Julio Fernandez-Cestau

Benoı ˆt Bertrand

Maria Blaya

Garth A Jones

Thomas J Penfold
email: tom.penfold@newcastle.ac.uk

Manfred Bochmann
email: m.bochmann@uea.ac.uk

Synthesis and luminescence modulation of pyrazine-based gold(III) pincer complexes †

Keywords: ray crystallography, photophysical properties, theoretical calculations

The first examples of pyrazine-based gold(III) pincer complexes are reported; their intense photoemissions can be modified by protonation, N-alkylation or metal ions, without the need for altering the ligand framework. Emissions shift from red (77 K) to blue (298 K) due to thermally activated delayed fluorescence (TADF).

Gold(III) complexes with bis-cyclometallated ligands are characterized by their chemical stability and resistance to reduction. The ligand scaffold based on 2,6-diphenylpyridine 1 has proved particularly useful in gold(III) chemistry and forms (C^N^C)AuX pincer complexes of type A (Chart 1). Such complexes have proved to be highly versatile and, in combination with strong carbon-based s-donor ligands (e.g. X = N-heterocyclic carbene, alkynyl), display interesting photophysical [2][3][4] properties. This C^N py ^C ligand system has also been successful in stabilizing types of compounds that have frequently been invoked as unstable intermediates in catalytic cycles or postulated in computer modelling of catalytic processes, such as gold(III) hydrido, alkene, CO and peroxo complexes. 5 Photoemissive materials as components of electronic devices such as flat screen displays should ideally be capable of covering the whole range of the visible spectrum. Such changes in emission colours can be induced by suitable modification of the ligand framework. For (C^N^C)Au(III) complexes a widely applied strategy for modulating the photoluminescence (PL) response and widening the range of emission wavelengths has been the introduction of electron donating or withdrawing substituents in the 4-position of the pyridine moiety. 3,4 There are however limitations in this approach: firstly, the synthesis of C^N^C gold complexes involves two C-H activation steps, each of which is sensitive to the ligand structure and needs to be optimized for each new ligand; secondly, the modulation of electronic characteristics of the central pyridine moiety that can be achieved by inductive or mesomeric substituent effects is limited.

Rather more profound electronic changes in C^N^C ligands can be introduced by replacing the central pyridine ring by other heterocycles, such as pyrazine, to give compounds of type B. The lowest-energy p-p* transition in pyrazine is about 0.95 eV smaller than in pyridine; 6,7 pyrazine ligands are therefore much better electron acceptors and are likely to produce a red-shift of their UV and photoemission wavelengths. In addition, pyrazine complexes B offer scope for further derivatisation by protonation or quaternisation of the non-coordinating N atom, to give salts of type C.

Complexes of aryl-substituted pyrazines and quinoxalines are of course well-known for iridium(III) and platinum(II), since for these metals they are readily accessible by direct cyclometallation of the neutral ligand precursors by noble metal halides. 8 By contrast, related gold(III) complexes have until now been inaccessible since the usual methods employed for the synthesis of pyridine complexes A fail for the analogous pyrazine derivatives. Here we report the first examples of cyclometallated Au(III) pyrazine complexes and the facile modification of their photoluminescence properties.

The mercuration of the pro-ligand 2,6-bis(4 0 -t-BuC 6 H 4 ) 2 pz (pz = pyrazine) requires forcing conditions but proceeds using Hg(tfa) 2 in Htfa (tfa = CF 3 CO 2 ) to give (C^N pz ^C)HgClÁ2Htfa (1). Transmetallation with KAuCl 4 affords (C^N pz ^C)AuCl (2) as a yellow crystalline powder in good yield (Scheme 1). Substitution of the chloride ligand in 2 with KCN gives the cyanide complex 3, while treatment with dimethylpyrazole affords the pyrazolato complex 4. The reaction with AgCRCR in CH 2 Cl 2 affords the acetylides (C^N pz ^C)AuCRCR (5a, R = Ph; 5b, R = Bu t ) in essentially quantitative yield.

