{X}_{\text{grad},h}^k := \left\{ \begin{array}{l} q_h = ((q_T)_{T \in \mathcal{T}_h}, (q_F)_{F \in \mathcal{F}_h}, q_{\varepsilon_h}) : \\ \qquad q_T \in \mathcal{P}^{k-1}(T) \text{ for all } T \in \mathcal{T}_h, \\ \qquad q_F \in \mathcal{P}^{k-1}(F) \text{ for all } F \in \mathcal{F}_h, \\ \qquad \text{and } q_{\varepsilon_h} \in \mathcal{P}_c^{k+1}(\varepsilon_h) \end{array} \right\}, \quad (3.1) $$

$$ \underline{X}_{\text{curl},h}^k := \left\{ \begin{array}{l} \underline{v}_h = ((\nu_{\mathcal{R},T}, \nu_{\mathcal{R},T}^c)_{T \in \mathcal{T}_h}, (\nu_{\mathcal{R},F}, \nu_{\mathcal{R},F}^c)_{F \in \mathcal{F}_h}, (v_E)_{E \in \varepsilon_h}) : \\ \qquad \nu_{\mathcal{R},T} \in \mathcal{R}^{k-1}(T) \text{ and } \nu_{\mathcal{R},T}^c \in \mathcal{R}^{c,k}(T) \text{ for all } T \in \mathcal{T}_h, \\ \qquad \nu_{\mathcal{R},F} \in \mathcal{R}^{k-1}(F) \text{ and } \nu_{\mathcal{R},F}^c \in \mathcal{R}^{c,k}(F) \text{ for all } F \in \mathcal{F}_h, \\ \qquad \text{and } v_E \in \mathcal{P}^k(E) \text{ for all } E \in \varepsilon_h \end{array} \right\}, \quad (3.2) $$

$$ \underline{X}_{\text{div},h}^k := \left\{ \underline{w}_h = ((\mathbf{w}_{\mathcal{G},T}, \mathbf{w}_{\mathcal{G},T}^c)_{T \in \mathcal{T}_h}, (w_F)_{F \in \mathcal{F}_h}) : \begin{array}{l} \mathbf{w}_{\mathcal{G},T} \in \mathcal{G}^{k-1}(T) \text{ and } \mathbf{w}_{\mathcal{G},T}^c \in \mathcal{G}^{c,k}(T) \text{ for all } T \in \mathcal{T}_h, \\ \text{and } w_F \in \mathcal{P}^k(F) \text{ for all } F \in \mathcal{F}_h \end{array} \right\}, \quad (3.3) $$

and

$$ \mathcal{P}^k(\mathcal{T}_h) := \{q_h \in L^2(\Omega) : (q_h)|_T \in \mathcal{P}^k(T) \text{ for all } T \in \mathcal{T}_h\}. $$

**Remark 5 (Component of $\underline{X}_{\text{grad},h}^k$ on the mesh edge skeleton)** By the isomorphism (2.2) with $\mathcal{E} = \varepsilon_h$, we can replace the space $\mathcal{P}_c^{k+1}(\varepsilon_h)$ in the definition of $\underline{X}_{\text{grad},h}^k$ by the Cartesian product space $(\times_{E \in \varepsilon_h} \mathcal{P}^{k-1}(E)) \times \mathbb{R}^{V_h}$. This product space is easier to manipulate in practical implementations of the DDR complex.
### 3 Discrete de Rham complex

We define a discrete counterpart of the de Rham complex (1.1). Throughout the rest of this section, we fix an integer $k \ge 0$ corresponding to the polynomial degree of the discrete sequence. The rules used in the notations are detailed in Appendix C, and the main DDR-related notations are summarised in Table 4.

#### 3.1 Discrete spaces

The DDR spaces are spanned by vectors of polynomials whose components, each attached to a mesh entity, are selected in order to:

1) enable the reconstruction of consistent local discrete vector calculus operators and (scalar or vector) potentials in full polynomial spaces of total degree $\le k$ (or $\le k + 1$ for the potentials associated with the gradient);

2) give rise to exact local sequences on mesh elements and faces.

Specifically, the discrete counterparts of $H^1(\Omega)$, $H(\text{curl};\Omega)$, $H(\text{div};\Omega)$ and $L^2(\Omega)$ are respectively defined as follows:

$$ \underline{\mathbf{X}}_{\text{grad},h}^k := \left\{ \begin{array}{l} q_h = ((q_T)_{T \in \mathcal{T}_h}, (q_F)_{F \in \mathcal{F}_h}, q_{\varepsilon_h}) : \\ \qquad q_T \in \mathcal{P}^{k-1}(T) \text{ for all } T \in \mathcal{T}_h, \\ \qquad q_F \in \mathcal{P}^{k-1}(F) \text{ for all } F \in \mathcal{F}_h, \\ \qquad \text{and } q_{\varepsilon_h} \in \mathcal{P}_c^{k+1}(\varepsilon_h) \end{array} \right\}, \quad (3.1) $$

$$ \underline{\mathbf{X}}_{\text{curl},h}^k := \left\{ \begin{array}{l} \underline{\mathbf{v}}_h = ((\mathbf{v}_{\mathcal{R},T}, \mathbf{v}_{\mathcal{R},T}^c)_{T \in \mathcal{T}_h}, (\mathbf{v}_{\mathcal{R},F}, \mathbf{v}_{\mathcal{R},F}^c)_{F \in \mathcal{F}_h}, (\nu_E)_{E \in \varepsilon_h}) : \\ \qquad \mathbf{v}_{\mathcal{R},T} \in \mathcal{R}^{k-1}(T) \text{ and } \mathbf{v}_{\mathcal{R},T}^c \in \mathcal{R}^{c,k}(T) \text{ for all } T \in \mathcal{T}_h, \\ \qquad \mathbf{v}_{\mathcal{R},F} \in \mathcal{R}^{k-1}(F) \text{ and } \mathbf{v}_{\mathcal{R},F}^c \in \mathcal{R}^{c,k}(F) \text{ for all } F \in \mathcal{F}_h, \\ \qquad \text{and } \nu_E \in \mathcal{P}^k(E) \text{ for all } E \in \varepsilon_h \end{array} \right\}, \quad (3.2) $$

$$ \underline{\mathbf{X}}_{\text{div},h}^k := \left\{ \underline{\mathbf{w}}_h = ((\mathbf{w}_{\mathcal{G},T}, \mathbf{w}_{\mathcal{G},T}^c)_{T \in \mathcal{T}_h}, (\mathbf{w}_F)_{F \in \mathcal{F}_h}) : \mathbf{w}_{\mathcal{G},T} \in \mathcal{G}^{k-1}(T) \text{ and } \mathbf{w}_{\mathcal{G},T}^c \in \mathcal{G}^{c,k}(T) \text{ for all } T \in \mathcal{T}_h, \text{ and } \mathbf{w}_F \in \mathcal{P}^k(F) \text{ for all } F \in \mathcal{F}_h \right\}, \quad (3.3) $$

and

$$ \mathcal{P}^k(\mathcal{T}_h) := \{q_h \in L^2(\Omega) : (q_h)|_T \in \mathcal{P}^k(T) \text{ for all } T \in \mathcal{T}_h\}. $$

**Remark 5 (Component of $\underline{\mathbf{X}}_{\text{grad},h}^k$ on the mesh edge skeleton)** By the isomorphism (2.2) with $\mathcal{E} = \varepsilon_h$, we can replace the space $\mathcal{P}_c^{k+1}(\varepsilon_h)$ in the definition of $\underline{\mathbf{X}}_{\text{grad},h}^k$ by the Cartesian product space $(\times_{E \in \varepsilon_h} \mathcal{P}^{k-1}(E)) \times \mathbb{R}^{V_h}$. This product space is easier to manipulate in practical implementations of the DDR complex.
<table><caption>Table 1: Polynomial components attached to each mesh vertex $V \in \mathcal{V}_h$, edge $E \in \mathcal{E}_h$, face $F \in \mathcal{F}_h$, and element $T \in \mathcal{T}_h$ for each of the DDR spaces. The space $\mathcal{P}_c^{k+1}(\mathcal{E}_h)$ in the definition of $X_{\text{grad},h}^k$ has been replaced by $(\times_{E \in \mathcal{E}_h} \mathcal{P}^{k-1}(E)) \times \mathbb{R}^{\mathcal{V}_h}$, see Remark 5.</caption><thead><tr><th>Index</th><th>Space</th><th>V</th><th>E</th><th>F</th><th>T</th></tr></thead><tbody><tr><td>0</td><td>X<sup>k</sup><sub>grad,h</sub></td><td>R = P<sup>k</sup>(V)</td><td>P<sup>k-1</sup>(E)</td><td>P<sup>k-1</sup>(F)</td><td>P<sup>k-1</sup>(T)</td></tr><tr><td>1</td><td>X<sup>k</sup><sub>curl,h</sub></td><td></td><td>P<sup>k</sup>(E)</td><td>R<sup>k-1</sup>(F) &times; R<sup>c,k</sup>(F)</td><td>R<sup>k-1</sup>(T) &times; R<sup>c,k</sup>(T)</td></tr><tr><td>2</td><td>X<sup>k</sup><sub>div,h</sub></td><td></td><td></td><td>P<sup>k</sup>(F)</td><td>G<sup>k-1</sup>(T) &times; G<sup>c,k</sup>(T)</td></tr><tr><td>3</td><td>P<sup>k</sup>(T<sub>h</sub>)</td><td></td><td></td><td></td><td>P<sup>k</sup>(T)</td></tr></tbody></table>

<table><caption>Table 2: Number of degrees of freedom of the local DDR spaces for tetrahedral and hexahedral elements, and comparison with Raviart-Thomas-Nédélec (RTN) finite element spaces (in parentheses). For the discrete L²-space, we have L<sup>k</sup>(T) = Q<sup>k</sup>(T) (d-variate polynomials of degree ≤ k in each variable) for hexahedral RTN finite elements, L<sup>k</sup>(T) = P<sup>k</sup>(T) otherwise.</caption><thead><tr><th rowspan="2">k</th><th colspan="2">X<sup>k</sup><sub>grad,T</sub></th><th colspan="2">X<sup>k</sup><sub>curl,T</sub></th><th colspan="2">X<sup>k</sup><sub>div,T</sub></th><th colspan="2">L<sup>k</sup>(T)</th></tr><tr><th>Tetra</th><th>Hexa</th><th>Tetra</th><th>Hexa</th><th>Tetra</th><th>Hexa</th><th>Tetra</th><th>Hexa</th></tr></thead><tbody><tr><td>0</td><td>4 (4)</td><td>8 (8)</td><td>6 (6)</td><td>12 (12)</td><td>4 (4)</td><td>6 (6)</td><td>1 (1)</td><td>1 (1)</td></tr><tr><td>1</td><td>15 (10)</td><td>27 (27)</td><td>28 (20)</td><td>46 (54)</td><td>18 (15)</td><td>24 (36)</td><td>4 (4)</td><td>4 (8)</td></tr><tr><td>2</td><td>32 (20)</td><td>54 (64)</td><td>65 (45)</td><td>99 (144)</td><td>44 (36)</td><td>56 (108)</td><td>10 (10)</td><td>10 (27)</td></tr></tbody></table>

Remark 6 (Components of $X_{\text{curl},h}^k$ and $X_{\text{div},h}^k$) For each mesh element or face $Y \in \mathcal{T}_h \cup \mathcal{F}_h$, the pair of components $(v_{\mathcal{R},Y}, v_{\mathcal{R},Y}^c)$ of a vector in $X_{\text{curl},h}^k$ defines an element in $\mathcal{R}\mathcal{T}^k(Y)$. Similarly, for any $T \in \mathcal{T}_h$, each pair of element components $(w_{\mathcal{G},T}, w_{\mathcal{G},T}^c)$ of a vector in $X_{\text{div},h}^k$ defines an element in $\mathcal{N}^k(T)$. In the exposition, we prefer to distinguish these components as they play very different roles in the construction.

The polynomial components attached to mesh vertices, edges, faces, and elements for each of the DDR spaces are summarised in Table 1 (notice that we have accounted for Remark 5 for $X_{\text{grad},h}^k$). An inspection of Table 1 reveals that its diagonal contains full polynomial spaces on the mesh entities of dimension corresponding to the index of the space in the sequence (with the convection that $\mathcal{P}^k(V) := \mathbb{R}$ for any vertex $V \in \mathcal{V}_h$). The components collected in the upper triangular portion of the table are non-zero only for $k \ge 1$, and encode additional information required for the reconstruction of high-order discrete vector calculus operators and potentials. In particular, the complements $\mathcal{R}^{c,k}(F)$, $\mathcal{R}^{c,k}(T)$, and $\mathcal{G}^{c,k}(T)$ complete the information contained, respectively, in the face curl, element curl and tangential trace, and element divergence to construct the corresponding face or element vector potentials; see Sections 3.3.2, 4.2, and 4.3.
<table><caption>Table 1: Polynomial components attached to each mesh vertex $V \in \mathcal{V}_h$, edge $E \in \mathcal{E}_h$, face $F \in \mathcal{F}_h$, and element $T \in \mathcal{T}_h$ for each of the DDR spaces. The space $\mathcal{P}_c^{k+1}(\mathcal{E}_h)$ in the definition of $X_{\text{grad},h}^k$ has been replaced by $(\times_{E \in \mathcal{E}_h} \mathcal{P}^{k-1}(E)) \times \mathbb{R}^{\mathcal{V}_h}$, see Remark 5.</caption><thead><tr><th>Index</th><th>Space</th><th>V</th><th>E</th><th>F</th><th>T</th></tr></thead><tbody><tr><td>0</td><td>X<sup>k</sup><sub>grad,h</sub></td><td>R = P<sup>k</sup>(V)</td><td>P<sup>k-1</sup>(E)</td><td>P<sup>k-1</sup>(F)</td><td>P<sup>k-1</sup>(T)</td></tr><tr><td>1</td><td>X<sup>k</sup><sub>curl,h</sub></td><td></td><td>P<sup>k</sup>(E)</td><td>R<sup>k-1</sup>(F) × R<sup>c,k</sup>(F)</td><td>R<sup>k-1</sup>(T) × R<sup>c,k</sup>(T)</td></tr><tr><td>2</td><td>X<sup>k</sup><sub>div,h</sub></td><td></td><td></td><td>P<sup>k</sup>(F)</td><td>G<sup>k-1</sup>(T) × G<sup>c,k</sup>(T)</td></tr><tr><td>3</td><td>P<sup>k</sup>(T<sub>h</sub>)</td><td></td><td></td><td></td><td>P<sup>k</sup>(T)</td></tr></tbody></table>

<table><caption>Table 2: Number of degrees of freedom of the local DDR spaces for tetrahedral and hexahedral elements, and comparison with Raviart-Thomas-Nédélec (RTN) finite element spaces (in parentheses). For the discrete L²-space, we have L<sup>k</sup>(T) = Q<sup>k</sup>(T) (d-variate polynomials of degree ≤ k in each variable) for hexahedral RTN finite elements, L<sup>k</sup>(T) = P<sup>k</sup>(T) otherwise.</caption><thead><tr><th rowspan="2">k</th><th colspan="2">X<sup>k</sup><sub>grad,T</sub></th><th colspan="2">X<sup>k</sup><sub>curl,T</sub></th><th colspan="2">X<sup>k</sup><sub>div,T</sub></th><th colspan="2">L<sup>k</sup>(T)</th></tr><tr><th>Tetra</th><th>Hexa</th><th>Tetra</th><th>Hexa</th><th>Tetra</th><th>Hexa</th><th>Tetra</th><th>Hexa</th></tr></thead><tbody><tr><td>0</td><td>4 (4)</td><td>8 (8)</td><td>6 (6)</td><td>12 (12)</td><td>4 (4)</td><td>6 (6)</td><td>1 (1)</td><td>1 (1)</td></tr><tr><td>1</td><td>15 (10)</td><td>27 (27)</td><td>28 (20)</td><td>46 (54)</td><td>18 (15)</td><td>24 (36)</td><td>4 (4)</td><td>4 (8)</td></tr><tr><td>2</td><td>32 (20)</td><td>54 (64)</td><td>65 (45)</td><td>99 (144)</td><td>44 (36)</td><td>56 (108)</td><td>10 (10)</td><td>10 (27)</td></tr></tbody></table>

Remark 6 (Components of $X_{\text{curl},h}^k$ and $X_{\text{div},h}^k$) For each mesh element or face $Y \in \mathcal{T}_h \cup \mathcal{F}_h$, the pair of components $(v_{\mathcal{R},Y}, v_{\mathcal{R},Y}^c)$ of a vector in $X_{\text{curl},h}^k$ defines an element in $\mathcal{R}\mathcal{T}^k(Y)$. Similarly, for any $T \in \mathcal{T}_h$, each pair of element components $(w_{\mathcal{G},T}, w_{\mathcal{G},T}^c)$ of a vector in $X_{\text{div},h}^k$ defines an element in $\mathcal{N}^k(T)$. In the exposition, we prefer to distinguish these components as they play very different roles in the construction.

