# Comptes Rendus
## Mécanique

Michele Botti, Daniele A. Di Pietro and Marwa Salah

A serendipity fully discrete div-div complex on polygonal meshes

Volume 351, Special Issue S1 (2023), p. 219-249

Online since: 7 March 2023

**Part of Special Issue:** The scientific legacy of Roland Glowinski

**Guest editors:** Grégoire Allaire (CMAP, Ecole Polytechnique, Institut Polytechnique de Paris, Palaiseau, France),

Jean-Michel Coron (Laboratoire Jacques-Louis Lions, Sorbonne Université)

and Vivette Girault (Laboratoire Jacques-Louis Lions, Sorbonne Université)

https://doi.org/10.5802/crmeca.150

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

This article is licensed under the

CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.

http://creativecommons.org/licenses/by/4.0/
# The scientific legacy of Roland Glowinski / *L'héritage scientifique de Roland Glowinski*

## A serendipity fully discrete div-div complex on polygonal meshes

### *Un complexe div-div discret avec sérendipité sur maillages polygonaux*

Michele Botti<sup>a</sup>, Daniele A. Di Pietro<sup>*,b</sup> and Marwa Salah<sup>b</sup>

<sup>a</sup> MOX, Politecnico di Milano, Italy

<sup>b</sup> IMAG, Univ Montpellier, CNRS, Montpellier, France

*E-mails*: michele.botti@polimi.it (M. Botti), daniele.di-pietro@umontpellier.fr (D. Di Pietro), marwa.salah@etu.umontpellier.fr (M. Salah)

**Abstract.** In this work we address the reduction of face degrees of freedom (DOFs) for discrete elasticity complexes. Specifically, using serendipity techniques, we develop a reduced version of a recently introduced two-dimensional complex arising from traces of the three-dimensional elasticity complex. The keystone of the reduction process is a new estimate of symmetric tensor-valued polynomial fields in terms of boundary values, completed with suitable projections of internal values for higher degrees. We prove an extensive set of new results for the original complex and show that the reduced complex has the same homological and analytical properties as the original one. This paper also contains an appendix with proofs of general Poincaré-Korn-type inequalities for hybrid fields.

**Résumé.** Dans cet article, nous abordons la réduction des degrés de liberté de face pour le complexe de l'élasticité discrète. Plus précisément, en utilisant des techniques de sérendipité, nous développons une version réduite d'un complexe bidimensionnel qui apparaît dans la discretisation des traces du complexe de l'élasticité tridimensionnel. La clé de voûte de la construction est une nouvelle estimation des fonctions polynomiales à valeurs tensorielles symétriques en termes de leur valeur au bord. Nous prouvons de nouveaux résultats pour le complexe original et montrons que le complexe réduit a les mêmes propriétés homologiques et analytiques que celui-ci. Cet article contient également une annexe avec des preuves d'inégalités de type Poincaré-Korn pour les champs hybrides.

**Keywords.** Discrete de Rham method, serendipity, compatible discretisations, mixed formulation, div-div complex, biharmonic equation, Kirchhoff-Love plates.

**Mots-clés.** Méthode de de Rham discrète, sérendipité, discrétisations compatibles, formulation mixte, complexe div-div, équation biharmonique, plaques de Kirchhoff-Love.

**Mathematical subject classification** (2010). 74K20, 74S05, 65N30.

Manuscript received 8 November 2022, accepted 15 November 2022.

<footnote>* Corresponding author.</footnote>
# 1. Introduction

The development of computationally viable discrete elasticity complexes is a long-standing problem in numerical analysis. Finite element versions of the elasticity complex typically require a large number of degrees of freedom (DOFs) to deal with the symmetry constraint on tensor-valued fields [1–6]; recently, advances on this topic have been made in [7] in the framework of Finite Element Systems [8]. Particularly critical are DOFs attached to mesh faces, that cannot be efficiently eliminated via static condensation. In this work, we study DOFs reduction through serendipity. Serendipity techniques exploit the information on the boundary to fix the values of (a subset of) internal DOFs while preserving polynomial consistency. When working with discrete complexes, this reduction must be carefully designed in order to preserve key properties of the original complex.

With face DOFs reduction in mind, we focus on the two-dimensional *div-div complex* [9] that arises when considering traces for the three-dimensional elasticity complex on polyhedra (see [5, Section 3.4]). Specifically, denoting by $\Omega \subset \mathbb{R}^2$ a bounded connected polygonal set and by $\mathcal{S}$ the set of symmetric $2 \times 2$ matrices, this complex reads:

$$ \mathcal{R}\mathcal{T}^1(\Omega) \longrightarrow \mathbf{H}^1(\Omega; \mathbb{R}^2) \xrightarrow{\text{sym curl}} \mathbf{H}(\text{div div}, \Omega; \mathcal{S}) \xrightarrow{\text{div div}} L^2(\Omega) \xrightarrow{0} 0, \quad (1) $$

where “sym” denotes the symmetric part of a space or an operator, $\mathcal{R}\mathcal{T}^1(\Omega) := \mathcal{P}^0(\Omega) + \mathcal{x}\mathcal{P}^0(\Omega)$ is the lowest-order Raviart–Thomas space [10], and a definition of the sym curl and div div operators in Cartesian coordinates is given in (2) below. A discrete version of the complex (1) has been recently obtained in [11] following the discrete de Rham (DDR) paradigm [12, 13]. A salient feature of DDR constructions is the native support of general polygonal/polyhedral meshes, which simplifies the discretisation of complicated domain geometries and/or the capture of fine-scale features of the solution. Alternative approaches to the use of polygonal/polyhedral meshes in the finite element framework include the fictitious domain method popularised by the work of Glowinski and coauthors; see, e.g., [14, 15]. Here, following the abstract framework of [16] (closely inspired, through the bridges constructed in [17], by the ideas originally developed in [18–20]), we derive a reduced version of the DDR complex of [11] that preserves both its homological and analytical properties. The keystone of this reduced version is the estimate of tensor-valued polynomials established in Lemma 11 below, which provides indications on which DOFs can be discarded while preserving polynomial consistency. A comparison of the number of DOFs between the full and serendipity div-div complexes for various element shapes is provided in Table 1, showing gains between 13% and 27% depending on the polynomial degree and element shape.

The rest of this work is organised as follows. In Section 2 we briefly recall the general setting. The construction underlying the full DDR div-div complex is briefly recalled in Section 3, where we also prove a complete set of analytical results (Poincaré inequalities, consistency, and adjoint consistency) that complement the ones established in [11]. The serendipity version of the DDR div-div complex is derived in Section 4. Through the sufficient conditions identified in [11], we establish, in Theorems 20 and 24 below, that the serendipity and full complexes have analogous homological and analytical properties. Finally, Appendix A focuses on Poincaré–Korn type inequalities for hybrid vector fields that are instrumental for the previous analysis.
<table><caption>**Table 1.** Number of DOFs for the full • serendipity discrete counterparts of the spaces $H^1(T; \mathbb{R}^2)$ and $H(\text{div div}, T; \mathfrak{S})$ on a triangle, quadrangle, and pentagon element $T$ for polynomial degrees $k$ ranging from 3 to 6. The relative DOFs reduction is in parenthesis. The parameter $\eta_T$ is defined in Assumption 10 below.</caption><thead><tr><th>Discrete space</th><th>k = 3</th><th>k = 4</th><th>k = 5</th><th>k = 6</th></tr></thead><tbody><tr><td colspan="5">Triangle, η<sub>T</sub> = 3</td></tr><tr><td><b>H</b><sup>1</sup>(T; ℝ<sup>2</sup>)</td><td>24 • 20 (-17%)</td><td>36 • 30 (-17%)</td><td>50 • 42 (-16%)</td><td>66 • 56 (-15%)</td></tr><tr><td><b>H</b>(div div, T; ℝ<sup>2</sup>)</td><td>24 • 20 (-17%)</td><td>39 • 33 (-15%)</td><td>57 • 49 (-14%)</td><td>78 • 68 (-13%)</td></tr><tr><td colspan="5">Quadrangle, η<sub>T</sub> = 4</td></tr><tr><td><b>H</b><sup>1</sup>(T; ℝ<sup>2</sup>)</td><td>30 • 24 (-20%)</td><td>44 • 34 (-23%)</td><td>60 • 46 (-23%)</td><td>78 • 60 (-23%)</td></tr><tr><td><b>H</b>(div div, T; ℝ<sup>2</sup>)</td><td>30 • 24 (-20%)</td><td>47 • 37 (-21%)</td><td>67 • 53 (-21%)</td><td>90 • 72 (-20%)</td></tr><tr><td colspan="5">Pentagon, η<sub>T</sub> = 5</td></tr><tr><td><b>H</b><sup>1</sup>(T; ℝ<sup>2</sup>)</td><td>36 • 30 (-17%)</td><td>52 • 40 (-23%)</td><td>70 • 52 (-26%)</td><td>90 • 66 (-27%)</td></tr><tr><td><b>H</b>(div div, T; ℝ<sup>2</sup>)</td><td>36 • 30 (-17%)</td><td>55 • 43 (-22%)</td><td>77 • 59 (-23%)</td><td>102 • 78 (-24%)</td></tr></tbody></table>

## 2. Setting

### 2.1. Two-dimensional vector calculus operators

Consider the real plane $\mathbb{R}^2$ endowed with the Cartesian coordinate system $(x_1, x_2)$, and denote by $\partial_i$ the weak partial derivative with respect to the $i$th coordinate. We need the following two-dimensional differential operators acting on smooth enough scalar-valued fields $q$, vector-valued fields $v = (v_1/v_2)$, or matrix-valued fields $\tau = (\tau_{11}/\tau_{21}, \tau_{12}/\tau_{22})$:

$$ \begin{align} \text{curl } q &:= \begin{pmatrix} \partial_2 q \\ -\partial_1 q \end{pmatrix}, & \text{div } v &:= \partial_1 v_1 + \partial_2 v_2, \nonumber \\ \text{grad } v &:= \begin{pmatrix} \partial_1 v_1 & \partial_2 v_1 \\ \partial_1 v_2 & \partial_2 v_2 \end{pmatrix}, & \text{sym curl } v &:= \begin{pmatrix} \partial_2 v_1 & -\frac{\partial_1 v_1 + \partial_2 v_2}{2} \\ -\frac{\partial_1 v_1 + \partial_2 v_2}{2} & -\partial_1 v_2 \end{pmatrix}, \tag{2} \\ \text{div } \tau &:= \begin{pmatrix} \partial_1 \tau_{11} + \partial_2 \tau_{12} \\ \partial_1 \tau_{21} + \partial_2 \tau_{22} \end{pmatrix}, & \text{rot } \tau &:= \begin{pmatrix} \partial_2 \tau_{11} - \partial_1 \tau_{12} \\ \partial_2 \tau_{21} - \partial_1 \tau_{22} \end{pmatrix}. \nonumber \end{align} $$

Defining the fourth-order tensor $\mathcal{C}$ such that

$$ \mathcal{C} \tau = \begin{pmatrix} \tau_{12} & -\frac{\tau_{11} + \tau_{22}}{2} \\ -\frac{\tau_{11} + \tau_{22}}{2} & -\tau_{21} \end{pmatrix} \quad \forall \tau = \begin{pmatrix} \tau_{11} & \tau_{12} \\ \tau_{21} & \tau_{22} \end{pmatrix} \in \mathbb{R}^{2 \times 2}, \quad (3) $$

we have $\text{sym curl } v = \mathcal{C} \text{grad } v$.

### 2.2. Mesh and notation for inequalities up to a constant

We denote by $\mathcal{M}_h = \mathcal{T}_h \cup \mathcal{E}_h \cup \mathcal{V}_h$ a polygonal mesh of $\Omega$ in the usual sense of [21], with $\mathcal{T}_h, \mathcal{E}_h$, and $\mathcal{V}_h$ collecting, respectively, the elements, edges, and vertices and $h$ denoting the meshsize. For all $Y \in \mathcal{M}_h$, we let $h_Y$ denote its diameter so that, in particular, $h = \max_{T \in \mathcal{T}_h} h_T$. $\mathcal{M}_h$ is assumed to belong to a refined mesh sequence $(\mathcal{M}_h)_{h>0}$ with regularity parameter bounded away from zero. We additionally assume that each element $T \in \mathcal{T}_h$ is contractible and denote by $x_T$ a point inside $T$ such that there exists a disk contained in $T$ centered in $x_T$ and of diameter comparable to $h_T$ uniformly in $h$. The sets of edges and vertices of $T$ are denoted by $\mathcal{E}_T$ and $\mathcal{V}_T$, respectively. By
mesh regularity, the number of edges (and vertices) of mesh elements are bounded uniformly in $h$: There exists $N_\partial \in \mathbb{N}$ such that, for all $h$,

$$ \mathrm{card}(\mathcal{E}_T) = \mathrm{card}(\mathcal{V}_T) \leq N_\partial \quad \forall T \in \mathcal{T}_h. $$

For each edge $E \in \mathcal{E}_h$, we denote by $\mathcal{V}_E$ the set of vertices corresponding to its endpoints and fix an orientation by prescribing a unit tangent vector $t_E$. This orientation determines two numbers $(\omega_{EV})_{V \in \mathcal{V}_E}$ in $\{-1, +1\}$ such that $\omega_{EV} = +1$ whenever $t_E$ points towards $V$. The corresponding unit normal vector $n_E$ is selected so that $(t_E, n_E)$ forms a right-handed system of coordinates, and, for each $T \in \mathcal{T}_h$ such that $E \in \mathcal{E}_T$, we denote by $\omega_{TE} \in \{-1, +1\}$ the orientation of $E$ relative to $T$, defined so that $\omega_{TE} n_E$ points out of $T$.

From this point on, $a \lesssim b$ means $a \leq Cb$ with $C$ only depending on $\Omega$, the mesh regularity parameter, and the polynomial degree $k$ of the complex (see (8) below). We also write $a \simeq b$ as a shorthand notation for "$a \lesssim b$ and $b \lesssim a$".

## 2.3. Polynomial spaces

Given $Y \in \mathcal{M}_h$ and an integer $m \geq 0$, we denote by $\mathcal{P}^m(Y)$ the space spanned by the restriction to $Y$ of two-variate polynomials of total degree $\leq m$, with the additional convention that $\mathcal{P}^{-1}(Y) = \{0\}$. The symbols $\mathcal{P}^m(Y; \mathbb{R}^2)$ and $\mathcal{P}^m(Y; \mathbb{S})$ denote, respectively, vector-valued and symmetric tensor-valued functions over $Y$ whose components are in $\mathcal{P}^m(Y)$. Finally, for each $T \in \mathcal{T}_h$, we denote by $\mathcal{P}^m(\mathcal{E}_T)$ the space of broken polynomials of total degree $\leq m$ on $\mathcal{E}_T$. Vector and tensor versions of this space are denoted in boldface and the codomain is specified.

Denoting by $\mathrm{sym}: \mathbb{R}^{2 \times 2} \ni \boldsymbol{\tau} \mapsto \frac{\boldsymbol{\tau}+\boldsymbol{\tau}^{\top}}{2} \in \mathbb{S}$ the symmetrisation operator, the following decompositions hold:

$$ \mathcal{P}^m(T; \mathbb{S}) = \mathcal{H}^m(T) \oplus \mathcal{H}^{c,m}(T) $$

$$ \text{with } \mathcal{H}^m(T) := \mathbf{hess} \mathcal{P}^{m+2}(T) \quad \text{and} \quad \mathcal{H}^{c,m}(T) := \mathrm{sym}((\mathbf{x}-\mathbf{x}_T)^{\perp} \otimes \mathcal{P}^{m-1}(T;\mathbb{R}^2)), $$

where, for all $\mathbf{v} = (\nu_1 \nu_2) \in \mathbb{R}^2$, $\mathbf{v}^{\perp} = (-\nu_2 \nu_1)$ denotes the vector obtained rotating $\mathbf{v}$ by an angle of $-\frac{\pi}{2}$ radians and

$$ \mathcal{P}^m(T; \mathbb{S}) = \mathcal{C}^m(T) \oplus \mathcal{C}^{c,m}(T) $$

$$ \text{with } \mathcal{C}^m(T) := \mathrm{sym} \mathrm{curl} \mathcal{P}^{m+1}(T; \mathbb{R}^2) \quad \text{and} \quad \mathcal{C}^{c,m}(T) := (\mathbf{x}-\mathbf{x}_T)(\mathbf{x}-\mathbf{x}_T)^{\top} \mathcal{P}^{m-2}(T). \quad (4) $$

The following result will be needed in the analysis.

**Proposition 1 (Continuity of the inverses of local isomorphisms).** Let $m \geq 1$ and set, for $\ell \geq 2$, $\mathfrak{P}^\ell(T) := \mathrm{span}\{(\frac{\mathbf{x}-\mathbf{x}_T}{h_T})^\alpha : \alpha = (\alpha_1, \alpha_2) \in \mathbb{N}^2, 2 \leq \alpha_1 + \alpha_2 \leq \ell\}$, where $\mathbf{y}^\alpha := y_1^{\alpha_1} y_2^{\alpha_2} \text{ if } \mathbf{y} = (y_1, y_2) \in \mathbb{R}^2$. Then, $\mathrm{rot}: \mathcal{H}^{c,m}(T) \to \mathcal{P}^{m-1}(T;\mathbb{R}^2)$, and, if $m \geq 2$, $\mathrm{div} \mathrm{div}: \mathcal{C}^{c,m}(T) \to \mathcal{P}^{m-2}(T)$ and $\mathbf{hess}: \mathfrak{P}^m(T) \to \mathcal{H}^{m-2}(T)$ are isomorphisms with continuous inverse, i.e.,

$$ \|\mathbf{v}\|_{L^2(T;\mathbb{R}^{2 \times 2})} \lesssim h_T \|\mathrm{rot}\mathbf{v}\|_{L^2(T;\mathbb{R}^2)} \quad \forall \mathbf{v} \in \mathcal{H}^{c,m}(T), \quad (5) $$

$$ \|\mathbf{v}\|_{L^2(T;\mathbb{R}^{2 \times 2})} \lesssim h_T^2 \|\mathrm{div} \mathrm{div} \mathbf{v}\|_{L^2(T)} \quad \forall \mathbf{v} \in \mathcal{C}^{c,m}(T), \quad (6) $$

$$ \|\mathbf{q}\|_{L^2(T)} \lesssim h_T^2 \|\mathbf{hess} \mathbf{q}\|_{L^2(T;\mathbb{R}^{2 \times 2})} \quad \forall \mathbf{q} \in \mathfrak{P}^m(T). \quad (7) $$

**Proof.** Inequality (7) is proved in [11, Lemma 9]. The proof of the other inequalities hinges on a similar scaling argument, not repeated here for the sake of brevity. $\square$

Given a polynomial (sub)space $\mathcal{X}^m(Y)$ on $Y \in \mathcal{M}_h$, the corresponding $L^2$-orthogonal projector is denoted by $\pi_{\mathcal{X},Y}^m$. Boldface fonts will be used when the elements of $\mathcal{X}^m(Y)$ are tensor- or vector-valued and, for $T \in \mathcal{T}_h$, we additionally denote by $\pi_{\mathcal{X},T}^{c,m}$, $\mathcal{X} \in \{\mathcal{H}, \mathcal{C}\}$, the $L^2$-orthogonal projector on $\mathcal{X}^{c,m}(T)$.
### 3. Full DDR complex, Poincaré inequalities, and consistency

In this section we briefly recall the discrete div-div complex of [11], for which we prove a complete panel of properties including Poincaré inequalities, consistency, and adjoint consistency results that complement the ones established in the previous reference.

#### 3.1. Spaces

Throughout the rest of the paper, the integer

$$k \ge 3 \qquad (8)$$

will denote the polynomial degree of the discrete complex. The discrete counterparts of the spaces $H^1(\Omega; \mathbb{R}^2)$ and $H(\text{div div}, \Omega; \mathbb{S})$ are, respectively,

$$ \begin{aligned} \underline{V}_h^k := \{ \underline{v}_h = ((v_T)_{T \in \mathcal{T}_h}, (v_E)_{E \in \mathcal{E}_h}, (v_V, G_{v,V})_{V \in \mathcal{V}_h}) : \\ v_T \in \mathcal{P}^{k-2}(T; \mathbb{R}^2) \text{ for all } T \in \mathcal{T}_h, \\ v_E \in \mathcal{P}^{k-4}(E; \mathbb{R}^2) \text{ for all } E \in \mathcal{E}_h, \\ v_V \in \mathbb{R}^2 \text{ and } G_{v,V} \in \mathbb{R}^{2 \times 2} \text{ for all } V \in \mathcal{V}_h \} \end{aligned} \qquad (9a) $$

$$ \begin{aligned} \underline{\Sigma}_h^{k-1} := \{ \underline{\tau}_h = \left( (\underline{\tau}_{\mathcal{H},T}, \underline{\tau}_{\mathcal{H},T}^c)_{T \in \mathcal{T}_h}, (\tau_E, D_{\tau,E})_{E \in \mathcal{E}_h}, (\tau_V)_{V \in \mathcal{V}_h} \right) : \\ \underline{\tau}_{\mathcal{H},T} \in \mathcal{H}^{k-4}(T) \text{ and } \underline{\tau}_{\mathcal{H},T}^c \in \mathcal{H}^{c,k-1}(T) \text{ for all } T \in \mathcal{T}_h, \\ \tau_E \in \mathcal{P}^{k-3}(E) \text{ and } D_{\tau,E} \in \mathcal{P}^{k-2}(E) \text{ for all } E \in \mathcal{E}_h, \\ \underline{\tau}_V \in \mathbb{S} \text{ for all } V \in \mathcal{V}_h \}. \end{aligned} \qquad (9b) $$

The interpolators $\underline{I}_{V,h}^k : C^1(\bar{\Omega}; \mathbb{R}^2) \to \underline{V}_h^k$ and $\underline{I}_{\Sigma,h}^{k-1} : H^2(\Omega; \mathbb{S}) \to \underline{\Sigma}_h^{k-1}$ are obtained collecting $L^2$-orthogonal projections on the polynomial components: For all $v \in C^1(\bar{\Omega}; \mathbb{R}^2)$ and all $\tau \in H^2(\Omega; \mathbb{S})$,

$$ \begin{aligned} \underline{I}_{V,h}^k v := \left( \left( \underline{\pi}_{\mathcal{P},T}^{k-2} v|_T \right)_{T \in \mathcal{T}_h}, \left( \underline{\pi}_{\mathcal{P},E}^{k-4} v|_E \right)_{E \in \mathcal{E}_T}, (v(x_V), \text{grad } v(x_V))_{V \in \mathcal{V}_T} \right), \\ \underline{I}_{\Sigma,h}^{k-1} \tau := \left( \left( \underline{\pi}_{\mathcal{H},T}^{k-4} \tau|_T, \underline{\pi}_{\mathcal{H},T}^{c,k-1} \tau|_T \right)_{T \in \mathcal{T}_h}, \left( \underline{\pi}_{\mathcal{P},E}^{k-3} (\tau|_E n_E \cdot n_E), \underline{\pi}_{\mathcal{P},E}^{k-2} \delta_E \tau \right)_{E \in \mathcal{E}_T}, (\tau(x_V))_{V \in \mathcal{V}_T} \right), \end{aligned} \qquad (10) $$

where $x_V$ denotes the coordinate vector of the vertex $V \in \mathcal{V}_T$ while, for all $E \in \mathcal{E}_T$, $\partial_{\mathbf{t}_E}$ denotes the derivative along the edge $E$ in the direction of $\mathbf{t}_E$ and we have set, for the sake of conciseness,

$$ \delta_E \tau := \partial_{\mathbf{t}_E} (\tau|_E n_E \cdot \mathbf{t}_E) + (\text{div } \tau)|_E \cdot n_E. $$

As customary for DDR methods, we denote the restrictions of spaces and operators to a mesh element or edge $Y \in \mathcal{T}_h \cup \mathcal{E}_h$ by replacing the subscript "h" with "Y". Such restrictions are obtained collecting the polynomial components on $Y$ and its boundary. Given $T \in \mathcal{T}_h$, for $\underline{V}_h^k$ we will also need its restriction $\underline{V}_{\partial T}^k$ to the boundary of $T$, obtained collecting all the polynomial components that lie thereon.