Pyrazine is only weakly basic (pK a 1.30, vs. 5.20 of pyridine). 9 However, protonation of 2 by HBF 4 ÁOEt 2 generated the corresponding salt 6. Remarkably, the protonation by dry HCl in Et 2 O proved to be reversible, and evaporation of the solvent from the HCl adduct regenerated neutral 2. The alkylation of the noncoordinating pyrazine N-atom was achieved using Meerwein's salt [Me 3 O]BF 4 to give 7 (Scheme 1) as a deep-red crystalline solid in 87% yield. The molecular structures 10 of 2, 4, 5a and 5b were determined by X-ray diffraction (ESI †). The interatomic distances and angles are as expected for square-planar pincer complexes of this type (see ESI † for details).

All the complexes show moderate to intense in the solid state and in solution (for a detailed summary see the ESI, † Tables S2 andS3). Both the UV-vis spectra and the PL bands are characterized by the vibronic progression of the C^N pz ^C pincer. As expected on the basis of the lower p-p* energy gap of the pyrazine compounds, the bands are red-shifted by about 40-50 nm compared with their C^N py ^C analogues (see Fig. S22,ESI †).

Whereas in the case of diphenylpyridine-based complexes of type A strongly s-donating ligands (such as aryls, acetylides or N-heterocyclic carbenes 3,4 ) are required to raise the d-orbital energy of Au(III) and increase the 3 LLCT contribution to the emissive state, the pyrazine derivatives 2-7 reported here are emissive even if ligands X are weaker heteroatom donors. This is best exemplified by the chlorides: whereas (C^N py ^C)AuCl is non-emissive at room temperature, 1 complex 2 shows noticeable naked eye emission in fluid media (CH 2 Cl 2 or 2-MeTHF) at 298 K that becomes very intense at 77 K (Fig. S20, ESI †). Similarly, the pyrazolato complex 4 is brightly emissive in the solid state or in fluid media at 298 K, while under identical conditions, the C^N py ^C analogue is only weakly emissive. 11 The pyrazine ligand framework of type B therefore significantly widens the range of photoemissive metal-ligand combinations.

In solution, the neutral complexes 2-5 all emit in the yellow to green region of the spectrum, mainly due to a triplet state based on the diphenylpyrazine pincer ligand. The excited state lifetimes show biexponential decay, with the fast component in the range of 5-20 ns.

At 77 K the emissions of 2, both in the solid state (l max em 563 nm) and in solution (l max em = 532 nm), agree reasonably well with the value calculated for a T 1 -S 0 transition (l calc em = 541 nm). ‡ However, at 298 K the emission shows a significant blue-shift, to 482 nm, a feature that is even more pronounced on addition of acid (vide infra).

One of the most important challenges in the design of photoluminescent devices is the ability to modulate the energy of the emitted light. In the pyrazine ligand system, the easiest way to achieve this is by making use of the non-coordinating nitrogen of the pyrazine ring. Strong Brønsted and Lewis acids do indeed produce dramatic changes in PL response. In order to eliminate any possible anion effects, initial protonation studies were carried out using the solid Brønsted acid [H(OEt 2 ) 2 ][H 2 N{B(C 6 F 5 ) 3 } 2 ] 12 (''HNB 2 ''). As shown in Fig. 1a, protonating 2 with this acid results in a blue-shift compared with the neutral complex, from 482 to 458 nm. On the other hand, at 77 K the emissions of both the neutral and protonated compounds are remarkable similar (as expected since both the HOMO and LUMO are based on the pyrazine ligand and are equally affected by protonation) and the emission is due to an intra-ligand charge transfer ( 3 ILCT) process. 11), Au-C4 2.078; N1-Au-C4 81.28( 8), C4-Au-Cl 98.72( 8), C4-Au-C4 0 162.56( 16), N1-Au-Cl 180. 4: Au-N1 1.979( 2), Au-C6 2.080(3), Au-C16 2.073(3), Au-N3 1.991( 2); C6-Au-N1 81.39( 10), C16-Au-N1 81.32( 11), C6-Au-N3 98.88 (10), C16-Au-N3 98.38(11), N1-Au-N3 178.33 (9). 5a: Au-N1 1.997( 5), Au-C6 2.070( 6), Au-C16 2.083( 7), Au-C25 1.979( 7), C25-C26 1.199( 9); N1-Au-C6 80.4( 2), N1-Au-C16 81.2( 2), C6-Au-C25 99.3(3), C16-Au-C25 99.0(3), C6-Au-C16 161.6(3), Au-C25-C26 175.9( 6), C25-C26-C27 177.4 (7).