The polynomial components attached to mesh vertices, edges, faces, and elements for each of the DDR spaces are summarised in Table 1 (notice that we have accounted for Remark 5 for $X_{\text{grad},h}^k$). An inspection of Table 1 reveals that its diagonal contains full polynomial spaces on the mesh entities of dimension corresponding to the index of the space in the sequence (with the convection that $\mathcal{P}^k(V) := \mathbb{R}$ for any vertex $V \in \mathcal{V}_h$). The components collected in the upper triangular portion of the table are non-zero only for $k \ge 1$, and encode additional information required for the reconstruction of high-order discrete vector calculus operators and potentials. In particular, the complements $\mathcal{R}^{c,k}(F)$, $\mathcal{R}^{c,k}(T)$, and $\mathcal{G}^{c,k}(T)$ complete the information contained, respectively, in the face curl, element curl and tangential trace, and element divergence to construct the corresponding face or element vector potentials; see Sections 3.3.2, 4.2, and 4.3.
In what follows, given $\bullet \in \{\text{grad}, \text{curl}, \text{div}\}$ and a mesh entity $Y$ of dimension greater than or equal to the index of $\underline{X}_{\bullet,h}^k$, we denote by $\underline{X}_{\bullet,Y}^k$ the restriction of this space to $Y$, i.e., $\underline{X}_{\bullet,Y}^k$ contains the polynomial components attached to $Y$ and to all the mesh entities that lie on its boundary.

**Remark 7 (Comparison with Raviart-Thomas-Nédélec finite elements)** When $T$ is a tetrahedron or a hexahedron, the local spaces in the DDR sequence can be compared to classical (Raviart-Thomas-Nédélec) FE spaces. The number of degrees of freedom in each case for polynomial degrees $k \in \{0, 1, 2\}$ (the most commonly used) is reported in Table 2. For $k \ge 1$, the DDR construction leads to slightly larger spaces on tetrahedra and to significantly smaller spaces on hexahedra. The number of degrees of freedom for the DDR spaces could be further reduced adapting the serendipity techniques of Virtual Elements [6]; this topic is left for a future work.

For codes aiming at general meshes, the implementation of the DDR spaces requires the local (element-by-element) computation of discrete vector operators and potentials, which is an additional cost with respect to traditional FE codes. It should be noticed, however, that: 1) these computations are an embarrassingly parallel task that scales with the number of mesh elements, and are therefore asymptotically less expensive than the resolution of the algebraic systems (see, e.g., Figure 3); 2) this cost can be substantially reduced when dealing with meshes composed of a finite number of element shapes using standard reference element techniques; 3) it possible to combine the FE and DDR approaches on a given mesh (using the former on elements of standard shape and the latter on elements of more general shape, possibly resulting from local mesh refinement).

## 3.2 Interpolators

In the following, for all $q_{\bullet,h} \in \underline{X}_{\text{grad},h}^k$, we set

$$q_E := (q_{\varepsilon_h})_{|E} \in \mathcal{P}^{k+1}(E). \qquad (3.4)$$

The interpolators on the DDR spaces are defined collecting component-wise L²-projections. Specifically $\underline{I}_{\text{grad},h}^k : C^0(\bar{\Omega}) \to \underline{X}_{\text{grad},h}^k$ is such that, for all $q \in C^0(\bar{\Omega})$,

$$ \begin{gathered} \underline{I}_{\text{grad},h}^k q := ((\pi_{\mathcal{P},T}^{k-1} q|_T)_{T \in \mathcal{T}_h}, (\pi_{\mathcal{P},F}^{k-1} q|_F)_{F \in \mathcal{F}_h}, q_{\varepsilon_h}) \in \underline{X}_{\text{grad},h}^k \\ \text{where } \pi_{\mathcal{P},E}^{k-1}(q_{\varepsilon_h})_{|E} = \pi_{\mathcal{P},E}^{k-1} q|_E \text{ for all } E \in \mathcal{E}_h \qquad (3.5) \\ \text{and } q_{\varepsilon_h}(\mathbf{x}_V) = q(\mathbf{x}_V) \text{ for all } V \in \mathcal{V}_h. \end{gathered} $$

$\underline{I}_{\text{curl},h}^k : C^0(\bar{\Omega}) \to \underline{X}_{\text{curl},h}^k$ is defined setting, for all $\mathbf{v} \in C^0(\bar{\Omega})$,

$$ \begin{aligned} \underline{I}_{\text{curl},h}^k \mathbf{v} := & ((\pi_{\mathcal{R},T}^{k-1} \mathbf{v}|_T, \pi_{\mathcal{R},T}^{c,k} \mathbf{v}|_T)_{T \in \mathcal{T}_h}, \\ & (\pi_{\mathcal{R},F}^{k-1} \mathbf{v}_{T,F}, \pi_{\mathcal{R},F}^{c,k} \mathbf{v}_{T,F})_{F \in \mathcal{F}_h}, \\ & (\pi_{\mathcal{P},E}^k (\mathbf{v}|_E \cdot \mathbf{t}_E)_{E \in \mathcal{E}_h}), \end{aligned} \qquad (3.6) $$
In what follows, given $\bullet \in \{\text{grad}, \text{curl}, \text{div}\}$ and a mesh entity $Y$ of dimension greater than or equal to the index of $\underline{X}_{\bullet,h}^k$, we denote by $\underline{X}_{\bullet,Y}^k$ the restriction of this space to $Y$, i.e., $\underline{X}_{\bullet,Y}^k$ contains the polynomial components attached to $Y$ and to all the mesh entities that lie on its boundary.

**Remark 7 (Comparison with Raviart-Thomas-Nédélec finite elements)** When $T$ is a tetrahedron or a hexahedron, the local spaces in the DDR sequence can be compared to classical (Raviart-Thomas-Nédélec) FE spaces. The number of degrees of freedom in each case for polynomial degrees $k \in \{0, 1, 2\}$ (the most commonly used) is reported in Table 2. For $k \ge 1$, the DDR construction leads to slightly larger spaces on tetrahedra and to significantly smaller spaces on hexahedra. The number of degrees of freedom for the DDR spaces could be further reduced adapting the serendipity techniques of Virtual Elements [6]; this topic is left for a future work.

For codes aiming at general meshes, the implementation of the DDR spaces requires the local (element-by-element) computation of discrete vector operators and potentials, which is an additional cost with respect to traditional FE codes. It should be noticed, however, that: 1) these computations are an embarrassingly parallel task that scales with the number of mesh elements, and are therefore asymptotically less expensive than the resolution of the algebraic systems (see, e.g., Figure 3); 2) this cost can be substantially reduced when dealing with meshes composed of a finite number of element shapes using standard reference element techniques; 3) it possible to combine the FE and DDR approaches on a given mesh (using the former on elements of standard shape and the latter on elements of more general shape, possibly resulting from local mesh refinement).

## 3.2 Interpolators

In the following, for all $q_{\bullet,h} \in \underline{X}_{\text{grad},h}^k$, we set

$$q_E := (q_{\varepsilon_h})_{|E} \in \mathcal{P}^{k+1}(E). \qquad (3.4)$$

The interpolators on the DDR spaces are defined collecting component-wise L²-projections. Specifically $\underline{I}_{\text{grad},h}^k : C^0(\bar{\Omega}) \to \underline{X}_{\text{grad},h}^k$ is such that, for all $q \in C^0(\bar{\Omega})$,

$$ \begin{gathered} \underline{I}_{\text{grad},h}^k q := ((\pi_{\mathcal{P},T}^{k-1} q|_T)_{T \in \mathcal{T}_h}, (\pi_{\mathcal{P},F}^{k-1} q|_F)_{F \in \mathcal{F}_h}, q_{\varepsilon_h}) \in \underline{X}_{\text{grad},h}^k \\ \text{where } \pi_{\mathcal{P},E}^{k-1}(q_{\varepsilon_h})_{|E} = \pi_{\mathcal{P},E}^{k-1} q|_E \text{ for all } E \in \mathcal{E}_h \qquad (3.5) \\ \text{and } q_{\varepsilon_h}(\mathbf{x}_V) = q(\mathbf{x}_V) \text{ for all } V \in \mathcal{V}_h. \end{gathered} $$

$\underline{I}_{\text{curl},h}^k : C^0(\bar{\Omega}) \to \underline{X}_{\text{curl},h}^k$ is defined setting, for all $\mathbf{v} \in C^0(\bar{\Omega})$,

$$ \begin{align} \underline{I}_{\text{curl},h}^k \mathbf{v} &:= ((\pi_{\mathcal{R},T}^{k-1} \mathbf{v}|_T, \pi_{\mathcal{R},T}^{c,k} \mathbf{v}|_T)_{T \in \mathcal{T}_h}, \nonumber \\ & \quad (\pi_{\mathcal{R},F}^{k-1} \mathbf{v}_{T,F}, \pi_{\mathcal{R},F}^{c,k} \mathbf{v}_{T,F})_{F \in \mathcal{F}_h}, \tag{3.6} \\ & \quad (\pi_{\mathcal{P},E}^k (\mathbf{v}|_E \cdot \mathbf{t}_E)_{E \in \mathcal{E}_h}), \nonumber \end{align} $$
where $v_{t,F} := n_F \times (v|_F \times n_F)$ denotes the tangent trace of $v$ over $F$. Finally, $\underline{I}_{\text{div},h}^k : \mathbf{H}^1(\Omega) \to \underline{X}_{\text{div},h}^k$ is such that, for all $w \in \mathbf{H}^1(\Omega)$,

$$ \underline{I}_{\text{div},h}^k w := ((\pi_{\mathcal{G},T}^{k-1} w|_T, \pi_{\mathcal{G},T}^{c,k} w|_T)_{T \in \mathcal{T}_h}, (\pi_{\mathcal{P},F}^k (w|_F \cdot n_F)_{F \in \mathcal{T}_h})). \quad (3.7) $$

The restriction of the above interpolators to a mesh entity $Y$ of dimension larger than or equal to the index of the corresponding space in the sequence (see Table 1) is denoted replacing the subscript $h$ by $Y$. Finally, we let $\pi_{\mathcal{P},h}^k : \mathbf{L}^2(\Omega) \to \mathcal{P}^k(\mathcal{T}_h)$ denote the global $\mathbf{L}^2$-orthogonal projector such that, for all $q \in \mathbf{L}^2(\Omega)$, $(\pi_{\mathcal{P},h}^k q)|_T = \pi_{\mathcal{P},T}^k q|_T$ for all $T \in \mathcal{T}_h$.

### 3.3 Discrete vector calculus operators

We define in this section the discrete vector calculus operators that appear in the DDR sequence, obtained collecting the $\mathbf{L}^2$-orthogonal projections of local discrete operators mapping on full polynomial spaces. In what follows, the operators that only appear in the discrete sequence (3.37) through projections are denoted in sans serif font, while those appearing verbatim (without projection) in the sequence are in standard font.

#### 3.3.1 Gradient

The discrete counterpart of the gradient operator in the DDR sequence maps on $\underline{X}_{\text{curl},h}^k$, and therefore requires to define local gradients on mesh edges, faces, and elements.

For any $E \in \mathcal{E}_h$, the edge gradient $G_E^k : \underline{X}_{\text{grad},E}^k \to \mathcal{P}^k(E)$ is defined as: For all $q_E \in \underline{X}_{\text{grad},E}^k = \mathcal{P}^{k+1}(E)$,

$$ G_E^k q_E := q'_E, \quad (3.8) $$

where the derivative is taken along $E$ according to the orientation of $t_E$.

For any $F \in \mathcal{T}_h$, the face gradient $\mathbf{G}_F^k : \underline{X}_{\text{grad},F}^k \to \mathcal{P}^k(F)$ is such that, for all $q_F = (q_F, q_{\mathcal{E}_F}) \in \underline{X}_{\text{grad},F}^k$ and all $w_F \in \mathcal{P}^k(F)$,

$$ \begin{aligned} \int_F \mathbf{G}_F^k q_F \cdot \mathbf{w}_F &= -\int_F q_F \operatorname{div}_F \mathbf{w}_F + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E q_{\mathcal{E}_F} (\mathbf{w}_F \cdot \mathbf{n}_{FE}) \quad (3.9) \\ &= \int_F \mathbf{grad}_F q_F \cdot \mathbf{w}_F + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E (q_{\mathcal{E}_F} - q_F)(\mathbf{w}_F \cdot \mathbf{n}_{FE}). \end{aligned} $$

The existence and uniqueness of $\mathbf{G}_F^k q_F$ in $\mathcal{P}^k(F)$ follow from the Riesz representation theorem applied to this space equipped with the usual $\mathbf{L}^2$-product. Similar considerations hold for the other discrete vector calculus operators defined below, and will not be repeated.
where $v_{t,F} := n_F \times (v|_F \times n_F)$ denotes the tangent trace of $v$ over $F$. Finally, $\underline{I}_{\text{div},h}^k : \mathbf{H}^1(\Omega) \to \underline{X}_{\text{div},h}^k$ is such that, for all $w \in \mathbf{H}^1(\Omega)$,

$$ \underline{I}_{\text{div},h}^k w := \left( (\pi_{\mathcal{G},T}^{k-1} w|_T, \pi_{\mathcal{G},T}^{c,k} w|_T)_{T \in \mathcal{T}_h}, (\pi_{\mathcal{P},F}^k (w|_F \cdot n_F)_{F \in \mathcal{T}_h}) \right). \quad (3.7) $$

The restriction of the above interpolators to a mesh entity $Y$ of dimension larger than or equal to the index of the corresponding space in the sequence (see Table 1) is denoted replacing the subscript $h$ by $Y$. Finally, we let $\pi_{\mathcal{P},h}^k : \mathbf{L}^2(\Omega) \to \mathcal{P}^k(\mathcal{T}_h)$ denote the global $\mathbf{L}^2$-orthogonal projector such that, for all $q \in \mathbf{L}^2(\Omega)$, $(\pi_{\mathcal{P},h}^k q)|_T = \pi_{\mathcal{P},T}^k q|_T$ for all $T \in \mathcal{T}_h$.

### 3.3 Discrete vector calculus operators

We define in this section the discrete vector calculus operators that appear in the DDR sequence, obtained collecting the $\mathbf{L}^2$-orthogonal projections of local discrete operators mapping on full polynomial spaces. In what follows, the operators that only appear in the discrete sequence (3.37) through projections are denoted in sans serif font, while those appearing verbatim (without projection) in the sequence are in standard font.

#### 3.3.1 Gradient

The discrete counterpart of the gradient operator in the DDR sequence maps on $\underline{X}_{\text{curl},h}^k$, and therefore requires to define local gradients on mesh edges, faces, and elements.

For any $E \in \mathcal{E}_h$, the edge gradient $G_E^k : \underline{X}_{\text{grad},E}^k \to \mathcal{P}^k(E)$ is defined as: For all $q_E \in \underline{X}_{\text{grad},E}^k = \mathcal{P}^{k+1}(E)$,

$$ G_E^k q_E := q'_E, \quad (3.8) $$

where the derivative is taken along $E$ according to the orientation of $t_E$.