#### 3.2. Reconstructions

Let a mesh element $T \in \mathcal{T}_h$ be fixed. The DDR method hinges on the reconstructions of differential operators and of the corresponding potentials described below.
### 3.2.1. Symmetric curl and vector potential

The key integration by parts formula to reconstruct discrete counterparts of the symmetric curl and of the corresponding vector potential is the following: For any $v : T \to \mathbb{R}^2$ and any $\tau : T \to \mathbb{S}$ smooth enough,

$$ \int_T v \cdot \mathrm{rot} \, \tau = - \int_T \mathrm{sym \, curl} \, v : \tau + \sum_{E \in \mathcal{E}_T} \omega_{TE} \int_E v \cdot (\tau \, t_E). \quad (11) $$

The full symmetric curl $\mathbf{C}_{\mathrm{sym},T}^{k-1} : \underline{\mathbf{v}}_T^k \to \mathcal{P}^{k-1}(T; \mathbb{S})$ is such that, for all $\underline{v}_T \in \underline{\mathbf{v}}_T^k$,

$$ \int_T \mathbf{C}_{\mathrm{sym},T}^{k-1} \underline{\mathbf{v}}_T : \boldsymbol{\tau}_T = - \int_T \mathbf{v}_T \cdot \mathrm{rot} \, \boldsymbol{\tau}_T + \sum_{E \in \mathcal{E}_T} \omega_{TE} \int_E \mathbf{v}_{\mathcal{E}_T} \cdot (\boldsymbol{\tau}_T \, \mathbf{t}_E) \quad \forall \boldsymbol{\tau}_T \in \mathcal{P}^{k-1}(T; \mathbb{S}), \quad (12) $$

where $\mathbf{v}_{\mathcal{E}_T} \in \mathcal{P}^k(\mathcal{E}_T; \mathbb{R}^2) \cap \mathbf{C}^0(\partial T; \mathbb{R}^2)$ is uniquely defined by the following conditions:

$$ \begin{align} \pi_{\mathcal{P},E}^{k-4}(\mathbf{v}_{\mathcal{E}_T})|_E &= \mathbf{v}_E && \text{for all } E \in \mathcal{E}_T, \nonumber \\ \partial_{\mathbf{t}_E}(\mathbf{v}_{\mathcal{E}_T})|_{E}(\mathbf{x}_V) &= \mathbf{G}_{\mathbf{v},V} \mathbf{t}_E && \text{for all } E \in \mathcal{E}_T \text{ and } V \in \mathcal{V}_E, \tag{13} \\ \text{and } \mathbf{v}_{\mathcal{E}_T}(\mathbf{x}_V) &= \mathbf{v}_V && \text{for all } V \in \mathcal{V}_T. \nonumber \end{align} $$

The discrete symmetric curl $\underline{\mathbf{C}}_{\mathrm{sym},T}^{k-1} : \underline{\mathbf{v}}_T^k \to \underline{\boldsymbol{\Sigma}}_T^{k-1}$, acting between the discrete spaces in the complex, is obtained setting, for all $\underline{\mathbf{v}}_T \in \underline{\mathbf{v}}_T^k$,

$$ \underline{\mathbf{C}}_{\mathrm{sym},T}^{k-1} \underline{\mathbf{v}}_T := \left( \pi_{\mathcal{H},T}^{k-4} \left( \mathbf{C}_{\mathrm{sym},T}^{k-1} \underline{\mathbf{v}}_T \right), \pi_{\mathcal{H},T}^{c,k-1} \left( \mathbf{C}_{\mathrm{sym},T}^{k-1} \underline{\mathbf{v}}_T \right), \right. \\ \left. \left( \pi_{\mathcal{P},E}^{k-3} (\partial_{\mathbf{t}_E} \mathbf{v}_{\mathcal{E}_T} \cdot \mathbf{n}_E), \partial_{\mathbf{t}_E}^2 \mathbf{v}_{\mathcal{E}_T} \cdot \mathbf{t}_E \right)_{E \in \mathcal{E}_T}, \quad (\mathcal{C}\mathbf{G}_{\mathbf{v},V})_{V \in \mathcal{V}_T} \right), \quad (14) $$

with $\mathcal{C}$ as in (3). The global symmetric curl $\underline{\mathbf{C}}_{\mathrm{sym},h}^{k-1} : \underline{\mathbf{v}}_h^k \to \underline{\boldsymbol{\Sigma}}_h^{k-1}$ is such that, for all $\underline{\mathbf{v}}_h \in \underline{\mathbf{v}}_h^k$,

$$ \left( \underline{\mathbf{C}}_{\mathrm{sym},h}^{k-1} \underline{\mathbf{v}}_h \right)|_{T} = \underline{\mathbf{C}}_{\mathrm{sym},T}^{k-1} \underline{\mathbf{v}}_T \quad \forall T \in \mathcal{T}_h. $$

Notice that this definition makes sense since the discrete curl components at vertices and edges are single-valued. The vector potential $\mathbf{P}_{\mathbf{v},T}^k : \underline{\mathbf{v}}_T^k \to \mathcal{P}^k(T; \mathbb{R}^2)$ is such that, for all $\underline{\mathbf{v}}_T \in \underline{\mathbf{v}}_T^k$,

$$ \int_T \mathbf{P}_{\mathbf{v},T}^k \underline{\mathbf{v}}_T \cdot \mathrm{rot} \, \boldsymbol{\tau}_T = - \int_T \mathbf{C}_{\mathrm{sym},T}^{k-1} \underline{\mathbf{v}}_T : \boldsymbol{\tau}_T + \sum_{E \in \mathcal{E}_T} \omega_{TE} \int_E \mathbf{v}_{\mathcal{E}_T} \cdot (\boldsymbol{\tau}_T \, \mathbf{t}_E) \quad \forall \boldsymbol{\tau}_T \in \mathcal{H}^{c,k+1}(T). \quad (15) $$

We recall the following polynomial consistency property from [11]:

$$ \mathbf{P}_{\mathbf{v},T}^k \underline{\mathbf{I}}_{\mathbf{v},T}^k \mathbf{v} = \mathbf{v} \quad \forall \mathbf{v} \in \mathcal{P}^k(T; \mathbb{R}^2). \quad (16) $$

**Remark 2 (Validity of (15)).** Relation (15) remains valid for all $\boldsymbol{\tau}_T \in \mathcal{H}^{k-1}(T) \oplus \mathcal{H}^{c,k+1}(T)$, as can be checked taking $\boldsymbol{\tau}_T = \mathbf{hess} \, q_T$ with $q_T \in \mathcal{P}^{k+1}(T)$ and noticing that both sides vanish (use **rothess** = 0 for the left-hand side and the definition (12) of $\mathbf{C}_{\mathrm{sym},T}^{k-1}$ with $\boldsymbol{\tau}_T = \mathbf{hess} \, q_T$ along with **rothess** = 0 for the right-hand side). This implies, in particular, that (15) holds for all $\boldsymbol{\tau}_T \in \mathcal{P}^{k-1}(T; \mathbb{S}) \subset \mathcal{H}^{k-1}(T) \oplus \mathcal{H}^{c,k+1}(T)$.

### 3.2.2. Div-div and tensor potential

The starting point for reconstructions in $\underline{\boldsymbol{\Sigma}}_T^{k-1}$ is the following integration by parts formula, corresponding to [22, Eq. (2.4)] (see also [23, Eq. (2.1)]) and valid for all tensor-valued functions $\boldsymbol{\tau} : T \to \mathbb{S}$ and all scalar-valued functions $q : T \to \mathbb{R}$ smooth enough:

$$ \begin{align} \int_T \mathrm{div} \, \mathbf{div} \, \boldsymbol{\tau} \, q = \int_T \boldsymbol{\tau} : \mathbf{hess} \, q & - \sum_{E \in \mathcal{E}_T} \omega_{TE} \left[ \int_E (\boldsymbol{\tau} \mathbf{n}_E \cdot \mathbf{n}_E) \partial_{\mathbf{n}_E} q - \int_E \delta_{E} \boldsymbol{\tau} \, q \right] \tag{17} \\ & - \sum_{E \in \mathcal{E}_T} \omega_{TE} \sum_{V \in \mathcal{V}_E} \omega_{EV} (\boldsymbol{\tau} \mathbf{n}_E \cdot \mathbf{t}_E)(\mathbf{x}_V) q(\mathbf{x}_V). \nonumber \end{align} $$
For all $\underline{\boldsymbol{\tau}}_T \in \underline{\boldsymbol{\Sigma}}_T^{k-1}$, the discrete div-div operator $DD_T^{k-2} : \underline{\boldsymbol{\Sigma}}_T^{k-1} \to \mathcal{P}^{k-2}(T)$ is such that

$$ \int_T DD_T^{k-2} \underline{\boldsymbol{\tau}}_T q_T = \int_T \boldsymbol{\tau}_{\mathcal{H},T} : \mathbf{hess} \, q_T - \sum_{E \in \mathcal{E}_T} \omega_{TE} \left( \int_E \tau_E \partial_{\boldsymbol{n}_E} q_T - \int_E D_{\boldsymbol{\tau},E} q_T \right) \\ - \sum_{E \in \mathcal{E}_T} \omega_{TE} \sum_{V \in \mathcal{V}_E} \omega_{EV} (\boldsymbol{\tau}_V \boldsymbol{n}_E \cdot \boldsymbol{t}_E) q_T(\boldsymbol{x}_V) \quad \forall q_T \in \mathcal{P}^{k-2}(T), \quad (18) $$

while the tensor potential $\mathbf{P}_{\underline{\boldsymbol{\Sigma}},T}^{k-1} : \underline{\boldsymbol{\Sigma}}_T^{k-1} \to \mathcal{P}^{k-1}(T;\mathbb{S})$ satisfies, for all $(q_T, \boldsymbol{v}_T) \in \mathcal{P}^{k+1}(T) \times \mathcal{H}^{c,k-1}(T)$,

$$ \int_T \mathbf{P}_{\underline{\boldsymbol{\Sigma}},T}^{k-1} \underline{\boldsymbol{\tau}}_T : (\mathbf{hess} \, q_T + \boldsymbol{v}_T) = \int_T DD_T^{k-2} \underline{\boldsymbol{\tau}}_T q_T + \sum_{E \in \mathcal{E}_T} \omega_{TE} \left( \int_E P_{\underline{\boldsymbol{\Sigma}},E}^{k-1} \underline{\boldsymbol{\tau}}_E \partial_{\boldsymbol{n}_E} q_T - \int_E D_{\boldsymbol{\tau},E} q_T \right) \\ + \sum_{E \in \mathcal{E}_T} \omega_{TE} \sum_{V \in \mathcal{V}_E} \omega_{EV} (\boldsymbol{\tau}_V \boldsymbol{n}_E \cdot \boldsymbol{t}_E) q_T(\boldsymbol{x}_V) + \int_T \boldsymbol{\tau}_{\mathcal{H},T}^c : \boldsymbol{v}_T. \quad (19) $$

Above, for all $E \in \mathcal{E}_T$, denoting by $\underline{\boldsymbol{\tau}}_E := (\tau_E, D_{\boldsymbol{\tau},E}, (\boldsymbol{\tau}_V)_{V \in \mathcal{V}_E})$ the restriction of $\underline{\boldsymbol{\tau}}_T$ to $E$, $P_{\underline{\boldsymbol{\Sigma}},E}^{k-1} \underline{\boldsymbol{\tau}}_E \in \mathcal{P}^{k-1}(E)$ uniquely defined by the following conditions:

$$ P_{\underline{\boldsymbol{\Sigma}},E}^{k-1} \underline{\boldsymbol{\tau}}_E(\boldsymbol{x}_V) = \boldsymbol{\tau}_V \boldsymbol{n}_E \cdot \boldsymbol{n}_E \quad \text{for all } V \in \mathcal{V}_E \text{ and } \pi_{\mathcal{P},E}^{k-3}(P_{\underline{\boldsymbol{\Sigma}},E}^{k-1} \underline{\boldsymbol{\tau}}_E) = \tau_E. $$

We recall, for future, use, the following result proved in [11, Lemma 4]:

$$ \mathbf{P}_{\underline{\boldsymbol{\Sigma}},T}^{k-1} \circ \underline{\mathbf{C}}_{\text{sym},T}^{k-1} = \mathbf{C}_{\text{sym},T}^{k-1}, \quad (20) $$

expressing the commutativity of the following diagram:

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

The global div-div operator $DD_h^{k-2} : \underline{\boldsymbol{\Sigma}}_h^{k-1} \to \mathcal{P}^{k-2}(T_h)$ acting between spaces in the discrete complex is such that, for all $\underline{\boldsymbol{\tau}}_h \in \underline{\boldsymbol{\Sigma}}_h^{k-1}$,

$$ (DD_h^{k-2} \underline{\boldsymbol{\tau}}_h)_{|T} := DD_T^{k-2} \underline{\boldsymbol{\tau}}_T \quad \forall T \in \mathcal{T}_h. $$

### 3.3. $L^2$-products and norms

The discrete $L^2$-products in $\underline{\mathbf{V}}_h^k$ and $\underline{\boldsymbol{\Sigma}}_h^{k-1}$ are defined setting: For all $\underline{\mathbf{w}}_h$, $\underline{\mathbf{v}}_h \in \underline{\mathbf{V}}_h^k$ and all $\underline{\mathbf{v}}_h$, $\underline{\boldsymbol{\tau}}_h \in \underline{\boldsymbol{\Sigma}}_h^{k-1}$,

$$ (\underline{\mathbf{w}}_h, \underline{\mathbf{v}}_h)_{V,h} := \sum_{T \in \mathcal{T}_h} (\underline{\mathbf{w}}_T, \underline{\mathbf{v}}_T)_{V,T}, \quad (\underline{\mathbf{v}}_h, \underline{\boldsymbol{\tau}}_h)_{\underline{\boldsymbol{\Sigma}},h} := \sum_{T \in \mathcal{T}_h} (\underline{\mathbf{v}}_T, \underline{\boldsymbol{\tau}}_T)_{\underline{\boldsymbol{\Sigma}},T}, $$

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

$$ (\underline{\mathbf{w}}_T, \underline{\mathbf{v}}_T)_{V,T} := \int_T \mathbf{P}_{V,T}^k \underline{\mathbf{w}}_T \cdot \mathbf{P}_{V,T}^k \underline{\mathbf{v}}_T + s_{V,T} (\underline{\mathbf{w}}_T, \underline{\mathbf{v}}_T), \quad (21) $$

$$ (\underline{\mathbf{v}}_T, \underline{\boldsymbol{\tau}}_T)_{\underline{\boldsymbol{\Sigma}},T} := \int_T \mathbf{P}_{\underline{\boldsymbol{\Sigma}},T}^{k-1} \underline{\mathbf{v}}_T : \mathbf{P}_{\underline{\boldsymbol{\Sigma}},T}^{k-1} \underline{\boldsymbol{\tau}}_T + s_{\underline{\boldsymbol{\Sigma}},T} (\underline{\mathbf{v}}_T, \underline{\boldsymbol{\tau}}_T). \quad (22) $$

Above, $s_{V,T} : \underline{\mathbf{V}}_T^k \times \underline{\mathbf{V}}_T^k \to \mathbb{R}$ and $s_{\underline{\boldsymbol{\Sigma}},T} : \underline{\boldsymbol{\Sigma}}_T^{k-1} \times \underline{\boldsymbol{\Sigma}}_T^{k-1} \to \mathbb{R}$ are local stabilisation bilinear forms. We refer to [11, Section 4.2] for the precise expression of $s_{\underline{\boldsymbol{\Sigma}},T}$ and we set

$$ s_{V,T}(\underline{\mathbf{w}}_T, \underline{\mathbf{v}}_T) := h_T \sum_{E \in \mathcal{E}_T} \int_E (\mathbf{P}_{V,T}^k \underline{\mathbf{w}}_T - \mathbf{w}_{\mathcal{E}_T}) \cdot (\mathbf{P}_{V,T}^k \underline{\mathbf{v}}_T - \mathbf{v}_{\mathcal{E}_T}). \quad (23) $$
By (16), this stabilisation bilinear form satisfies the following polynomial consistency property:

$$s_{V,T}(\underline{\mathbf{I}}_{V,T}^k \mathbf{w}, \underline{\mathbf{v}}_T) = 0 \quad \forall (\mathbf{w}, \underline{\mathbf{v}}_T) \in \mathcal{P}^k(T; \mathbb{R}^2) \times \underline{\mathbf{V}}_T^k,$$

so that

$$(\underline{\mathbf{I}}_{V,T}^k \mathbf{w}, \underline{\mathbf{v}}_T)_{V,T} = \int_T \mathbf{w} \cdot \mathbf{P}_{V,T}^k \underline{\mathbf{v}}_T \quad \forall (\mathbf{w}, \underline{\mathbf{v}}_T) \in \mathcal{P}^k(T; \mathbb{R}^2) \times \underline{\mathbf{V}}_T^k. \quad (24)$$

We define the following $L^2$-product norms: For $\bullet \in \mathcal{T}_h \cup \{h\}$ and all $(\underline{\mathbf{v}}_\bullet, \underline{\mathbf{r}}_\bullet) \in \underline{\mathbf{V}}_\bullet^k \times \underline{\mathbf{\Sigma}}_\bullet^{k-1}$,

$$\|\underline{\mathbf{v}}_\bullet\|_{V,\bullet} := (\underline{\mathbf{v}}_\bullet, \underline{\mathbf{v}}_\bullet)_{V,\bullet}^{1/2}, \quad \|\underline{\mathbf{r}}_\bullet\|_{\underline{\mathbf{\Sigma}}_\bullet} := (\underline{\mathbf{r}}_\bullet, \underline{\mathbf{r}}_\bullet)_{\underline{\mathbf{\Sigma}}_\bullet}^{1/2}. \quad (25)$$

Given $T \in \mathcal{T}_h$, we also define the local component norms $\|\cdot\|_{V,T}$ on $\underline{\mathbf{V}}_T^k$ and $\|\cdot\|_{\underline{\mathbf{\Sigma}}_T}$ on $\underline{\mathbf{\Sigma}}_T^{k-1}$ such that, for all $(\underline{\mathbf{v}}_T, \underline{\mathbf{r}}_T) \in \underline{\mathbf{V}}_T^k \times \underline{\mathbf{\Sigma}}_T^{k-1}$,

$$\|\underline{\mathbf{v}}_T\|_{V,T}^2 := \|\mathbf{v}_T\|_{L^2(T;\mathbb{R}^2)}^2 + \sum_{E \in \mathcal{E}_T} h_T \|\mathbf{v}_E\|_{L^2(E;\mathbb{R}^2)}^2 + \sum_{V \in \mathcal{V}_T} (h_T^2 |\mathbf{v}_V|^2 + h_T^4 |\mathbf{G}_{\mathbf{v},V}|^2), \quad (26)$$

$$\|\underline{\mathbf{r}}_T\|_{\underline{\mathbf{\Sigma}}_T}^2 := \|\underline{\mathbf{r}}_{\mathcal{H},T}\|_{L^2(T;\mathbb{R}^{2\times2})}^2 + \|\underline{\mathbf{r}}_{\mathcal{H},T}^c\|_{L^2(T;\mathbb{R}^{2\times2})}^2 + \sum_{E \in \mathcal{E}_T} (h_T \|\underline{\mathbf{r}}_E\|_{L^2(E)}^2 + h_T^3 \|D_{\underline{\mathbf{r}},E}\|_{L^2(E)}^2) \quad (27) \\ + \sum_{V \in \mathcal{V}_T} h_T^2 |\underline{\mathbf{r}}_V|^2.$$

The corresponding global component norms, respectively denoted by $\|\cdot\|_{V,h}$ and $\|\cdot\|_{\underline{\mathbf{\Sigma}}_h}$, are obtained summing the squares of the local norms on every $T \in \mathcal{T}_h$ and taking the square root of the result. The following equivalences hold uniformly in $h$: For all $\bullet \in \mathcal{T}_h \cup \{h\}$ and all $(\underline{\mathbf{v}}_\bullet, \underline{\mathbf{r}}_\bullet) \in \underline{\mathbf{V}}_\bullet^k \times \underline{\mathbf{\Sigma}}_\bullet^{k-1}$,

$$\|\underline{\mathbf{v}}_\bullet\|_{V,\bullet} \approx \|\underline{\mathbf{v}}_\bullet\|_{V,\bullet}, \quad \|\underline{\mathbf{r}}_\bullet\|_{\underline{\mathbf{\Sigma}}_\bullet} \approx \|\underline{\mathbf{r}}_\bullet\|_{\underline{\mathbf{\Sigma}}_\bullet}. \quad (28)$$

The second equivalence has been proved in [11, Lemma 9]. The first one follows from similar arguments, not detailed here for the sake of conciseness.

For future use, we note the following boundedness properties of the local interpolators, that can be proved using trace inequalities: For all $(\mathbf{v}, \mathbf{r}) \in \mathbf{H}^3(T; \mathbb{R}^2) \times \mathbf{H}^2(T; \mathbb{S})$,

$$\|\underline{\mathbf{I}}_{V,T}^k \mathbf{v}\|_{V,T} \lesssim \|\mathbf{v}\|_{L^2(T;\mathbb{R}^2)} + h_T |\mathbf{v}|_{\mathbf{H}^1(T;\mathbb{R}^2)} + h_T^2 |\mathbf{v}|_{\mathbf{H}^2(T;\mathbb{R}^2)} + h_T^3 |\mathbf{v}|_{\mathbf{H}^3(T;\mathbb{R}^2)} \quad (29a)$$

$$\|\underline{\mathbf{I}}_{\underline{\mathbf{\Sigma}}_T}^{k-1} \mathbf{r}\|_{\underline{\mathbf{\Sigma}}_T} \lesssim \|\underline{\mathbf{r}}\|_{L^2(T;\mathbb{R}^{2\times2})} + h_T |\underline{\mathbf{r}}|_{\mathbf{H}^1(T;\mathbb{R}^{2\times2})} + h_T^2 |\underline{\mathbf{r}}|_{\mathbf{H}^2(T;\mathbb{R}^{2\times2})} \quad (29b)$$

The details of the proof of (29b) are given in [11, Proposition 12].

### 3.4. Poincaré inequalities

The goal of this section is to prove the following result.

**Lemma 3 (Poincaré inequalities).** The following properties hold:

(1) For all $\underline{\mathbf{v}}_h \in \underline{\mathbf{V}}_h^k$ such that

$$\sum_{T \in \mathcal{T}_h} \int_T \mathbf{P}_{V,T}^k \underline{\mathbf{v}}_T \cdot \mathbf{w} = 0 \quad \forall \mathbf{w} \in \mathcal{R}\mathcal{T}^1(\Omega), \quad (30)$$

it holds, with hidden constant independent of $\underline{\mathbf{v}}_h$,

$$\|\underline{\mathbf{v}}_h\|_{V,h} \lesssim \|\underline{\mathbf{C}}_{\text{sym},h}^{k-1} \underline{\mathbf{v}}_h\|_{\underline{\mathbf{\Sigma}}_h}; \quad (31)$$

(2) Denote by $[\cdot, \cdot]_{\underline{\mathbf{\Sigma}}_h}$ an inner product in $\underline{\mathbf{\Sigma}}_h^{k-1}$ with induced norm equivalent to $\|\cdot\|_{\underline{\mathbf{\Sigma}}_h}$ uniformly in $h$. Then, for all $\underline{\mathbf{r}}_h \in \underline{\mathbf{\Sigma}}_h^{k-1}$ such that

$$[\underline{\mathbf{r}}_h, \underline{\mathbf{\eta}}_h]_{\underline{\mathbf{\Sigma}}_h} = 0 \quad \forall \underline{\mathbf{\eta}}_h \in \ker \mathrm{DD}_h^{k-2},$$
it holds, with hidden constant independent of $\underline{\mathbf{\tau}}_h$,

$$ \|\underline{\mathbf{\tau}}_h\|_{\Sigma,h} \lesssim \|DD_h^{k-2} \underline{\mathbf{\tau}}_h\|_{L^2(\Omega)}. \quad (32) $$

**Remark 4 (Poincaré inequality for the symmetric curl).** The condition (30) can be reformulated in terms of the discrete $L^2$-product in $\underline{V}_h^k$ observing that the left-hand side of this expression is in fact equal to $(\underline{v}_h, \underline{I}_{V,h}^k w)_{V,h}$ by (24). Considering a more general product $[\cdot, \cdot]_{V,h}$ then amounts to replacing the $L^2$-product in (30) with a weighted version with positive and uniformly bounded weight function.

### 3.4.1. Preliminary results

This section contains preliminary results required in the proof of Lemma 3.

For all $E \in \mathcal{E}_h$, recalling the estimate of the $L^2$-norm of functions on the unit segment $[0, 1]$ corresponding to the first display equation in the proof of [11, Eq. (58), Proposition 10] and using the isomorphism $[0, 1] \ni s \mapsto x_{V_1} + s(x_{V_2} - x_{V_1})$ (with $V_1$ and $V_2$ respectively denoting the first and second vertices of $E$ in the direction of $t_E$), it is inferred that: For all $m \in \mathbb{N}$ and all $\varphi \in \mathcal{P}^m(E)$,

$$ \|\varphi\|_{L^2(E)} \lesssim \|\pi_{\mathcal{P},E}^{m-2}\varphi\|_{L^2(E)} + h_E^{1/2} \sum_{V \in \mathcal{V}_E} |\varphi(x_V)|. \quad (33) $$

Let now $\underline{v}_{\partial T} \in \underline{V}_{\partial T}^k$ and let $\underline{v}_{\mathcal{E}_T}$ be given by (13). For all $E \in \mathcal{E}_T$, we decompose it into its tangential and normal components as $\underline{v}_{\mathcal{E}_T|E} = v_{n,E}n_E + v_{t,E}t_E$ and, for $\bullet \in \{n, t\}$, we let $v_{\bullet, \partial T} \in \mathcal{P}^k(\mathcal{E}_T)$ be such that $(v_{\bullet, \partial T})_{|E} = \omega_{TE}v_{\bullet,E}$ for all $E \in \mathcal{E}_T$. We additionally denote by $\partial_{t_{\partial T}}$ the piecewise tangential derivative on $\partial T$ such that $(\partial_{t_{\partial T}})_{|E} := \omega_{TE}\partial_{t_E}$ for all $E \in \mathcal{E}_T$.

**Proposition 5 (Estimate of the tangential derivative of the boundary reconstruction).** Let $T \in \mathcal{T}_h$, $\underline{v}_{\partial T} \in \underline{V}_{\partial T}^k$, and $\underline{v}_{\mathcal{E}_T}$ given by (13) be such that $\int_{\partial T} \partial_{t_{\partial T}} \underline{v}_{t, \partial T} = 0$. Then,

$$ \|\partial_{t_{\partial T}} \underline{v}_{\mathcal{E}_T}\|_{L^2(\partial T; \mathbb{R}^2)} \lesssim \|\pi_{\mathcal{P},\mathcal{E}_T}^{k-3}(\partial_{t_{\partial T}} \underline{v}_{n,\partial T})\|_{L^2(\partial T)} + h_T \|\partial_{t_{\partial T}}^2 \underline{v}_{t,\partial T}\|_{L^2(\partial T)} + h_T^{1/2} \sum_{V \in \mathcal{V}_T} |\mathcal{C}\underline{G}_{v,V}|, \quad (34) $$

where $\pi_{\mathcal{P},\mathcal{E}_T}^{k-3}$ denotes the $L^2$-orthogonal projector on $\mathcal{P}^{k-3}(\mathcal{E}_T)$.