This journal is © The Royal Society of Chemistry 2015 Chem. Commun.

The temperature-dependence of the photoluminescence response is in agreement with a TADF process, 13 where the energy difference between the T 1 and S 1 excited states is sufficiently small to allow thermally driven repopulation of the S 1 state from the T 1 state, with the faster decay occurring via the S 1 -S 0 transition. In agreement with this mechanism, the quantum yields for the protonated pyrazine systems are about an order of magnitude higher than those of the neutral complexes.

via a TADF mechanism are common for copper(I); 14 but have not been previously reported for Au(III).

The alkynyl complexes 5a and 5b show similar behaviour: the gradual addition of triflic acid results in the progressive growth of a blue emission (l max em = 450 nm for 5a, Fig. 1b). Emissions involve a mixture of states, with both blue and green components, which combine to appear white.

The same effect can be achieved with the Lewis acid B(C 6 F 5 ) 3 . In this case the high energy component appears at slightly lower energy (l max em = 481 nm), to give a more pronounced blue effect. Whereas in the case of the chloride 2 both HOMO and LUMO were based on the pincer ligand, for 5a the excitation involves metal-perturbed ligand-to-ligand p(CRCPh)p*(C^N pz ^C) charge transfer ( 3 LLCT, also involving HOMOÀ2). The effect of adding H + or B(C 6 F 5 ) 3 to the pyrazine N-atom is therefore pronounced. DFT calculations predict a strong red shift on forming the 5a-H + cation since on protonation the HOMO-LUMO gap is reduced (e.g. calculated HOMO-LUMO difference for 5a = 5.963 eV, compared to 4.878 eV after H + addition). In agreement with this, solutions of 5aÁB(C 6 F 5 ) 3 at 77 K display an intense emission at 569 nm (Fig. S26, ESI †) which is gradually deactivated on warming, while at the same time a high energy component increases, resulting in the blue emission at l max em = 481 nm at 298 K. As can be seen (Fig. S29, ESI †), these reactions also occur in the solid state.

Different metal ions also lead to distinctive modulations of the PL emission, arising from a mixture of states. The addition of ZnCl 2 , CdCl 2 and HgCl 2 (1 : 1 molar ratio) results in white, turquoise and light green emissions, respectively (Fig. 1c). The gradual addition of ZnCl 2 to a solution of the complex 5a in 2-MeTHF, results in the activation of the blue TADF component at the expense of the green phosphorescence (see Fig. S27, ESI †). Cooling to 77 K recovers the phosphorescent emission.

The addition of AgOTf or CuOTf also enhances the emission via the TADF mechanism (see Fig. 1d for Ag + and Fig. S28, ESI † for Cu + ). Freezing 2-MeTHF solutions of 5a/AgOTf to 77 K reveals the presence of two different systems; complex 5a (l max em = 521 nm), and a second complex characterized by a band with maximum at 545 nm. The mixture of 5a with two equivalents and with an excess of Ag + both show the same emission band, while 1 equivalent of Ag + gives two bands of about equal intensity. For these reasons, we tentatively suggest the formation of an aggregate of 5a with two Ag + ions (e.g. coordination of two Ag + ions to the CRC bond and the pyrazine-N atom may be envisaged).

The N-methylated complex 7 mirrors the behaviour of the H + adducts and shows blue emission in solution (l max em = 460 nm) but is dark-red in the solid state (l em = 623 max , 680sh).

In summary, cyclometallated gold(III) pincer complexes based on pyrazine provide a new family of photoluminescent compounds which allow facile modulation of the emission characteristics by protonation, alkylation, Lewis acids or metal ions, without the need for modifying the pincer ligand framework. The modulation arises from the coexistence of high energy TADF and 3 IL (C^N pz ^C)/ 3 LLCT (X -C^N pz ^C) transitions.