For any $F \in \mathcal{T}_h$, the face gradient $\mathbf{G}_F^k : \underline{X}_{\text{grad},F}^k \to \mathcal{P}^k(F)$ is such that, for all $q_F = (q_F, q_{\mathcal{E}_F}) \in \underline{X}_{\text{grad},F}^k$ and all $w_F \in \mathcal{P}^k(F)$,

$$ \begin{aligned} \int_F \mathbf{G}_F^k q_F \cdot \mathbf{w}_F &= -\int_F q_F \operatorname{div}_F \mathbf{w}_F + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E q_{\mathcal{E}_F} (\mathbf{w}_F \cdot \mathbf{n}_{FE}) \quad (3.9) \\ &= \int_F \operatorname{grad}_F q_F \cdot \mathbf{w}_F + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E (q_{\mathcal{E}_F} - q_F)(\mathbf{w}_F \cdot \mathbf{n}_{FE}). \end{aligned} $$

The existence and uniqueness of $\mathbf{G}_F^k q_F$ in $\mathcal{P}^k(F)$ follow from the Riesz representation theorem applied to this space equipped with the usual $\mathbf{L}^2$-product. Similar considerations hold for the other discrete vector calculus operators defined below, and will not be repeated.
The scalar trace $\gamma_F^{k+1} : \underline{\mathcal{X}}_{\text{grad},F}^k \to \mathcal{P}^{k+1}(F)$ is such that, for all $\underline{q}_F \in \underline{\mathcal{X}}_{\text{grad},F}^k$,

$$ \int_F \gamma_F^{k+1} \underline{q}_F \operatorname{div}_F \mathbf{v}_F = - \int_F \mathbf{G}_F^k \underline{q}_F \cdot \mathbf{v}_F + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E q_{\mathcal{E}_F} (\mathbf{v}_F \cdot \mathbf{n}_{FE}) \\ \forall \mathbf{v}_F \in \mathcal{R}^{c,k+2}(F). \quad (3.10) $$

This relation defines $\gamma_F^{k+1} \underline{q}_F$ uniquely in $\mathcal{P}^{k+1}(F)$ owing to the isomorphism property (2.9) with $l = k + 2$.

**Remark 8 (Validity of (3.10))** The relation (3.10) holds, in fact, for any $\mathbf{v}_F \in \mathcal{R}^k(F) \oplus \mathcal{R}^{c,k+2}(F)$. To check it, take $\mathbf{v}_F \in \mathcal{R}^k(F)$ and notice that the left-hand side vanishes owing to $\operatorname{div}_F \mathbf{v}_F = 0$, while the right-hand side vanishes owing to the definition (3.9) of $\mathbf{G}_F^k \underline{q}_F$ and again $\operatorname{div}_F \mathbf{v}_F = 0$. This means, in particular, that (3.10) holds for any $\mathbf{v}_F \in \mathcal{R}\mathcal{T}^{k+1}(F) \subset \mathcal{R}^k(F) \oplus \mathcal{R}^{c,k+2}(F)$ (see Remark 1).

For all $T \in \mathcal{T}_h$, the element gradient $\mathbf{G}_T^k : \underline{\mathcal{X}}_{\text{grad},T}^k \to \mathcal{P}^k(T)$ is defined such that, for all $\underline{q}_T \in \underline{\mathcal{X}}_{\text{grad},T}^k$ and all $\mathbf{w}_T \in \mathcal{P}^k(T)$,

$$ \int_T \mathbf{G}_T^k \underline{q}_T \cdot \mathbf{w}_T = - \int_T q_T \operatorname{div} \mathbf{w}_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \gamma_F^{k+1} \underline{q}_F (\mathbf{w}_T \cdot \mathbf{n}_F) \quad (3.11) \\ = \int_T \operatorname{grad} q_T \cdot \mathbf{w}_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F (\gamma_F^{k+1} \underline{q}_F - q_T)(\mathbf{w}_T \cdot \mathbf{n}_F), $$

where we have performed an integration by parts on the first term in the right-hand side to pass to the second line.

**Lemma 3 (Consistency properties)** The edge, face, and element gradients, and scalar trace satisfy the following consistency properties:

$$ \forall E \in \mathcal{E}_h \qquad G_E^k(\underline{I}_{\text{grad},E}^k q) = \pi_{\mathcal{P},E}^k(q') \qquad \forall q \in \mathbb{H}^1(E), \quad (3.12) $$

$$ \forall F \in \mathcal{F}_h \qquad \mathbf{G}_F^k(\underline{I}_{\text{grad},F}^k q) = \operatorname{grad}_F q \qquad \forall q \in \mathcal{P}^{k+1}(F), \quad (3.13) $$

$$ \forall F \in \mathcal{F}_h \qquad \gamma_F^{k+1}(\underline{I}_{\text{grad},F}^k q) = q \qquad \forall q \in \mathcal{P}^{k+1}(F), \quad (3.14) $$

$$ \forall F \in \mathcal{F}_h \qquad \pi_{\mathcal{P},F}^{k-1}(\gamma_F^{k+1} \underline{q}_F) = q_F \qquad \forall \underline{q}_F \in \underline{\mathcal{X}}_{\text{grad},F}^k, \quad (3.15) $$

$$ \forall T \in \mathcal{T}_h \qquad \mathbf{G}_T^k(\underline{I}_{\text{grad},T}^k q) = \operatorname{grad} q \qquad \forall q \in \mathcal{P}^{k+1}(T). \quad (3.16) $$

**Proof** Let us prove (3.12). Take $q \in \mathbb{H}^1(E)$. For all $r_E \in \mathcal{P}^k(E)$, denoting by $\mathbf{x}_{V_1}$ and $\mathbf{x}_{V_2}$ the coordinates of the vertices $V_1$ and $V_2$ of $E$, oriented so that $\mathbf{t}_E$ points from $V_1$ to $V_2$, we have

$$ \int_E (\underline{I}_{\text{grad},E}^k q)' r_E = (\underline{I}_{\text{grad},E}^k q r_E)(\mathbf{x}_{V_2}) - (\underline{I}_{\text{grad},E}^k q r_E)(\mathbf{x}_{V_1}) - \int_E (\underline{I}_{\text{grad},E}^k q) r'_E \\ = (q r_E)(\mathbf{x}_{V_2}) - (q r_E)(\mathbf{x}_{V_1}) - \int_E q r'_E = \int_E q' r_E, $$
The scalar trace $\gamma_F^{k+1} : \underline{\mathcal{X}}_{\text{grad},F}^k \to \mathcal{P}^{k+1}(F)$ is such that, for all $q \in \underline{\mathcal{X}}_{\text{grad},F}^k$,

$$ \int_F \gamma_F^{k+1} \underline{q}_F \operatorname{div}_F \mathbf{v}_F = - \int_F \mathbf{G}_F^k \underline{q}_F \cdot \mathbf{v}_F + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E q_{\mathcal{E}_F} (\mathbf{v}_F \cdot \mathbf{n}_{FE}) \\ \forall \mathbf{v}_F \in \mathcal{R}^{c,k+2}(F). \quad (3.10) $$

This relation defines $\gamma_F^{k+1} \underline{q}_F$ uniquely in $\mathcal{P}^{k+1}(F)$ owing to the isomorphism property (2.9) with $l = k + 2$.

**Remark 8 (Validity of (3.10))** The relation (3.10) holds, in fact, for any $\mathbf{v}_F \in \mathcal{R}^k(F) \oplus \mathcal{R}^{c,k+2}(F)$. To check it, take $\mathbf{v}_F \in \mathcal{R}^k(F)$ and notice that the left-hand side vanishes owing to $\operatorname{div}_F \mathbf{v}_F = 0$, while the right-hand side vanishes owing to the definition (3.9) of $\mathbf{G}_F^k \underline{q}_F$ and again $\operatorname{div}_F \mathbf{v}_F = 0$. This means, in particular, that (3.10) holds for any $\mathbf{v}_F \in \mathcal{R}\mathcal{T}^{k+1}(F) \subset \mathcal{R}^k(F) \oplus \mathcal{R}^{c,k+2}(F)$ (see Remark 1).

For all $T \in \mathcal{T}_h$, the element gradient $\mathbf{G}_T^k : \underline{\mathcal{X}}_{\text{grad},T}^k \to \mathcal{P}^k(T)$ is defined such that, for all $\underline{q}_T \in \underline{\mathcal{X}}_{\text{grad},T}^k$ and all $\mathbf{w}_T \in \mathcal{P}^k(T)$,

$$ \int_T \mathbf{G}_T^k \underline{q}_T \cdot \mathbf{w}_T = - \int_T q_T \operatorname{div} \mathbf{w}_T + \sum_{F \in \mathcal{T}_h} \omega_{TF} \int_F \gamma_F^{k+1} \underline{q}_F (\mathbf{w}_T \cdot \mathbf{n}_F) \quad (3.11) \\ = \int_T \operatorname{grad} q_T \cdot \mathbf{w}_T + \sum_{F \in \mathcal{T}_h} \omega_{TF} \int_F (\gamma_F^{k+1} \underline{q}_F - q_T)(\mathbf{w}_T \cdot \mathbf{n}_F), $$

where we have performed an integration by parts on the first term in the right-hand side to pass to the second line.

**Lemma 3 (Consistency properties)** The edge, face, and element gradients, and scalar trace satisfy the following consistency properties:

$$ \forall E \in \mathcal{E}_h \qquad G_E^k (\underline{I}_{\text{grad},E}^k q) = \pi_{\mathcal{P},E}^k (q') \qquad \forall q \in \mathbb{H}^1(E), \quad (3.12) $$

$$ \forall F \in \mathcal{F}_h \qquad \mathbf{G}_F^k (\underline{I}_{\text{grad},F}^k q) = \operatorname{grad}_F q \qquad \forall q \in \mathcal{P}^{k+1}(F), \quad (3.13) $$

$$ \forall F \in \mathcal{F}_h \qquad \gamma_F^{k+1} (\underline{I}_{\text{grad},F}^k q) = q \qquad \forall q \in \mathcal{P}^{k+1}(F), \quad (3.14) $$

$$ \forall F \in \mathcal{F}_h \qquad \pi_{\mathcal{P},F}^{k-1} (\gamma_F^{k+1} \underline{q}_F) = q_F \qquad \forall q_F \in \underline{\mathcal{X}}_{\text{grad},F}^k, \quad (3.15) $$

$$ \forall T \in \mathcal{T}_h \qquad \mathbf{G}_T^k (\underline{I}_{\text{grad},T}^k q) = \operatorname{grad} q \qquad \forall q \in \mathcal{P}^{k+1}(T). \quad (3.16) $$

**Proof** Let us prove (3.12). Take $q \in \mathbb{H}^1(E)$. For all $r_E \in \mathcal{P}^k(E)$, denoting by $\mathbf{x}_{V_1}$ and $\mathbf{x}_{V_2}$ the coordinates of the vertices $V_1$ and $V_2$ of $E$, oriented so that $\mathbf{t}_E$ points from $V_1$ to $V_2$, we have

$$ \int_E (\underline{I}_{\text{grad},E}^k q)' r_E = (\underline{I}_{\text{grad},E}^k q r_E)(\mathbf{x}_{V_2}) - (\underline{I}_{\text{grad},E}^k q r_E)(\mathbf{x}_{V_1}) - \int_E (\underline{I}_{\text{grad},E}^k q) r'_E \\ = (q r_E)(\mathbf{x}_{V_2}) - (q r_E)(\mathbf{x}_{V_1}) - \int_E q r'_E = \int_E q' r_E, $$
where we have used an integration by parts in the first line, obtained the second equality applying the definition of $I_{\text{grad},E}^k q \in \mathcal{P}^{k+1}(E)$ (which satisfies $(I_{\text{grad},E}^k q)(\mathbf{x}_V) = q(\mathbf{x}_V)$ for all $V \in \mathcal{V}_E$ and $\pi_{\mathcal{P},E}^{k-1}(I_{\text{grad},E}^k q) = \pi_{\mathcal{P},E}^{k-1} q$) together with $r'_E \in \mathcal{P}^{k-1}(E)$, and used another integration by parts to conclude. This proves that $(I_{\text{grad},E}^k q)' = \pi_{\mathcal{P},E}^k (q')'$.

Relation (3.13) can be deduced as in [31, Proposition 4.1]. To prove (3.14), we write (3.10) for $q_F = I_{\text{grad},F}^k q$ with $q \in \mathcal{P}^{k+1}(T)$, use (3.13), and notice that $q_{\mathcal{E}_F} = q|_{\partial F}$ (since $q|_{\partial F} \in \mathcal{P}_c^{k+1}(\mathcal{E}_F)$) to get, for all $\mathbf{v}_F \in \mathcal{R}^{c,k+2}(F)$,

$$ \int_F \gamma_F^{k+1} (I_{\text{grad},F}^k q) \text{div}_F \mathbf{v}_F = - \int_F \text{grad}_F q \cdot \mathbf{v}_F + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E q|_{\partial F} (\mathbf{v}_F \cdot \mathbf{n}_{FE}) \\ = \int_F q \text{div}_F \mathbf{v}_F. $$

The isomorphism property (2.9) with $\ell = k + 2$ then concludes the proof that $\gamma_F^{k+1}(I_{\text{grad},F}^k q) = q$.

The equality (3.15) follows from (3.10) written for $\mathbf{v}_F \in \mathcal{R}^{c,k}(F)$ (this choice is made possible by (2.7)) after replacing the full face gradient $\mathbf{G}_F^k$ by its definition (3.9), simplifying the boundary terms, and invoking again the isomorphism property (2.9), this time with $\ell = k$.

Finally, (3.16) can be established from (3.14) following the ideas in [31, Lemma 5.1]. □

The following proposition contains a stronger version of [31, Eq. (5.16)], with test function taken in the Nédélec space $\mathcal{N}^{k+1}(T)$ instead of $\mathcal{P}^k(T)$.

**Proposition 1 (Link between element and face gradients)** For all $T \in \mathcal{T}_h$ and all $(q_T, z_T) \in \underline{\mathcal{X}}_{\text{grad},T} \times \mathcal{N}^{k+1}(T)$,

$$ \int_T \mathbf{G}_T^k q_T \cdot \text{curl} z_T = - \sum_{F \in \mathcal{T}_T} \omega_{TF} \int_F \mathbf{G}_F^k q_F \cdot (z_T \times \mathbf{n}_F). \quad (3.17) $$

*Proof* Writing (3.11) with $w_T = \text{curl} z_T \in \mathcal{P}^k(T)$ and recalling the relation $\text{div} \text{curl} z_T = 0$, we have

$$ \int_T \mathbf{G}_T^k q_T \cdot \text{curl} z_T = \sum_{F \in \mathcal{T}_T} \omega_{TF} \int_F \gamma_F^{k+1} q_F (\text{curl} z_T \cdot \mathbf{n}_F) \\ = \sum_{F \in \mathcal{T}_T} \omega_{TF} \int_F \gamma_F^{k+1} q_F \text{div}_F (z_T \times \mathbf{n}_F), $$

the last equality being a consequence of [31, Eq. (3.7)]. To conclude, we invoke (3.10) with $\mathbf{v}_F = (z_T)|_F \times \mathbf{n}_F \in \mathcal{R}\mathcal{T}^{k+1}(F)$ (cf. (A.5) and Remark 8) and cancel the edge terms using [31, Eqs. (5.13) and (5.14)]. □
where we have used an integration by parts in the first line, obtained the second equality applying the definition of $I_{\text{grad},E}^k q \in \mathcal{P}^{k+1}(E)$ (which satisfies $(I_{\text{grad},E}^k q)(\mathbf{x}_V) = q(\mathbf{x}_V)$ for all $V \in \mathcal{V}_E$ and $\pi_{\mathcal{P},E}^{k-1}(I_{\text{grad},E}^k q) = \pi_{\mathcal{P},E}^{k-1} q$) together with $r'_E \in \mathcal{P}^{k-1}(E)$, and used another integration by parts to conclude. This proves that $(I_{\text{grad},E}^k q)' = \pi_{\mathcal{P},E}^k (q)'$.

Relation (3.13) can be deduced as in [31, Proposition 4.1]. To prove (3.14), we write (3.10) for $q_F = I_{\text{grad},F}^k q$ with $q \in \mathcal{P}^{k+1}(T)$, use (3.13), and notice that $q_{\mathcal{E}_F} = q|_{\partial F}$ (since $q|_{\partial F} \in \mathcal{P}_c^{k+1}(\mathcal{E}_F)$) to get, for all $\mathbf{v}_F \in \mathcal{R}^{c,k+2}(F)$,

$$ \int_F \gamma_F^{k+1} (I_{\text{grad},F}^k q) \text{div}_F \mathbf{v}_F = - \int_F \text{grad}_F q \cdot \mathbf{v}_F + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E q|_{\partial F} (\mathbf{v}_F \cdot \mathbf{n}_{FE}) \\ = \int_F q \text{div}_F \mathbf{v}_F. $$

The isomorphism property (2.9) with $\ell = k + 2$ then concludes the proof that $\gamma_F^{k+1}(I_{\text{grad},F}^k q) = q$.