**Proof.** Denote, for the sake of brevity, by $\mathcal{N}_{\partial T}(\underline{v}_{\mathcal{E}_T})$ the quantity in the right-hand side of (34). We start using a triangle inequality along with Hölder inequalities and the fact that $t_E$ and $n_E$ are unit vectors to write

$$ \|\partial_{t_{\partial T}} \underline{v}_{\mathcal{E}_T}\|_{L^2(\partial T; \mathbb{R}^2)} \lesssim \|\partial_{t_{\partial T}} \underline{v}_{n,\partial T}\|_{L^2(\partial T)} + \|\partial_{t_{\partial T}} \underline{v}_{t,\partial T}\|_{L^2(\partial T)} := \mathcal{T}_1 + \mathcal{T}_2. \quad (35) $$

By (33) applied to each $E \in \mathcal{E}_T$ with $(\varphi, m) = (\partial_{t_E} \underline{v}_{n,E}, k-1)$, it is readily inferred that

$$ \mathcal{T}_1 \lesssim \|\pi_{\mathcal{P},\mathcal{E}_T}^{k-3}(\partial_{t_{\partial T}} \underline{v}_{n,\partial T})\|_{L^2(\partial T)} + h_T^{1/2} \sum_{E \in \mathcal{E}_T} \sum_{V \in \mathcal{V}_E} |\partial_{t_E} \underline{v}_{n,E}(x_V)| \lesssim \mathcal{N}_{\partial T}(\underline{v}_{\mathcal{E}_T}), \quad (36) $$

where the conclusion follows noticing that, for all $E \in \mathcal{E}_T$ and all $V \in \mathcal{V}_E$, $|\partial_{t_E} \underline{v}_{n,E}(x_V)| = |\underline{G}_{v,V}t_E \cdot n_E| = |\mathcal{C}\underline{G}_{v,V}n_E \cdot n_E| \lesssim |\mathcal{C}\underline{G}_{v,V}|$ and using card$(\mathcal{E}_T) \lesssim 1$.

Let us now turn to $\mathcal{T}_2$. Let $\varphi \in \mathcal{P}^k(\mathcal{E}_T)$ be such that $\int_{\partial T} \varphi = 0$. For all $V \in \mathcal{V}_T$ shared by the edges $E_1, E_2 \in \mathcal{E}_T$ numbered so that $E_2$ follows $E_1$ travelling along $\partial T$ according to its orientation, define the jump $[\varphi]_V := \varphi|_{E_2} - \varphi|_{E_1}$. Then, it holds

$$ \|\varphi\|_{L^2(\partial T)} \lesssim h_T \|\partial_{t_{\partial T}} \varphi\|_{L^2(\partial T)} + h_T^{1/2} \sum_{V \in \mathcal{V}_T} |[\varphi]_V|. \quad (37) $$

Apply this inequality to $\varphi = \partial_{t_{\partial T}} \underline{v}_{t, \partial T}$ and denote by $\mathcal{T}_{2,1}$ and $\mathcal{T}_{2,2}$ the terms in the right-hand side. Clearly, $\mathcal{T}_{2,1} = \|\partial_{t_{\partial T}}^2 \underline{v}_{t, \partial T}\|_{L^2(\partial T)} \le \mathcal{N}_{\partial T}(\underline{v}_{\mathcal{E}_T})$. For the second contribution, we start by noticing that, for all $E \in \mathcal{E}_T$ and all $V \in \mathcal{V}_E$, $\partial_{t_{\partial T}} \underline{v}_{t, \partial T}(x_V) = -\mathcal{C}\underline{G}_{v,V}n_E \cdot t_E + \frac{1}{2} \text{tr } \underline{G}_{v,V}$ so that, in particular, for all $V \in \mathcal{V}_T$, $[\partial_{t_{\partial T}} \underline{v}_{t, \partial T}]_V = \mathcal{C}\underline{G}_{v,V}n_{E_1} \cdot t_{E_1} - \mathcal{C}\underline{G}_{v,V}n_{E_2} \cdot t_{E_2}$. Using this fact along with card$(\mathcal{E}_T) \lesssim 1$, we conclude that $|\mathcal{T}_{2,2}| \lesssim h_T^{1/2} \sum_{V \in \mathcal{V}_T} |\mathcal{C}\underline{G}_{v,V}| \le \mathcal{N}_{\partial T}(\underline{v}_{\mathcal{E}_T})$. Gathering the above estimates on $\mathcal{T}_{2,1}$ and $\mathcal{T}_{2,2}$ finally gives $\mathcal{T}_2 \lesssim \mathcal{N}_{\partial T}(\underline{v}_{\mathcal{E}_T})$ which, combined with (36), yields (34). $\square$
**Proposition 6 (Estimate of the discrete sym-curl norm of the vector potential).** For all $T \in \mathcal{T}_h$ and all $\underline{v}_T \in \underline{V}_T^k$, it holds

$$ \| \mathrm{sym \ curl} \mathbf{P}_{V,T}^k \underline{v}_T \|_{L^2(T; \mathbb{R}^{2 \times 2})} + \sum_{E \in \mathcal{E}_T} h_T^{-1/2} \| \mathbf{P}_{V,T}^k \underline{v}_T - \underline{v}_{\mathcal{E}_T} \|_{L^2(E; \mathbb{R}^2)} \lesssim \| \underline{\mathbf{C}}_{\mathrm{sym},T}^{k-1} \underline{v}_T \|_{\underline{\mathbf{\Sigma}},T}. \quad (38) $$

**Proof.** Recalling Remark 2 to write (15) for $\tau \in \mathcal{P}^{k-1}(T; \mathcal{S})$ and using the integration by parts formula (11) for the left-hand side of the resulting expression, we have

$$ \int_T \mathrm{sym \ curl} \mathbf{P}_{V,T}^k \underline{v}_T : \tau = \int_T \mathbf{C}_{\mathrm{sym},T}^{k-1} \underline{v}_T : \tau + \sum_{E \in \mathcal{E}_T} \omega_{TE} \int_E (\mathbf{P}_{V,T}^k \underline{v}_T - \underline{v}_{\mathcal{E}_T}) \cdot (\tau \, t_E). $$

Taking $\tau = \mathrm{sym \ curl} \mathbf{P}_{V,T}^k \underline{v}_T$, using Cauchy-Schwarz and discrete trace inequalities in the right-hand side, and simplifying, we infer that

$$ \begin{aligned} \| \mathrm{sym \ curl} \mathbf{P}_{V,T}^k \underline{v}_T \|_{L^2(T; \mathbb{R}^{2 \times 2})} &\lesssim \| \mathbf{C}_{\mathrm{sym},T}^{k-1} \underline{v}_T \|_{L^2(T; \mathbb{R}^{2 \times 2})} + \sum_{E \in \mathcal{E}_T} h_T^{-1/2} \| \mathbf{P}_{V,T}^k \underline{v}_T - \underline{v}_{\mathcal{E}_T} \|_{L^2(E; \mathbb{R}^2)} \\ &=: \mathcal{T}_1 + \mathcal{T}_2. \end{aligned} \quad (39) $$

We proceed to estimate the terms in the right-hand side.

(i) *Estimate of* $\mathcal{T}_1$. Using, in this order, (20), the definitions (25) of $\| \cdot \|_{\underline{\mathbf{\Sigma}},T}$ and (22) of the discrete $L^2$-product in $\underline{\mathcal{S}}_T^{k-1}$, and the norm equivalence (28), we can write: For all $\underline{v}_T \in \underline{V}_T^k$,

$$ \mathcal{T}_1 = \| \mathbf{P}_{\underline{\mathbf{\Sigma}},T}^{k-1} \underline{\mathbf{C}}_{\mathrm{sym},T}^{k-1} \underline{v}_T \|_{L^2(T; \mathbb{R}^{2 \times 2})} \le \| \underline{\mathbf{C}}_{\mathrm{sym},T}^{k-1} \underline{v}_T \|_{\underline{\mathbf{\Sigma}},T} \lesssim \| \underline{\mathbf{C}}_{\mathrm{sym},T}^{k-1} \underline{v}_T \|_{\underline{\mathbf{\Sigma}},T}. \quad (40) $$

(ii) *Estimate of* $\mathcal{T}_2$. Let $w \in \mathcal{R}\mathcal{T}^1(T)$ be such that

$$ \int_{\partial T} w = \int_{\partial T} \underline{v}_{\mathcal{E}_T} \quad \text{and} \quad \int_{\partial T} \partial_{t_{\partial T}} w_{t,\partial T} = \int_{\partial T} \partial_{t_{\partial T}} \underline{v}_{t_{\partial T}}. \quad (41) $$

To check that it is possible to match these conditions, write $w(x) = z + (x - x_{\partial T})q$ with $z \in \mathbb{R}^2$, $q \in \mathbb{R}$, and $x_{\partial T} := \frac{1}{|\partial T|} \int_{\partial T} x$, and notice that the first condition in (41) yields $z = \frac{1}{|\partial T|} \int_{\partial T} \underline{v}_{\mathcal{E}_T}$, while the second one is fulfilled taking $q = \frac{1}{|\partial T|} \int_{\partial T} \partial_{t_{\partial T}} \underline{v}_{t_{\partial T}}$.

Using triangle inequalities, we have

$$ \mathcal{T}_2 \lesssim \sum_{E \in \mathcal{E}_T} h_T^{-1/2} \| \underline{w} - \underline{v}_{\mathcal{E}_T} \|_{L^2(E; \mathbb{R}^2)} + \sum_{E \in \mathcal{E}_T} h_T^{-1/2} \| \mathbf{P}_{V,T}^k \underline{v}_T - \underline{w} \|_{L^2(E; \mathbb{R}^2)} =: \mathcal{T}_{2,1} + \mathcal{T}_{2,2}. $$

Noticing that $w - \underline{v}_{\mathcal{E}_T} \in \mathcal{C}^0(\partial T; \mathbb{R}^2)$ has zero average on $\partial T$, applying a Poincaré-Wiretiger inequality on $\partial T$ as in [16, Lemma 15], and concluding with Proposition 5 gives

$$ \mathcal{T}_{2,1} \lesssim h_T^{1/2} \| \partial_{t_{\partial T}} (\underline{v}_{\mathcal{E}_T} - \underline{w}) \|_{L^2(\partial T; \mathbb{R}^2)} \lesssim \| \underline{\mathbf{C}}_{\mathrm{sym},T}^{k-1} (\underline{v}_T - \underline{I}_{V,T}^k \underline{w}) \|_{\underline{\mathbf{\Sigma}},T} = \| \underline{\mathbf{C}}_{\mathrm{sym},T}^{k-1} \underline{v}_T \|_{\underline{\mathbf{\Sigma}},T}, \quad (42) $$

where, in the second step, we have additionally used the consistency of the boundary reconstruction (13) applied to $\underline{I}_{V,T}^k w$, while the conclusion follows recalling that $\underline{\mathbf{C}}_{\mathrm{sym},T}^{k-1} \underline{I}_{V,T}^k w = \underline{0}$ by the local complex property for the DDR sequence.

Let us now consider $\mathcal{T}_{2,2}$. By polynomial consistency (16) of $\mathbf{P}_{V,T}^k$, it holds $\mathbf{P}_{V,T}^k \underline{I}_{V,T}^k w = w$, hence

$$ \mathcal{T}_{2,2} = \sum_{E \in \mathcal{E}_T} h_T^{-1/2} \| \mathbf{P}_{V,T}^k (\underline{v}_T - \underline{I}_{V,T}^k \underline{w}) \|_{L^2(E; \mathbb{R}^2)} \lesssim h_T^{-1} \| \mathbf{P}_{V,T}^k (\underline{v}_T - \underline{I}_{V,T}^k \underline{w}) \|_{L^2(T; \mathbb{R}^2)}, \quad (43) $$

where the conclusion follows from discrete trace inequalities along with $\mathrm{card}(\mathcal{E}_T) \lesssim 1$. Taking, in the definition (15) of $\mathbf{P}_{V,T}^k$, $\tau \in \mathcal{H}^{c,k+1}(T)$ such that $\mathrm{rot} \, \tau = \mathbf{P}_{V,T}^k (\underline{v}_T - \underline{I}_{V,T}^k \underline{w})$ (this is possible since $\mathrm{rot} : \mathcal{H}^{c,k+1}(T) \to \mathcal{P}^k(T; \mathbb{R}^2)$ is surjective by Proposition 1) and using Cauchy-Schwarz and discrete trace inequalities in the right-hand side along with (5) to write $\| \tau \|_{L^2(T; \mathbb{R}^{2 \times 2})} \lesssim h_T \| \mathrm{rot} \, \tau \|_{L^2(T; \mathbb{R}^2)} = h_T \| \mathbf{P}_{V,T}^k (\underline{v}_T - \underline{I}_{V,T}^k \underline{w}) \|_{L^2(T; \mathbb{R}^2)}$, we obtain, after simplification,
$$
\begin{aligned}
\| \mathbf{P}_{V,T}^{k} \left( \underline{\mathbf{v}}_{T} - \underline{\mathbf{I}}_{V,T}^{k} \mathbf{w} \right) \|_{L^2(T; \mathbb{R}^2)} & \lesssim h_T \left( \| \mathbf{C}_{\text{sym},T}^{k-1} \underline{\mathbf{v}}_{T} \|_{L^2(T; \mathbb{R}^{2 \times 2})} + h_T^{-1/2} \| \mathbf{v}_{\mathcal{E}_T} - \mathbf{w} \|_{L^2(\partial T; \mathbb{R}^2)} \right) \\
& \lesssim h_T \| \mathbf{C}_{\text{sym},T}^{k-1} \underline{\mathbf{v}}_{T} \|_{\Sigma, T'}
\end{aligned}
$$

where the conclusion follows using, respectively, (40) and (42) to estimate the terms in parentheses. Plugging this estimate into (43), we finally get $\mathfrak{T}_{2,2} \lesssim \|\underline{\mathbf{C}}_{\text{sym},T}^{k-1} \underline{\mathbf{v}}_T\|_{\Sigma,T}$, which, combined with (42), gives

$$ \mathfrak{T}_2 \lesssim \| \underline{\mathbf{C}}_{\text{sym},T}^{k-1} \underline{\mathbf{v}}_T \|_{\Sigma,T}. \quad (44) $$

(iii) **Conclusion.** Plug (40) and (44) into (39) to estimate the first term in the left-hand side of (38) and notice that the estimate of the second term in the left-hand side of (38) is precisely (44). $\square$

### 3.4.2. *Proof of the discrete Poincaré inequalities*

**Proof of Lemma 3.**

(i) **Poincaré inequality** (31) for $\underline{\mathbf{C}}_{\text{sym},h}^{k-1}$.

Let $\underline{\mathbf{v}}_h \in \underline{\mathbf{v}}_h^k$ be such that $\int_{\Omega} \mathbf{P}_{V,h}^k \underline{\mathbf{v}}_h \cdot \mathbf{w} = 0$ for all $\mathbf{w} \in \mathcal{R}\mathcal{T}^{-1}(\Omega)$, with the global reconstruction operator $\mathbf{P}_{V,h}^k$ defined such that $(\mathbf{P}_{V,h}^k \underline{\mathbf{v}}_h)|_{T} = \mathbf{P}_{V,T}^k \underline{\mathbf{v}}_T$ for all $T \in \mathcal{T}_h$. Owing to the uniform norm equivalence (28), the definitions (25) of the $\|\cdot\|_{V,T}$-norm and (23) of the stabilisation bilinear form, and the fact that $h_T \lesssim 1$ for all $T \in \mathcal{T}_h$, we infer

$$
\begin{aligned}
\|\underline{\mathbf{v}}_h\|_{V,h}^2 \lesssim \|\underline{\mathbf{v}}_h\|_{V,h}^2 &= \|\mathbf{P}_{V,h}^k \underline{\mathbf{v}}_h\|_{L^2(\Omega; \mathbb{R}^2)}^2 + \sum_{T \in \mathcal{T}_h} s_{V,T}(\underline{\mathbf{v}}_T, \underline{\mathbf{v}}_T) \\
&\lesssim \|\mathbf{P}_{V,h}^k \underline{\mathbf{v}}_h\|_{L^2(\Omega; \mathbb{R}^2)}^2 + \sum_{T \in \mathcal{T}_h} \sum_{E \in \mathcal{E}_T} h_T^{-1} \|\mathbf{P}_{V,T}^k \underline{\mathbf{v}}_T - \mathbf{v}_{\mathcal{E}_T}\|_{L^2(E; \mathbb{R}^2)}^2.
\end{aligned}
\quad (45)
$$

We notice that, for all neighboring elements $T_1, T_2 \in \mathcal{T}_h$ sharing the internal edge $E$, we have $(\mathbf{v}_{\mathcal{E}_{T_1}})|_E = (\mathbf{v}_{\mathcal{E}_{T_2}})|_E := \hat{\mathbf{v}}_E$. Letting, for any boundary edge $E \subset \partial T \cap \partial \Omega$, $\hat{\mathbf{v}}_E := (\mathbf{v}_{\mathcal{E}_T})|_E$ and applying the second inequality of Proposition 27 below to the hybrid vector field $\underline{\mathbf{u}}_h = ((\mathbf{P}_{V,T}^k \underline{\mathbf{v}}_T)|_{T \in \mathcal{T}_h}, (\hat{\mathbf{v}}_E)|_{E \in \mathcal{E}_h})$, we obtain

$$
\begin{aligned}
\| \mathbf{P}_{V,h}^k \underline{\mathbf{v}}_h \|_{L^2(\Omega; \mathbb{R}^2)}^2 & \lesssim \sum_{T \in \mathcal{T}_h} \left( \| \operatorname{sym curl} \mathbf{P}_{V,T}^k \underline{\mathbf{v}}_T \|_{L^2(\Omega; \mathbb{R}^{2 \times 2})}^2 + \sum_{E \in \mathcal{E}_T} h_T^{-1} \| \mathbf{P}_{V,T}^k \underline{\mathbf{v}}_T - \mathbf{v}_{\mathcal{E}_T} \|_{L^2(E; \mathbb{R}^2)}^2 \right).
\end{aligned}
$$

Plugging the previous bound into (45) and using Proposition 6, (31) follows.

(ii) **Poincaré inequality** (32) for $DD_h^{k-2}$.

Let $\underline{\mathbf{v}}_h \in (\ker DD_h^{k-2})^{\perp}$, where $(\ker DD_h^{k-2})^{\perp} \subset \underline{\mathbf{\Sigma}}_h^{k-1}$ denotes the orthogonal of $\ker DD_h^{k-2}$ with respect to the inner product $[\cdot, \cdot]_{\underline{\mathbf{\Sigma}}_h}$. Owing to the surjectivity of the operator $\operatorname{div} \operatorname{div} : \mathbf{H}^2(\Omega; \mathcal{S}) \to L^2(\Omega)$ (cf. [24, Theorem 3.25]), the commutation property stated in [11, Eq. (19)], and the boundedness of the global interpolator $\underline{\mathbf{I}}_{\underline{\mathbf{\Sigma}}_h}^{k-1}$ resulting from (29b), we infer the existence of a tensor field $\underline{\mathbf{\tau}} \in \mathbf{H}^2(\Omega; \mathcal{S})$ such that

$$
DD_h^{k-2} \underline{\mathbf{\tau}}_h = \operatorname{div} \operatorname{div} \underline{\mathbf{\tau}} = DD_h^{k-2} (\underline{\mathbf{I}}_{\underline{\mathbf{\Sigma}}_h}^{k-1} \underline{\mathbf{\tau}}) \\
\text{and} \quad \left\| \underline{\mathbf{I}}_{\underline{\mathbf{\Sigma}}_h}^{k-1} \underline{\mathbf{\tau}} \right\|_{\underline{\mathbf{\Sigma}}_h} \lesssim \| \underline{\mathbf{\tau}} \|_{\mathbf{H}^2(\Omega; \mathbb{R}^{2 \times 2})} \lesssim \| DD_h^{k-2} \underline{\mathbf{\tau}}_h \|_{L^2(\Omega)} \quad (46)
$$

Therefore, we have that $\underline{\mathbf{\tau}}_h - \underline{\mathbf{I}}_{\underline{\mathbf{\Sigma}}_h}^{k-1} \underline{\mathbf{\tau}} \in \ker DD_h^{k-2}$, i.e.,

$$ \left[ \underline{\mathbf{\tau}}_h - \underline{\mathbf{I}}_{\underline{\mathbf{\Sigma}}_h}^{k-1} \underline{\mathbf{\tau}}, \underline{\mathbf{v}}_h \right]_{\underline{\mathbf{\Sigma}}_h} = 0 \quad \forall \underline{\mathbf{v}}_h \in (\ker DD_h^{k-2})^{\perp}, $$
namely, $\underline{\tau}_h$ can be regarded as the $[\cdot, \cdot]_{\Sigma,h}$-orthogonal projection of $\underline{I}_{\Sigma,h}^{k-1}\tau$ on $(\ker DD_h^{k-2})^{\perp}$. Thus, the norm induced by $[\cdot, \cdot]_{\Sigma,h}$ of $\underline{\tau}_h$ is bounded by that of $\underline{I}_{\Sigma,h}^{k-1}\tau$, and the assumed uniform equivalence between the induced norm and $\|\cdot\|_{\Sigma,h}$ along with the inequality in (46) yields the result. $\square$

### 3.5. Consistency of the discrete $L^2$-products

**Lemma 7 (Consistency of the discrete $L^2$-products).** The discrete $L^2$-products satisfy the following consistency properties:

(1) For all $\mathbf{w} \in \mathbf{H}^3(\Omega; \mathbb{R}^2)$, define the linear form $\mathfrak{E}_{V,h}(\mathbf{w}; \cdot) : \underline{V}_h^k \to \mathbb{R}$ such that

$$ \mathfrak{E}_{V,h}(\mathbf{w}; \underline{\mathbf{v}}_h) := \sum_{T \in \mathcal{T}_h} \int_T \mathbf{w} \cdot \mathbf{P}_{V,T}^k \underline{\mathbf{v}}_T - (\underline{I}_{V,h}^k \mathbf{w}, \underline{\mathbf{v}}_h)_{V,h} \quad \forall \underline{\mathbf{v}}_h \in \underline{V}_h^k. $$

Then, under the additional regularity $\mathbf{w} \in \mathbf{H}^{k+1}(\mathcal{T}_h; \mathbb{R}^2)$, it holds

$$ \sup_{\underline{\mathbf{v}}_h \in \underline{V}_h^k \setminus \{\underline{0}\}} \frac{\mathfrak{E}_{V,h}(\mathbf{w}; \underline{\mathbf{v}}_h)}{\|\underline{\mathbf{v}}_h\|_{V,h}} \lesssim h^{k+1} |\mathbf{v}|_{\mathbf{H}^{k+1}(\mathcal{T}_h; \mathbb{R}^2)}. \quad (47) $$

(2) For all $\mathbf{v} \in \mathbf{H}^2(\Omega; \mathfrak{S})$, define the linear form $\mathfrak{E}_{\Sigma,h}(\mathbf{v}; \cdot) : \underline{\Sigma}_h^{k-1} \to \mathbb{R}$ such that

$$ \mathfrak{E}_{\Sigma,h}(\mathbf{v}; \underline{\tau}_h) := \sum_{T \in \mathcal{T}_h} \int_T \mathbf{v} : \mathbf{P}_{\Sigma,T}^{k-1} \underline{\tau}_T - (\underline{I}_{\Sigma,h}^{k-1} \mathbf{v}, \underline{\tau}_h)_{\Sigma,h} \quad \forall \underline{\tau}_h \in \underline{\Sigma}_h^{k-1}. $$

Then, under the additional regularity $\mathbf{v} \in \mathbf{H}^k(\mathcal{T}_h; \mathfrak{S})$, it holds

$$ \sup_{\underline{\tau}_h \in \underline{\Sigma}_h^{k-1} \setminus \{\underline{0}\}} \frac{\mathfrak{E}_{\Sigma,h}(\mathbf{v}; \underline{\tau}_h)}{\|\underline{\tau}_h\|_{\Sigma,h}} \lesssim h^k |\mathbf{v}|_{\mathbf{H}^k(\mathcal{T}_h; \mathbb{R}^{2 \times 2})}. \quad (48) $$

**Proof.**

(i) *Proof of (47).* By the polynomial consistency (24) of the discrete $L^2$-product in $\underline{V}_T^k$, we can write

$$ \mathfrak{E}_{V,h}(\mathbf{w}; \underline{\mathbf{v}}_h) = \sum_{T \in \mathcal{T}_h} (\tilde{\mathfrak{F}}_1(T) + \tilde{\mathfrak{F}}_2(T)), \quad (49) $$

where, recalling that $\underline{\pi}_{\mathcal{P},T}^k$ denotes the $L^2$-orthogonal projector on $\mathcal{P}^k(T; \mathbb{R}^2)$,

$$ \tilde{\mathfrak{F}}_1(T) := \int_T (\mathbf{w} - \underline{\pi}_{\mathcal{P},T}^k \mathbf{w}_{|T}) \cdot \mathbf{P}_{V,T}^k \underline{\mathbf{v}}_T, \quad \tilde{\mathfrak{F}}_2(T) := (\underline{I}_{V,T}^k (\mathbf{w} - \underline{\pi}_{\mathcal{P},T}^k \mathbf{w}_{|T}), \underline{\mathbf{v}}_T)_{V,T}. $$

For the first term, a Cauchy–Schwarz inequality followed by the approximation properties of $\underline{\pi}_{\mathcal{P},T}^k$ (see, e.g., [25, Lemma 3.4] or [21, Section 1.3.3]) and the definition (25) of the $\|\cdot\|_{V,T}$-norm give

$$ |\tilde{\mathfrak{F}}_1(T)| \le \|\mathbf{w} - \underline{\pi}_{\mathcal{P},T}^k \mathbf{w}_{|T}\|_{L^2(T;\mathbb{R}^2)} \|\mathbf{P}_{V,T}^k \underline{\mathbf{v}}_T\|_{L^2(T;\mathbb{R}^2)} \lesssim h_T^{k+1} |\mathbf{w}|_{\mathbf{H}^{k+1}(T;\mathbb{R}^2)} \|\underline{\mathbf{v}}_T\|_{V,T}. \quad (50) $$

For the second term, a Cauchy–Schwarz inequality, the local norm equivalence expressed by (28) with $\cdot = T$ along with the boundedness (29a) of $\underline{I}_{V,T}^k$, and again the approximation properties of $\underline{\pi}_{\mathcal{P},T}^k$ give

$$ \begin{align} |\tilde{\mathfrak{F}}_2(T)| &\le \|\underline{I}_{V,T}^k (\mathbf{w} - \underline{\pi}_{\mathcal{P},T}^k \mathbf{w}_{|T})\|_{V,T} \|\underline{\mathbf{v}}_T\|_{V,T} \nonumber \\ &\lesssim \left(\sum_{i=0}^3 h_T^i |\mathbf{w} - \underline{\pi}_{\mathcal{P},T}^k \mathbf{w}_{|T}|_{H^1(T;\mathbb{R}^2)}\right) \|\underline{\mathbf{v}}_T\|_{V,T} \tag{51} \\ &\lesssim h_T^{k+1} |\mathbf{w}|_{\mathbf{H}^{k+1}(T;\mathbb{R}^2)} \|\underline{\mathbf{v}}_T\|_{V,T}. \nonumber \end{align} $$

Using (50) and (51) to bound the terms in the right-hand side of (49), we obtain (47) after applying a discrete Cauchy–Schwarz inequality on the sum over $T \in \mathcal{T}_h$.
(ii) *Proof of (48)*. The proof coincides with the estimate the term $\mathfrak{T}_1$ in the proof of [16, Lemma 15] with $\ell = k-1$, to which we refer for further details. $\square$

### 3.6. Adjoint consistency of the discrete differential operators

Adjoint consistency is a notion relevant in the context of nonconforming methods, and measures the failure to satisfy a global integration by parts formula. In the analysis of numerical schemes, such integration by parts formulas are typically the ones used to derive the weak formulation. To state the following theorem, in which adjoint consistency errors are estimated, we denote by **n** the normal vector field on $\partial\Omega$ pointing out of $\Omega$ and by **t** the tangent vector field oriented so that $(\mathbf{t}, \mathbf{n})$ forms a right-handed coordinate system.