This work was supported by the European Research Council. M. B. is an ERC Advanced Investigator Award holder (grant no. 338944-GOCAT).

Notes and references ‡ In the solid state at room temperature, complex 2 is non-emissive, most probably due to close intermolecular interactions in the crystal lattice which quench the luminescence. This notion is supported by the observed higher solid-state emission intensity of 5b (l max = 523 nm, j = 8.3) compared to 5a (l max = 523 nm, j = 4.5); the crystal packing shows that for 5b intermolecular p-stacking is disfavoured on steric grounds. In solution at 298 K the PL intensities of 5a (l max = 526 nm, j = 0.462) and 5b (l max = 526 nm, j = 0.512) are essentially identical (see Fig. S19, ESI †).

Scheme 1

 1 Scheme 1 Conditions: (i) KAuCl 4 , CH 3 CN/H 2 O 1 : 1, reflux 72 h. (ii) KCN, CH 2 Cl 2 , 20 1C, 24 h. (iii) KO t Bu, 3,5-dimethylpyrazole, toluene, 20 1C, 6 h. (iv) AgCRCR, CH 2 Cl 2 , 20 1C, 24 h. (v) HBF 4 ÁEt 2 O, Et 2 O, 20 1C, 30 min. (6), or [Me 3 O]BF 4 , toluene/1,2-C 6 H 4 F 2 (5 : 1 v/v), 12 h. (7). Molecular structures [selected bond distances (Å) and angles (1)]: 2: Au-N1 1.972(3), Au-Cl 2.2651(11), Au-C4 2.078; N1-Au-C4 81.28(8), C4-Au-Cl 98.72(8), C4-Au-C4 0 162.56(16), N1-Au-Cl 180. 4: Au-N1 1.979(2), Au-C6 2.080(3), Au-C16 2.073(3), Au-N3 1.991(2); C6-Au-N1 81.39(10), C16-Au-N1 81.32(11), C6-Au-N3 98.88(10), C16-Au-N3 98.38(11), N1-Au-N3 178.33(9). 5a: Au-N1 1.997(5), Au-C6 2.070(6), Au-C16 2.083(7), Au-C25 1.979(7), C25-C26 1.199(9); N1-Au-C6 80.4(2), N1-Au-C16 81.2(2), C6-Au-C25 99.3(3), C16-Au-C25 99.0(3), C6-Au-C16 161.6(3), Au-C25-C26 175.9(6), C25-C26-C27 177.4(7).


Daniele A Di

Marien-Lorenzo Hanot
email: marien-lorenzo.hanot@umontpellier.fr

Uniform Poincaré inequalities for the Discrete de Rham complex on general domains

Keywords: Discrete de Rham complex, polytopal methods, Poincaré inequalities MSC2020. 65N30, 65N99, 14F40

In this paper we prove Poincaré inequalities for the Discrete de Rham (DDR) sequence on a general connected polyhedral domain Ω of R 3 . We unify the ideas behind the inequalities for all three operators in the sequence, deriving new proofs for the Poincaré inequalities for the gradient and the divergence, and extending the available Poincaré inequality for the curl to domains with arbitrary second Betti numbers. A key preliminary step consists in deriving "mimetic" Poincaré inequalities giving the existence and stability of the solutions to topological balance problems useful in general discrete geometric settings. As an example of application, we study the stability of a novel DDR scheme for the magnetostatics problem on domains with general topology.