The equality (3.15) follows from (3.10) written for $\mathbf{v}_F \in \mathcal{R}^{c,k}(F)$ (this choice is made possible by (2.7)) after replacing the full face gradient $\mathbf{G}_F^k$ by its definition (3.9), simplifying the boundary terms, and invoking again the isomorphism property (2.9), this time with $\ell = k$.

Finally, (3.16) can be established from (3.14) following the ideas in [31, Lemma 5.1]. □

The following proposition contains a stronger version of [31, Eq. (5.16)], with test function taken in the Nédélec space $\mathcal{N}^{k+1}(T)$ instead of $\mathcal{P}^k(T)$.

**Proposition 1 (Link between element and face gradients)** For all $T \in \mathcal{T}_h$ and all $(\underline{q}_T, z_T) \in \underline{\mathcal{X}}_{\text{grad},T} \times \mathcal{N}^{k+1}(T)$,

$$ \int_T \mathbf{G}_T^k \underline{q}_T \cdot \text{curl} z_T = - \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \mathbf{G}_F^k \underline{q}_F \cdot (z_T \times \mathbf{n}_F). \quad (3.17) $$

*Proof* Writing (3.11) with $w_T = \text{curl} z_T \in \mathcal{P}^k(T)$ and recalling the relation $\text{div} \text{curl} z_T = 0$, we have

$$ \int_T \mathbf{G}_T^k \underline{q}_T \cdot \text{curl} z_T = \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \gamma_F^{k+1} \underline{q}_F (\text{curl} z_T \cdot \mathbf{n}_F) \\ = \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \gamma_F^{k+1} \underline{q}_F \text{div}_F (z_T \times \mathbf{n}_F), $$

the last equality being a consequence of [31, Eq. (3.7)]. To conclude, we invoke (3.10) with $\mathbf{v}_F = (z_T)|_F \times \mathbf{n}_F \in \mathcal{R}\mathcal{T}^{k+1}(F)$ (cf. (A.5) and Remark 8) and cancel the edge terms using [31, Eqs. (5.13) and (5.14)]. □
The global discrete gradient $\underline{\underline{G}}_h^k : \underline{\underline{X}}_{\underline{\underline{grad}},h}^k \to \underline{\underline{X}}_{\underline{\underline{curl}},h}^k$ is obtained collecting the projections of each local gradient on the space attached to the corresponding mesh entity: For all $q_h \in \underline{\underline{X}}_{\underline{\underline{grad}},h}^k$,

$$ \underline{\underline{G}}_h^k q_h := \left( (\pi_{\mathcal{R},T}^{k-1}(\underline{\underline{G}}_T^k q_T), \pi_{\mathcal{R},T}^{c,k}(\underline{\underline{G}}_T^k q_T))_{T \in \mathcal{T}_h}, \right. \\ \left. (\pi_{\mathcal{R},F}^{k-1}(\underline{\underline{G}}_F^k q_F), \pi_{\mathcal{R},F}^{c,k}(\underline{\underline{G}}_F^k q_F))_{F \in \mathcal{F}_h}, \right. \\ \left. (G_E^k q_E)_{E \in \mathcal{E}_h} \right). \qquad (3.18) $$

**Remark 9 (Practical implementation)** In schemes based on the DDR sequence, the discrete gradient (3.18) only appears as an argument of the discrete L²-product on $\underline{\underline{X}}_{\underline{\underline{curl}},h}^k$ (see (4.15) below), that is, composed with the scalar trace and potential reconstruction on this space. Thus, leveraging (4.29) below, one never has to implement $\underline{\underline{G}}_h^k$, as only the full element gradients $(\underline{\underline{G}}_T^k)_{T \in \mathcal{T}_h}$ are required. Similar considerations hold for the discrete curl defined by (3.31) below (see also [29, Remark 7] on this matter).

Notice that this strategy differs from the one often pursued in the context of Virtual Elements, which consists in directly taking the appropriate components of $\underline{\underline{G}}_h^k$ as degrees of freedom. This difference is linked to the fact that the present construction embeds what could be interpreted in Virtual Element terms as an *enhancement*, enabling us to reduce the degree of certain internal polynomial components.

### 3.3.2 Curl

We next consider the DDR counterpart of the curl operator, which maps on $\underline{\underline{X}}_{\underline{\underline{div}},h}^k$ and therefore has components at mesh faces and inside mesh elements. For all $F \in \mathcal{F}_h$, the face curl $C_F^k : \underline{\underline{X}}_{\underline{\underline{curl}},F}^k \to \mathcal{P}^k(F)$ is such that, for all $\underline{\underline{v}}_F = (\underline{\underline{v}}_{\mathcal{R},F}, \underline{\underline{v}}_{\mathcal{R},F}^c, (v_E)_{E \in \mathcal{E}_F}) \in \underline{\underline{X}}_{\underline{\underline{curl}},F}^k$,

$$ \int_F C_F^k \underline{\underline{v}}_F r_F = \int_F \underline{\underline{v}}_{\mathcal{R},F} \cdot \text{rot}_F r_F - \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E v_E r_F \quad \forall r_F \in \mathcal{P}^k(F). \quad (3.19) $$

Reasoning as in [31, Proposition 4.3], we get

$$ C_F^k(\underline{\underline{I}}_{\underline{\underline{curl}},F}^k \underline{\underline{v}}) = \pi_{\mathcal{R},F}^k(\text{rot}_F \underline{\underline{v}}) \quad \forall \underline{\underline{v}} \in \mathbf{H}^1(F). \qquad (3.20) $$

**Proposition 2 (Local complex property)** Let $F \in \mathcal{F}_h$ and denote by $\underline{\underline{G}}_F^k : \underline{\underline{X}}_{\underline{\underline{grad}},F}^k \to \underline{\underline{X}}_{\underline{\underline{curl}},F}^k$ the restriction to $F$ of the global gradient $\underline{\underline{G}}_h^k$ defined by (3.18). Then, it holds

$$ \mathrm{Im} \, \underline{\underline{G}}_F^k \subset \mathrm{Ker} \, C_F^k \quad \forall F \in \mathcal{F}_h. \qquad (3.21) $$
The global discrete gradient $\underline{\mathbf{G}}_h^k : \underline{\mathbf{X}}_{\underline{\text{grad}},h}^k \to \underline{\mathbf{X}}_{\text{curl},h}^k$ is obtained collecting the projections of each local gradient on the space attached to the corresponding mesh entity: For all $\underline{\mathbf{q}}_h \in \underline{\mathbf{X}}_{\underline{\text{grad}},h}^k$,

$$ \underline{\mathbf{G}}_h^k \underline{\mathbf{q}}_h := \left( (\pi_{\mathcal{R},T}^{k-1}(\mathbf{G}_T^k \underline{\mathbf{q}}_T), \pi_{\mathcal{R},T}^{c,k}(\mathbf{G}_T^k \underline{\mathbf{q}}_T))_{T \in \mathcal{T}_h}, \right. \\ \left. (\pi_{\mathcal{R},F}^{k-1}(\mathbf{G}_F^k \underline{\mathbf{q}}_F), \pi_{\mathcal{R},F}^{c,k}(\mathbf{G}_F^k \underline{\mathbf{q}}_F))_{F \in \mathcal{F}_h}, \right. \\ \left. (\mathbf{G}_E^k \underline{\mathbf{q}}_E)_{E \in \mathcal{E}_h} \right). \quad (3.18) $$

**Remark 9 (Practical implementation)** In schemes based on the DDR sequence, the discrete gradient (3.18) only appears as an argument of the discrete L²-product on $\underline{\mathbf{X}}_{\text{curl},h}^k$ (see (4.15) below), that is, composed with the scalar trace and potential reconstruction on this space. Thus, leveraging (4.29) below, one never has to implement $\underline{\mathbf{G}}_h^k$, as only the full element gradients $(\mathbf{G}_T^k)_{T \in \mathcal{T}_h}$ are required. Similar considerations hold for the discrete curl defined by (3.31) below (see also [29, Remark 7] on this matter).

Notice that this strategy differs from the one often pursued in the context of Virtual Elements, which consists in directly taking the appropriate components of $\underline{\mathbf{G}}_h^k$ as degrees of freedom. This difference is linked to the fact that the present construction embeds what could be interpreted in Virtual Element terms as an *enhancement*, enabling us to reduce the degree of certain internal polynomial components.

### 3.3.2 Curl

We next consider the DDR counterpart of the curl operator, which maps on $\underline{\mathbf{X}}_{\text{div},h}^k$ and therefore has components at mesh faces and inside mesh elements. For all $F \in \mathcal{F}_h$, the face curl $C_F^k : \underline{\mathbf{X}}_{\text{curl},F}^k \to \mathcal{P}^k(F)$ is such that, for all $\underline{\mathbf{v}}_F = (\underline{\mathbf{v}}_{\mathcal{R},F}, \underline{\mathbf{v}}_{\mathcal{R},F}^c, (v_E)_{E \in \mathcal{E}_F}) \in \underline{\mathbf{X}}_{\text{curl},F}^k$,

$$ \int_F C_F^k \underline{\mathbf{v}}_F r_F = \int_F \underline{\mathbf{v}}_{\mathcal{R},F} \cdot \text{rot}_F r_F - \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E v_E r_F \quad \forall r_F \in \mathcal{P}^k(F). \quad (3.19) $$

Reasoning as in [31, Proposition 4.3], we get

$$ C_F^k(\underline{\mathbf{I}}_{\text{curl},F}^k \underline{\mathbf{v}}) = \pi_{\mathcal{R},F}^k(\text{rot}_F \underline{\mathbf{v}}) \quad \forall \underline{\mathbf{v}} \in \mathbf{H}^1(F). \quad (3.20) $$

**Proposition 2 (Local complex property)** Let $F \in \mathcal{F}_h$ and denote by $\underline{\mathbf{G}}_F^k : \underline{\mathbf{X}}_{\text{grad},F}^k \to \underline{\mathbf{X}}_{\text{curl},F}^k$ the restriction to $F$ of the global gradient $\underline{\mathbf{G}}_h^k$ defined by (3.18). Then, it holds

$$ \mathrm{Im} \, \underline{\mathbf{G}}_F^k \subset \mathrm{Ker} \, C_F^k \quad \forall F \in \mathcal{F}_h. \quad (3.21) $$
**Remark 10 (Two-dimensional complex)** The relations (3.13) and (3.21) show that the following two-dimensional sequence forms a complex:

$$ \mathbb{R} \xrightarrow{\underline{l}_{\text{grad},F}^k} \underline{X}_{\text{grad},F}^k \xrightarrow{\underline{G}_F^k} \underline{X}_{\text{curl},F}^k \xrightarrow{\underline{C}_F^k} \mathcal{P}^k(F) \xrightarrow{\underline{0}} \{0\}. $$

Having assumed $F$ simply connected, adapting the arguments of [31, Theorem 4.1], one can additionally prove that this complex is exact, that is, $\text{Ker } \underline{G}_F^k = \underline{l}_{\text{grad},F}^k \mathbb{R}$, $\text{Im } \underline{G}_F^k = \text{Ker } C_F^k$, and $\text{Im } C_F^k = \mathcal{P}^k(F)$.

*Proof (Proposition 2)* Let $\underline{q}_F \in \underline{X}_{\text{grad},F}^k$. Using the definition (3.19) of $C_F^k$ and (3.18) of $\underline{G}_h^k$ we have, for all $r_F \in \mathcal{P}^k(F)$,

$$ \begin{aligned} \int_F C_F^k (\underline{G}_F^k \underline{q}_F) r_F &= \int_F \pi_{\mathcal{R},F}^{k-1} (\underline{G}_F^k \underline{q}_F) \cdot \text{rot}_F r_F - \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E G_E^k \underline{q}_F r_F \\ &= \int_F \underline{G}_F^k \underline{q}_F \cdot \text{rot}_F r_F - \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E G_E^k \underline{q}_F r_F \\ &= \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E [q_{\mathcal{E}_F}(\text{rot}_F r_F \cdot \mathbf{n}_{FE}) - q'_E r_F] = 0, \end{aligned} $$

where the suppression of $\pi_{\mathcal{R},F}^{k-1}$ in the second line is possible since $\text{rot}_F r_F \in \mathcal{R}^{k-1}(F)$, the third line is obtained using the definitions (3.9) of $\underline{G}_F^k$ with $\underline{w}_F = \text{rot}_F r_F$ (additionally noticing that $\text{div}_F(\text{rot}_F r_F) = 0$) and (3.8) of $G_E^k$, while the conclusion is obtained reasoning as in [31, Point 2. of Proposition 4.4] (see in particular Eq. (4.19) therein). $\square$

The tangential trace $\gamma_{t,F}^k : \underline{X}_{\text{curl},F}^k \to \mathcal{P}^k(F)$ is such that, for all $\underline{v}_F \in \underline{X}_{\text{curl},F}^k$, recalling the notation (2.17),

$$ \gamma_{t,F}^k \underline{v}_F := \mathfrak{R}_{\mathcal{R},F}^k (\gamma_{t,\mathcal{R},F}^k \underline{v}_F, \underline{v}_{\mathcal{R},F}^c), \quad (3.22) $$

where $\gamma_{t,\mathcal{R},F}^k \underline{v}_F \in \mathcal{R}^k(F)$ is defined, using the isomorphism property (2.8) with $\ell = k + 1$, by

$$ \int_F \gamma_{t,\mathcal{R},F}^k \underline{v}_F \cdot \text{rot}_F r_F = \int_F C_F^k \underline{v}_F r_F + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E \nu_E r_F \\ \forall r_F \in \mathcal{P}^{0,k+1}(F). \quad (3.23) $$

**Remark 11 (Validity of (3.23))** Observing that both sides of (3.23) vanish when $r_F \in \mathcal{P}^0(F)$, it is inferred that this relation holds in fact for any $r_F \in \mathcal{P}^{k+1}(F)$. We also notice that, since $\pi_{\mathcal{R},F}^k(\gamma_{t,F}^k \underline{v}_F) = \gamma_{t,\mathcal{R},F}^k \underline{v}_F$ (by virtue of (3.22) and (2.14)), $\gamma_{t,\mathcal{R},F}^k$ can be replaced by $\gamma_{t,F}^k$ in the left-hand side of (3.23).
Remark 10 (Two-dimensional complex) The relations (3.13) and (3.21) show that the following two-dimensional sequence forms a complex:

$$ \mathbb{R} \xrightarrow{\underline{l}_{\text{grad},F}^k} \underline{X}_{\text{grad},F}^k \xrightarrow{\underline{G}_F^k} \underline{X}_{\text{curl},F}^k \xrightarrow{\underline{C}_F^k} \mathcal{P}^k(F) \xrightarrow{\underline{0}} \{0\}. $$

Having assumed $F$ simply connected, adapting the arguments of [31, Theorem 4.1], one can additionally prove that this complex is exact, that is, $\text{Ker } \underline{G}_F^k = \underline{l}_{\text{grad},F}^k \mathbb{R}$, $\text{Im } \underline{G}_F^k = \text{Ker } C_F^k$, and $\text{Im } C_F^k = \mathcal{P}^k(F)$.