**Lemma 8 (Adjoint consistency).** The discrete differential operators defined in Section 3.2 satisfy the following adjoint consistency properties:

(1) Given $\mathbf{v} \in H^2(\Omega; \mathfrak{S})$ such that $\mathbf{v}\mathbf{t} = \mathbf{0}$ on $\partial\Omega$, we define the sym curl adjoint consistency error $\mathcal{E}_{\text{symcurl},h} : \underline{\mathbf{V}}_h^k \to \mathbb{R}$ by: For all $\underline{\mathbf{v}}_h \in \underline{\mathbf{V}}_h^k$,

$$ \mathcal{E}_{\text{symcurl},h}(\mathbf{v}; \underline{\mathbf{v}}_h) := (\underline{\mathbf{I}}_{\Sigma,h}^{k-1} \mathbf{v}, \underline{\mathbf{C}}_{\text{sym},h}^{k-1} \underline{\mathbf{v}}_h)_{\Sigma,h} + \sum_{T \in \mathcal{T}_h} \int_T \mathbf{rot} \mathbf{v} \cdot \mathbf{P}_{V,T}^k \underline{\mathbf{v}}_T. \quad (52) $$

Then, further assuming $\mathbf{v} \in H^k(\mathcal{T}_h; \mathfrak{S})$, it holds: For all $\underline{\mathbf{v}}_h \in \underline{\mathbf{V}}_h^k$,

$$ |\mathcal{E}_{\text{symcurl},h}(\mathbf{v}; \underline{\mathbf{v}}_h)| \lesssim h^k |\mathbf{v}|_{H^k(\mathcal{T}_h; \mathbb{R}^{2 \times 2})} \left\| \underline{\mathbf{C}}_{\text{sym},h}^{k-1} \underline{\mathbf{v}}_h \right\|_{\Sigma, h}. \quad (53) $$

(2) Given $q \in H^2(\Omega)$ such that $q = \partial_n q = 0$ on $\partial\Omega$, we define the div-div adjoint consistency error $\mathcal{E}_{\text{divdiv},h} : \underline{\mathbf{\Sigma}}_h^{k-1} \to \mathbb{R}$ by: For all $\underline{\mathbf{\tau}}_h \in \underline{\mathbf{\Sigma}}_h^{k-1}$,

$$ \mathcal{E}_{\text{divdiv},h}(q; \underline{\mathbf{\tau}}_h) := \int_{\Omega} q \mathrm{DD}^{k-2}_h \underline{\mathbf{\tau}}_h - \sum_{T \in \mathcal{T}_h} \int_T \mathbf{hess} q : \mathbf{P}_{\underline{\mathbf{\Sigma}},T}^{k-1} \underline{\mathbf{\tau}}_T. \quad (54) $$

Then, further assuming $q \in H^{k+2}(\Omega)$, it holds: For all $\underline{\mathbf{\tau}}_h \in \underline{\mathbf{\Sigma}}_h^{k-1}$,

$$ |\mathcal{E}_{\text{divdiv},h}(q; \underline{\mathbf{\tau}}_h)| \lesssim h^k |q|_{H^{k+2}(\mathcal{T}_h)} \|\underline{\mathbf{\tau}}_h\|_{\Sigma, h}. \quad (55) $$

**Remark 9 (Interpretation of adjoint consistency errors).** The adjoint consistency errors (52) and (54) respectively measure the failure to satisfy at the discrete level the following global integration by parts formulas (with $L^2$-products replaced by discrete counterparts): For $\mathbf{v} \in H^1(\Omega; \mathfrak{S})$ such that $\mathbf{v}\mathbf{t} = \mathbf{0}$ on $\partial\Omega$ and all $\mathbf{v} \in C^\infty(\bar{\Omega}; \mathbb{R}^2)$,

$$ \int_{\Omega} \mathbf{v} : \mathrm{symcurl} \mathbf{v} + \int_{\Omega} \mathbf{rot} \mathbf{v} \cdot \mathbf{v} = 0 $$

and, for all $q \in H^2(\Omega)$ such that $q = \partial_n q = 0$ on $\partial\Omega$ and all $\mathbf{\tau} \in C^\infty(\bar{\Omega}; \mathfrak{S})$,

$$ \int_{\Omega} q \mathrm{div} \mathrm{div} \mathbf{\tau} - \int_{\Omega} \mathbf{hess} q : \mathbf{\tau} = 0. $$

**Proof of Lemma 8.**

(i) *Proof of (53).* By definition (22) of the local discrete $L^2$-product in $\underline{\mathbf{\Sigma}}_h^{k-1}$ and the commutation property (20), it holds that

$$ \mathcal{E}_{\text{symcurl},h}(\mathbf{v}; \underline{\mathbf{v}}_h) = \sum_{T \in \mathcal{T}_h} \left[ \int_T \mathbf{P}_{\underline{\mathbf{\Sigma}},T}^{k-1} \underline{\mathbf{v}}_T : \mathbf{C}_{\text{sym},T}^{k-1} \underline{\mathbf{v}}_T + s_{\underline{\mathbf{\Sigma}},T} \left( \underline{\mathbf{I}}_{\underline{\mathbf{\Sigma}},T}^{k-1} \underline{\mathbf{v}}_T |_{T} , \underline{\mathbf{C}}_{\text{sym},T}^{k-1} \underline{\mathbf{v}}_T \right) + \int_T \mathbf{rot} \mathbf{v} \cdot \mathbf{P}_{V,T}^k \underline{\mathbf{v}}_T \right]. \quad (56) $$
Accounting for Remark 2, it holds, for all $(\tau_T)_{T \in \mathcal{T}_h} \in \mathcal{X}_{T \in \mathcal{T}_h} \mathcal{P}^{k-1}(T; \mathfrak{S})$,

$$ \sum_{T \in \mathcal{T}_h} \left[ \int_T \mathbf{P}_{\mathbf{V},T}^k \underline{\mathbf{v}}_T \cdot \mathbf{rot} \, \boldsymbol{\tau}_T + \int_T \mathbf{C}_{\text{sym},T}^{k-1} \underline{\mathbf{v}}_T : \boldsymbol{\tau}_T - \sum_{E \in \mathcal{E}_T} \omega_{TE} \int_E \mathbf{v}_{\mathcal{E}_T} \cdot (\boldsymbol{\tau}_T \mathbf{t}_E) \right] = 0. $$

Subtracting this expression from (56), we obtain

$$ \mathcal{E}_{\text{sym curl},h}(\mathbf{v}; \underline{\mathbf{v}}_h) = \sum_{T \in \mathcal{T}_h} \left[ \int_T \left( \mathbf{P}_{\boldsymbol{\Sigma},T}^{k-1} \underline{\mathbf{I}}_{\boldsymbol{\Sigma},T}^{k-1} \mathbf{v}_{|T} - \boldsymbol{\tau}_T \right) : \mathbf{C}_{\text{sym},T}^{k-1} \underline{\mathbf{v}}_T + s_{\boldsymbol{\Sigma},T} \left( \underline{\mathbf{I}}_{\boldsymbol{\Sigma},T}^{k-1} \mathbf{v}_{|T}, \underline{\mathcal{C}}_{\text{sym},T}^{k-1} \underline{\mathbf{v}}_T \right) \right. \\ \left. + \sum_{T \in \mathcal{T}_h} \left[ \int_T \mathbf{rot}(\mathbf{v} - \boldsymbol{\tau}_T) \cdot \mathbf{P}_{\mathbf{V},T}^k \underline{\mathbf{v}}_T + \sum_{E \in \mathcal{E}_T} \omega_{TE} \int_E \mathbf{v}_{\mathcal{E}_T} \cdot (\boldsymbol{\tau}_T - \mathbf{v}) \mathbf{t}_E \right] \right], $$

where we have additionally introduced $\mathbf{v}_{|E} \mathbf{t}_E$ into the boundary term using the fact that this quantity is single-valued if $E$ is an internal edge while it vanishes if $E \subset \partial \Omega$. Applying the integration by parts formula (11) to the third term leads to

$$ \mathcal{E}_{\text{sym curl},h}(\mathbf{v}; \underline{\mathbf{v}}_h) = \sum_{T \in \mathcal{T}_h} \left[ \int_T \left( \mathbf{P}_{\boldsymbol{\Sigma},T}^{k-1} \underline{\mathbf{I}}_{\boldsymbol{\Sigma},T}^{k-1} \mathbf{v}_{|T} - \boldsymbol{\tau}_T \right) \cdot \mathbf{C}_{\text{sym},T}^{k-1} \underline{\mathbf{v}}_T + s_{\boldsymbol{\Sigma},T} \left( \underline{\mathbf{I}}_{\boldsymbol{\Sigma},T}^{k-1} \mathbf{v}_{|T}, \underline{\mathcal{C}}_{\text{sym},T}^{k-1} \underline{\mathbf{v}}_T \right) \right. \\ \left. - \sum_{T \in \mathcal{T}_h} \left[ \int_T (\mathbf{v} - \boldsymbol{\tau}_T) \cdot \mathbf{sym curl} \mathbf{P}_{\mathbf{V},T}^k \underline{\mathbf{v}}_T - \sum_{E \in \mathcal{E}_T} \omega_{TE} \int_E (\mathbf{v}_{\mathcal{E}_T} - \mathbf{P}_{\mathbf{V},T}^k \underline{\mathbf{v}}_T) \cdot (\boldsymbol{\tau}_T - \mathbf{v}) \cdot \mathbf{t}_E \right] \right]. \quad (57) $$

Take now $\boldsymbol{\tau}_T = \boldsymbol{\pi}_{\mathcal{P},T}^{k-1} \mathbf{v}_{|T}$ for all $T \in \mathcal{T}_h$. Using Cauchy-Schwarz inequalities in the right-hand side of (57) followed by the approximation properties of $\mathbf{P}_{\boldsymbol{\Sigma},T}^{k-1}$ and $\boldsymbol{\pi}_{\mathcal{P},T}^{k-1}$ (see, respectively, [11, Proposition 14] and [21, Theorem 1.45]) as well as the consistency property of the stabilisation term proved in [11, Proposition 14], we get

$$ \left| \mathcal{E}_{\text{sym curl},h}(\mathbf{v}; \underline{\mathbf{v}}_h) \right| \\ \lesssim h^k |\mathbf{v}|_{H^k(\mathcal{T}_h; \mathbb{R}^{2 \times 2})} \left[ \sum_{T \in \mathcal{T}_h} \left( \left\| \mathbf{C}_{\text{sym},T}^{k-1} \underline{\mathbf{v}}_T \right\|_{L^2(T; \mathbb{R}^{2 \times 2})}^2 + s_{\boldsymbol{\Sigma},T} \left( \underline{\mathbf{C}}_{\text{sym},T}^{k-1} \underline{\mathbf{v}}_T, \underline{\mathbf{C}}_{\text{sym},T}^{k-1} \underline{\mathbf{v}}_T \right) \right. \right. \\ \left. + \left\| \mathbf{sym curl} \mathbf{P}_{\mathbf{V},T}^k \underline{\mathbf{v}}_T \right\|_{L^2(T; \mathbb{R}^{2 \times 2})}^2 + h_T^{-1} \sum_{E \in \mathcal{E}_T} \left\| \mathbf{v}_{\mathcal{E}_T} - \mathbf{P}_{\mathbf{V},T}^k \underline{\mathbf{v}}_T \right\|_{L^2(E; \mathbb{R}^2)}^2 \right]^{1/2} . $$

Let us consider the factor in square brackets. Using, respectively, (20) along with (40) for the first term, the definition (25) of the $L^2$-product norm on $\boldsymbol{\Sigma}_h^{k-1}$ for the second term, and (38) for the third and fourth terms, this factor is $\lesssim \| \underline{\mathbf{C}}_{\text{sym},h}^{k-1} \underline{\mathbf{v}}_h \|_{\boldsymbol{\Sigma},h}$, thus concluding the proof of (53).

(ii) *Proof of* (55). Combining the definitions (54) of the adjoint consistency error and (19) of the tensor potential, it is inferred that, for all $(q_T)_{T \in \mathcal{T}_h} \in \mathcal{X}_{T \in \mathcal{T}_h} \mathcal{P}^{k+1}(T)$,

$$ \mathcal{E}_{\text{div div},h}(q, \underline{\tau}_h) = \sum_{T \in \mathcal{T}_h} \left[ \int_T (q - q_T) DD_T^{k-2} \underline{\tau}_T - \int_T \mathbf{hess}(q - q_T) : \mathbf{P}_{\boldsymbol{\Sigma},T}^{k-1} \underline{\tau}_T \right] \\ + \sum_{T \in \mathcal{T}_h} \omega_{TE} \sum_{E \in \mathcal{E}_T} \left[ \int_E \mathbf{P}_{\boldsymbol{\Sigma},E}^{k-1} \underline{\mathbf{\tau}}_E \partial_{n_E} (q - q_T) - \int_E D_{\boldsymbol{\tau},E} (q - q_T) \right] \\ + \sum_{T \in \mathcal{T}_h} \sum_{E \in \mathcal{E}_T} \omega_{EV} (\boldsymbol{\tau}_V \mathbf{n}_E \cdot \mathbf{t}_E) (q - q_T) (\mathbf{x}_V), $$

where the insertion of $q$ and $\partial_{n_E} q$ into the boundary integrals is possible since these quantities are continuous at internal edges and vanish on boundary edges. Taking $q_T = \pi_{\mathcal{P},T}^{k+1} q$ for all $T \in \mathcal{T}_h$
and using Cauchy-Schwarz inequalities followed by the approximation properties of $\pi_{\mathcal{P},T}^{k+1}$, it is inferred that

$$ |\mathcal{E}_{\text{div div},h}(q, \underline{\tau}_h)| \lesssim h^k |q|_{H^{k+2}(\mathcal{T}_h)} \left\{ \sum_{T \in \mathcal{T}_h} \left[ h_T^4 \|DD_T^{k-2} \underline{\tau}_T\|_{L^2(T)}^2 + \|\mathbf{P}_{\underline{\Sigma},T}^{k-1} \underline{\tau}_T\|_{L^2(T; \mathbb{R}^{2 \times 2})}^2 \right. \right. \\ \left. \left. + \sum_{E \in \mathcal{E}_T} \left( h_T \|P_{\underline{\Sigma},E}^{k-1} \underline{\tau}_E\|_{L^2(E)}^2 + h_T^3 \|D\tau_E\|_{L^2(E)}^2 + \sum_{V \in \mathcal{V}_E} h_T^2 |\tau_V|^2 \right) \right] \right\}^{\frac{1}{2}}. $$

Using [11, Eq. (57)–(59)] for the first three terms and the definition (27) for the last two, we infer that the quantity in braces is $\lesssim \|\underline{\tau}_T\|_{\underline{\Sigma},T}$, hence $\lesssim \|\underline{\tau}_T\|_{\underline{\Sigma},T}$ by the norm equivalence (28) written for $\bullet = T$, thus concluding the proof of (55). □

## 4. Serendipity DDR complex

In this section, we design the serendipity version of the discrete div-div complex. The construction aims at obtaining a significant reduction in the number of DOFs while preserving the homological and analytical properties of the original complex described in the previous section. Denote, as before, by $k \ge 3$ the polynomial degree of the discrete complex. Following [16], we consider the construction illustrated in the following diagram:

$$ \begin{tikzcd}[column sep=2.8em, row sep=2.8em]
    \mathcal{R}\mathcal{T}^1(\Omega) \arrow[r, "\underline{I}_{V,h}^k"] & \underline{V}_h^k \arrow[r, "\underline{C}_{\text{sym},h}^{k-1}"] & \underline{\Sigma}_h^{k-1} \arrow[r, "DD_h^{k-2}"] & \mathcal{P}^{k-2}(\mathcal{T}_h) \arrow[r, "0"] & 0 \\
    \mathcal{R}\mathcal{T}^1(\Omega) \arrow[r, "\underline{I}_{V,h}^k"] & \widehat{\underline{V}}_h^k \arrow[r, "\underline{C}_{\text{sym},h}^{k-1}"] & \widehat{\underline{\Sigma}}_h^{k-1} \arrow[r, "\overline{DD}_h^{k-2}"] & \mathcal{P}^{k-2}(\mathcal{T}_h) \arrow[r, "0"] & 0,
\end{tikzcd} \quad (58) $$

where, according to [16, Eqs. (2.2) and (2.4)], we have set

$$ \underline{\hat{I}}_{V,h}^k := \underline{\hat{R}}_{V,h} \underline{I}_{V,h}^k, \quad \underline{\hat{C}}_{\text{sym},h}^{k-1} := \underline{\hat{R}}_{\underline{\Sigma},h} \underline{C}_{\text{sym},h}^{k-1} \underline{E}_{V,h}, \quad \overline{\hat{D}}_h^{k-2} := \overline{\hat{D}}_h^{k-2} \underline{E}_{\underline{\Sigma},h}. \quad (59) $$

The purpose of the rest of this section is to:

* provide a precise definition of the extension and reduction operators $\underline{E}_{V,h}$, $\underline{\hat{R}}_{V,h}$, $\underline{E}_{\underline{\Sigma},h}$, $\underline{\hat{R}}_{\underline{\Sigma},h}$, as well as the spaces and operators that appear in the bottom (serendipity) complex;

* prove that the properties of the top complex are inherited by the bottom complex.

This latter point makes the object of Theorems 20 and 24 below, which are therefore the main results of this section.

As most of the developments are local, in what follows we denote by $T \in \mathcal{T}_h$ a generic mesh element without necessarily specifying this fact at each occurrence. As usual, a local version of diagram (58) on $T$ is obtained taking the restriction of the spaces and operators collecting the components attached to $T$ and, when present, to the edges and nodes that lie on its boundary.

### 4.1. Estimate of symmetric tensor-valued polynomials

Throughout the rest of this section, we work under the following assumption:

**Assumption 10 (Boundaries selection for serendipity spaces).** For each $T \in \mathcal{T}_h$ element of the mesh, we select a set $\hat{\mathcal{E}}_T$ of $\eta_T \ge 2$ edges that are not pairwise aligned and such that, for all $E \in \hat{\mathcal{E}}_T$, $T$ lies entirely on one side of the line $H_E$ containing $E$. For all $E \in \hat{\mathcal{E}}_T$, denoting by $\mathbf{x}_E$ its middle point and defining the scaled distance function to $H_E$ by $d_E(\mathbf{x}) = h_E^{-1}\omega_{TE}(\mathbf{x} - \mathbf{x}_E) \cdot \mathbf{n}_E$, we assume the existence of a real number $\theta > 0$ such that $d_E(\mathbf{x}_{E'}) \ge \theta$ for all $E, E' \in \hat{\mathcal{E}}_T, E \ne E'$.
From this point on, the hidden constant in $a \lesssim b$ (see Section 2.2) will possibly depend also on the boundaries selection regularity parameter $\theta$.

**Lemma 11 (Estimate of symmetric tensor-valued polynomials).** Let $m \ge 0$ and let Assumption 10 hold. Let $T \in \mathcal{T}_h$ be a mesh element. Then, for all $\boldsymbol{\tau} \in \mathcal{P}^m(T; \mathcal{S})$, it holds

$$ \|\boldsymbol{\tau}\|_{L^2(T; \mathbb{R}^{2 \times 2})} \lesssim \|\boldsymbol{\pi}_{\mathcal{H},T}^{m-3} \boldsymbol{\tau}\|_{L^2(T; \mathbb{R}^{2 \times 2})} + \|\boldsymbol{\pi}_{\mathcal{H},T}^{c,m+2-\eta_T} \boldsymbol{\tau}\|_{L^2(T; \mathbb{R}^{2 \times 2})} \\ + \sum_{E \in \mathcal{E}_T} \left( h_T^{1/2} \|\boldsymbol{\pi}_{\mathcal{P},E}^{m-2} (\boldsymbol{\tau}|_E \boldsymbol{n}_E \cdot \boldsymbol{n}_E)\|_{L^2(E)} + h_T^{3/2} \|\delta_E \boldsymbol{\tau}\|_{L^2(E)} \right) + h_T \sum_{V \in \mathcal{V}_T} |\boldsymbol{\tau}(\boldsymbol{x}_V)|. \quad (60) $$

**Remark 12 (Reduction by serendipity).** Lemma 11 clearly shows which polynomial components $\sum_{T}^{k-1}$ can be reduced by serendipity, namely the ones in $\mathcal{H}^{c,k-1}(T)$. As it will become clear in what follows, in order to preserve the homological properties, a corresponding reduction of the components of $\underline{v}_T^k$ in $\mathcal{P}^{k-2}(T)$ is required; see Remark 15 below.

**Proof of Lemma 11.** Let $\boldsymbol{\tau} \in \mathcal{P}^m(T; \mathcal{S})$ and denote, for the sake of brevity, by $\mathcal{N}_T(\boldsymbol{\tau})$ the right-hand side of (60). We start by estimating $\|\mathrm{div}\,\mathbf{div}\,\boldsymbol{\tau}\|_{L^2(T)}$. Using the integration by parts formula (17) with $q \in \mathcal{P}^{m-2}(T)$, inserting $\boldsymbol{\pi}_{\mathcal{H},T}^{m-3}$ in front of $\boldsymbol{\tau}$ in the first term in the right-hand side (since hess $q \in \mathcal{H}^{m-4}(T) \subset \mathcal{H}^{m-3}(T)$) and $\boldsymbol{\pi}_{\mathcal{P},E}^{m-2}$ in front of $\boldsymbol{\tau} \boldsymbol{n}_E \cdot \boldsymbol{n}_E$ in the second term (since $\partial_{\boldsymbol{n}_E} q \in \mathcal{P}^{m-3}(E) \subset \mathcal{P}^{m-2}(E)$), and using Cauchy-Schwarz along with discrete trace and inverse inequalities, we infer $\int_T \mathrm{div}\,\mathbf{div}\,\boldsymbol{\tau} \,q \lesssim h_T^{-2} \mathcal{N}_T(\boldsymbol{\tau}) \|q\|_{L^2(T)}$. Taking $q = \mathrm{div}\,\mathrm{div}\,\boldsymbol{\tau}$, simplifying, and multiplying both sides by $h_T^2$ yields

$$ h_T^2 \|\mathrm{div}\,\mathbf{div}\,\boldsymbol{\tau}\|_{L^2(T)} \lesssim \mathcal{N}_T(\boldsymbol{\tau}). \quad (61) $$

By (4), $\boldsymbol{\tau}$ can be decomposed as follows:

$$ \boldsymbol{\tau} = \mathrm{sym}\,\mathrm{curl}\,\boldsymbol{v} + \boldsymbol{v}, \quad (62) $$

with $\boldsymbol{v} \in \mathcal{P}^{m+1}(T; \mathbb{R}^2)$ and $\boldsymbol{v} \in \mathcal{C}^{c,m}(T)$. Since $\boldsymbol{v}$ is defined up to a function in $\mathcal{RT}^1(T)$, we can assume that

$$ \int_{\partial T} \boldsymbol{v} = \mathbf{0} \quad \text{and} \quad \int_{\partial T} \partial_{t_{\partial T}} \boldsymbol{v} t_{\partial T} = 0, \quad (63) $$

where we remind the reader that, as in Section 3.4.1, $\partial_{t_{\partial T}}$ and $v_{t,\partial T}$ are, respectively, the broken tangential derivative and tangential component of $\boldsymbol{v}$ on $\partial T$.