Introduction

Poincaré inequalities are a key tool to prove the well-posedness of many common partial differential equation problems. Mimicking them at the discrete level is typically required for the stability of numerical approximations. Poincaré inequalities for conforming Finite Element de Rham complexes can be derived through bounded cochain projections as described, e.g., in [2, Chapter 5]; see also [START_REF] Christiansen | Poincaré-Friedrichs inequalities of complexes of discrete distributional differential forms[END_REF] for a recent generalisation. In the context of Virtual Element de Rham complexes [START_REF] Beirão Da Veiga | A family of three-dimensional virtual elements with applications to magnetostatics[END_REF], similar results typically hinge on non-trivial norm comparison results, examples of which can be found in [START_REF] Beirão Da Veiga | Virtual elements for Maxwell's equations[END_REF]. Discrete Poincaré-type inequalities in the context of the (non-compatible) Hybrid High-Order methods have been derived, e.g., in [START_REF] Di Pietro | The Hybrid High-Order method for polytopal meshes[END_REF] (gradient), [START_REF] Botti | A Hybrid High-Order method for nonlinear elasticity[END_REF] (symmetric gradient) and [START_REF] Chave | A discrete Weber inequality on three-dimensional hybrid spaces with application to the HHO approximation of magnetostatics[END_REF][START_REF] Lemaire | Discrete Weber inequalities and related Maxwell compactness for hybrid spaces over polyhedral partitions of domains with general topology[END_REF] (curl).

The focus of the present work is on the derivation of Poincaré inequalities for the Discrete de Rham (DDR) sequence of [START_REF] Di Pietro | An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness, Poincaré inequalities, and consistency[END_REF] on domains with general topology. Unlike Finite and Virtual Elements, DDR formulations are fully discrete, with spaces spanned by vectors of polynomials and continuous vector calculus operators replaced by discrete counterparts. Discrete Poincaré inequalities thus require to bound 2 -like norms of vectors of polynomials with 2 -like norms of suitable discrete operators applied to them. To establish such bounds, we take inspiration from [START_REF] Di Pietro | Cohomology of the discrete de Rham complex on domains of general topology[END_REF], where it was noticed that the topological information is fully contained in the lowest-order DDR subsequence, and [START_REF] Di Pietro | An arbitrary-order method for magnetostatics on polyhedral meshes based on a discrete de Rham sequence[END_REF], where a Poincaré inequality for the curl on topologically trivial domains of R 3 was derived. The lowest-order DDR sequence is strongly linked to Mimetic Finite Differences and related methods [START_REF] Beirão Da Veiga | The mimetic finite difference method for elliptic problems[END_REF][START_REF] Bonelle | Low-order reconstruction operators on polyhedral meshes: Application to Compatible Discrete Operator schemes[END_REF][START_REF] Bonelle | Analysis of compatible discrete operator schemes for elliptic problems on polyhedral meshes[END_REF][START_REF] Brezzi | Mimetic finite differences for elliptic problems[END_REF][START_REF] Brezzi | Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes[END_REF][START_REF] Codecasa | A new set of basis functions for the discrete geometric approach[END_REF]. The first step to prove discrete Poincaré inequalities in DDR spaces is thus precisely to establish the mimetic counterparts stated in Theorems 4, 6, and 7 below. Their proofs require to work at the global level, with conditions accounting for the topology of the domain appearing for the curl. The discrete Poincaré inequalities for arbitrary-order DDR spaces collected in Section 2.7 below are then obtained combining the mimetic Poincaré inequalities with local estimates of the higher-order components.

We next briefly discuss the links between the present work and previous results for DDR methods. Fully general Poincaré inequalities for the gradient and the divergence had already been obtained, respectively, in [START_REF] Di Pietro | An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness, Poincaré inequalities, and consistency[END_REF]Theorem 3] and [START_REF] Di Pietro | An arbitrary-order method for magnetostatics on polyhedral meshes based on a discrete de Rham sequence[END_REF] using different techniques. The main novelty of the proofs provided here is that they are better suited to generalisations in the framework of the Polytopal Exterior Calculus recently introduced in [START_REF] Bonaldi | An exterior calculus framework for polytopal methods[END_REF]. A Poincaré inequality for the curl on topologically trivial domains had been obtained in [START_REF] Di Pietro | An arbitrary-order method for magnetostatics on polyhedral meshes based on a discrete de Rham sequence[END_REF]Theorem 20]. The main novelty with respect to this result consists in the extension to domains encapsulating voids. The interest of the material in this paper is additionally that it contains preliminary results to establish discrete Poincaré inequalities for advanced complexes, such as the three-dimensional discrete div-div complex recently introduced in [START_REF] Di Pietro | A discrete three-dimensional divdiv complex on polyhedral meshes with application to a mixed formulation of the biharmonic problem[END_REF].