*Proof (Proposition 2)* Let $\underline{q}_F \in \underline{X}_{\text{grad},F}^k$. Using the definition (3.19) of $C_F^k$ and (3.18) of $\underline{G}_h^k$ we have, for all $r_F \in \mathcal{P}^k(F)$,

$$ \begin{aligned} \int_F C_F^k (\underline{G}_F^k \underline{q}_F) r_F &= \int_F \pi_{\mathcal{R},F}^{k-1} (\underline{G}_F^k \underline{q}_F) \cdot \text{rot}_F r_F - \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E G_E^k \underline{q}_F r_F \\ &= \int_F \underline{G}_F^k \underline{q}_F \cdot \text{rot}_F r_F - \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E G_E^k \underline{q}_F r_F \\ &= \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E [q_{\mathcal{E}_F}(\text{rot}_F r_F \cdot \mathbf{n}_{FE}) - q'_E r_F] = 0, \end{aligned} $$

where the suppression of $\pi_{\mathcal{R},F}^{k-1}$ in the second line is possible since $\text{rot}_F r_F \in \mathcal{R}^{k-1}(F)$, the third line is obtained using the definitions (3.9) of $\underline{G}_F^k$ with $\underline{w}_F = \text{rot}_F r_F$ (additionally noticing that $\text{div}_F(\text{rot}_F r_F) = 0$) and (3.8) of $G_E^k$, while the conclusion is obtained reasoning as in [31, Point 2. of Proposition 4.4] (see in particular Eq. (4.19) therein). $\square$

The tangential trace $\gamma_{t,F}^k : \underline{X}_{\text{curl},F}^k \to \mathcal{P}^k(F)$ is such that, for all $\underline{v}_F \in \underline{X}_{\text{curl},F}^k$, recalling the notation (2.17),

$$ \gamma_{t,F}^k \underline{v}_F := \mathfrak{R}_{\mathcal{R},F}^k (\gamma_{t,\mathcal{R},F}^k \underline{v}_F, \underline{v}_{\mathcal{R},F}^c), \quad (3.22) $$

where $\gamma_{t,\mathcal{R},F}^k \underline{v}_F \in \mathcal{R}^k(F)$ is defined, using the isomorphism property (2.8) with $\ell = k + 1$, by

$$ \int_F \gamma_{t,\mathcal{R},F}^k \underline{v}_F \cdot \text{rot}_F r_F = \int_F C_F^k \underline{v}_F r_F + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E \nu_E r_F \\ \forall r_F \in \mathcal{P}^{0,k+1}(F). \quad (3.23) $$

Remark 11 (Validity of (3.23)) Observing that both sides of (3.23) vanish when $r_F \in \mathcal{P}^0(F)$, it is inferred that this relation holds in fact for any $r_F \in \mathcal{P}^{k+1}(F)$. We also notice that, since $\pi_{\mathcal{R},F}^k (\gamma_{t,F}^k \underline{v}_F) = \gamma_{t,\mathcal{R},F}^k \underline{v}_F$ (by virtue of (3.22) and (2.14)), $\gamma_{t,\mathcal{R},F}^k$ can be replaced by $\gamma_{t,F}^k$ in the left-hand side of (3.23).
The actual computation of $\gamma_{t,F}^k$ does not require the implementation of the recovery operator in the right-hand side of (3.22), but rather hinges on the solution of the following equation: For all $(r_F, w_F) \in \mathcal{P}^{0,k+1}(F) \times \mathcal{R}^{c,k}(F)$,

$$ \int_F \gamma_{t,F}^k \underline{v}_F \cdot (\mathrm{rot}_F r_F + w_F) = \int_F C_F^k \underline{v}_F r_F + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E \underline{v}_E r_F + \int_F \underline{v}_{\mathcal{R},F}^c \cdot w_F. $$

Indeed, the test functions of the form $(r_F, 0)$ with $r_F$ spanning $\mathcal{P}^{0,k+1}(F)$ enforce that $\pi_{\mathcal{R},F}^k(\gamma_{t,F}^k \underline{v}_F) = \gamma_{t,\mathcal{R},F}^k \underline{v}_F$ satisfies (3.23), while the test functions of the form $(0, w_F)$ with $w_F$ spanning $\mathcal{R}^{c,k}(F)$ enforce that $\pi_{\mathcal{R},F}^{c,k}(\gamma_{t,F}^k \underline{v}_F) = \underline{v}_{\mathcal{R},F}^c$. These two conditions combined yield (3.22). Similar considerations hold for the three-dimensional potential reconstructions defined in Sections 4.2 and 4.3 below.

**Proposition 3 (Properties of the tangential trace) It holds**

$$ \pi_{\mathcal{R},F}^{k-1}(\gamma_{t,F}^k \underline{v}_F) = v_{\mathcal{R},F} \quad \text{and} \quad \pi_{\mathcal{R},F}^{c,k}(\gamma_{t,F}^k \underline{v}_F) = v_{\mathcal{R},F}^c \quad \forall \underline{v}_F \in \underline{X}_{\mathrm{curl},F}^k, \quad (3.24) $$

$$ \gamma_{t,F}^k (\underline{I}_{\mathrm{curl},F}^k \underline{v}) = \pi_{\mathcal{P},F}^k \underline{v} \quad \forall \underline{v} \in \mathcal{N}^{k+1}(F), \quad (3.25) $$

$$ \gamma_{t,F}^k (\underline{G}_F^k \underline{q}_F) = \underline{G}_F^k \underline{q}_F \quad \forall \underline{q}_F \in \underline{X}_{\mathrm{grad},F}^k. \quad (3.26) $$

*Proof 1. Proof of (3.24).* Since $\mathcal{R}^{k-1}(F) \subset \mathcal{R}^k(F)$, we have $\pi_{\mathcal{R},F}^{k-1} = \pi_{\mathcal{R},F}^{k-1} \pi_{\mathcal{R},F}^k$ and thus, using (2.14) and Remark 11, we obtain

$$ \pi_{\mathcal{R},F}^{k-1}(\gamma_{t,F}^k \underline{v}_F) = \pi_{\mathcal{R},F}^{k-1}(\pi_{\mathcal{R},F}^k \gamma_{t,F}^k \underline{v}_F) = \pi_{\mathcal{R},F}^{k-1}(\gamma_{t,\mathcal{R},F}^k \underline{v}_F). $$

Applying the definitions (3.23) of $\gamma_{t,\mathcal{R},F}^k$ and (3.19) of $C_F^k$ with a generic $r_F \in \mathcal{P}^{0,k}(F)$ leads to $\int_F \gamma_{t,\mathcal{R},F}^k \underline{v}_F \cdot \mathrm{rot}_F r_F = \int_F \underline{v}_{\mathcal{R},F} \cdot \mathrm{rot}_F r_F$, hence

$$ \pi_{\mathcal{R},F}^{k-1}(\gamma_{t,\mathcal{R},F}^k \underline{v}_F) = \underline{v}_{\mathcal{R},F}. $$

This proves the first relation in (3.24). The second relation is a straightforward consequence of (3.22) and (2.14).

2. *Proof of (3.25).* Let $\underline{v} \in \mathcal{N}^{k+1}(F)$. Writing (3.23) for $\underline{v}_F = \underline{I}_{\mathrm{curl},F}^k \underline{v}$, observing that $C_F^k (\underline{I}_{\mathrm{curl},F}^k \underline{v}) = \mathrm{rot}_F \underline{v} \in \mathcal{P}^k(F)$ by (3.20) and that $\underline{v}_E = \underline{v}_{|E} \cdot \underline{t}_E$ for all $E \in \mathcal{E}_F$ by (A.1) with $\ell = k+1$, and integrating by parts the right-hand side, it is inferred that $\gamma_{t,\mathcal{R},F}^k (\underline{I}_{\mathrm{curl},F}^k \underline{v}) = \pi_{\mathcal{R},F}^k \underline{v}$. Thus, by (3.22), $\gamma_{t,F}^k (\underline{I}_{\mathrm{curl},F}^k \underline{v}) = \mathfrak{R}_{\mathcal{R},F}^k (\pi_{\mathcal{R},F}^k \underline{v}, \pi_{\mathcal{R},F}^{c,k} \underline{v}) = \pi_{\mathcal{P},F}^k \underline{v}$, where the conclusion results from (2.11) with $(\underline{X}, \underline{Y}, \ell) = (\mathcal{R}, F, k)$ followed by (2.15).

3. *Proof of (3.26).* Let $\underline{q}_F \in \underline{X}_{\mathrm{grad},F}^k$. For all $r_F \in \mathcal{P}^{k+1}(F)$, it holds

$$ \begin{aligned} \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E G_E^k q_E r_F &= \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E q_{\mathcal{E}_F} (\mathrm{rot}_F r_F \cdot \underline{n}_{FE}) \\ &= \int_F \underline{G}_F^k \underline{q}_F \cdot \mathrm{rot}_F r_F, \end{aligned} \quad (3.27) $$
The actual computation of $\gamma_{t,F}^k$ does not require the implementation of the recovery operator in the right-hand side of (3.22), but rather hinges on the solution of the following equation: For all $(r_F, w_F) \in \mathcal{P}^{0,k+1}(F) \times \mathcal{R}^{c,k}(F)$,

$$ \int_F \gamma_{t,F}^k \underline{v}_F \cdot (\mathrm{rot}_F r_F + w_F) = \int_F C_F^k \underline{v}_F r_F + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E \underline{v}_E r_F + \int_F \underline{v}_{\mathcal{R},F}^c \cdot w_F. $$

Indeed, the test functions of the form $(r_F, 0)$ with $r_F$ spanning $\mathcal{P}^{0,k+1}(F)$ enforce that $\pi_{\mathcal{R},F}^k(\gamma_{t,F}^k \underline{v}_F) = \gamma_{t,\mathcal{R},F}^k \underline{v}_F$ satisfies (3.23), while the test functions of the form $(0, w_F)$ with $w_F$ spanning $\mathcal{R}^{c,k}(F)$ enforce that $\pi_{\mathcal{R},F}^{c,k}(\gamma_{t,F}^k \underline{v}_F) = \underline{v}_{\mathcal{R},F}^c$. These two conditions combined yield (3.22). Similar considerations hold for the three-dimensional potential reconstructions defined in Sections 4.2 and 4.3 below.

**Proposition 3 (Properties of the tangential trace) It holds**

$$ \pi_{\mathcal{R},F}^{k-1}(\gamma_{t,F}^k \underline{v}_F) = \underline{v}_{\mathcal{R},F} \quad \text{and} \quad \pi_{\mathcal{R},F}^{c,k}(\gamma_{t,F}^k \underline{v}_F) = \underline{v}_{\mathcal{R},F}^c \qquad \forall \underline{v}_F \in \underline{X}_{\mathrm{curl},F}^k, \quad (3.24) $$

$$ \gamma_{t,F}^k (\underline{I}_{\mathrm{curl},F}^k \underline{v}) = \pi_{\mathcal{P},F}^k \underline{v} \qquad \forall \underline{v} \in \mathcal{N}^{k+1}(F), \quad (3.25) $$

$$ \gamma_{t,F}^k (\underline{G}_F^k \underline{q}_F) = \underline{G}_F^k \underline{q}_F \qquad \forall \underline{q}_F \in \underline{X}_{\mathrm{grad},F}^k. \quad (3.26) $$

*Proof 1. Proof of (3.24).* Since $\mathcal{R}^{k-1}(F) \subset \mathcal{R}^k(F)$, we have $\pi_{\mathcal{R},F}^{k-1} = \pi_{\mathcal{R},F}^{k-1} \pi_{\mathcal{R},F}^k$ and thus, using (2.14) and Remark 11, we obtain

$$ \pi_{\mathcal{R},F}^{k-1}(\gamma_{t,F}^k \underline{v}_F) = \pi_{\mathcal{R},F}^{k-1}(\pi_{\mathcal{R},F}^k \gamma_{t,F}^k \underline{v}_F) = \pi_{\mathcal{R},F}^{k-1}(\gamma_{t,\mathcal{R},F}^k \underline{v}_F). $$

Applying the definitions (3.23) of $\gamma_{t,\mathcal{R},F}^k$ and (3.19) of $C_F^k$ with a generic $r_F \in \mathcal{P}^{0,k}(F)$ leads to $\int_F \gamma_{t,\mathcal{R},F}^k \underline{v}_F \cdot \mathrm{rot}_F r_F = \int_F \underline{v}_{\mathcal{R},F} \cdot \mathrm{rot}_F r_F$, hence

$$ \pi_{\mathcal{R},F}^{k-1}(\gamma_{t,\mathcal{R},F}^k \underline{v}_F) = \underline{v}_{\mathcal{R},F}. $$

This proves the first relation in (3.24). The second relation is a straightforward consequence of (3.22) and (2.14).

2. *Proof of (3.25).* Let $\underline{v} \in \mathcal{N}^{k+1}(F)$. Writing (3.23) for $\underline{v}_F = \underline{I}_{\mathrm{curl},F}^k \underline{v}$, observing that $C_F^k (\underline{I}_{\mathrm{curl},F}^k \underline{v}) = \mathrm{rot}_F \underline{v} \in \mathcal{P}^k(F)$ by (3.20) and that $\underline{v}_E = \underline{v}_{|E} \cdot \underline{t}_E$ for all $E \in \mathcal{E}_F$ by (A.1) with $\ell = k+1$, and integrating by parts the right-hand side, it is inferred that $\gamma_{t,\mathcal{R},F}^k (\underline{I}_{\mathrm{curl},F}^k \underline{v}) = \pi_{\mathcal{R},F}^k \underline{v}$. Thus, by (3.22), $\gamma_{t,F}^k (\underline{I}_{\mathrm{curl},F}^k \underline{v}) = \mathfrak{R}_{\mathcal{R},F}^k (\pi_{\mathcal{R},F}^k \underline{v}, \pi_{\mathcal{R},F}^{c,k} \underline{v}) = \pi_{\mathcal{P},F}^k \underline{v}$, where the conclusion results from (2.11) with $(\underline{X}, \underline{Y}, \ell) = (\mathcal{R}, F, k)$ followed by (2.15).

3. *Proof of (3.26).* Let $\underline{q}_F \in \underline{X}_{\mathrm{grad},F}^k$. For all $r_F \in \mathcal{P}^{k+1}(F)$, it holds

$$ \begin{align} \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E G_E^k q_E r_F &= \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E q_{\mathcal{E}_F} (\mathrm{rot}_F r_F \cdot \underline{n}_{FE}) \tag{3.27} \\ &= \int_F \underline{G}_F^k \underline{q}_F \cdot \mathrm{rot}_F r_F, \nonumber \end{align} $$
where the first equality follows recalling that $G_E^k q_E = q'_E$ on $E$, integrating by parts on each edge, noting that $(r_F)'_E = -\text{rot}_F r_F \cdot n_{FE}$ (see [31, Eq. (4.20)]), and cancelling out the vertex values that appear twice with opposite sign, while the conclusion is obtained recalling the definition (3.9) of $G_F^k$ and observing that $\text{div}_F(\text{rot}_F r_F) = 0$. Writing (3.23) for $\underline{v}_F = \underline{G}_F^k q_F$, we obtain

$$ 
\begin{aligned}
\int_F \gamma_{t, \mathcal{R}, F}^k (\underline{G}_F^k \underline{q}_F) \cdot \text{rot}_F r_F &= \int_F C_F^k (\underline{G}_F^k \underline{q}_F) r_F + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E G_E^k q_E r_F \\
&= \int_F \underline{G}_F^k \underline{q}_F \cdot \text{rot}_F r_F,
\end{aligned}
$$

where we have used the inclusion (3.21) in the cancellation, while the conclusion follows from (3.27). This implies $\gamma_{t, \mathcal{R}, F}^k (\underline{G}_F^k \underline{q}_F) = \pi_{\mathcal{R}, F}^k (\underline{G}_F^k \underline{q}_F)$. By definition, the component of $\underline{G}_F^k \underline{q}_F$ on $\mathcal{R}^{c,k}(F)$ is $\pi_{\mathcal{R}, F}^{c,k}(\underline{G}_F^k \underline{q}_F)$. Plugging the above results into (3.22) with $\underline{v}_F = \underline{G}_F^k \underline{q}_F$ and using the recovery formula (2.15) with $(S, S^c) = (\mathcal{R}^k(F), \mathcal{R}^{c,k}(F))$ and $\underline{a} = \underline{G}_F^k \underline{q}_F$ concludes the proof. $\square$

For all $T \in \mathcal{T}_h$, the element curl $\mathbf{C}_T^k : \underline{X}_{\text{curl},T}^k \to \mathcal{P}^k(T)$ is defined such that, for all $\underline{v}_T = (\underline{v}_{\mathcal{R},T}, \underline{v}_{\mathcal{R},T}^c, (\underline{v}_{\mathcal{R},F}, \underline{v}_{\mathcal{R},F}^c)_{F \in \mathcal{F}_T}, (v_E)_{E \in \mathcal{E}_T} \in \underline{X}_{\text{curl},T}^k$,

$$ 
\begin{aligned}
\int_T \mathbf{C}_T^k \underline{v}_T \cdot \mathbf{w}_T &= \int_T \underline{v}_{\mathcal{R},T} \cdot \text{curl} \, \mathbf{w}_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \gamma_{t,F}^k \underline{v}_F \cdot (\mathbf{w}_T \times \mathbf{n}_F) \\
&\qquad \forall \mathbf{w}_T \in \mathcal{P}^k(T). \quad (3.28)
\end{aligned}
$$