We next proceed to estimate the $L^2$-norms of the terms in the right-hand side of (62). To estimate $\|\boldsymbol{v}\|_{L^2(T; \mathbb{R}^{2 \times 2})}$, we start with (6), notice that $\mathrm{div}\,\mathbf{div}\,\boldsymbol{v} = \mathrm{div}\,\mathrm{div}\,\boldsymbol{\tau}$ (since $\mathrm{div}\,\mathrm{div}\,\mathrm{sym}\,\mathrm{curl} = 0$), then invoke (61) to write

$$ \|\boldsymbol{v}\|_{L^2(T; \mathbb{R}^{2 \times 2})} \lesssim h_T^2 \|\mathrm{div}\,\mathbf{div}\,\boldsymbol{v}\|_{L^2(T)} = h_T^2 \|\mathrm{div}\,\mathbf{div}\,\boldsymbol{\tau}\|_{L^2(T)} \lesssim \mathcal{N}_T(\boldsymbol{\tau}). \quad (64) $$

To estimate $\|\mathrm{sym}\,\mathrm{curl}\,\boldsymbol{v}\|_{L^2(T; \mathbb{R}^{2 \times 2})}$, we start by using a discrete inverse inequality followed by [16, Lemma 13] to write

$$ \|\mathrm{sym}\,\mathrm{curl}\,\boldsymbol{v}\|_{L^2(T; \mathbb{R}^{2 \times 2})} \lesssim h_T^{-1} \|\boldsymbol{v}\|_{L^2(T; \mathbb{R}^2)} \lesssim h_T^{-1} \left( \|\boldsymbol{\pi}_{\mathcal{P},T}^{m+1-\eta_T} \boldsymbol{v}\|_{L^2(T; \mathbb{R}^2)} + h_T^{1/2} \|\boldsymbol{v}\|_{L^2(\partial T; \mathbb{R}^2)} \right). \quad (65) $$

We next proceed to estimate the terms in parentheses, starting with $\|\boldsymbol{v}\|_{L^2(\partial T; \mathbb{R}^2)}$. Since $\boldsymbol{v}$ has zero average on $\partial T$, by a Poincaré-Wireinger inequality on $\partial T$ we infer

$$ \|\boldsymbol{v}\|_{L^2(\partial T; \mathbb{R}^2)} \lesssim h_T \|\partial_{t_{\partial T}} \boldsymbol{v}\|_{L^2(\partial T; \mathbb{R}^2)}. \quad (66) $$

Decomposing $\boldsymbol{v}_{\partial T}$ into its normal and tangential components, and using triangle and Hölder inequalities along with the fact that $\boldsymbol{n}_E$ and $\boldsymbol{t}_E$ are unit vectors, we get

$$ \|\partial_{t_{\partial T}} \boldsymbol{v}\|_{L^2(\partial T; \mathbb{R}^2)} \le \|\partial_{t_{\partial T}} \boldsymbol{v}_{\boldsymbol{n},\partial T}\|_{L^2(\partial T)} + \|\partial_{t_{\partial T}} \boldsymbol{v}_{\boldsymbol{t},\partial T}\|_{L^2(\partial T)}. \quad (67) $$
Since, for all $E \in \mathcal{E}_T$, $\partial_{t_E} v_{n,E} = (\text{sym curl } v) n_E \cdot n_E = (\boldsymbol{\tau} - v) n_E \cdot n_E$ (cf., respectively, [23, Eq. (3)]) and (62)), we can use a triangle inequality to write

$$ 
\begin{aligned}
\| \partial_{t_{\partial T}} v_{n, \partial T} \|_{L^2(\partial T)}^2 & \lesssim \sum_{E \in \mathcal{E}_T} \left( \| \boldsymbol{\tau}_{|E} n_E \cdot n_E \|_{L^2(E)}^2 + \| v_{|E} n_E \cdot n_E \|_{L^2(E)}^2 \right) \\
& \lesssim \sum_{E \in \mathcal{E}_T} \left( \left\| \pi_{\mathcal{P},E}^{m-2} (\boldsymbol{\tau}_{|E} n_E \cdot n_E) \right\|_{L^2(E)}^2 + h_E \sum_{V \in \mathcal{V}_E} |\boldsymbol{\tau}(\mathbf{x}_V) n_E \cdot n_E|^2 + \| v_{|E} n_E \cdot n_E \|_{L^2(E)}^2 \right) \\
& \lesssim h_T^{-1} \mathcal{N}_T(\boldsymbol{\tau})^2,
\end{aligned} 
\quad (68) $$

where the second line follows from (33) applied to $\boldsymbol{\varphi} = \boldsymbol{\tau}_{|E} n_E \cdot n_E$, while the conclusion follows using the definition of $\mathcal{N}_T(\boldsymbol{\tau})$ for the first two terms and a discrete trace inequality followed by (64) for the last term. To estimate $\|\partial_{t_{\partial T}} v_{t, \partial T} \|_{L^2(\partial T)}$, we proceed in a similar way as for the estimate of $\mathcal{I}_2$ in Proposition 5 (using the fact that $\partial_{t_{\partial T}} v_{t, \partial T}$ has zero average on $T$) to infer

$$ 
\begin{aligned}
\| \partial_{t_{\partial T}} v_{t, \partial T} \|_{L^2(\partial T)} & \lesssim h_T \| \partial_{t_{\partial T}}^2 v_{t, \partial T} \|_{L^2(\partial T)} + h_T^{1/2} \sum_{V \in \mathcal{V}_T} |\text{sym curl } v(\mathbf{x}_V)| \\
& = h_T \left( \sum_{E \in \mathcal{E}_T} \| \delta_E (\boldsymbol{\tau} - v) \|_{L^2(E)}^2 \right)^{1/2} + h_T^{1/2} \sum_{V \in \mathcal{V}_T} |\boldsymbol{\tau}(\mathbf{x}_V) - v(\mathbf{x}_V)| \\
& \lesssim h_T \left( \sum_{E \in \mathcal{E}_T} \| \delta_E \boldsymbol{\tau} \|_{L^2(E)}^2 \right)^{1/2} + h_T^{1/2} \sum_{V \in \mathcal{V}_T} |\boldsymbol{\tau}(\mathbf{x}_V)| + h_T^{-1/2} \| v \|_{L^2(T; \mathbb{R}^{2 \times 2})},
\end{aligned} 
$$

where, to pass to the second line, we have used [23, Eq. (4)] to write $\partial_{t_E}^2 v_{t,E} = \delta_E \text{sym curl } v = \delta_E(\boldsymbol{\tau} - v)$ for the first term and (62) for the second, while, to pass to the third line, we have used triangle inequalities followed by discrete inverse and trace inequalities along with $\text{card}(\mathcal{E}_T) = \text{card}(\mathcal{V}_T) \lesssim 1$ to treat the terms containing $v$. Combining the definition of $\mathcal{N}_T(\boldsymbol{\tau})$ with (64), we conclude that

$$ \| \partial_{t_{\partial T}} v_{t, \partial T} \|_{L^2(\partial T)} \lesssim h_T^{-1/2} \mathcal{N}_T(\boldsymbol{\tau}). \quad (69) $$

Plugging (68) and (69) into (67) and the resulting inequality into (66), we conclude that

$$ \| v \|_{L^2(\partial T; \mathbb{R}^2)} \lesssim h_T^{1/2} \mathcal{N}_T(\boldsymbol{\tau}). \quad (70) $$

It only remains to estimate $\| \boldsymbol{\pi}_{\mathcal{P},T}^{m+1-\eta_T} v \|_{L^2(T; \mathbb{R}^2)}$ in (65). To this end, we start using the integration by parts formula (11) to write, for all $\boldsymbol{\phi} \in \mathcal{H}^{c,m+2-\eta_T}(T)$,

$$ 
\begin{aligned}
\int_T v \cdot \text{rot} \boldsymbol{\phi} &= - \int_T \text{sym curl } v : \boldsymbol{\phi} + \sum_{E \in \mathcal{E}_T} \omega_{TE} \int_E v \cdot (\boldsymbol{\phi} t_E) \\
&= - \int_T \boldsymbol{\pi}_{\mathcal{H},T}^{c,m+2-\eta_T} \boldsymbol{\tau} : \boldsymbol{\phi} + \int_T v : \boldsymbol{\phi} + \sum_{E \in \mathcal{E}_T} \omega_{TE} \int_E v \cdot (\boldsymbol{\phi} t_E),
\end{aligned} 
$$

where we have used (62) to pass to the second line and invoked its definition to insert $\boldsymbol{\pi}_{\mathcal{H},T}^{c,m+2-\eta_T}$ into the first term. We then apply Cauchy-Schwarz and discrete trace inequalities to get

$$ 
\begin{aligned}
\left| \int_T v \cdot \text{rot} \boldsymbol{\phi} \right| & \lesssim \left( \left\| \boldsymbol{\pi}_{\mathcal{H},T}^{c,m+2-\eta_T} \boldsymbol{\tau} \right\|_{L^2(T; \mathbb{R}^{2 \times 2})} + \| v \|_{L^2(T; \mathbb{R}^{2 \times 2})} + h_T^{-1/2} \| v \|_{L^2(\partial T; \mathbb{R}^2)} \right) \| \boldsymbol{\phi} \|_{L^2(T; \mathbb{R}^{2 \times 2})} \\
& \lesssim h_T \mathcal{N}_T(\boldsymbol{\tau}) \| \text{rot} \boldsymbol{\phi} \|_{L^2(T; \mathbb{R}^2)},
\end{aligned} 
$$

where the conclusion follows using the definition of $\mathcal{N}_T(\boldsymbol{\tau})$ along with (64) and (70) for the first factor and (5) for the second. Taking the supremum over $\boldsymbol{\phi} \in \mathcal{H}^{c,m+2-\eta_T}(T)$ such that $\| \text{rot} \boldsymbol{\phi} \|_{L^2(T; \mathbb{R}^2)} = 1$ finally yields

$$ \| \boldsymbol{\pi}_{\mathcal{P},T}^{m+1-\eta_T} v \|_{L^2(T; \mathbb{R}^2)} \lesssim h_T \mathcal{N}_T(\boldsymbol{\tau}). $$
Plugging this result and (70) into (65) gives $\|\mathrm{sym} \mathrm{curl} \mathbf{v}\|_{L^2(T;\mathbb{R}^{2 \times 2})} \lesssim \mathcal{N}_T(\boldsymbol{\tau})$ which, combined with (64), gives (60) after taking the $L^2$-norm of (62) and using a triangle inequality in the right-hand side. $\square$

## 4.2. Serendipity problem

Recalling Assumption 10, we let

$$ \ell_T := \max(k - \eta_T, -1) \le k - 2; \qquad (71) $$

**Remark 13 (Selective use of serendipity).** Should the identification of the set $\hat{\mathcal{E}}_T$ in Assumption 10 be difficult for some $T \in \mathcal{T}_h$ (e.g., because of a particularly complicated element geometry), one can always take $\eta_T = 2$, which corresponds to the absence of serendipity on that element (see Remark 15).

Given a linear form $\mathcal{L}_T : \mathcal{P}^{k-1}(T;\mathbb{S}) \times \mathcal{H}^{c,\ell_T+1}(T) \to \mathbb{R}$, we consider the following problem: Find $(\boldsymbol{\sigma}, \boldsymbol{\lambda}) \in \mathcal{P}^{k-1}(T;\mathbb{S}) \times \mathcal{H}^{c,\ell_T+1}(T)$ such that

$$ \mathcal{A}_T((\boldsymbol{\sigma}, \boldsymbol{\lambda}),(\boldsymbol{\tau}, \boldsymbol{\mu})) = \mathcal{L}_T(\boldsymbol{\tau}, \boldsymbol{\mu}) \quad \forall (\boldsymbol{\tau}, \boldsymbol{\mu}) \in \mathcal{P}^{k-1}(T;\mathbb{S}) \times \mathcal{H}^{c,\ell_T+1}(T), \qquad (72) $$

where the bilinear form $\mathcal{A}_T : [\mathcal{P}^{k-1}(T;\mathbb{S}) \times \mathcal{H}^{c,\ell_T+1}(T)]^2 \to \mathbb{R}$ is such that

$$ \begin{aligned} \mathcal{A}_T((\mathbf{v}, \mathbf{v}),(\boldsymbol{\tau}, \boldsymbol{\mu})) := & h_T^4 \int_T \mathrm{div} \, \mathrm{div} \, \mathbf{v} \, \mathrm{div} \, \mathrm{div} \, \boldsymbol{\tau} \\ & + h_T \sum_{E \in \mathcal{E}_T} \int_E \pi_{\mathcal{P},E}^{k-3} (\mathbf{v} \mathbf{n}_E \cdot \mathbf{n}_E) \pi_{\mathcal{P},E}^{k-3} (\boldsymbol{\tau} \mathbf{n}_E \cdot \mathbf{n}_E) + h_T^3 \sum_{E \in \mathcal{E}_T} \int_E \delta_E \mathbf{v} \, \delta_E \boldsymbol{\tau} \\ & + h_T^2 \sum_{V \in \mathcal{V}_T} \mathbf{v}(\mathbf{x}_V) : \boldsymbol{\tau}(\mathbf{x}_V) + \int_T \mathbf{v} : \boldsymbol{\mu} - \int_T \boldsymbol{\tau} : \mathbf{v}. \end{aligned} \qquad (73) $$

**Lemma 14 (Inf-sup condition and well-posedness of the serendipity problem).** The following inf-sup condition holds: For all $(\mathbf{v}, \mathbf{v}) \in \mathcal{P}^{k-1}(T;\mathbb{S}) \times \mathcal{H}^{c,\ell_T+1}(T)$,

$$ \| (\mathbf{v}, \mathbf{v}) \|_{T} \lesssim \sup_{(\boldsymbol{\tau}, \boldsymbol{\mu}) \in \mathcal{P}^{k-1}(T;\mathbb{S}) \times \mathcal{H}^{c,\ell_T+1}(T) \setminus (\mathbf{0}, \mathbf{0})} \frac{\mathcal{A}_T((\mathbf{v}, \mathbf{v}), (\boldsymbol{\tau}, \boldsymbol{\mu}))}{\|(\boldsymbol{\tau}, \boldsymbol{\mu})\|_T} =: \$, \qquad (74) $$

where $\|(\mathbf{v}, \mathbf{v})\|_T := \|\mathbf{v}\|_{L^2(T;\mathbb{R}^{2 \times 2})} + \|\mathbf{v}\|_{L^2(T;\mathbb{R}^{2 \times 2})}$. Hence, denoting by $\|\mathcal{L}_T\|_T$ the norm of $\mathcal{L}_T$ dual to $\|\cdot\|_T$, problem (72) admits a unique solution that satisfies

$$ \|(\boldsymbol{\sigma}, \boldsymbol{\lambda})\|_{T} \lesssim \|\mathcal{L}_{T}\|_{T}. \qquad (75) $$

**Proof.** The existence and uniqueness of a solution to (72) as well as the a priori estimate (75) classically follow from (74). Let us establish the latter condition for a given $(\mathbf{v}, \mathbf{v}) \in \mathcal{P}^{k-1}(T;\mathbb{S}) \times \mathcal{H}^{c,\ell_T+1}(T)$. Taking $(\boldsymbol{\tau}, \boldsymbol{\mu}) = (\mathbf{v}, \mathbf{v})$ in the expression (73) of $\mathcal{A}_T$, we obtain

$$ \begin{aligned} & h_T^4 \|\mathrm{div}\,\mathrm{div}\,\mathbf{v}\|_{L^2(T)}^2 + \sum_{E \in \mathcal{E}_T} \left( h_T \|\pi_{\mathcal{P},E}^{k-3}(\mathbf{v}\mathbf{n}_E \cdot \mathbf{n}_E)\|_{L^2(E)}^2 + h_T^3 \|\delta_E \mathbf{v}\|_{L^2(E)}^2 \right) \\ & + h_T^2 \sum_{V \in \mathcal{V}_T} |\mathbf{v}(\mathbf{x}_V)|^2 = \mathcal{A}_T((\mathbf{v}, \mathbf{v}), (\mathbf{v}, \mathbf{v})) \le \|\mathbf{v}, \mathbf{v}\|_T. \end{aligned} \qquad (76) $$

We next observe that, for any $q \in \mathcal{P}^{k-2}(T)$, writing the integration by parts formula (17) and inserting the appropriate $L^2$-orthogonal projectors according to their definition, it holds

$$ \int_T \pi_{\mathcal{H},T}^{k-4} \mathbf{v} : \mathbf{hess} \, \mathbf{q} = \int_T \mathrm{div} \, \mathrm{div} \, \mathbf{v} \, \mathbf{q} + \sum_{E \in \mathcal{E}_T} \omega_{TE} \left( \int_E \pi_{\mathcal{P},T}^{k-3} (\mathbf{v} \mathbf{n}_E \cdot \mathbf{n}_E) \partial_{n_E} \mathbf{q} - \int_E \delta_E \mathbf{v} \, \mathbf{q} \right) \\ + \sum_{E \in \mathcal{E}_T} \omega_{TE} \sum_{V \in \mathcal{V}_E} \omega_{EV} (\mathbf{v}(\mathbf{x}_V) \mathbf{n}_E \cdot \mathbf{t}_E) \, \mathbf{q}(\mathbf{x}_V). $$
By Proposition 1, we can select $q$ such that $\mathbf{hess} q = \pi_{\mathcal{H},T}^{k-4} \mathbf{v}$ and $\|q\|_{L^2(T)} \lesssim h_T^2 \|\mathbf{hess} q\|_{L^2(T;\mathbb{R}^{2 \times 2})}$. Applying Cauchy–Schwarz and discrete trace and inverse inequalities to estimate the right hand side of the resulting expression, simplifying, and raising to the square, we obtain

$$ \| \boldsymbol{\pi}_{\mathcal{H},T}^{k-4} \boldsymbol{v} \|_{L^2(T;\mathbb{R}^{2 \times 2})}^2 \lesssim h_T^4 \| \operatorname{div} \boldsymbol{\operatorname{div}} \boldsymbol{v} \|_{L^2(T)}^2 + \sum_{E \in \mathcal{E}_T} \left( h_T \| \boldsymbol{\pi}_{\mathcal{P},E}^{k-3} (\boldsymbol{v} \boldsymbol{n}_E \cdot \boldsymbol{n}_E) \|_{L^2(E)}^2 + h_T^3 \| \delta_E \boldsymbol{v} \|_{L^2(E)}^2 \right) \\ + h_T^2 \sum_{V \in \mathcal{V}_T} |\boldsymbol{v}(\boldsymbol{x}_V)|^2 \lesssim \| \boldsymbol{v}(\boldsymbol{v}) \|_T, \quad (77) $$

where the conclusion follows from (76).

Finally, writing the definition (73) of $A_T$ with $(\tau, \lambda) = (0, \boldsymbol{\pi}_{\mathcal{H},T}^{c, \ell_T+1} \boldsymbol{v})$, we get

$$ \| \boldsymbol{\pi}_{\mathcal{H},T}^{c, \ell_T+1} \boldsymbol{v} \|_{L^2(T;\mathbb{R}^{2 \times 2})}^2 = A_T((\boldsymbol{v}, \boldsymbol{v}), (0, \boldsymbol{\pi}_{\mathcal{H},T}^{c, \ell_T+1} \boldsymbol{v})) \le \| \| (0, \boldsymbol{\pi}_{\mathcal{H},T}^{c, \ell_T+1} \boldsymbol{v}) \|_T \| \le \| (\boldsymbol{v}, \boldsymbol{v}) \|_T, \quad (78) $$

where the conclusion follows from the uniform $L^2$-boundedness of $\boldsymbol{\pi}_{\mathcal{H},T}^{c, \ell_T+1}$. Summing (76), (77), and (78) and using (60) with $m = k-1$, we infer

$$ \| \boldsymbol{v} \|_{L^2(T;\mathbb{R}^{2 \times 2})}^2 \lesssim \| \boldsymbol{v} \|_T. \quad (79) $$

To estimate the $L^2$-norm of $\boldsymbol{v}$, we take $(\boldsymbol{\tau}, \boldsymbol{\mu}) = (-\boldsymbol{v}, \mathbf{0})$ in the expression (73) of $A_T$ (this is possible since $\boldsymbol{v} \in \mathcal{H}^{c, \ell_T+1}(T) \subset \mathcal{P}^{k-1}(T;\mathbb{S})$ owing to (71)) and, after using Cauchy–Schwarz, discrete trace, and inverse inequalities, simplifying, and raising to the square, we obtain

$$ \| \boldsymbol{v} \|_{L^2(T;\mathbb{R}^{2 \times 2})}^2 \lesssim \| \boldsymbol{v} \|_T. \quad (80) $$

Summing (79) and (80), using Young's inequality for the rightmost term in (80), and taking the square root of the resulting expression gives (74). $\square$

### 4.3. Serendipity spaces

Recalling the definition (71) of $\ell_T$, the local serendipity spaces are:

$$ \begin{align*} \hat{\underline{\boldsymbol{V}}}_T^k &:= \left\{ \hat{\underline{\boldsymbol{v}}}_T = (\hat{\underline{\boldsymbol{v}}}_T, (\hat{\underline{\boldsymbol{v}}}_E)_{E \in \mathcal{E}_h}, (\hat{\underline{\boldsymbol{v}}}_V, \mathbf{G}\hat{\underline{\boldsymbol{v}}}_V)_{V \in \mathcal{V}_h}) : \right. \\ & \phantom{:=} \hat{\underline{\boldsymbol{v}}}_T \in \mathcal{P}^{\ell_T} (T; \mathbb{R}^2), \\ & \phantom{:=} \hat{\underline{\boldsymbol{v}}}_E \in \mathcal{P}^{k-4} (E; \mathbb{R}^2) \text{ for all } E \in \mathcal{E}_T, \\ & \phantom{:=} \hat{\underline{\boldsymbol{v}}}_V \in \mathbb{R}^2 \text{ and } \mathbf{G}\hat{\underline{\boldsymbol{v}}}_V \in \mathbb{R}^{2 \times 2} \text{ for all } V \in \mathcal{V}_T, \\ \hat{\underline{\boldsymbol{\Sigma}}}_T^{k-1} &:= \left\{ \underline{\boldsymbol{\tau}}_T = \left( (\hat{\underline{\boldsymbol{\tau}}}_{\mathcal{H},T}, \hat{\underline{\boldsymbol{\tau}}}_{\mathcal{H},T}^c, (\hat{\underline{\boldsymbol{\tau}}}_E, D\hat{\underline{\boldsymbol{\tau}}}_E)_{E \in \mathcal{E}_h}, (\hat{\underline{\boldsymbol{\tau}}}_V)_{V \in \mathcal{V}_h} \right) : \right. \\ & \phantom{:=} \hat{\underline{\boldsymbol{\tau}}}_{\mathcal{H},T} \in \mathcal{H}^{k-4}(T) \text{ and } \hat{\underline{\boldsymbol{\tau}}}_{\mathcal{H},T}^c \in \mathcal{H}^{c, \ell_T+1}(T), \\ & \phantom{:=} \hat{\underline{\boldsymbol{\tau}}}_E \in \mathcal{P}^{k-3}(E) \text{ and } D\hat{\underline{\boldsymbol{\tau}}}_E \in \mathcal{P}^{k-2}(E) \text{ for all } E \in \mathcal{E}_T, \\ & \phantom{:=} \hat{\underline{\boldsymbol{\tau}}}_V \in \mathbb{S} \text{ for all } V \in \mathcal{V}_T \right\}. \end{align*} $$

Global spaces on $M_h$ are obtained enforcing the single-valuedness of polynomial components located at internal edges and nodes.

**Remark 15 (Serendipity DOFs reduction).** Comparing the above expressions with those of the corresponding full spaces (i.e., the restrictions of (9a) and (9b) to $T$) shows that the serendipity DOFs reduction acts on the components $\hat{\underline{\boldsymbol{v}}}_T$ and $\hat{\underline{\boldsymbol{\tau}}}_{\mathcal{H},T}^c$, whose polynomial degrees are reduced from $(k-2, k-1)$ to $(\ell_T, \ell_T+1)$. Recalling (71), the choice $\eta_T = 2$ therefore corresponds to no serendipity.
In what follows, the component norms defined in Section 3.3 are applied to the elements of the serendipity spaces $\hat{\mathbf{V}}_h^k$ and $\hat{\mathbf{\Sigma}}_h^{k-1}$ after observing that the latter inject in the full spaces $\mathbf{V}_h^k$ and $\mathbf{\Sigma}_h^{k-1}$ (notice that, by (71), $\mathcal{P}^{\ell_T}(T; \mathbb{R}^2) \subset \mathcal{P}^{k-2}(T; \mathbb{R}^2)$ and $\mathcal{H}^{c, \ell_T+1}(T) \subset \mathcal{H}^{c, k-1}(T)$ for all $T \in \mathcal{T}_h$).