The following polynomial consistency property is proved as in [31, Lemma 5.2] (recall the shift of exponent in the notation of the Nédélec space with respect to this reference):

$$ \forall T \in \mathcal{T}_h \quad \mathbf{C}_T^k (\underline{I}_{\text{curl},T}^k \underline{v}) = \text{curl} \, \underline{v} \quad \forall \underline{v} \in \mathcal{N}^{k+1}(T). \quad (3.29) $$

**Proposition 4 (Link between element and face curls)** For all $(\underline{v}_T, r_T) \in \underline{X}_{\text{curl},T}^k \times \mathcal{P}^{k+1}(T)$, it holds

$$ \int_T \mathbf{C}_T^k \underline{v}_T \cdot \text{grad} \, r_T = \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F C_F^k \underline{v}_F \, r_T. \quad (3.30) $$

*Proof* For any $r_T \in \mathcal{P}^{k+1}(T)$, writing (3.28) for $\mathbf{w}_T = \text{grad} \, r_T \in \mathcal{P}^k(T)$ and using the fact that $\text{curl}(\text{grad} \, r_T) = 0$ and that $(\text{grad} \, r_T)_{|F} \times \mathbf{n}_F = \text{rot}_F(r_{T|F})$ for all $F \in \mathcal{F}_T$ (see [31, Eq. (3.6)]), we infer that

$$ \int_T \mathbf{C}_T^k \underline{v}_T \cdot \text{grad} \, r_T = \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \gamma_{t,F}^k \underline{v}_F \cdot \text{rot}_F (r_{T|F}). $$

Using Remark 11, we arrive at

$$ \int_T \mathbf{C}_T^k \underline{v}_T \cdot \text{grad} \, r_T = \sum_{F \in \mathcal{F}_T} \omega_{TF} \left[ \int_F C_F^k \underline{v}_F r_T + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E \underline{v}_E r_T \right]. $$
where the first equality follows recalling that $G_E^k q_E = q'_E$ on $E$, integrating by parts on each edge, noting that $(r_F)'_E = -\mathbf{rot}_F r_F \cdot \mathbf{n}_{FE}$ (see [31, Eq. (4.20)]), and cancelling out the vertex values that appear twice with opposite sign, while the conclusion is obtained recalling the definition (3.9) of $\mathbf{G}_F^k$ and observing that $\mathrm{div}_F(\mathbf{rot}_F r_F) = 0$. Writing (3.23) for $\mathbf{v}_F = \underline{\mathbf{G}_F^k q}_F$, we obtain

$$ 
\begin{aligned}
\int_F \gamma_{t, \mathcal{R}, F}^k (\underline{\mathbf{G}_F^k q}_F) \cdot \mathbf{rot}_F r_F &= \int_F C_F^k (\underline{\mathbf{G}_F^k q}_F) r_F + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E G_E^k q_E r_F \\
&= \int_F \underline{\mathbf{G}_F^k q}_F \cdot \mathbf{rot}_F r_F,
\end{aligned}
$$

where we have used the inclusion (3.21) in the cancellation, while the conclusion follows from (3.27). This implies $\gamma_{t, \mathcal{R}, F}^k (\underline{\mathbf{G}_F^k q}_F) = \pi_{\mathcal{R}, F}^k (\underline{\mathbf{G}_F^k q}_F)$. By definition, the component of $\underline{\mathbf{G}_F^k q}_F$ on $\mathcal{R}^{c,k}(F)$ is $\pi_{\mathcal{R}, F}^{c,k}(\underline{\mathbf{G}_F^k q}_F)$. Plugging the above results into (3.22) with $\mathbf{v}_F = \underline{\mathbf{G}_F^k q}_F$ and using the recovery formula (2.15) with $(S, S^c) = (\mathcal{R}^k(F), \mathcal{R}^{c,k}(F))$ and $\mathbf{a} = \underline{\mathbf{G}_F^k q}_F$ concludes the proof. $\square$

For all $T \in \mathcal{T}_h$, the element curl $\mathbf{C}_T^k : \underline{\mathbf{X}}_{\mathrm{curl},T}^k \to \mathcal{P}^k(T)$ is defined such that, for all $\mathbf{v}_T = (\mathbf{v}_{\mathcal{R},T}, \mathbf{v}_{\mathcal{R},T}^c, (\mathbf{v}_{\mathcal{R},F}, \mathbf{v}_{\mathcal{R},F}^c)_{F \in \mathcal{F}_T}, (v_E)_{E \in \mathcal{E}_T} \in \underline{\mathbf{X}}_{\mathrm{curl},T}^k$,

$$ 
\begin{aligned}
\int_T \mathbf{C}_T^k \underline{\mathbf{v}}_T \cdot \mathbf{w}_T &= \int_T \mathbf{v}_{\mathcal{R},T} \cdot \mathbf{curl} \mathbf{w}_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \gamma_{t,F}^k \underline{\mathbf{v}}_F \cdot (\mathbf{w}_T \times \mathbf{n}_F) \\
&\qquad \forall \mathbf{w}_T \in \mathcal{P}^k(T). \quad (3.28)
\end{aligned}
$$

The following polynomial consistency property is proved as in [31, Lemma 5.2] (recall the shift of exponent in the notation of the Nédélec space with respect to this reference):

$$ \forall T \in \mathcal{T}_h \qquad \mathbf{C}_T^k (\underline{\mathbf{I}}_{\mathrm{curl},T}^k \mathbf{v}) = \mathbf{curl} \mathbf{v} \qquad \forall \mathbf{v} \in \mathcal{N}^{k+1}(T). \quad (3.29) $$

**Proposition 4 (Link between element and face curls)** For all $(\mathbf{v}_T, r_T) \in \underline{\mathbf{X}}_{\mathrm{curl},T}^k \times \mathcal{P}^{k+1}(T)$, it holds

$$ \int_T \mathbf{C}_T^k \underline{\mathbf{v}}_T \cdot \mathbf{grad} r_T = \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F C_F^k \underline{\mathbf{v}}_F r_T. \quad (3.30) $$

*Proof* For any $r_T \in \mathcal{P}^{k+1}(T)$, writing (3.28) for $\mathbf{w}_T = \mathbf{grad} r_T \in \mathcal{P}^k(T)$ and using the fact that $\mathbf{curl}(\mathbf{grad} r_T) = \mathbf{0}$ and that $(\mathbf{grad} r_T)_{|F} \times \mathbf{n}_F = \mathbf{rot}_F(r_{T|F})$ for all $F \in \mathcal{F}_T$ (see [31, Eq. (3.6)]), we infer that

$$ \int_T \mathbf{C}_T^k \underline{\mathbf{v}}_T \cdot \mathbf{grad} r_T = \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \gamma_{t,F}^k \underline{\mathbf{v}}_F \cdot \mathbf{rot}_F (r_{T|F}). $$

Using Remark 11, we arrive at

$$ \int_T \mathbf{C}_T^k \underline{\mathbf{v}}_T \cdot \mathbf{grad} r_T = \sum_{F \in \mathcal{F}_T} \omega_{TF} \left[ \int_F C_F^k \underline{\mathbf{v}}_F r_T + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E \mathbf{v}_E r_T \right]. $$
By [31, Eq. (5.13)], the edge terms in the above expression can be cancelled, thereby proving (3.30). □

The global discrete curl $\underline{\mathcal{C}}_h^k : \underline{\mathcal{X}}_{\text{curl},h}^k \to \underline{\mathcal{X}}_{\text{div},h}^k$ is such that, for all $\underline{\mathbf{v}}_h \in \underline{\mathcal{X}}_{\text{curl},h}^k$,

$$ \underline{\mathcal{C}}_h^k \underline{\mathbf{v}}_h := ((\pi_{\mathcal{G},T}^{k-1}(\mathcal{C}_T^k \underline{\mathbf{v}}_T), \pi_{\mathcal{G},T}^{c,k}(\mathcal{C}_T^k \underline{\mathbf{v}}_T))_{T \in \mathcal{T}_h}, (C_F^k \underline{\mathbf{v}}_F)_{F \in \mathcal{F}_h}). \quad (3.31) $$

### 3.3.3 Divergence

For all $T \in \mathcal{T}_h$, the element divergence $D_T^k : \underline{\mathcal{X}}_{\text{div},T}^k \to \mathcal{P}^k(T)$ is defined by: For all $\underline{\mathbf{w}}_T = (\mathbf{w}_{\mathcal{G},T}, \mathbf{w}_{\mathcal{G},T}^c, (w_F)_{F \in \mathcal{T}_T}) \in \underline{\mathcal{X}}_{\text{div},T}^k$,

$$ \int_T D_T^k \underline{\mathbf{w}}_T q_T = - \int_T \mathbf{w}_{\mathcal{G},T} \cdot \text{grad} q_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F w_F q_T \quad \forall q_T \in \mathcal{P}^k(T). \quad (3.32) $$

The global discrete divergence $D_h^k : \underline{\mathcal{X}}_{\text{div},h}^k \to \mathcal{P}^k(\mathcal{T}_h)$ is obtained setting, for all $\underline{\mathbf{w}}_h \in \underline{\mathcal{X}}_{\text{div},h}^k$,

$$ (D_h^k \underline{\mathbf{w}}_h)|_T := D_T^k \underline{\mathbf{w}}_T \quad \forall T \in \mathcal{T}_h, \quad (3.33) $$

**Proposition 5 (Local exactness property) It holds, for all $T \in \mathcal{T}_h$,**

$$ \mathrm{Im} \, \underline{\mathcal{C}}_T^k = \mathrm{Ker} \, D_T^k, \quad (3.34) $$

where $\underline{\mathcal{C}}_T^k$ denotes the restriction to $T$ of the global curl $\underline{\mathcal{C}}_h^k$ defined by (3.31)

*Proof* Let us start by proving that $D_T^k(\underline{\mathcal{C}}_T^k \underline{\mathbf{v}}_T) = 0$ for all $\underline{\mathbf{v}}_T \in \underline{\mathcal{X}}_{\text{curl},T}^k$, that is, $\mathrm{Im} \, \underline{\mathcal{C}}_T^k \subset \mathrm{Ker}(D_T^k)$. By Proposition 4, for all $q_T \in \mathcal{P}^k(T)$,

$$ \int_T \pi_{\mathcal{G},T}^{k-1}(\mathcal{C}_T^k \underline{\mathbf{v}}_T) \cdot \text{grad} q_T = \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F C_F^k \underline{\mathbf{v}}_F q_T, \quad (3.35) $$

where we have used **grad** $q_T \in \mathcal{G}^{k-1}(T)$ to introduce the projector $\pi_{\mathcal{G},T}^{k-1}$. Hence, using the definition (3.32) of $D_T^k$, we have, for all $q_T \in \mathcal{P}^k(T)$,

$$ \int_T D_T^k(\underline{\mathcal{C}}_T^k \underline{\mathbf{v}}_T) q_T = - \int_T \pi_{\mathcal{G},T}^{k-1}(\mathcal{C}_T^k \underline{\mathbf{v}}_T) \cdot \text{grad} q_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F C_F^k \underline{\mathbf{v}}_F q_T = 0. $$

Since $q_T$ is arbitrary in $\mathcal{P}^k(T)$, this shows that $D_T^k(\underline{\mathcal{C}}_T^k \underline{\mathbf{v}}_T) = 0$.

Let us now prove the inclusion $\mathrm{Ker}(D_T^k) \subset \mathrm{Im} \, \underline{\mathcal{C}}_T^k$. We fix an element $\underline{\mathbf{w}}_T \in \underline{\mathcal{X}}_{\text{div},T}^k$ such that $D_T^k \underline{\mathbf{w}}_T = 0$ and prove the existence of $\underline{\mathbf{v}}_T \in \underline{\mathcal{X}}_{\text{curl},T}^k$ such that $\underline{\mathbf{w}}_T = \underline{\mathcal{C}}_T^k \underline{\mathbf{v}}_T$. Enforcing $D_T^k \underline{\mathbf{w}}_T = 0$ in (3.32) with $q_T = 1$, we infer that $\sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F w_F = 0$. Thus, [31, Lemma 5.3], which remains valid in the present context, provides $(\underline{\mathbf{v}}_{\mathcal{R},F}, \underline{\mathbf{v}}_{\mathcal{R},F}^c)_{F \in \mathcal{F}_T}$ and $(\underline{v}_E)_{E \in \mathcal{E}_T}$ such that, for all $F \in \mathcal{F}_T$, letting $\underline{\mathbf{v}}_F := (\underline{\mathbf{v}}_{\mathcal{R},F}, \underline{\mathbf{v}}_{\mathcal{R},F}^c, (\underline{v}_E)_{E \in \mathcal{E}_F})$, it holds $w_F = C_F^k \underline{\mathbf{v}}_F$. Enforcing again $D_T^k \underline{\mathbf{w}}_T = 0$ in (3.32), this time for a generic test function $q_T \in \mathcal{P}^k(T)$,
By [31, Eq. (5.13)], the edge terms in the above expression can be cancelled, thereby proving (3.30). □

The global discrete curl $\underline{\mathcal{C}}_h^k : \underline{\mathcal{X}}_{\text{curl},h}^k \to \underline{\mathcal{X}}_{\text{div},h}^k$ is such that, for all $\underline{\mathbf{v}}_h \in \underline{\mathcal{X}}_{\text{curl},h}^k$,

$$ \underline{\mathcal{C}}_h^k \underline{\mathbf{v}}_h := ((\pi_{\mathcal{G},T}^{k-1}(\mathcal{C}_T^k \underline{\mathbf{v}}_T), \pi_{\mathcal{G},T}^{c,k}(\mathcal{C}_T^k \underline{\mathbf{v}}_T))_{T \in \mathcal{T}_h}, (C_F^k \underline{\mathbf{v}}_F)_{F \in \mathcal{F}_h}). \quad (3.31) $$

### 3.3.3 Divergence

For all $T \in \mathcal{T}_h$, the element divergence $D_T^k : \underline{\mathcal{X}}_{\text{div},T}^k \to \mathcal{P}^k(T)$ is defined by: For all $\underline{\mathbf{w}}_T = (\mathbf{w}_{\mathcal{G},T}, \mathbf{w}_{\mathcal{G},T}^c, (w_F)_{F \in \mathcal{T}_T}) \in \underline{\mathcal{X}}_{\text{div},T}^k$,

$$ \int_T D_T^k \underline{\mathbf{w}}_T q_T = - \int_T \mathbf{w}_{\mathcal{G},T} \cdot \text{grad} q_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F w_F q_T \quad \forall q_T \in \mathcal{P}^k(T). \quad (3.32) $$

The global discrete divergence $D_h^k : \underline{\mathcal{X}}_{\text{div},h}^k \to \mathcal{P}^k(\mathcal{T}_h)$ is obtained setting, for all $\underline{\mathbf{w}}_h \in \underline{\mathcal{X}}_{\text{div},h}^k$,

$$ (D_h^k \underline{\mathbf{w}}_h)|_T := D_T^k \underline{\mathbf{w}}_T \quad \forall T \in \mathcal{T}_h, \quad (3.33) $$

**Proposition 5 (Local exactness property) It holds, for all $T \in \mathcal{T}_h$,**

$$ \mathrm{Im} \, \underline{\mathcal{C}}_T^k = \mathrm{Ker} \, D_T^k, \quad (3.34) $$

where $\underline{\mathcal{C}}_T^k$ denotes the restriction to $T$ of the global curl $\underline{\mathcal{C}}_h^k$ defined by (3.31)

*Proof* Let us start by proving that $D_T^k(\underline{\mathcal{C}}_T^k \underline{\mathbf{v}}_T) = 0$ for all $\underline{\mathbf{v}}_T \in \underline{\mathcal{X}}_{\text{curl},T}^k$, that is, $\mathrm{Im} \, \underline{\mathcal{C}}_T^k \subset \mathrm{Ker}(D_T^k)$. By Proposition 4, for all $q_T \in \mathcal{P}^k(T)$,

$$ \int_T \pi_{\mathcal{G},T}^{k-1}(\mathcal{C}_T^k \underline{\mathbf{v}}_T) \cdot \text{grad} q_T = \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F C_F^k \underline{\mathbf{v}}_F q_T, \quad (3.35) $$

where we have used **grad** $q_T \in \mathcal{G}^{k-1}(T)$ to introduce the projector $\pi_{\mathcal{G},T}^{k-1}$. Hence, using the definition (3.32) of $D_T^k$, we have, for all $q_T \in \mathcal{P}^k(T)$,

$$ \int_T D_T^k(\underline{\mathcal{C}}_T^k \underline{\mathbf{v}}_T) q_T = - \int_T \pi_{\mathcal{G},T}^{k-1}(\mathcal{C}_T^k \underline{\mathbf{v}}_T) \cdot \text{grad} q_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F C_F^k \underline{\mathbf{v}}_F q_T = 0. $$

Since $q_T$ is arbitrary in $\mathcal{P}^k(T)$, this shows that $D_T^k(\underline{\mathcal{C}}_T^k \underline{\mathbf{v}}_T) = 0$.

Let us now prove the inclusion $\mathrm{Ker}(D_T^k) \subset \mathrm{Im} \, \underline{\mathcal{C}}_T^k$. We fix an element $\underline{\mathbf{w}}_T \in \underline{\mathcal{X}}_{\text{div},T}^k$ such that $D_T^k \underline{\mathbf{w}}_T = 0$ and prove the existence of $\underline{\mathbf{v}}_T \in \underline{\mathcal{X}}_{\text{curl},T}^k$ such that $\underline{\mathbf{w}}_T = \underline{\mathcal{C}}_T^k \underline{\mathbf{v}}_T$. Enforcing $D_T^k \underline{\mathbf{w}}_T = 0$ in (3.32) with $q_T = 1$, we infer that $\sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F w_F = 0$. Thus, [31, Lemma 5.3], which remains valid in the present context, provides $(\underline{\mathbf{v}}_{\mathcal{R},F}, \underline{\mathbf{v}}_{\mathcal{R},F}^c)_{F \in \mathcal{F}_T}$ and $(\underline{v}_E)_{E \in \mathcal{E}_T}$ such that, for all $F \in \mathcal{F}_T$, letting $\underline{\mathbf{v}}_F := (\underline{\mathbf{v}}_{\mathcal{R},F}, \underline{\mathbf{v}}_{\mathcal{R},F}^c, (\underline{v}_E)_{E \in \mathcal{E}_F})$, it holds $w_F = C_F^k \underline{\mathbf{v}}_F$. Enforcing again $D_T^k \underline{\mathbf{w}}_T = 0$ in (3.32), this time for a generic test function $q_T \in \mathcal{P}^k(T)$,
and accounting for the previous result, we can write, for all $\underline{v}_T \in \underline{X}_{\text{curl},T}^k$ with boundary values as above,

$$ \int_T w_{G,T} \cdot \text{grad} q_T = \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F C_F^k \underline{v}_F q_T = \int_T \pi_{G,T}^{k-1}(\mathbf{C}_T^k \underline{v}_T) \cdot \text{grad} q_T, $$

where the conclusion follows from the relation (3.35) linking volume and face curls. Since $\text{grad} q_T$ spans $\mathcal{G}^{k-1}(T)$ as $q_T$ spans $\mathcal{P}^k(T)$, this proves that $\pi_{G,T}^{k-1}(\mathbf{C}_T^k \underline{v}_T) = w_{G,T}$. Finally, we select $\underline{v}_{\mathcal{R},T} \in \mathcal{R}^{k-1}(T)$ in such a way as to have $\pi_{G,T}^{c,k}(\mathbf{C}_T^k \underline{v}_T) = w_{G,T}^c$, that is, recalling (3.28),

$$ \int_T \underline{v}_{\mathcal{R},T} \cdot \text{curl} z_T = \int_T w_{G,T}^c \cdot z_T - \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \gamma_{t,F}^k \underline{v}_F \cdot (z_T \times n_F) \\ \forall z_T \in \mathcal{G}^{c,k}(T). \quad (3.36) $$

By the isomorphism (2.10), this condition defines $\underline{v}_{\mathcal{R},T}$ uniquely. $\square$

## 3.4 Discrete sequence

Recalling the definitions (3.5), (3.18), (3.31), and (3.33) of the global discrete operators, the DDR sequence reads:

$$ \mathbb{R} \xrightarrow{\underline{I}_{\text{grad},h}^k} \underline{X}_{\text{grad},h}^k \xrightarrow{\underline{G}_h^k} \underline{X}_{\text{curl},h}^k \xrightarrow{\underline{C}_h^k} \underline{X}_{\text{div},h}^k \xrightarrow{\underline{D}_h^k} \mathcal{P}^k(\mathcal{T}_h) \xrightarrow{0} \{0\}. \quad (3.37) $$

*Remark 12 (Variations)* In the spirit of [7, Section 9], one could consider alternatives of the DDR sequence (3.37) obtained varying certain couples of polynomial degrees in such a way as to preserve the exactness properties. Thus one could, e.g., replace $\mathcal{R}^{k-1}(T)$ with $\mathcal{R}^k(T)$ in the definition (3.2) of $\underline{X}_{\text{curl},h}^k$ and, correspondingly, $\mathcal{G}^{c,k}(T)$ with $\mathcal{G}^{c,k+1}(T)$ in the definition (3.3) of $\underline{X}_{\text{div},h}^k$. With these changes, the results of Proposition 5 (and, in particular, (3.36)) remain valid. Assessing the impact such and similar changes on the consistency is, however, more delicate. These developments are left for a future work.

## 3.5 Commutation properties

**Lemma 4 (Local commutation properties)** It holds, for all $T \in \mathcal{T}_h$,

$$ \underline{G}_T^k(\underline{I}_{\text{grad},T}^k q) = \underline{I}_{\text{curl},T}^k(\text{grad } q) \quad \forall q \in \mathbb{C}^1(\bar{T}), \quad (3.38) $$

$$ \underline{C}_T^k(\underline{I}_{\text{curl},T}^k \underline{v}) = \underline{I}_{\text{div},T}^k(\text{curl } \underline{v}) \quad \forall \underline{v} \in \mathbf{H}^2(T), \quad (3.39) $$

$$ D_T^k(\underline{I}_{\text{div},T}^k w) = \pi_{\mathcal{P},T}^k(\text{div } w) \quad \forall w \in \mathbf{H}^1(T). \quad (3.40) $$
and accounting for the previous result, we can write, for all $\underline{v}_T \in \underline{X}_{\text{curl},T}^k$ with boundary values as above,

$$ \int_T w_{G,T} \cdot \text{grad} q_T = \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F C_F^k \underline{v}_F q_T = \int_T \pi_{G,T}^{k-1}(\mathbf{C}_T^k \underline{v}_T) \cdot \text{grad} q_T, $$

where the conclusion follows from the relation (3.35) linking volume and face curls. Since $\text{grad} q_T$ spans $\mathcal{G}^{k-1}(T)$ as $q_T$ spans $\mathcal{P}^k(T)$, this proves that $\pi_{G,T}^{k-1}(\mathbf{C}_T^k \underline{v}_T) = w_{G,T}$. Finally, we select $\nu_{R,T} \in \mathcal{R}^{k-1}(T)$ in such a way as to have $\pi_{G,T}^{c,k}(\mathbf{C}_T^k \underline{v}_T) = w_{G,T}^c$, that is, recalling (3.28),

$$ \int_T \nu_{R,T} \cdot \text{curl} z_T = \int_T w_{G,T}^c \cdot z_T - \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \gamma_{t,F}^k \underline{v}_F \cdot (z_T \times n_F) \\ \forall z_T \in \mathcal{G}^{c,k}(T). \quad (3.36) $$

By the isomorphism (2.10), this condition defines $\nu_{R,T}$ uniquely. $\square$

### 3.4 Discrete sequence

Recalling the definitions (3.5), (3.18), (3.31), and (3.33) of the global discrete operators, the DDR sequence reads:

$$ \mathbb{R} \xrightarrow{\underline{I}_{\text{grad},h}^k} \underline{X}_{\text{grad},h}^k \xrightarrow{\underline{G}_h^k} \underline{X}_{\text{curl},h}^k \xrightarrow{\underline{C}_h^k} \underline{X}_{\text{div},h}^k \xrightarrow{\underline{D}_h^k} \mathcal{P}^k(\mathcal{T}_h) \xrightarrow{0} \{0\}. \quad (3.37) $$

*Remark 12 (Variations)* In the spirit of [7, Section 9], one could consider alternatives of the DDR sequence (3.37) obtained varying certain couples of polynomial degrees in such a way as to preserve the exactness properties. Thus one could, e.g., replace $\mathcal{R}^{k-1}(T)$ with $\mathcal{R}^k(T)$ in the definition (3.2) of $\underline{X}_{\text{curl},h}^k$ and, correspondingly, $\mathcal{G}^{c,k}(T)$ with $\mathcal{G}^{c,k+1}(T)$ in the definition (3.3) of $\underline{X}_{\text{div},h}^k$. With these changes, the results of Proposition 5 (and, in particular, (3.36)) remain valid. Assessing the impact such and similar changes on the consistency is, however, more delicate. These developments are left for a future work.

### 3.5 Commutation properties

**Lemma 4 (Local commutation properties)** It holds, for all $T \in \mathcal{T}_h$,

$$ \underline{G}_T^k(\underline{I}_{\text{grad},T}^k q) = \underline{I}_{\text{curl},T}^k(\text{grad } q) \quad \forall q \in \mathbb{C}^1(\bar{T}), \quad (3.38) $$

$$ \underline{C}_T^k(\underline{I}_{\text{curl},T}^k \nu) = \underline{I}_{\text{div},T}^k(\text{curl } \nu) \quad \forall \nu \in \mathbf{H}^2(T), \quad (3.39) $$

$$ D_T^k(\underline{I}_{\text{div},T}^k w) = \pi_{\mathcal{P},T}^k(\text{div } w) \quad \forall w \in \mathbf{H}^1(T). \quad (3.40) $$
**Remark 13 (Global commutation properties)** Global commutation properties can be readily inferred from the local ones stated in Lemma 4 when interpolating functions that have sufficient global regularity.

**Remark 14 (Role of commutation properties in the design of robust methods)** The commutation properties of Lemma 4 play a key role in the design of discretisation methods robust with respect to the variations of physical parameters. See, e.g., [30] concerning a DDR method for the Reissner–Mindlin plate bending problem robust with respect to plate thickness.

**Proof (Lemma 4)** We start by noticing that all the interpolates defined in (3.38)–(3.40) are well-defined under the assumed regularities.

1. *Proof of (3.38).* By (3.12) it holds, for all $E \in \mathcal{E}_T$,

$$G_E^k(\underline{I}_{\text{grad},E}^k q|_E) = \pi_{\mathcal{P},E}^k(q'_{|E}) = \pi_{\mathcal{P},E}^k((\text{grad } q)|_E \cdot \mathbf{t}_E).$$

Let now $F \in \mathcal{F}_T$. Writing the definition (3.9) of $\mathbf{G}_F^k$ with $q_{|F} = \underline{I}_{\text{grad},F}^k q|_F$ and $\mathbf{w}_F \in \mathcal{R}\mathcal{T}^k(F)$, and recalling (A.2) to replace $q_{\mathcal{E}_F}$ with $\pi_{\mathcal{P},E}^{k-1}(q_{\mathcal{E}_F})|_E = \pi_{\mathcal{P},E}^{k-1}q|_E$ (see (3.5)) in each edge integral, we infer

$$\begin{aligned}
\int_F \mathbf{G}_F^k (\underline{I}_{\text{grad},F}^k q|_F) \cdot \mathbf{w}_F &= - \int_F \pi_{\mathcal{P},F}^{k-1} q|_F \operatorname{div}_F \mathbf{w}_F \\
&\quad + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E \pi_{\mathcal{P},E}^{k-1} q|_E (\mathbf{w}_F \cdot \mathbf{n}_{FE}) \\
&= - \int_F q \operatorname{div}_F \mathbf{w}_F + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E q (\mathbf{w}_F \cdot \mathbf{n}_{FE}) \\
&= \int_F \text{grad}_F q|_F \cdot \mathbf{w}_F,
\end{aligned}$$

where we have removed the projectors using their definition in the second equality and we have integrated by parts to conclude. Recalling the definition (2.12) of $\mathcal{R}\mathcal{T}^k(F)$, we can first let $\mathbf{w}_F$ span $\mathcal{R}^{k-1}(F)$ to infer

$$\pi_{\mathcal{R},F}^{k-1}[\mathbf{G}_F^k(\underline{I}_{\text{grad},F}^k q|_F)] = \pi_{\mathcal{R},F}^{k-1}(\text{grad}_F q|_F),$$

and then $\mathcal{R}^{c,k}(F)$ to infer

$$\pi_{\mathcal{R},F}^{c,k}[\mathbf{G}_F^k(\underline{I}_{\text{grad},F}^k q|_F)] = \pi_{\mathcal{R},F}^{c,k}(\text{grad}_F q|_F).$$

The proof that $\pi_{\mathcal{R},T}^{k-1}[\mathbf{G}_T^k(\underline{I}_{\text{grad},T}^k q)] = \pi_{\mathcal{R},T}^{k-1}(\text{grad } q)$ and $\pi_{\mathcal{R},T}^{c,k}[\mathbf{G}_T^k(\underline{I}_{\text{grad},T}^k q)] = \pi_{\mathcal{R},T}^{c,k}(\text{grad } q)$ is similar: we write the definition (3.11) of $\mathbf{G}_T^k$ for $q_T = \underline{I}_{\text{grad},T}^k q$ and $\mathbf{w}_T \in \mathcal{R}\mathcal{T}^k(T)$, use property (A.4) along with (3.15) to replace the trace $\gamma_F^{k+1}(\underline{I}_{\text{grad},F}^k q|_F)$ with $\pi_{\mathcal{P},F}^{k-1}[\gamma_F^{k+1}(\underline{I}_{\text{grad},F}^k q|_F)] = \pi_{\mathcal{P},F}^{k-1}q|_F$ in each face integral, remove the projectors using their definitions, and integrate by parts. This concludes the proof of (3.38).
**Remark 13 (Global commutation properties)** Global commutation properties can be readily inferred from the local ones stated in Lemma 4 when interpolating functions that have sufficient global regularity.

**Remark 14 (Role of commutation properties in the design of robust methods)** The commutation properties of Lemma 4 play a key role in the design of discretisation methods robust with respect to the variations of physical parameters. See, e.g., [30] concerning a DDR method for the Reissner–Mindlin plate bending problem robust with respect to plate thickness.

**Proof (Lemma 4)** We start by noticing that all the interpolates defined in (3.38)–(3.40) are well-defined under the assumed regularities.