### 4.4. Serendipity operators

The serendipity operators $\mathbf{S}_{\mathbf{V},T}^{k-1}: \hat{\mathbf{V}}_T^k \to \mathcal{P}^{k-1}(T; \mathfrak{S})$ and $\mathbf{S}_{\mathbf{\Sigma},T}^{k-1}: \hat{\mathbf{\Sigma}}_T^{k-1} \to \mathcal{P}^{k-1}(T; \mathfrak{S})$ are such that, for all $(\hat{\mathbf{v}}_T, \hat{\mathbf{\tau}}_T) \in \hat{\mathbf{V}}_T^k \times \hat{\mathbf{\Sigma}}_T^{k-1}$, $\mathbf{S}_{\mathbf{V},T}^{k-1} \hat{\mathbf{v}}_T$ and $\mathbf{S}_{\mathbf{\Sigma},T}^{k-1} \hat{\mathbf{\tau}}_T$ are the first components of the solutions of problem (72) with right-hand side linear form $\mathcal{L}_T$ respectively equal to

$$ 
\begin{aligned}
\mathcal{L}_{\mathbf{V},T}(\hat{\mathbf{v}}_T; \mathbf{\tau}, \mathbf{\mu}) = & h_T \sum_{E \in \mathcal{E}_T} \int_E \pi_{\mathcal{P},E}^{k-3}(\partial_{\mathbf{t}_E} \hat{\mathbf{v}}_{\mathcal{E}_T} \cdot \mathbf{n}_E) \pi_{\mathcal{P},E}^{k-3}(\mathbf{\tau} \mathbf{n}_E \cdot \mathbf{n}_E) \\
& + h_T^3 \sum_{E \in \mathcal{E}_T} \int_E (\partial_{\mathbf{t}_E}^2 \hat{\mathbf{v}}_{\mathcal{E}_T|E} \cdot \mathbf{t}_E) \delta_E \mathbf{\tau} \\
& + h_T^2 \sum_{V \in \mathcal{V}_T} \mathfrak{C} \hat{\mathbf{v}}_{V,T}: \mathbf{\tau}(\mathbf{x}_V) - \int_T \hat{\mathbf{v}}_T \cdot \mathbf{rot} \mathbf{\mu} + \sum_{E \in \mathcal{E}_T} \omega_{TE} \int_E \hat{\mathbf{v}}_{\mathcal{E}_T} \cdot (\mathbf{\mu} \mathbf{t}_E)
\end{aligned} 
\quad (81) $$

and

$$ 
\begin{aligned}
\mathcal{L}_{\mathbf{\Sigma},T}(\hat{\mathbf{v}}_T; \mathbf{\tau}, \mathbf{\mu}) = & h_T^4 \left[ \int_T \hat{\mathbf{v}}_{\mathcal{H},T}: \mathbf{hess} \operatorname{div} \operatorname{div} \mathbf{\tau} \right. \\
& \left. - \sum_{E \in \mathcal{E}_T} \omega_{TE} \left( \int_E \hat{\mathbf{v}}_E \partial_{\mathbf{n}_E}(\operatorname{div} \operatorname{div} \mathbf{\tau}) - \int_E D_{\hat{\mathbf{v}},E} \operatorname{div} \operatorname{div} \mathbf{\tau} \right) \right. \\
& \left. - \sum_{E \in \mathcal{E}_T} \omega_{TE} \sum_{V \in \mathcal{V}_E} \omega_{EV}(\hat{\mathbf{v}}_V \mathbf{n}_E \cdot \mathbf{t}_E) \operatorname{div} \operatorname{div} \mathbf{\tau}(\mathbf{x}_V) \right] \\
& + h_T \sum_{E \in \mathcal{E}_T} \int_E \hat{\mathbf{v}}_E \pi_{\mathcal{P},E}^{k-3}(\mathbf{\tau} \mathbf{n}_E \cdot \mathbf{n}_E) + h_T^3 \sum_{E \in \mathcal{E}_T} \int_E D_{\hat{\mathbf{v}},E} \delta_E \mathbf{\tau} \\
& + h_T^2 \sum_{V \in \mathcal{V}_T} \hat{\mathbf{v}}_V: \mathbf{\tau}(\mathbf{x}_V) + \int_T \hat{\mathbf{v}}_{\mathcal{H},T}^c: \mathbf{\mu}.
\end{aligned} 
\quad (82) $$

We remark that the serendipity operators $\mathbf{S}_{\mathbf{V},T}^{k-1}$ and $\mathbf{S}_{\mathbf{\Sigma},T}^{k-1}$ respectively act as a discrete reconstructions of sym curl and a tensor potential, both regarded as elements of $\mathcal{H}(\operatorname{div} \operatorname{div}, \Omega; \mathfrak{S})$.

**Remark 16 (Alternative expression for $\mathcal{L}_{\mathbf{\Sigma},T}(\hat{\mathbf{v}}_T; \cdot)$).** Using the injection $\hat{\mathbf{\Sigma}}_T^{k-1} \hookrightarrow \mathbf{\Sigma}_T^{k-1}$ to apply the operator $\mathbf{DD}_T^{k-2}$ defined by (18) to elements of $\hat{\mathbf{\Sigma}}_T^{k-1}$, we have the following equivalent reformulation of $\mathcal{L}_{\mathbf{\Sigma},T}(\hat{\mathbf{v}}_T; \cdot)$: For all $(\mathbf{\tau}, \mathbf{\mu}) \in \mathcal{P}^{k-1}(T; \mathfrak{S}) \times \mathcal{H}^{c, \ell_T+1}(T)$,

$$ 
\begin{aligned}
\mathcal{L}_{\mathbf{\Sigma},T}(\hat{\mathbf{v}}_T; \mathbf{\tau}, \mathbf{\mu}) = & h_T^4 \int_T \mathbf{DD}_T^{k-2} \hat{\mathbf{v}}_T \operatorname{div} \operatorname{div} \mathbf{\tau} \\
& + h_T \sum_{E \in \mathcal{E}_T} \int_E \hat{\mathbf{v}}_E \pi_{\mathcal{P},E}^{k-3}(\mathbf{\tau} \mathbf{n}_E \cdot \mathbf{n}_E) + h_T^3 \sum_{E \in \mathcal{E}_T} \int_E D_{\hat{\mathbf{v}},E} \delta_E \mathbf{\tau} \\
& + h_T^2 \sum_{V \in \mathcal{V}_T} \hat{\mathbf{v}}_V: \mathbf{\tau}(\mathbf{x}_V) + \int_T \hat{\mathbf{v}}_{\mathcal{H},T}^c: \mathbf{\mu}.
\end{aligned} 
\quad (83) $$

### 4.5. Reduction and extension operators

The restriction operators $\hat{\mathbf{R}}_{\mathbf{V},T}: \mathbf{V}_T^k \to \hat{\mathbf{V}}_T^k$ and $\hat{\mathbf{R}}_{\mathbf{\Sigma},T}: \mathbf{\Sigma}_T^{k-1} \to \hat{\mathbf{\Sigma}}_T^{k-1}$ are defined taking $\mathcal{L}^2$-orthogonal projections on the reduced component spaces: For all $(\mathbf{v}_T, \mathbf{\tau}_T) \in \mathbf{V}_T^k \times \mathbf{\Sigma}_T^{k-1}$,

$$ \hat{\mathbf{R}}_{\mathbf{V},T} \mathbf{v}_T := (\pi_{\mathcal{P},T}^{\ell_T} \mathbf{v}_T, (\mathbf{v}_E)_{E \in \mathcal{E}_T}, (\mathbf{v}_V, \mathbf{G}_{v,V})_{V \in \mathcal{V}_T}), \quad (84) $$

$$ \hat{\mathbf{R}}_{\mathbf{\Sigma},T} \mathbf{\tau}_T := (\mathbf{\tau}_{\mathcal{H},T}, \pi_{\mathcal{H},T}^{c, \ell_T+1} \mathbf{\tau}_{\mathcal{H},T}, (\tau_E, D_{\mathbf{\tau},E})_{E \in \mathcal{E}_T}, (\mathbf{\tau}_V)_{V \in \mathcal{V}_T}). \quad (85) $$
According to [16, Eq. (2.4)], the interpolators on the spaces $\hat{\mathbf{V}}_T^k$ and $\hat{\mathbf{\Sigma}}_T^{k-1}$ are respectively given by

$$ \hat{\mathbf{I}}_{V,T}^k := \hat{\mathbf{R}}_{V,T} \hat{\mathbf{I}}_{V,T}^k \quad \text{and} \quad \hat{\mathbf{I}}_{\Sigma,T}^{k-1} := \hat{\mathbf{R}}_{\Sigma,T} \hat{\mathbf{I}}_{\Sigma,T}^{k-1}. \qquad (86) $$

The extension operators $\underline{E}_{V,T} : \hat{\mathbf{V}}_T^k \to \mathbf{V}_T^k$ and $\underline{E}_{\Sigma,T} : \hat{\mathbf{\Sigma}}_T^{k-1} \to \mathbf{\Sigma}_T^{k-1}$ are such that, for all $(\hat{\mathbf{v}}_T, \hat{\mathbf{t}}_T) \in \hat{\mathbf{V}}_T^k \times \hat{\mathbf{\Sigma}}_T^{k-1}$,

$$ \underline{E}_{V,T} \hat{\mathbf{v}}_T := \left( \underline{E}_{\mathcal{P},T}^{k-2} \hat{\mathbf{v}}_T, (\hat{\mathbf{v}}_E)_{E \in \mathcal{E}_T}, (\hat{\mathbf{v}}_V, \mathbf{G}_{\hat{\mathbf{v}},V})_{V \in \mathcal{V}_T} \right), \qquad (87) $$

$$ \underline{E}_{\Sigma,T} \hat{\mathbf{t}}_T := \left( \hat{\mathbf{\tau}}_{\mathcal{H},T}, \pi_{\mathcal{H},T}^{c,k-1} \mathbf{S}_{\Sigma,T}^{k-1} \hat{\mathbf{t}}_T, (\hat{\mathbf{\tau}}_E, D_{\hat{\mathbf{t}},E})_{E \in \mathcal{E}_T}, (\hat{\mathbf{t}}_V)_{V \in \mathcal{V}_T} \right), \qquad (88) $$

where $\underline{E}_{\mathcal{P},T}^{k-2} : \hat{\mathbf{\Sigma}}_T^{k-1} \to \mathcal{P}^{k-2}(T)$ is such that, for all $\hat{\mathbf{v}}_T \in \hat{\mathbf{V}}_T^k$,

$$ \int_T \underline{E}_{\mathcal{P},T}^{k-2} \hat{\mathbf{v}}_T \cdot \mathbf{rot} \, \boldsymbol{\tau} = - \int_T \mathbf{S}_{V,T}^{k-1} \hat{\mathbf{v}}_T : \boldsymbol{\tau} + \sum_{E \in \mathcal{E}_T} \omega_{TE} \int_E \hat{\mathbf{v}}_{\mathcal{E}_T} \cdot (\boldsymbol{\tau} \, \mathbf{t}_E) \quad \forall \boldsymbol{\tau} \in \mathcal{H}^{c,k-1}(T). \qquad (89) $$

The fact that $\underline{E}_{\mathcal{P},T}^{k-2} \hat{\mathbf{v}}_T$ is uniquely defined by the above equation follows from the fact that $\mathbf{rot} : \mathcal{H}^{c,k-1}(T) \to \mathcal{P}^{k-2}(T;\mathbb{R}^2)$ is an isomorphism (see Proposition 1). Owing to the definition of the serendipity operator $\mathbf{S}_{V,T}^{k-1}$ and the bound (75), it can be checked that, for all $T \in \mathcal{T}_h$,

$$ \| \underline{E}_{V,T} \hat{\mathbf{v}}_T \|_{V,T} \lesssim \| \hat{\mathbf{v}}_T \|_{\hat{\mathbf{v}}_T}. \qquad (90) $$

## 4.6. Preliminary results

**Lemma 17 (Polynomial consistency of the serendipity and extension operators).** It holds:

$$ \mathbf{S}_{V,T}^{k-1} \hat{\mathbf{I}}_{V,T}^k \mathbf{v} = \operatorname{sym curl} \mathbf{v} \quad \forall \mathbf{v} \in \mathcal{P}^k(T;\mathbb{R}^2), \qquad (91) $$

$$ \underline{E}_{V,T} \hat{\mathbf{I}}_{V,T}^k \mathbf{v} = \underline{I}_{V,T}^k \mathbf{v} \quad \forall \mathbf{v} \in \mathcal{P}^k(T;\mathbb{R}^2), \qquad (92) $$

$$ \mathbf{S}_{\Sigma,T}^{k-1} \hat{\mathbf{I}}_{\Sigma,T}^{k-1} \mathbf{v} = \mathbf{v} \quad \forall \mathbf{v} \in \mathcal{P}^{k-1}(T;\mathbb{S}), \qquad (93) $$

$$ \underline{E}_{\Sigma,T} \hat{\mathbf{I}}_{\Sigma,T}^{k-1} \mathbf{v} = \underline{I}_{\Sigma,T}^{k-1} \mathbf{v} \quad \forall \mathbf{v} \in \mathcal{P}^{k-1}(T;\mathbb{S}). \qquad (94) $$

**Proof.**

(i) *Proof of (91).* Let $\hat{\mathbf{v}}_T := \hat{\mathbf{I}}_{V,T}^k \mathbf{v}$. It suffices to show that $(\operatorname{sym curl} \mathbf{v}, 0)$ solves the problem defining $\mathbf{S}_{V,T}^{k-1} \hat{\mathbf{v}}_T$, i.e., (72) with linear form $\mathcal{L}_T(\cdot) = \mathcal{L}_{V,T}(\hat{\mathbf{v}}_T; \cdot)$ given by (81). Recalling the definition (73) of the bilinear form $\mathcal{A}_T$, we have, for all $(\boldsymbol{\tau}, \boldsymbol{\mu}) \in \mathcal{P}^{k-1}(T;\mathbb{S}) \times \mathcal{H}^{c,\ell_T+1}(T)$,

$$ \begin{aligned} \mathcal{A}_T((\operatorname{sym \, curl} \mathbf{v}, 0), (\boldsymbol{\tau}, \boldsymbol{\mu})) = & h_T^4 \int_T \operatorname{div} \operatorname{div} \operatorname{sym \, curl} \mathbf{v} \operatorname{div} \operatorname{div} \boldsymbol{\tau} \\ & + h_T \sum_{E \in \mathcal{E}_T} \int_E \pi_{\mathcal{P},E}^{k-3} \underbrace{(\operatorname{sym \, curl} \mathbf{v} \mathbf{n}_E \cdot \mathbf{n}_E)}_{\partial_{\mathbf{t}_E} \mathbf{v}_{|E} \cdot \mathbf{n}_E} \pi_{\mathcal{P},E}^{k-3} (\boldsymbol{\tau} \mathbf{n}_E \cdot \mathbf{n}_E) \\ & + h_T^3 \sum_{E \in \mathcal{E}_T} \int_E \underbrace{\delta_E \operatorname{sym \, curl} \mathbf{v}}_{\partial_{\mathbf{t}_E}^2 \mathbf{v}_{|E} \cdot \mathbf{t}_E} \delta_E \boldsymbol{\tau} + h_T^2 \sum_{V \in \mathcal{V}_T} \operatorname{sym \, curl} \mathbf{v}(\mathbf{x}_V) : \boldsymbol{\tau}(\mathbf{x}_V), \\ & + \int_T \operatorname{sym \, curl} \mathbf{v} : \boldsymbol{\mu}. \end{aligned} $$

where we have used [23, Lemma 2.2] for the second and third term. Using the integration by parts formula (11), observing that $\mathbf{rot} \boldsymbol{\mu} \in \mathcal{P}^{\ell_T}(T;\mathbb{R}^2)$ to insert $\boldsymbol{\pi}_{\mathcal{P},T}^{\ell_T}$ into the first term and that $\mathbf{v}_{|\partial T} = \hat{\mathbf{v}}_{\mathcal{E}_T}$ by polynomial consistency of this trace reconstruction,

$$ \int_T \operatorname{sym \, curl} \mathbf{v} : \boldsymbol{\mu} = - \int_T \underbrace{\boldsymbol{\pi}_{\mathcal{P},T}^{\ell_T} \mathbf{v} \cdot \mathbf{rot} \boldsymbol{\mu}}_{= \hat{\mathbf{v}}_T} + \sum_{E \in \mathcal{E}_T} \omega_{TE} \int_E \hat{\mathbf{v}}_{\mathcal{E}_T} \cdot (\boldsymbol{\mu} \, \mathbf{t}_E). $$
Using the above relation, again $v|_{\partial T} = \hat{v}_{\mathcal{E}_T}$, and further noticing that $\text{sym} \text{curl } v(x_V) = C G_{\hat{v},V}$ for all $V \in \mathcal{V}_T$ by definition of the interpolator, we have, recalling the definition (81) of $\mathcal{L}_{V,T}(\hat{v}_T; \cdot)$,

$$ \mathcal{A}_T((\text{sym} \text{curl } v, 0), (\tau, \mu)) = \mathcal{L}_{V,T}(\hat{v}_T; (\tau, \mu)) \quad \forall (\tau, \mu) \in \mathcal{P}^{k-1}(T; \mathfrak{S}) \times \mathcal{H}^{c, \ell_T+1}(T). $$

Since problem (72) is well-posed, this shows that $(\text{sym} \text{curl } v, 0)$ is its unique solution and, as a result, (91) holds.

(ii) *Proof of* (92). Set again $\hat{v}_T := \hat{\underline{I}}_{V,T}^k v$. Starting from (89), using (91) to write $S_{V,T}^{k-1} \hat{v}_T = \text{sym} \text{curl } v$ along with the polynomial consistency of the trace to write $\hat{v}_{\mathcal{E}_T} = v|_{\partial T}$, and concluding applying the integration by parts formula (11) to the right-hand side of the resulting expression, we have

$$ \int_T E_{\mathcal{P},T}^{k-2} \hat{v}_T \cdot \text{rot} \tau = \int_T v \cdot \text{rot} \tau \quad \forall \tau \in \mathcal{H}^{c,k-1}(T). $$

Recalling that $\text{rot} : \mathcal{H}^{c,k-1}(T) \to \mathcal{P}^{k-2}(T; \mathbb{R}^2)$ is an isomorphism, this shows that $E_{\mathcal{P},T}^{k-2} \hat{v}_T = \pi_{\mathcal{P},T}^{k-2} v$. Noticing that the other components of the local interpolator are not affected by the serendipity reduction process, (92) follows.

(iii) *Proof of* (93). It suffices to show that $(v, 0)$ solves the problem defining $S_{\Sigma,T}^{k-1} \hat{\underline{I}}_{\Sigma,T}^{k-1} v$, i.e., (72) with linear form $\mathcal{L}_T(\cdot) = \mathcal{L}_{\Sigma,T}(\hat{\underline{I}}_{\Sigma,T}^{k-1} v; \cdot)$ given by (82). To this end, we use the alternative expression (83) of $\mathcal{L}_{\Sigma,T}(\hat{v}_T; \cdot)$ based on the restriction of the operator $\text{DD}_T^{k-2}$ to $\hat{\underline{\Sigma}}_T^{k-1}$ resulting from the injection $\hat{\underline{\Sigma}}_T^{k-1} \hookrightarrow \underline{\Sigma}_T^{k-1}$. Since this operator only depends on the polynomial components of $\hat{\underline{\Sigma}}_T^{k-1}$ left unchanged by the serendipity reduction, by [16, Eq. (19)] it holds $\text{DD}_T^{k-2} \underline{\Sigma}_{\Sigma,T}^{k-1} v = \text{div} \text{div} v$. Plugging this relation into (83) and recalling the definition (86) of $\hat{\underline{I}}_{\Sigma,T}^{k-1}$, we obtain: For all $(\tau, \mu) \in \mathcal{P}^{k-1}(T; \mathfrak{S}) \times \mathcal{H}^{c, \ell_T+1}(T)$,

$$ \mathcal{L}_{\Sigma,T}(\hat{\underline{I}}_{\Sigma,T}^{k-1} v; \tau, \mu) = h_T^4 \int_T \operatorname{div} \text{div} v \operatorname{div} \text{div} \tau \\ + h_T \sum_{E \in \mathcal{E}_T} \int_E \pi_{\mathcal{P},E}^{k-3} (v n_E \cdot n_E) \pi_{\mathcal{P},E}^{k-3} (\tau n_E \cdot n_E) + h_T^3 \sum_{E \in \mathcal{E}_T} \int_E \pi_{\mathcal{P},E}^{k-2} \delta_E v \delta_E \tau \\ + h_T^2 \sum_{V \in \mathcal{V}_T} v(x_V) : \tau(x_V) + \int_T \pi_{\mathcal{H},T}^{c, \ell_T+1} v : \mu, $$

where the cancellation of the projectors is made possible by their definition. Comparing with the definition (73) of $\mathcal{A}_T$, we have thus proved that

$$ \mathcal{A}_T((v, 0), (\tau, \mu)) = \mathcal{L}_{\Sigma,T}(\hat{\underline{I}}_{\Sigma,T}^{k-1} v; (\tau, \mu)) \quad \forall (\tau, \mu) \in \mathcal{P}^{k-1}(T; \mathfrak{S}) \times \mathcal{H}^{c, \ell_T+1}(T). $$

By uniqueness of the solution to (72), this proves the assertion.

(iv) *Proof of* (94). Immediate consequence of (93) along with the definition (88) of $\underline{E}_{\Sigma,T}$. $\square$

**Lemma 18 (Projections of extension and serendipity operators).** It holds, for all $T \in \mathcal{T}_h$,

$$ \pi_{\mathcal{P},T}^{\ell_T} E_{\mathcal{P},T}^{k-2} \hat{v}_T = \hat{v}_T \quad \forall \hat{v}_T \in \hat{\underline{V}}_T^k, \quad (95) $$

$$ \pi_{\mathcal{H},T}^{c,k-1} \mathbf{C}_{\text{sym},T}^{k-1} \underline{E}_{V,T} \hat{v}_T = \pi_{\mathcal{H},T}^{c,k-1} \mathbf{S}_{V,T}^{k-1} \underline{v}_T, \quad \forall \hat{v}_T \in \hat{\underline{V}}_T^k, \quad (96) $$

$$ \pi_{\mathcal{H},T}^{c,\ell_T+1} \mathbf{S}_{\Sigma,T}^{k-1} \underline{\tau}_T = \hat{\tau}_T^c \quad \forall \hat{\underline{\tau}}_T \in \hat{\underline{\Sigma}}_T^{k-1}. \quad (97) $$

**Proof.**

(i) *Proof of* (95). For any $\mu \in \mathcal{H}^{c, \ell_T+1}(T)$, taking tests functions of the form $(0, \mu)$ with $\mu \in \mathcal{H}^{c, \ell_T+1}(T)$ in the problem defining $\mathbf{S}_{V,T}^{k-1}$ (i.e., (72) with $\mathcal{L}_T(\cdot) = \mathcal{L}_T(\hat{v}_T; \cdot)$ given by (81)), it is inferred that

$$ \int_T \mathbf{S}_{V,T}^{k-1} \hat{v}_T : \mu = - \int_T \hat{v}_T \cdot \text{rot} \mu + \sum_{E \in \mathcal{E}_T} \omega_{TE} \int_E \hat{v}_{\mathcal{E}_T} \cdot (\mu t_E). \quad (98) $$
On the other hand, by definition (89) of $E_{\mathcal{P},T}^{k-2}$, and since $\mu \in \mathcal{H}^{c,\ell_T+1}(T) \subset \mathcal{H}^{c,k-1}(T)$ (recall that $\ell_T+1 \le k-1$ by (71)), we have

$$ - \int_E E_{\mathcal{P},T}^{k-2} \hat{\nu}_T \cdot \mathbf{rot} \, \mu = \int_T S_{\mathbf{V},T}^{k-1} \hat{\nu}_T : \mu - \sum_{E \in \mathcal{E}_T} \omega_{TE} \int_E \hat{\nu}_{\mathcal{E}_T} \cdot (\mu \mathbf{t}_E). \quad (99) $$

Summing (98) and (99), (95) follows recalling that $\mathbf{rot} : \mathcal{H}^{c,\ell_T+1}(T) \to \mathcal{P}^{\ell_T}(T; \mathbb{R}^2)$ is an isomorphism.

(ii) *Proof of* (96). Using the definition (12) of $C_{\text{sym},T}^{k-1}$ for $\underline{\nu}_T = \underline{E}_{V,T} \hat{\nu}_T$ and recalling the definition (87) of $\underline{E}_{V,T}$, we can write, for any $\tau \in \mathcal{H}^{c,k-1}(T)$,

$$ \int_T \mathbf{C}_{\text{sym},T}^{k-1} \underline{E}_{V,T} \hat{\nu}_T : \tau = - \int_E E_{\mathcal{P},T}^{k-2} \hat{\nu}_T \cdot \mathbf{rot} \, \tau + \sum_{E \in \mathcal{E}_T} \omega_{TE} \int_E \hat{\nu}_{\mathcal{E}_T} \cdot (\tau \mathbf{t}_E) = \int_E S_{\mathbf{V},T}^{k-1} \hat{\nu}_T : \tau, $$

where the conclusion follows from the definition (89) of $E_{\mathcal{P},T}^{k-2}$. Then, (96) follows by definition of the $L^2$-orthogonal projector on $\mathcal{H}^{c,k-1}(T)$.

(iii) *Proof of* (97). It suffices to take test functions of the form $(0, \mu)$ with $\mu$ spanning $\mathcal{H}^{c,\ell_T+1}(T)$ in the problem defining $S_{\Sigma,T}^{k-1}$, that is (72) with linear form $\mathcal{L}_T(\cdot) = \mathcal{L}_{\Sigma,T}(\hat{\tau}_T; \cdot)$ given by (82). $\square$

## 4.7. Commutation property for the serendipity operators

**Lemma 19 (Commutation property for the serendipity operators).** *Recalling that, according to (59), $\underline{\mathcal{C}}_{\text{sym},T}^{k-1} = \underline{\mathcal{R}}_{\Sigma,T} \underline{\mathcal{C}}_{\text{sym},T}^{k-1} \underline{E}_{V,T}$, it holds*

$$ \mathbf{S}_{\Sigma,T}^{k-1} \underline{\mathcal{C}}_{\text{sym},T}^{k-1} \underline{\hat{\nu}}_T = \mathbf{S}_{\mathbf{V},T}^{k-1} \underline{\hat{\nu}}_T \quad \forall \underline{\hat{\nu}}_T \in \underline{\hat{\nu}}_T^k, \quad (100) $$

so that the following diagram commutes:

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

**Proof.** Let $\underline{\hat{\nu}}_T \in \underline{\hat{\nu}}_T^k$ and set $\underline{\nu}_T := \underline{E}_{V,T} \underline{\hat{\nu}}_T$. Recalling (87), we have $\underline{\nu}_E = \underline{\hat{\nu}}_E$ for all $E \in \mathcal{E}_T$ and $(\underline{\nu}_V, \underline{G}_{\nu,V}) = (\underline{\hat{\nu}}_V, \underline{G}_{\hat{\nu},V})$ for all $V \in \mathcal{V}_T$. We next analyse the expression (82) of $\mathcal{L}_{\Sigma,T}(\underline{\hat{\nu}}_T; \cdot)$ when $\underline{\hat{\nu}}_T := \underline{\mathcal{C}}_{\text{sym},T}^{k-1} \underline{\hat{\nu}}_T = \underline{\mathcal{R}}_{\Sigma,T} \underline{\mathcal{C}}_{\text{sym},T}^{k-1} \underline{\nu}_T$ with the aim of showing that

$$ \mathcal{L}_{\Sigma,T}(\underline{\hat{\nu}}_T; (\tau, \mu)) = \mathcal{L}_{V,T}(\underline{\hat{\nu}}_T; (\tau, \mu)) \quad \forall (\tau, \mu) \in \mathcal{P}^{k-1}(T; \mathbb{S}) \times \mathcal{H}^{c,\ell_T+1}(T). \quad (101) $$

The conclusion follows from this relation proceeding as in [16, Lemma 20].

We start by observing that, for all $q \in \mathcal{P}^{k-2}(T)$,

$$ \begin{align} & \int_T \underline{\hat{\nu}}_{\mathcal{H},T} : \mathbf{hess} \, q - \sum_{E \in \mathcal{E}_T} \omega_{TE} \left( \int_E \underline{\hat{\nu}}_E \partial_{n_E} q - \int_E D\underline{\hat{\nu}}_E q \sum_{V \in \mathcal{V}_E} \omega_{EV} (\underline{\hat{\nu}}_V n_E \cdot \mathbf{t}_E) q(\mathbf{x}_V) \right) \nonumber \\ &= \int_T \pi_{\mathcal{H},T}^{k-4} \mathbf{C}_{\text{sym},T}^{k-1} \underline{\nu}_T : \mathbf{hess} \, q \nonumber \\ & \quad - \sum_{E \in \mathcal{E}_T} \omega_{TE} \left( \int_E \pi_{\mathcal{P},E}^{k-3} (\partial_{\mathbf{t}_E} \underline{\nu}_{\mathcal{E}_T} \cdot \mathbf{n}_E) \partial_{n_E} q - \int_E (\partial_{\mathbf{t}_E}^2 \underline{\nu}_{\mathcal{E}_T} \cdot \mathbf{t}_E) q \right) \tag{102} \\ & \quad - \sum_{E \in \mathcal{E}_T} \omega_{TE} \sum_{V \in \mathcal{V}_E} \omega_{EV} (\mathcal{C}\underline{G}_{\nu,V} \mathbf{n}_E \cdot \mathbf{t}_E) q(\mathbf{x}_V) \nonumber \\ &= \int_T \mathbf{D}\mathbf{D}_T^{k-2} \underline{\mathcal{C}}_{\text{sym},T}^{k-1} \underline{\nu}_T q = 0, \nonumber \end{align} $$
where the second equality follows from the definitions (14) of $\underline{C}_{\text{sym},T}^{k-1}$ and (18) of $\overline{DD}_T^{k-2}$, while the conclusion is a consequence of the fact that (58) defines a complex. This implies that the terms in the first three lines of (82) vanish since $\text{div}\,\text{div}\,\boldsymbol{\tau} \in \mathcal{P}^{k-3}(T) \subset \mathcal{P}^{k-2}(T)$. Additionally, from property (96) it follows that $\hat{\nu}_{\mathcal{H},T}^c = \boldsymbol{\pi}_{\mathcal{H},T}^{c,\ell_T+1} \underline{C}_{\text{sym},T}^{k-1} \underline{E}_{V,T} \hat{\nu}_T = \boldsymbol{\pi}_{\mathcal{H},T}^{c,\ell_T+1} \overline{S}_{V,T}^{k-1} \hat{\nu}_T$. Hence, for all $\boldsymbol{\mu} \in \mathcal{H}^{c,\ell_T+1}(T)$,

$$ \int_T \hat{\nu}_{\mathcal{H},T}^c : \boldsymbol{\mu} = \int_T \overline{S}_{V,T}^{k-1} \underline{\hat{\nu}}_T : \boldsymbol{\mu} = - \int_T \hat{\nu}_T \cdot \text{rot} \boldsymbol{\mu} + \sum_{E \in \mathcal{E}_T} \omega_{TE} \int_E \hat{\nu}_E \cdot (\boldsymbol{\mu} \boldsymbol{t}_E), \quad (103) $$

where the conclusion follows from the definition of $\overline{S}_{V,T}^{k-1}$. Plugging (102)–(103) into (82) and comparing with (81) proves (101). □

## 4.8. Homological properties of the serendipity DDR sequence

**Theorem 20 (Homological properties of the serendipity DDR sequence).** The following properties hold:

(1) Complex properties:

$$ \underline{E}_{V,h} \hat{\underline{R}}_{V,h} \underline{v}_h = \underline{v}_h \qquad \forall \underline{v}_h \in \text{Ker}(\underline{C}_{\text{sym},h}^{k-1}), \quad (104) $$

$$ \underline{E}_{\Sigma,h} \hat{\underline{R}}_{\Sigma,h} \underline{\tau}_h - \underline{\tau}_h \in \text{Im}(\underline{C}_{\text{sym},h}^{k-1}) \qquad \forall \underline{\tau}_h \in \underline{\Sigma}_h^{k-1}; \quad (105) $$

(2) Cochain map properties for the reduction and extension maps:

$$ \underline{E}_{V,h} \hat{\underline{I}}_{V,h}^k \underline{v} = \underline{I}_{V,h}^k \underline{v} \qquad \forall \underline{v} \in \mathcal{R}\mathcal{T}^1(\Omega), \quad (106) $$

$$ \underline{C}_{\text{sym},h}^{k-1} \hat{\underline{R}}_{V,h} \underline{v}_h = \hat{\underline{R}}_{\Sigma,h} \underline{C}_{\text{sym},h}^{k-1} \underline{v}_h \qquad \forall \underline{v}_h \in \underline{V}_h^k, \quad (107) $$

$$ \underline{E}_{\Sigma,h} \hat{\underline{C}}_{\text{sym},h}^{k-1} \underline{\hat{v}}_h = \underline{C}_{\text{sym},h}^{k-1} \underline{E}_{V,h} \underline{\hat{v}}_h \qquad \forall \underline{\hat{v}}_h \in \underline{\hat{V}}_h^k; \quad (108) $$

(3) Isomorphism properties for the cohomology groups:

$$ \hat{\underline{R}}_{V,h} \underline{E}_{V,h} \underline{\hat{v}}_h = \underline{\hat{v}}_h \qquad \forall \underline{\hat{v}}_h \in \underline{\hat{V}}_h^k, \quad (109) $$

$$ \hat{\underline{R}}_{\Sigma,h} \underline{E}_{\Sigma,h} \underline{\tau}_h = \underline{\tau}_h \qquad \forall \underline{\tau}_h \in \underline{\Sigma}_h^{k-1}. \quad (110) $$

Hence, the cohomologies of the top and bottom complexes in (58) are isomorphic.

**Remark 21 (Homological properties).** The respective role of the above properties is the following: the complex properties ensure that the serendipity DDR sequence is a cochain complex; thanks to the cochain map properties, the reduction and extension maps are cochain maps; finally, the isomorphism properties guarantee that the cohomology groups of the DDR and serendipity DDR complexes are isomorphic. We additionally notice, in passing, that:

* It would suffice for property (105) to hold for all $\underline{\tau}_h \in \text{Ker}(\overline{DD}_h^{k-2})$ to ensure that the serendipity DDR sequence is a cochain complex;

* The cochain property for $\hat{\underline{R}}_{V,h}$ (i.e., $\hat{\underline{R}}_{V,h} \underline{I}_{V,h}^k \underline{w} = \underline{I}_{V,h}^k \underline{w}$ for all $\underline{w} \in \mathcal{R}\mathcal{T}^1(\Omega)$), holds by definition (59) of $\underline{I}_{V,h}^k$, and is therefore not listed in point 2.;

* Property (109) (resp., (110)) could be restricted to $\underline{\hat{v}}_h \in \text{Ker}(\underline{\hat{C}}_{\text{sym},h}^{k-1})$ (resp., $\underline{\hat{\tau}}_h \in \text{Ker}(\overline{\hat{DD}}_h^{k-2}))$ for the isomorphism in cohomology to hold.

**Proof of Theorem 20.** The isomorphism between the cohomologies of the top and bottom complexes in (58) is a straightforward consequence of [13, Proposition 2] once we prove properties (104)–(110), which we do next.

(i) *Proof of (104).* We notice that $\underline{C}_{\text{sym},h}^{k-1} \underline{v}_h = \underline{0}$ implies $\underline{C}_{\text{sym},T}^{k-1} \underline{v}_T = \underline{0}$ for all $T \in \mathcal{T}_h$. The exactness of the local DDR complex proved in [11, Theorem 3] then implies, for any $T \in \mathcal{T}_h$, the existence
of $w_T \in \mathcal{R}\mathcal{T}^1(T)$ such that $\underline{v}_T = \underline{I}_{V,T}^k w_T$. We can then write $\underline{E}_{V,T} \hat{\underline{R}}_{V,T} \underline{v}_T = \underline{E}_{V,T} \hat{\underline{R}}_{V,T} \underline{I}_{V,T}^k w_T = \underline{E}_{V,T} \underline{I}_{V,T}^k w_T = \underline{I}_{V,T}^k w_T$, where we have used the definition (59) of $\underline{\hat{I}}_{V,T}^k$ in the second step and the polynomial consistency property (92) (after observing that $w_T \in \mathcal{P}^k(T; \mathbb{R}^2)$) to conclude.

(ii) *Proof of* (105). Let $\underline{\tau}_h \in \underline{\Sigma}_h^{k-1}$ and set $\underline{\hat{\tau}}_h := \underline{\hat{R}}_{\underline{\Sigma}_h,h} \underline{\tau}_h$. The components of $\underline{\tau}_h$ and $\underline{E}_{\underline{\Sigma}_h,h} \underline{\hat{\tau}}_h$ on the mesh vertices and edges, as well as on $\mathcal{H}^{k-4}(T)$, $T \in \mathcal{T}_h$, coincide by definition of the restriction and extension operators (see (88) and (85)). Since $\mathrm{DD}_T^{k-2}$ only depends on these components (see (18)), this implies $\mathrm{DD}_h^{k-2} \underline{E}_{\underline{\Sigma}_h,h} \underline{\hat{\tau}}_h = \mathrm{DD}_h^{k-2} \underline{\tau}_h$, i.e.

$$ \underline{E}_{\underline{\Sigma}_h,h} \underline{\hat{\tau}}_h - \underline{\tau}_h = \left( \left( \underline{0}, \underline{\pi}_{\underline{\mathcal{H}},T}^{c,k-1} \underline{S}_{\underline{\Sigma}_h,T}^{k-1} \underline{\hat{\tau}}_T - \underline{\tau}_{\underline{\mathcal{H}},T}^{c} \right)_{T \in \mathcal{T}_h}, (0,0)_{E \in \mathcal{E}_h}, (0)_{V \in \mathcal{V}_h} \right) \in \mathrm{Ker} \left( \mathrm{DD}_h^{k-2} \right). \quad (111) $$

By exactness of the local DDR complex (see [11, Theorem 4]), for all $T \in \mathcal{T}_h$ there exists $\underline{v}_T \in \underline{V}_T^k$, defined up to an element of $\underline{I}_{V,T}^k \mathcal{R}\mathcal{T}^1(T)$, such that $\underline{E}_{\underline{\Sigma}_h,h} \underline{\hat{\tau}}_T - \underline{\tau}_T = \underline{C}_{\mathrm{sym},T}^{k-1} \underline{v}_T$ which additionally satisfies, by (111),

$$ \begin{gathered} \pi_{\mathcal{P},E}^{k-3} \partial_{\underline{t}_E} (\underline{v}_{\mathcal{E}_T} \cdot \underline{n}_E) = 0 \quad \text{and} \quad \partial_{\underline{t}_E}^2 (\underline{v}_{\mathcal{E}_T} \cdot \underline{t}_E) = 0 \quad \text{for all } E \in \mathcal{E}_T \\ \text{and } \mathcal{C}\underline{G}_{\underline{v},V} = \underline{0} \quad \text{for all } V \in \mathcal{V}_T. \end{gathered} $$

Under these conditions, [11, Point 1. of Theorem 3] yields the existence of $w_T \in \mathcal{R}\mathcal{T}^1(T)$ such that $\underline{v}_{\mathcal{E}_T} = w_{T|\partial T}$. Up to the substitution $\underline{v}_T \leftarrow \underline{v}_T - \underline{I}_{V,T}^k w_T$, we can therefore assume that $\underline{v}_{\mathcal{E}_T} = \underline{0}$. Hence, the $\underline{v}_T$, $T \in \mathcal{T}_h$, can be patched together on internal edges to form an element of $\underline{V}_h^k$. This concludes the proof of (105).

(iii) *Proof of* (106). The cochain map property (106) for $\underline{E}_{V,h}$ immediately follows from (92) applied to polynomials in $\mathcal{R}\mathcal{T}^1(T) \subset \mathcal{P}^k(T; \mathbb{R}^2)$ for all $T \in \mathcal{T}_h$.

(iv) *Proof of* (107). Let $\underline{v}_h \in \underline{V}_h^k$ and set, for the sake of brevity $\underline{w}_h := \underline{E}_{V,h} \hat{\underline{R}}_{V,h} \underline{v}_h$. By (59), $\underline{\underline{C}}_{\mathrm{sym},h}^{k-1} \hat{\underline{R}}_{V,h} \underline{v}_h = \underline{\underline{R}}_{\underline{\Sigma}_h,h} \underline{\underline{C}}_{\mathrm{sym},h}^{k-1} \underline{w}_h$. The components of $\underline{w}_h$ and $\underline{v}_h$ on the mesh vertices and edges coincide by definitions (87) of $\underline{E}_{V,h}$ and (84) of $\underline{\hat{R}}_{V,h}$, hence so do the components of their discrete symmetric curls on the edges and vertices, as well as those on $\mathcal{H}^{k-4}(T)$, $T \in \mathcal{T}_h$ (notice that the first term in the right-hand side of (12) vanishes for $\underline{\tau} \in \mathcal{H}^{k-4}(T)$ since *rothess* = $\underline{0}$). It only remains to prove the equality of the components on $\mathcal{H}^{c,\ell_T+1}(T)$, $T \in \mathcal{T}_h$, which follows if we prove that:

$$ \pi_{\mathcal{H},T}^{c,\ell_T+1} \underline{\underline{C}}_{\mathrm{sym},T}^{k-1} \underline{w}_T = \pi_{\mathcal{H},T}^{c,\ell_T+1} \underline{\underline{C}}_{\mathrm{sym},T}^{k-1} \underline{v}_T \quad \text{for all } T \in \mathcal{T}_h. \quad (112) $$

Set $\underline{\hat{v}}_T := \underline{\hat{R}}_{V,T} \underline{v}_T$. By virtue of (96), it suffices to prove that $\underline{\pi}_{\mathcal{H},T}^{c,\ell_T+1} \underline{S}_{V,T}^{k-1} \underline{\hat{v}}_T = \underline{\pi}_{\mathcal{H},T}^{c,\ell_T+1} \underline{C}_{\mathrm{sym},T}^{k-1} \underline{v}_T$. This relation can be established taking test functions of the form $(\underline{0}, \underline{\mu})$ with $\underline{\mu} \in \mathcal{H}^{c,\ell_T+1}(T)$ in the problem defining $\underline{S}_{V,T}^{k-1} \underline{v}_T$ (i.e., (72) with linear form $\mathcal{L}_T(\cdot) = \mathcal{L}_{V,T}(\underline{\hat{v}}_T; \cdot)$) to write

$$ \int_T \underline{S}_{V,T}^{k-1} \underline{\hat{v}}_T : \underline{\mu} = - \int_T \underline{\pi}_{\mathcal{P},T}^{\ell_T} \underline{v}_T \cdot \underline{\mathrm{rot}} \underline{\mu} + \sum_{E \in \mathcal{E}_T} \omega_{TE} \int_E \underline{v}_T \cdot (\underline{\mu} \underline{t}_E) = \int_T \underline{\mathrm{C}}_{\mathrm{sym},T}^{k-1} \underline{v}_T : \underline{\mu}, $$

where we have used the fact that, by (84), $\underline{\hat{v}}_T = \underline{\pi}_{\mathcal{P},T}^{\ell_T} \underline{v}_T$ and $\underline{\hat{v}}_{\mathcal{E}_T} = \underline{v}_{\mathcal{E}_T}$ in the first step (and also cancelled the projector since $\underline{\mathrm{rot}} \underline{\mu} \in \mathcal{P}^{\ell_T}(T; \mathbb{R}^2)$), while the conclusion follows from the definition (12) of $\underline{\mathrm{C}}_{\mathrm{sym},T}^{k-1}$. This concludes the proof of (112) and, therefore, of (107).

(v) *Proof of* (108). By (59), (108) amounts to proving that $\underline{E}_{\underline{\Sigma}_h,h} \underline{\hat{R}}_{\underline{\Sigma}_h,h} \underline{\underline{C}}_{\mathrm{sym},h}^{k-1} \underline{E}_{V,h} \underline{\hat{v}}_h = \underline{\underline{C}}_{\mathrm{sym},h}^{k-1} \underline{E}_{V,h} \underline{\hat{v}}_h$. Since $\underline{E}_{\underline{\Sigma}_h,h}$ and $\underline{\hat{R}}_{\underline{\Sigma}_h,h}$ leave the components on mesh vertices, edges, as well as those on $\mathcal{H}^{k-4}(T)$, $T \in \mathcal{T}_h$, unaltered, the equality of this components in (108) is immediate. It only remains to prove the equality of the components on $\mathcal{H}^{c,k-1}(T)$, $T \in \mathcal{T}_h$. To this purpose, it suffices to invoke (100) and (96) to write: For all $T \in \mathcal{T}_h$,

$$ \underline{\pi}_{\mathcal{H},T}^{c,k-1} \underline{S}_{\underline{\Sigma}_h,T}^{k-1} \underline{\hat{C}}_{\mathrm{sym},T}^{k-1} \underline{\hat{v}}_T = \underline{\pi}_{\mathcal{H},T}^{c,k-1} \underline{S}_{V,T}^{k-1} \underline{\hat{v}}_T = \underline{\pi}_{\mathcal{H},T}^{c,k-1} \underline{\mathrm{C}}_{\mathrm{sym},T}^{k-1} \underline{E}_{V,T} \underline{\hat{v}}_T. $$
(vi) *Proof of (109) and (110).* These relations are immediate consequences of, respectively, (95) and (97) along with the definitions (84) and (85) of the restrictions. $\square$

### 4.9. Analytical properties of the serendipity complex

Following [16, Eq. (2.3)], for $\bullet \in \mathcal{T}_h \cup \{h\}$, the discrete $L^2$-products and norms on $\hat{\mathbf{V}}_\bullet^k$ and $\hat{\mathbf{\Sigma}}_\bullet^{k-1}$ are defined setting, for all $\hat{\mathbf{w}}_\bullet, \hat{\mathbf{v}}_\bullet \in \hat{\mathbf{V}}_\bullet^k$ and $\hat{\mathbf{u}}_\bullet, \hat{\mathbf{t}}_\bullet \in \hat{\mathbf{\Sigma}}_\bullet^{k-1}$,

$$ (\hat{\mathbf{w}}_\bullet, \hat{\mathbf{v}}_\bullet)_{\hat{\mathbf{V}}_\bullet} := (\underline{\mathbf{E}}_{\mathbf{V},\bullet} \hat{\mathbf{w}}_\bullet, \underline{\mathbf{E}}_{\mathbf{V},\bullet} \hat{\mathbf{v}}_\bullet)_{\mathbf{V},\bullet} \quad \text{and} \quad \|\hat{\mathbf{v}}_\bullet\|_{\hat{\mathbf{V}}_\bullet} := \|\underline{\mathbf{E}}_{\mathbf{V},\bullet} \hat{\mathbf{v}}_\bullet\|_{\mathbf{V},\bullet}, \quad (113) $$

$$ (\underline{\mathbf{v}}_\bullet, \underline{\mathbf{t}}_\bullet)_{\underline{\mathbf{\Sigma}}_\bullet} := (\underline{\mathbf{E}}_{\underline{\mathbf{\Sigma}}_\bullet} \underline{\mathbf{v}}_\bullet, \underline{\mathbf{E}}_{\underline{\mathbf{\Sigma}}_\bullet} \underline{\mathbf{t}}_\bullet)_{\underline{\mathbf{\Sigma}}_\bullet} \quad \text{and} \quad \|\underline{\mathbf{t}}_\bullet\|_{\underline{\mathbf{\Sigma}}_\bullet} := \|\underline{\mathbf{E}}_{\underline{\mathbf{\Sigma}}_\bullet} \underline{\mathbf{v}}_\bullet\|_{\underline{\mathbf{\Sigma}}_\bullet}. \quad (114) $$

**Lemma 22 (Equivalence of norms on $\hat{\mathbf{V}}_T^k$).** It holds $\|\cdot\|_{\hat{\mathbf{V}}_T} \approx \|\cdot\|_{\mathbf{V},T}$ on $\hat{\mathbf{V}}_T^k$.

**Proof.** For all $\hat{\mathbf{v}}_T \in \hat{\mathbf{V}}_T^k$, we have

$$ \|\underline{\mathbf{v}}_T\|_{\hat{\mathbf{V}}_T} = \|\underline{\mathbf{E}}_{\mathbf{V},T} \underline{\mathbf{v}}_T\|_{\mathbf{V},T} \lesssim \|\underline{\mathbf{E}}_{\mathbf{V},T} \underline{\mathbf{v}}_T\|_{\mathbf{V},T} \lesssim \|\underline{\mathbf{v}}_T\|_{\mathbf{V},T}, $$

where the first inequality comes from the norm equivalence (28), while the conclusion is (90).

To prove the converse inequality, we use (95) to write:

$$ \begin{aligned} \|\underline{\mathbf{v}}_T\|_{\mathbf{V},T}^2 &= \|\boldsymbol{\pi}_{\mathcal{P},T}^{\ell_T} \mathbf{E}_{\mathcal{P},T}^{k-2} \underline{\mathbf{v}}_T\|_{\mathbf{L}^2(T;\mathbb{R}^2)}^2 + \sum_{E \in \mathcal{E}_T} h_T \|\underline{\mathbf{v}}_E\|_{\mathbf{L}^2(E;\mathbb{R}^2)}^2 + \sum_{V \in \mathcal{V}_T} (h_T^2 |\underline{\mathbf{v}}_V|^2 + h_T^4 |\mathbf{G}_{\underline{\mathbf{v}},V}|^2) \\ &\le \|\mathbf{E}_{\mathcal{P},T}^{k-2} \underline{\mathbf{v}}_T\|_{\mathbf{L}^2(T;\mathbb{R}^2)}^2 + \sum_{E \in \mathcal{E}_T} h_T \|\underline{\mathbf{v}}_E\|_{\mathbf{L}^2(E;\mathbb{R}^2)}^2 + \sum_{V \in \mathcal{V}_T} (h_T^2 |\underline{\mathbf{v}}_V|^2 + h_T^4 |\mathbf{G}_{\underline{\mathbf{v}},V}|^2) \\ &= \|\underline{\mathbf{E}}_{\mathbf{V},T} \underline{\mathbf{v}}_T\|_{\mathbf{V},T}^2, \end{aligned} $$

where the inequality follows from the $L^2$-boundedness of $\boldsymbol{\pi}_{\mathcal{P},T}^{\ell_T}$, while the conclusion is an immediate consequence of the definitions (26) of $\|\cdot\|_{\mathbf{V},T}$ and (87) of $\underline{\mathbf{E}}_{\mathbf{V},T}$. We then continue with the equivalence of norms (28) and with (113) to write $\|\underline{\mathbf{E}}_{\mathbf{V},T} \underline{\mathbf{v}}_T\|_{\hat{\mathbf{V}}_T}^2 \lesssim \|\underline{\mathbf{E}}_{\mathbf{V},T} \underline{\mathbf{v}}_T\|_{\mathbf{V},T}^2 = \|\underline{\mathbf{v}}_T\|_{\hat{\mathbf{V}}_T}^2$. $\square$

**Remark 23 (Equivalence of norms on $\hat{\mathbf{\Sigma}}_T^{k-1}$).** The uniform equivalence of $\|\cdot\|_{\hat{\mathbf{\Sigma}}_T}$ defined in (114) and $\|\cdot\|_{\mathbf{\Sigma}_T}$ can be established in a similar way. Since this result is not needed in what follows, the details are left to the reader.