1. *Proof of (3.38).* By (3.12) it holds, for all $E \in \mathcal{E}_T$,

$$G_E^k(\underline{I}_{\text{grad},E}^k q|_E) = \pi_{\mathcal{P},E}^k(q'_{|E}) = \pi_{\mathcal{P},E}^k((\text{grad } q)|_E \cdot \mathbf{t}_E).$$

Let now $F \in \mathcal{F}_T$. Writing the definition (3.9) of $\mathbf{G}_F^k$ with $q_{|F} = \underline{I}_{\text{grad},F}^k q|_F$ and $\mathbf{w}_F \in \mathcal{R}\mathcal{T}^k(F)$, and recalling (A.2) to replace $q_{\mathcal{E}_F}$ with $\pi_{\mathcal{P},E}^{k-1}(q_{\mathcal{E}_F})|_E = \pi_{\mathcal{P},E}^{k-1}q|_E$ (see (3.5)) in each edge integral, we infer

$$\begin{aligned}
\int_F \mathbf{G}_F^k (\underline{I}_{\text{grad},F}^k q|_F) \cdot \mathbf{w}_F &= - \int_F \pi_{\mathcal{P},F}^{k-1} q|_F \operatorname{div}_F \mathbf{w}_F \\
&\quad + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E \pi_{\mathcal{P},E}^{k-1} q|_E (\mathbf{w}_F \cdot \mathbf{n}_{FE}) \\
&= - \int_F q \operatorname{div}_F \mathbf{w}_F + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E q (\mathbf{w}_F \cdot \mathbf{n}_{FE}) \\
&= \int_F \text{grad}_F q|_F \cdot \mathbf{w}_F,
\end{aligned}$$

where we have removed the projectors using their definition in the second equality and we have integrated by parts to conclude. Recalling the definition (2.12) of $\mathcal{R}\mathcal{T}^k(F)$, we can first let $\mathbf{w}_F$ span $\mathcal{R}^{k-1}(F)$ to infer

$$\pi_{\mathcal{R},F}^{k-1}[\mathbf{G}_F^k(\underline{I}_{\text{grad},F}^k q|_F)] = \pi_{\mathcal{R},F}^{k-1}(\text{grad}_F q|_F),$$

and then $\mathcal{R}^{c,k}(F)$ to infer

$$\pi_{\mathcal{R},F}^{c,k}[\mathbf{G}_F^k(\underline{I}_{\text{grad},F}^k q|_F)] = \pi_{\mathcal{R},F}^{c,k}(\text{grad}_F q|_F).$$

The proof that $\pi_{\mathcal{R},T}^{k-1}[\mathbf{G}_T^k(\underline{I}_{\text{grad},T}^k q)] = \pi_{\mathcal{R},T}^{k-1}(\text{grad } q)$ and $\pi_{\mathcal{R},T}^{c,k}[\mathbf{G}_T^k(\underline{I}_{\text{grad},T}^k q)] = \pi_{\mathcal{R},T}^{c,k}(\text{grad } q)$ is similar: we write the definition (3.11) of $\mathbf{G}_T^k$ for $q_T = \underline{I}_{\text{grad},T}^k q$ and $\mathbf{w}_T \in \mathcal{R}\mathcal{T}^k(T)$, use property (A.4) along with (3.15) to replace the trace $\gamma_F^{k+1}(\underline{I}_{\text{grad},F}^k q|_F)$ with $\pi_{\mathcal{P},F}^{k-1}[\gamma_F^{k+1}(\underline{I}_{\text{grad},F}^k q|_F)] = \pi_{\mathcal{P},F}^{k-1}q|_F$ in each face integral, remove the projectors using their definitions, and integrate by parts. This concludes the proof of (3.38).
2. *Proof of (3.39).* For all $F \in \mathcal{F}_T$, by (3.20) it holds

$$C_F^k(\underline{\mathbf{I}}_{\mathrm{curl},F}^k \mathbf{v}|_F) = \pi_{\mathcal{P},F}^k ((\mathrm{curl} \mathbf{v})|_F \cdot \mathbf{n}_F),$$

where we have used $\mathrm{rot}_F \mathbf{v}_{t,F} = (\mathrm{curl} \mathbf{v})|_F \cdot \mathbf{n}_F$, see [31, Eq. (3.7)]. Writing the definition (3.28) for $\mathbf{w}_T \in \mathcal{N}^k(T)$, we have

$$\int_T \mathbf{C}_T^k(\underline{\mathbf{I}}_{\mathrm{curl},T}^k \mathbf{v}) \cdot \mathbf{w}_T = \int_T \pi_{\mathcal{R},T}^k \mathbf{v} \cdot \mathrm{curl} \mathbf{w}_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \pi_{\mathcal{R},T,F}^k [\gamma_{t,F}^k(\underline{\mathbf{I}}_{\mathrm{curl},F}^k \underline{\mathbf{v}}_{t,F})] \cdot (\mathbf{w}_T \times \mathbf{n}_F), \quad (3.41)$$

where we have removed $\pi_{\mathcal{R},T}^{k-1}$ using its definition and, recalling (A.5), we have introduced the L²-orthogonal projector $\pi_{\mathcal{R},T,F}^k$ on $\mathcal{R}\mathcal{T}^k(F)$ in the boundary integral. By (2.12) together with (2.15) written with the choices $(E, S, S^c) = (\mathcal{R}\mathcal{T}^k(F), \mathcal{R}^{k-1}(F), \mathcal{R}^{c,k}(F))$ and (3.24),

$$\pi_{\mathcal{R},T,F}^k [\gamma_{t,F}^k(\underline{\mathbf{I}}_{\mathrm{curl},F}^k \mathbf{v}_{t,F})] = \mathfrak{R}_{\mathcal{R}^{k-1}(F), \mathcal{R}^{c,k}(F)} (\pi_{\mathcal{R},F}^{k-1} \mathbf{v}_{t,F}, \pi_{\mathcal{R},F}^{c,k} \mathbf{v}_{t,F}) = \pi_{\mathcal{R},T,F}^k \mathbf{v}_{t,F}.$$

Plugging this relation into (3.41), we infer

$$\int_T \mathbf{C}_T^k(\underline{\mathbf{I}}_{\mathrm{curl},T}^k \mathbf{v}) \cdot \mathbf{w}_T = \int_T \mathbf{v} \cdot \mathrm{curl} \mathbf{w}_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \pi_{\mathcal{R},T,F}^k \mathbf{v}_{t,F} \cdot (\mathbf{w}_T \times \mathbf{n}_F) \\ = \int_T \mathrm{curl} \mathbf{v} \cdot \mathbf{w}_T,$$

where we have used again (A.5) to remove the projector in the boundary term and we have integrated by parts to conclude. Letting $\mathbf{w}_T$ span $\mathcal{G}^{k-1}(T)$ (respectively $\mathcal{G}^{c,k}(T)$), this yields $\pi_{\mathcal{G},T}^{k-1}[\mathbf{C}_T^k(\underline{\mathbf{I}}_{\mathrm{curl},T}^k \mathbf{v})] = \pi_{\mathcal{G},T}^{k-1}(\mathrm{curl} \mathbf{v})$ (respectively $\pi_{\mathcal{G},T}^{c,k}[\mathbf{C}_T^k(\underline{\mathbf{I}}_{\mathrm{curl},T}^k \mathbf{v})] = \pi_{\mathcal{G},T}^{c,k}(\mathrm{curl} \mathbf{v}))$, thus concluding the proof of (3.39).

3. *Proof of (3.40).* The proof is done as in [31, Lemma 5.4], noticing that the cancellation of the component in the complement of $\mathcal{G}^k(T)$, obtained therein by orthogonality of this complement, is not required here since this component is absent from the definition (3.32) of $D_T^k$. $\square$

## 3.6 Complex and exactness properties

The properties collected in the following theorem show that the sequence (3.37) forms a (cochain) complex.

**Theorem 1 (Complex property) It holds**

$$\underline{\mathbf{I}}_{\mathrm{grad},h}^k \mathbb{R} = \mathrm{Ker} \underline{\mathbf{G}}_h^k, \quad (3.42)$$

$$\mathrm{Im} \underline{\mathbf{G}}_h^k \subset \mathrm{Ker} \underline{\mathbf{C}}_h^k, \quad (3.43)$$

$$\mathrm{Im} \underline{\mathbf{C}}_h^k \subset \mathrm{Ker} D_h^k, \quad (3.44)$$

$$\mathrm{Im} D_h^k = \mathcal{P}^k(\mathcal{J}_h). \quad (3.45)$$
2. *Proof of (3.39).* For all $F \in \mathcal{F}_T$, by (3.20) it holds

$$C_F^k(\underline{I}_{\text{curl},F}^k \mathbf{v}|_F) = \pi_{\mathcal{P},F}^k ((\text{curl} \mathbf{v})|_F \cdot \mathbf{n}_F),$$

where we have used $\text{rot}_F \mathbf{v}_{t,F} = (\text{curl} \mathbf{v})|_F \cdot \mathbf{n}_F$, see [31, Eq. (3.7)]. Writing the definition (3.28) for $\mathbf{w}_T \in \mathcal{N}^k(T)$, we have

$$\int_T \mathbf{C}_T^k(\underline{I}_{\text{curl},T}^k \mathbf{v}) \cdot \mathbf{w}_T = \int_T \pi_{\mathcal{R},T}^k \mathbf{v} \cdot \text{curl} \mathbf{w}_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \pi_{\mathcal{R},T,F}^k [\gamma_{t,F}^k(\underline{I}_{\text{curl},F}^k \underline{\mathbf{v}}_{t,F})] \cdot (\mathbf{w}_T \times \mathbf{n}_F), \quad (3.41)$$

where we have removed $\pi_{\mathcal{R},T}^{k-1}$ using its definition and, recalling (A.5), we have introduced the L²-orthogonal projector $\pi_{\mathcal{R},T,F}^k$ on $\mathcal{R}\mathcal{T}^k(F)$ in the boundary integral. By (2.12) together with (2.15) written with the choices $(E, S, S^c) = (\mathcal{R}\mathcal{T}^k(F), \mathcal{R}^{k-1}(F), \mathcal{R}^{c,k}(F))$ and (3.24),

$$\pi_{\mathcal{R},T,F}^k [\gamma_{t,F}^k (\underline{I}_{\text{curl},F}^k \mathbf{v}_{t,F})] = \mathfrak{R}_{\mathcal{R}^{k-1}(F), \mathcal{R}^{c,k}(F)} (\pi_{\mathcal{R},F}^{k-1} \mathbf{v}_{t,F}, \pi_{\mathcal{R},F}^{c,k} \mathbf{v}_{t,F}) = \pi_{\mathcal{R},T,F}^k \mathbf{v}_{t,F}.$$

Plugging this relation into (3.41), we infer

$$\int_T \mathbf{C}_T^k(\underline{I}_{\text{curl},T}^k \mathbf{v}) \cdot \mathbf{w}_T = \int_T \mathbf{v} \cdot \text{curl} \mathbf{w}_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F \pi_{\mathcal{R},T,F}^k \mathbf{v}_{t,F} \cdot (\mathbf{w}_T \times \mathbf{n}_F) \\ = \int_T \text{curl} \mathbf{v} \cdot \mathbf{w}_T,$$

where we have used again (A.5) to remove the projector in the boundary term and we have integrated by parts to conclude. Letting $\mathbf{w}_T$ span $\mathcal{G}^{k-1}(T)$ (respectively $\mathcal{G}^{c,k}(T)$), this yields $\pi_{\mathcal{G},T}^{k-1}[\mathbf{C}_T^k(\underline{I}_{\text{curl},T}^k \mathbf{v})] = \pi_{\mathcal{G},T}^{k-1}(\text{curl} \mathbf{v})$ (respectively $\pi_{\mathcal{G},T}^{c,k}[\mathbf{C}_T^k(\underline{I}_{\text{curl},T}^k \mathbf{v})] = \pi_{\mathcal{G},T}^{c,k}(\text{curl} \mathbf{v}))$, thus concluding the proof of (3.39).

3. *Proof of (3.40).* The proof is done as in [31, Lemma 5.4], noticing that the cancellation of the component in the complement of $\mathcal{G}^k(T)$, obtained therein by orthogonality of this complement, is not required here since this component is absent from the definition (3.32) of $D_T^k$. $\square$

## 3.6 Complex and exactness properties

The properties collected in the following theorem show that the sequence (3.37) forms a (cochain) complex.

**Theorem 1 (Complex property) It holds**

$$\underline{I}_{\text{grad},h}^k \mathbb{R} = \text{Ker } \underline{\mathcal{G}}_h^k, \quad (3.42)$$

$$\text{Im } \underline{\mathcal{G}}_h^k \subset \text{Ker } \underline{\mathcal{C}}_h^k, \quad (3.43)$$

$$\text{Im } \underline{\mathcal{C}}_h^k \subset \text{Ker } D_h^k, \quad (3.44)$$

$$\text{Im } D_h^k = \mathcal{P}^k(\mathcal{J}_h). \quad (3.45)$$
**Proof 1.** *Proof of (3.42).* From the consistency properties (3.12), (3.13) and (3.16) of the full gradients and the definition (3.18) of $\underline{\mathbf{G}}_h^k$, it is readily inferred that $\underline{\mathbf{G}}_h^k(\underline{I}_{\text{grad},h}^k C) = \underline{0}$ for all $C \in \mathbb{R}$, hence $\underline{I}_{\text{grad},h}^k \mathbb{R} \subset \text{Ker } \underline{\mathbf{G}}_h^k$.

To prove converse inclusion Ker $\underline{\mathbf{G}}_h^k \subset \underline{I}_{\text{grad},h}^k \mathbb{R}$, let $q_h \in \underline{\mathcal{X}}_{\text{grad},h}^k$ be such that $\underline{\mathbf{G}}_h^k q_h = \underline{0}$. By the definitions (3.18) of $\underline{\mathbf{G}}_h^k$ and (3.8) of $\mathcal{G}_E^k$, this means that $q'_E = 0$ for all $E \in \mathcal{E}_h$, that is, $(q_{\mathcal{E}_h})|_E$ is constant over $E$. Since $\Omega$ has only one connected component, accounting for the single-valuedness of $q_{\mathcal{E}_h}$ at vertices, we thus infer the existence of $C \in \mathbb{R}$ such that $q_{\mathcal{E}_h} = C$. Let now $F \in \mathcal{F}_h$ and $\mathbf{w}_F \in \mathcal{R}^{c,k}(F)$. We have $\pi_{\mathcal{R},F}^{c,k}(\underline{\mathbf{G}}_F^k q_F) = \underline{0}$, and thus

$$ 
\begin{aligned}
0 &= \int_F \underline{\mathbf{G}}_F^k q_F \cdot \mathbf{w}_F \\
&= -\int_F q_F \operatorname{div}_F \mathbf{w}_F + \sum_{E \in \mathcal{E}_F} \omega_{FE} \int_E q_{\mathcal{E}_F} (\mathbf{w}_F \cdot \mathbf{n}_{FE}) \\
&= \int_F (C - q_F) \operatorname{div}_F \mathbf{w}_F,
\end{aligned}
$$

where the second equality comes from the definition (3.9) of $\underline{\mathbf{G}}_F^k q_F$, and the conclusion is obtained accounting for the fact that $q_{\mathcal{E}_F} = C$ and integrating by parts. Since $\mathbf{w}_F$ is generic in $\mathcal{R}^{c,k}(F)$, recalling the isomorphism (2.9) this implies $\pi_{\mathcal{P},F}^{k-1}(q_F - C) = 0$, and thus $q_F = \pi_{\mathcal{P},F}^{k-1} C$. As, for all $F \in \mathcal{F}_h$, the previous results give $q_F = (q_F, q_{\mathcal{E}_F}) = \underline{I}_{\text{grad},F}^k C$, we also have $\gamma_F^{k+1} q_F = C$ by (3.14). Similarly, let $T \in \mathcal{T}_h$ and $\mathbf{w}_T \in \mathcal{R}^{c,k}(T)$. Writing the definition (3.11) of $\underline{\mathbf{G}}_T^k q_T$ for $\mathbf{w}_T \in \mathcal{R}^{c,k}(T)$, and accounting for $\pi_{\mathcal{R},T}^{c,k}(\underline{\mathbf{G}}_T^k q_T) = \underline{0}$ and $\gamma_F^{k+1} q_F = C$, it is inferred

$$ 
\begin{aligned}
0 &= \int_T \underline{\mathbf{G}}_T^k q_T \cdot \mathbf{w}_T \\
&= -\int_T q_T \operatorname{div} \mathbf{w}_T + \sum_{F \in \mathcal{F}_T} \omega_{TF} \int_F C(\mathbf{w}_T \cdot \mathbf{n}_F) \\
&= \int_F (C - q_T) \operatorname{div} \mathbf{w}_T,
\end{aligned}
$$

which implies, invoking the isomorphism (2.9), $\pi_{\mathcal{P},T}^{k-1}(q_T - C) = 0$ since $\mathbf{w}_T$ is generic in $\mathcal{R}^{c,k}(T)$. Hence $q_T = \pi_{\mathcal{P},T}^{k-1} C$ for all $T \in \mathcal{T}_h$, which concludes the proof that $q_T = \underline{I}_{\text{grad},h}^k C$.

**2.** *Proof of (3.43).* The inclusion (3.43) follows from the local property:

$$ \operatorname{Im} \underline{\mathbf{G}}_T^k \subset \operatorname{Ker} \underline{\mathbf{C}}_T^k \quad \forall T \in \mathcal{T}_h, \qquad (3.46) $$

i.e., $\underline{\mathbf{C}}_T^k(\underline{\mathbf{G}}_T^k q_T) = \underline{0}$ for all $q_T \in \underline{\mathcal{X}}_{\text{grad},T}$. Let $T \in \mathcal{T}_h$. The relation (3.21) implies $\underline{C}_F^k(\underline{\mathbf{G}}_F^k q_F) = \underline{0}$ for all $F \in \mathcal{F}_T$. The fact that $\pi_{\underline{\mathbf{G}},T}^{k-1}[\underline{\mathbf{C}}_T^k(\underline{\mathbf{G}}_T^k q_T)] = \underline{0}$ then follows