**Theorem 24 (Analytical properties of the serendipity DDR complex).** The following properties hold:

(1) Continuity of the reductions:

$$ \|\underline{\mathbf{R}}_{\mathbf{V},h} \underline{\mathbf{v}}_h\|_{\hat{\mathbf{V}}_h} \lesssim \|\underline{\mathbf{v}}_h\|_{\mathbf{V},h} \qquad \forall \underline{\mathbf{v}}_h \in \underline{\mathbf{V}}_h^k, \quad (115) $$

$$ \|\underline{\mathbf{R}}_{\underline{\mathbf{\Sigma}}_h, h} \underline{\mathbf{t}}_h\|_{\underline{\mathbf{\Sigma}}_h} \lesssim \|\underline{\mathbf{t}}_h\|_{\underline{\mathbf{\Sigma}}_h} \qquad \forall \underline{\mathbf{t}}_h \in \underline{\mathbf{\Sigma}}_h^{k-1}; \quad (116) $$

(2) Polynomial consistency: For all $T \in \mathcal{T}_h$,

$$ \underline{\mathbf{E}}_{\mathbf{V},T} \underline{\mathbf{R}}_{\mathbf{V},T} \underline{\mathbf{I}}_{\mathbf{V},T}^k \underline{\mathbf{v}} = \underline{\mathbf{v}} \qquad \forall \underline{\mathbf{v}} \in \mathcal{P}^k(T; \mathbb{R}^2), \quad (117) $$

$$ \underline{\mathbf{E}}_{\underline{\mathbf{\Sigma}}_T} \underline{\mathbf{R}}_{\underline{\mathbf{\Sigma}}_T} \underline{\mathbf{I}}_{\underline{\mathbf{\Sigma}}_T}^{k-1} \underline{\mathbf{\tau}} = \underline{\mathbf{\tau}} \qquad \forall \underline{\mathbf{\tau}} \in \mathcal{P}^{k-1}(T; \mathbb{S}). \quad (118) $$

Hence, Lemmas 3, 7, and 8 hold with $(\underline{\mathbf{V}}_T^k, \underline{\mathbf{\Sigma}}_T^{k-1})$ replaced by $(\hat{\mathbf{V}}_T^k, \hat{\mathbf{\Sigma}}_T^{k-1})$.
**Proof.** The fact that Lemmas 3, 7, and 8 hold with $(\underline{V}_T^k, \underline{\Sigma}_T^{k-1})$ replaced by $(\hat{\underline{V}}_T^k, \hat{\underline{\Sigma}}_T^{k-1})$ is a consequence of Theorem 20 along with the continuity of the interpolators (29) and [16, Propositions 4–10] once properties (115)–(118) have been proved. We therefore turn to the latter.

(i) Proof of (115) and (116). Using the norm equivalence in Lemma 22 and the definitions (26) of the component norm $|||\underline{v}_T, \underline{v}_T|||$ and (84) of $\hat{\underline{R}}_{\underline{v}_T, \underline{v}_T}$, we infer

$$ \begin{aligned} \| \hat{\underline{R}}_{\underline{v}_T, \underline{v}_T} \|_{\underline{V}, T}^2 &\lesssim \| \underline{\hat{R}}_{\underline{v}_T, \underline{v}_T} \|_{\underline{V}, T}^2 \\ &= \| \underline{\pi}_{\underline{P}, T}^{\ell_T} \underline{v}_T \|_{\underline{L}^2(\underline{T}; \mathbb{R}^2)}^2 + \sum_{E \in \mathcal{E}_T} h_T \| \underline{v}_E \|_{\underline{L}^2(\underline{E}; \mathbb{R}^2)}^2 + \sum_{V \in \mathcal{V}_T} (h_T^2 |\underline{v}_V|^2 + h_T^4 |(\underline{G}_{\underline{v}, V})|^2) \\ &\lesssim \| \underline{v}_T \|_{\underline{L}^2(\underline{T}; \mathbb{R}^2)}^2 + \sum_{E \in \mathcal{E}_T} h_T \| \underline{v}_E \|_{\underline{L}^2(\underline{E}; \mathbb{R}^2)}^2 + \sum_{V \in \mathcal{V}_T} (h_T^2 |\underline{v}_V|^2 + h_T^4 |(\underline{G}_{\underline{v}, V})|^2) \end{aligned} $$

where the second line results from the $L^2$-boundedness of $\underline{\pi}_{\underline{P}, T}^{\ell_T}$. Noticing that the expression in the last line is precisely $\|\underline{v}_T\|_{\underline{L}^2(\underline{V}, T)}^2$ and invoking the uniform norm equivalence (28) with $\bullet = T$ concludes the proof of (115). The proof of (116) is similar and we omit the details for the sake of conciseness.

(ii) Proof of (117) and (118). Recalling the definition (86) of the interpolators on the serendipity spaces, properties (117) and (118) are nothing but (92) and (94), respectively. $\square$

## 4.10. Use of the serendipity DDR complex

Theorems 20 and 24 show that the serendipity complex at the bottom of (58) inherits all the relevant homological and analytical properties of the full complex at the top. As a consequence, given a scheme based on the full complex, a serendipity version is simply obtained replacing $\underline{V}_h^k$ with $\hat{\underline{V}}_h^k$ and $\underline{\Sigma}_h^{k-1}$ with $\hat{\underline{\Sigma}}_h^{k-1}$. This procedure can be applied, e.g., to the numerical scheme for Kirchhoff-Love plates corresponding to [11, Eq. (48)]. The stability analysis of this scheme carried out in [11, Section 4.5] hinges on the surjectivity of the discrete div div operator. This property is inherited at the discrete level by the serendipity DDR complex as a result of Theorem 20. The convergence analysis carried out in [11, Section 4.6] is based, on the other hand, on the consistency of the tensor potential $\underline{P}_{\underline{\Sigma}_T^{k-1}}$ and on the adjoint consistency of the discrete div div operator. Both properties are inherited by the corresponding operators for the serendipity DDR complex in view of Theorem 24, and thus the serendipity version of the numerical scheme [11, Eq. (48)] has the same convergence properties as the full version.

# Appendix A. Poincaré-Korn type inequalities in hybrid spaces

The proof of the functional inequality for hybrid vector fields that is used to establish point (i) of Lemma 3 (see Section 3.4.2) is presented below. First, we introduce some additional notations concerning tensor calculus in three dimensions. For a bounded, Lipschitz domain $D \subset \mathbb{R}^3$ and for a sufficiently regular tensor field $\underline{P}: D \to \mathbb{R}^{3 \times 3}$, we define

$$ \mathrm{Curl} \underline{P} := \begin{pmatrix} \partial_2 P_{13} - \partial_3 P_{12} & \partial_3 P_{11} - \partial_1 P_{13} & \partial_1 P_{12} - \partial_2 P_{11} \\ \partial_2 P_{23} - \partial_3 P_{22} & \partial_3 P_{21} - \partial_1 P_{23} & \partial_1 P_{22} - \partial_2 P_{21} \\ \partial_2 P_{33} - \partial_3 P_{32} & \partial_3 P_{31} - \partial_1 P_{33} & \partial_1 P_{32} - \partial_2 P_{31} \end{pmatrix}. $$
For later use, we also introduce the spaces of rigid-body motions $\mathcal{RM}_2 := \{a(x_2, -x_1)^\top + \mathbf{b} : a \in \mathbb{R}, \mathbf{b} \in \mathbb{R}^2\}$ and $\mathcal{RM}_3 := \{\mathbf{a} \times \mathbf{x} + \mathbf{b} : \mathbf{a}, \mathbf{b} \in \mathbb{R}^3\}$ respectively for $d = 2$ and $d = 3$, and the operator $\text{Anti}: \mathbb{R}^3 \to \mathbb{R}^{3 \times 3}$ given by

$$ \text{Anti} \mathbf{a} := \begin{pmatrix} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0 \end{pmatrix} \quad \forall \mathbf{a} \in \mathbb{R}^3. $$

Additionally, we remark that the planar rotation $\xi \mapsto \xi^\perp = (\xi_2^\perp - \xi_1^\perp)$ defines an isomorphism between the lowest-order Raviart-Thomas space and the space $\mathcal{RM}_2$ of two-dimensional rigid-body motions.

The discrete functional inequalities below hinge on [26, Theorem 3.3], which the authors refer to as *incompatible Korn type inequality for $L^p$-regular tensor fields*. For the sake of clarity, we recall the statement of this key result.

**Lemma 25 (Incompatible Korn type inequality).** Let $D \subset \mathbb{R}^3$ be a bounded, Lipschitz domain and let $p \in (1, \infty)$. Then, there exists $C_{IK} > 0$ depending only on $D$ and $p$ such that, for all $\mathbf{P} \in L^p(D; \mathbb{R}^{3 \times 3})$,

$$ \inf_{\mathbf{w} \in \mathcal{RM}_3} \|\mathbf{P} - \text{Anti} \mathbf{w}\|_{L^p(D; \mathbb{R}^{3 \times 3})} \le C_{IK} \left( \|\text{sym } \mathbf{P}\|_{L^p(D; \mathbb{R}^{3 \times 3})} + \|\text{sym } \text{Curl } \mathbf{P}\|_{W^{-1,p}(D; \mathbb{R}^{3 \times 3})} \right). \quad (119) $$

It has been observed in [26, 27] that the previous result can be seen as a generalisation of both the Poincaré-Wirtinger and Korn's second inequalities. In the following Proposition, we apply Lemma 25 to some particular cases in which the tensor field $\mathbf{P}$ is skew-symmetric and assuming $p = 2$.

**Proposition 26 (Poincaré-Korn inequalities for $L^2$-regular vector fields).** Let $D \subset \mathbb{R}^n$, with $n \in \{2, 3\}$ be a bounded, Lipschitz domain. Then, the following inequalities hold:

$$ \inf_{\bar{\mathbf{u}} \in \mathcal{P}^0(D; \mathbb{R}^d)} \|\mathbf{u} - \bar{\mathbf{u}}\|_{L^2(D; \mathbb{R}^d)} \lesssim C_{IK} \|\text{grad } \mathbf{u}\|_{H^{-1}(D; \mathbb{R}^{n \times d})} \quad \forall \mathbf{u} \in L^2(D; \mathbb{R}^d) \text{ with } 1 \le d \le n; \quad (120) $$

$$ \inf_{\bar{\mathbf{v}} \in \mathcal{R}\mathcal{T}^1(D)} \|\mathbf{v} - \bar{\mathbf{v}}\|_{L^2(D; \mathbb{R}^2)} \lesssim C_{IK} \|\text{sym } \text{curl } \mathbf{v}\|_{H^{-1}(D; \mathbb{R}^{2 \times 2})} \quad \forall \mathbf{v} \in L^2(D; \mathbb{R}^2) \text{ with } n=2; \quad (121) $$

$$ \inf_{\bar{\mathbf{w}} \in \mathcal{RM}_d} \|\mathbf{w} - \bar{\mathbf{w}}\|_{L^2(D; \mathbb{R}^d)} \lesssim C_{IK} \|\text{sym } \text{grad } \mathbf{w}\|_{H^{-1}(D; \mathbb{R}^{d \times d})} \quad \forall \mathbf{w} \in L^2(D; \mathbb{R}^d) \text{ with } d=n. \quad (122) $$

**Proof.** In order to establish (120) for $1 \le d \le n \le 3$, it suffices to consider the case $n=3$ and $d=1$. Hence, we let $u: D \to \mathbb{R}$ and apply Lemma 25 with $\mathbf{P}$ such that $P_{3,2} = -P_{2,3} = u$ and all the other components set to zero. Therefore, we clearly have $\text{sym } \mathbf{P} = \mathbf{0}$ and

$$ \inf_{\mathbf{w} \in \mathcal{RM}_3} \|\mathbf{P} - \text{Anti} \mathbf{w}\|_{L^2(D; \mathbb{R}^{3 \times 3})} = \inf_{\bar{\mathbf{u}} \in \mathbb{R}} \|\mathbf{P} - \text{Anti}(\bar{\mathbf{u}}, 0, 0)\|_{L^2(D; \mathbb{R}^{3 \times 3})} = \sqrt{2} \inf_{\bar{\mathbf{u}} \in \mathbb{R}} \|\mathbf{u} - \bar{\mathbf{u}}\|_{L^2(D)}. $$

Moreover, it is observed that

$$ \text{sym } \text{Curl } \mathbf{P} = \frac{1}{2} \begin{pmatrix} 0 & -\partial_2 u & -\partial_3 u \\ -\partial_2 u & 2\partial_1 u & 0 \\ -\partial_3 u & 0 & 2\partial_1 u \end{pmatrix} \implies \|\text{sym } \text{Curl } \mathbf{P}\|_{H^{-1}(D; \mathbb{R}^{3 \times 3})} \le 2 \|\text{grad } u\|_{H^{-1}(D; \mathbb{R}^3)}. $$

As a result, we get the conclusion.

We now proceed with the proof of (121). We let $n=2$, $I := [-\frac{1}{2}, \frac{1}{2}]$, $\mathbf{v} \in L^2(D; \mathbb{R}^2)$, and define a skew-symmetric tensor field $\mathbf{P}$ such that

$$ \mathbf{P} = \begin{pmatrix} 0 & 0 & v_1 \\ 0 & 0 & v_2 \\ -v_1 & -v_2 & 0 \end{pmatrix} \implies \text{sym } \text{Curl } \mathbf{P} = \begin{pmatrix} \text{sym } \text{curl } \mathbf{v} & \mathbf{0} \\ \mathbf{0} & \text{tr}(\text{sym } \text{curl } \mathbf{v}) \end{pmatrix}. $$
Thus, it is readily inferred that $\|\mathrm{sym}\,\mathbf{Curl}\,\mathbf{P}\|_{H^{-1}(D \times I; \mathbb{R}^{3 \times 3})} \le 2 \|\mathrm{sym}\,\mathbf{curl}\,\mathbf{v}\|_{H^{-1}(D; \mathbb{R}^{2 \times 2})}$. Additionally, since $\mathbf{v}$ does not depend on $x_3$ and due to the position of the non-zero entries in $\mathbf{P}$, it follows that

$$
\begin{align*}
\inf_{\mathbf{w} \in \mathcal{RM}_3} \|\mathbf{P} - \text{Anti}\,\mathbf{w}\|_{L^2(D \times I; \mathbb{R}^{3 \times 3})} &= \inf_{\bar{\mathbf{w}} \in \mathcal{RM}_2} \|\mathbf{P} - \text{Anti}(\bar{\mathbf{w}}, 0)\|_{L^2(D \times I; \mathbb{R}^{3 \times 3})} \\
&= \inf_{\bar{\mathbf{v}} \in \mathcal{RT}^1(D)} \|\mathbf{P} - \text{Anti}(\bar{\mathbf{v}}^{\perp}, 0)\|_{L^2(D; \mathbb{R}^{3 \times 3})} \\
&= \sqrt{2} \inf_{\bar{\mathbf{v}} \in \mathcal{RT}^1(D)} \|\mathbf{v} - \bar{\mathbf{v}}\|_{L^2(D; \mathbb{R}^2)}.
\end{align*}
$$

Hence, the conclusion follows again by using (119) on the domain $D \times I \subset \mathbb{R}^3$.

The proof of (121) is obtained with similar arguments by using Lemma 25 with

$$
\mathbf{P} = \begin{pmatrix} 0 & 0 & w_2 \\ 0 & 0 & -w_1 \\ -w_2 & w_1 & 0 \end{pmatrix} \quad \text{and} \quad \mathbf{P} = \begin{pmatrix} 0 & -w_3 & w_2 \\ w_3 & 0 & -w_1 \\ -w_2 & w_1 & 0 \end{pmatrix},
$$

for the case $n = d = 2$ and $n = d = 3$, respectively. $\square$

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

We are now ready to establish the main result of this Section. For the sake of simplicity, we detail the result only for the two dimensional case, but we refer to Remark 28 for some possible generalisations.

**Proposition 27 (Poincaré–Korn inequalities for hybrid vector fields).** Let

$$
\underline{\mathbf{u}}_h^k := \left\{ \underline{\mathbf{u}}_h = \left( (\mathbf{u}_T)_{T \in \mathcal{T}_h}, (\mathbf{u}_E)_{E \in \mathcal{E}_h} \right) : \mathbf{u}_T \in \mathcal{P}^k(T; \mathbb{R}^2) \quad \forall T \in \mathcal{T}_h, \mathbf{u}_E \in \mathcal{P}^k(E; \mathbb{R}^2) \quad \forall E \in \mathcal{E}_h \right\}
$$

and, for all $\underline{\mathbf{u}}_h \in \underline{\mathbf{U}}_h^k$, denote by $\mathbf{u}_h \in \mathcal{P}^k(\mathcal{T}_h; \mathbb{R}^2)$ the piecewise polynomial field on $\mathcal{T}_h$ such that $(\mathbf{u}_h)_{|T} := \mathbf{u}_T$ for all $T \in \mathcal{T}_h$. Then, there is a constant $C_{\text{PK}} > 0$, only depending on $\Omega$ and the mesh regularity parameter, such that

(1) For all $\underline{\mathbf{u}}_h \in \underline{\mathbf{U}}_h^k$ satisfying $\int_{\Omega} \mathbf{u}_h = \mathbf{0}$,

$$
\| \mathbf{u}_h \|_{L^2(\Omega; \mathbb{R}^2)}^2 \le C_{\text{PK}} \sum_{T \in \mathcal{T}_h} \left( \| \operatorname{grad} \mathbf{u}_T \|_{L^2(T; \mathbb{R}^{2 \times 2})}^2 + \sum_{E \in \mathcal{E}_T} h_T^{-1} \| \mathbf{u}_T - \mathbf{u}_E \|_{L^2(E; \mathbb{R}^2)}^2 \right); \quad (123)
$$

(2) For all $\underline{\mathbf{u}}_h \in \underline{\mathbf{U}}_h^k$ satisfying $\int_{\Omega} \mathbf{u}_h \cdot \mathbf{w} = 0$ for all $\mathbf{w} \in \mathcal{RT}^1(\Omega)$,

$$
\| \mathbf{u}_h \|_{L^2(\Omega; \mathbb{R}^2)}^2 \le C_{\text{PK}} \sum_{T \in \mathcal{T}_h} \left( \| \mathrm{sym} \, \mathbf{curl} \, \mathbf{u}_T \|_{L^2(T; \mathbb{R}^{2 \times 2})}^2 + \sum_{E \in \mathcal{E}_T} h_T^{-1} \| \mathbf{u}_T - \mathbf{u}_E \|_{L^2(E; \mathbb{R}^2)}^2 \right); \quad (124)
$$

(3) For all $\underline{\mathbf{u}}_h \in \underline{\mathbf{U}}_h^k$ satisfying $\int_{\Omega} \mathbf{u}_h \cdot \mathbf{w} = 0$ for all $\mathbf{w} \in \mathcal{RM}_2$,

$$
\| \mathbf{u}_h \|_{L^2(\Omega; \mathbb{R}^2)}^2 \le C_{\text{PK}} \sum_{T \in \mathcal{T}_h} \left( \| \mathrm{sym} \, \mathbf{grad} \, \mathbf{u}_T \|_{L^2(T; \mathbb{R}^{2 \times 2})}^2 + \sum_{E \in \mathcal{E}_T} h_T^{-1} \| \mathbf{u}_T - \mathbf{u}_E \|_{L^2(E; \mathbb{R}^2)}^2 \right). \quad (125)
$$

**Proof.** We only detail the proof of (124), which is used in the proof of Lemma 3, since (123) and (125) can be obtained by reasoning in a similar way. Let $\underline{\mathbf{u}}_h \in \underline{\mathbf{U}}_h^k$ and observe that the condition $\int_{\Omega} \mathbf{u}_h \cdot \mathbf{w} = 0$ for all $\mathbf{w} \in \mathcal{RT}^1(\Omega)$ implies

$$
\inf_{\bar{\mathbf{v}} \in \mathcal{RT}^1(\Omega)} \| \mathbf{u}_h - \bar{\mathbf{v}} \|_{L^2(\Omega; \mathbb{R}^2)} = \| \mathbf{u}_h \|_{L^2(\Omega; \mathbb{R}^2)}.
$$
Therefore, applying (121), it follows that

$$
\begin{align*}
\|\mathbf{u}_h\|_{L^2(\Omega; \mathbb{R}^2)} &\lesssim \|\mathrm{sym\,curl}\,\mathbf{u}_h\|_{H^{-1}(\Omega; \mathbb{R}^{2\times2})} \\
&= \sup_{\eta \in H_0^1(\Omega; \mathbb{S})} \sup_{\|\eta\|_{H^1(\Omega; \mathbb{R}^{2\times2})} = 1} \int_{\Omega} \mathbf{u}_h \cdot (\mathrm{rot}\,\eta) \\
&= \sup_{\eta \in H_0^1(\Omega; \mathbb{S})} \sum_{T \in \mathcal{T}_h} \left( -\int_T \mathrm{sym\,curl}\,\mathbf{u}_T : \eta + \sum_{E \in \mathcal{E}_T} \omega_{TE} \int_E \mathbf{u}_T \cdot (\eta \,\mathbf{t}_E) \right) \\
&= \sup_{\eta \in H_0^1(\Omega; \mathbb{S})} \sum_{T \in \mathcal{T}_h} \left( -\int_T \mathrm{sym\,curl}\,\mathbf{u}_T : \eta + \sum_{E \in \mathcal{E}_T} \omega_{TE} \int_E (\mathbf{u}_T - \mathbf{u}_E) \cdot \eta \,\mathbf{t}_E \right),
\end{align*}
$$

where we have integrated by parts element by element and used the fact that $\eta$ has continuous
tangential traces across interedges and vanishing tangential traces on the boundary in order to
insert $\mathbf{u}_E$ into the boundary term. Applying a Cauchy–Schwarz inequality on the integrals and
invoking a discrete Cauchy–Schwarz inequality on the sum over $T \in \mathcal{T}_h$, we infer that

$$
\begin{align*}
\|\mathbf{u}_h\|_{L^2(\Omega; \mathbb{R}^2)} &\lesssim \sup_{\eta \in H_0^1(\Omega; \mathbb{S}), \|\eta\|_{H^1(\Omega; \mathbb{R}^{2\times2})} = 1} \left( \sum_{T \in \mathcal{T}_h} \|\mathrm{sym\,curl}\,\mathbf{u}_T\|_{L^2(T; \mathbb{R}^{2\times2})}^2 \right)^{\frac{1}{2}} \|\eta\|_{L^2(\Omega; \mathbb{R}^{2\times2})} \\
&\quad + \left( \sum_{T \in \mathcal{T}_h} \sum_{E \in \mathcal{E}_T} h_T^{-1} \|\mathbf{u}_T - \mathbf{u}_E\|_{L^2(E; \mathbb{R}^2)}^2 \right)^{\frac{1}{2}} \left( \sum_{T \in \mathcal{T}_h} \sum_{E \in \mathcal{E}_T} h_T \|\eta \cdot \mathbf{t}_E\|_{L^2(E; \mathbb{R}^2)} \right)^{\frac{1}{2}} \\
&\lesssim \left[ \sum_{T \in \mathcal{T}_h} \left( \|\mathrm{sym\,curl}\,\mathbf{u}_T\|_{L^2(T; \mathbb{R}^{2\times2})}^2 + \sum_{E \in \mathcal{E}_T} h_T^{-1} \|\mathbf{u}_T - \mathbf{u}_E\|_{L^2(E; \mathbb{R}^2)}^2 \right) \right]^{\frac{1}{2}} \\
&\qquad \times \sup_{\eta \in H_0^1(\Omega; \mathbb{S}), \|\eta\|_{H^1(\Omega; \mathbb{R}^{2\times2})} = 1} \|\eta\|_{H^1(\Omega; \mathbb{S})}.
\end{align*}
$$

where, in the second inequality, we have used the continuous trace inequality [21, Lemma 1.31]. □

**Remark 28 (Generalisations).** The results of Proposition 27 admit several extensions that we have decided not to include for the sake of brevity. First, (123) and (125) can also be established in the three-dimensional case simply by replacing the interedges with interfaces. Second, since the starting argument given by Lemma 25 holds for all Lebesgue indices $p \in (1, \infty)$, we can generalise the discrete Poincaré–Korn inequalities to the Banach setting. The main modification required in the proof consists in replacing Cauchy–Schwarz inequalities with suitable versions of Hölder inequalities. Finally, we notice that in the proof of Proposition 27 we are not using any inverse inequality requiring the hybrid vector fields to be polynomials. Thus, the previous Poincaré–Korn inequalities can be extended to vector fields with piecewise Sobolev regularity.

Acknowledgments

Michele Botti acknowledges funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 896616 "PDGeoFF".
Daniele Di Pietro acknowledges the partial support of Agence Nationale de la Recherche through grants ANR-20-MRS2-0004 "NEMESIS" and ANR-16-IDEX-0006 "RHAMNUS".

Conflicts of interest

The authors have no conflict of interest to declare.
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