The rest of the paper is organized as follows. The definitions of the relevant DDR spaces and operators are briefly recalled in Section 2. Mimetic Poincarés inequalities are derived in Section 3, and then used to prove discrete Poincarés inequalities for the DDR complex in Section 4. The latter are used in Section 5 to carry out the stability analysis of a DDR scheme for the magnetostatics problem on domains with general topology. Some arguments in the proofs of mimetic Poincaré inequalities rely on specific shape functions for Finite Element spaces on a submesh, whose definitions and properties are summarised in Appendix A.

Discrete de Rham construction

Domain and mesh

Let Ω ⊂ R 3 denote a connected polyhedral domain. We consider a polyhedral mesh M ℎ ≔ T ℎ ∪ F ℎ ∪ E ℎ ∪V ℎ , where T ℎ gathers the elements, F ℎ the faces, E ℎ the edges, and V ℎ the vertices. For all ∈ M ℎ , we denote by ℎ its diameter and set ℎ ≔ max ∈ T ℎ ℎ . For each face ∈ F ℎ , we fix a unit normal to and, for each edge ∈ E ℎ , a unit tangent . For ∈ T ℎ , F gathers the faces on the boundary of and E the edges in ; if ∈ F ℎ , E is the set of edges contained in the boundary of . For ∈ F , ∈ {-1, +1} is such that is the outer normal on to . Each face ∈ F ℎ is oriented counter-clockwise with respect to and, for ∈ E , we let ∈ {-1, +1} be such that = +1 if points along the boundary of in the clockwise sense, and = -1 otherwise; we also denote by the unit normal vector to , in the plane spanned by , such that points outside . We denote by grad and div the tangent gradient and divergence operators acting on smooth enough functions. Moreover, for any : → R and : → R 2 smooth enough, we let rot ≔ (grad ) ⊥ and rot = div ( ⊥ ), with ⊥ denoting the rotation of angle -2 in the oriented tangent space to .

We further assume that (T ℎ , F ℎ ) belongs to a regular mesh sequence in the sense of [18, Definition 1.9], with mesh regularity parameter > 0. This implies that, for each ∈ T ℎ ∪ F ℎ ∪ E ℎ , there exists a point ∈ such that the ball centered at and of radius ℎ is contained in . Throughout the paper, (resp., ) stands for ≤ (resp., ≥ ) with depending only on Ω, the mesh regularity parameter and, when polynomial functions are involved, the corresponding polynomial degree. We also write when both and hold.

Polynomial spaces and 2 -orthogonal projectors

For any ∈ M ℎ and an integer ℓ ≥ 0, we denote by P ℓ ( ) the space spanned by the restriction to of polynomial functions of the space variables. Let, for ∈ T ℎ ∪ F ℎ , P ℓ ( ) ≔ P ℓ ( ) with denoting the dimension of . We have the following direct decompositions: For all ∈ F ℎ ,

P ℓ ( ) = R ℓ ( ) ⊕ R c,ℓ ( ) with R ℓ ( ) ≔ rot P ℓ+1 ( ) and R c,ℓ ( ) ≔ ( -)P ℓ -1 ( )
and, for all ∈ T ℎ ,

P ℓ ( ) = G ℓ ( ) ⊕ G c,ℓ ( ) with G ℓ ( ) ≔ grad P ℓ+1 ( ) and G c,ℓ ( ) ≔ ( -) × P ℓ -1 ( ) = R ℓ ( ) ⊕ R c,ℓ ( ) with R ℓ ( ) ≔ curl P ℓ+1 ( ) and R c,ℓ ( ) ≔ ( -)P ℓ -1 ( ).
We extend the above notations to negative exponents ℓ by setting all the spaces appearing in the decompositions equal to the trivial vector space {0}. Given a polynomial (sub)space X ℓ ( ) on ∈ M ℎ , the corresponding 2 -orthogonal projector is denoted by ℓ X, . Boldface font will be used when the elements of X ℓ ( ) are vector-valued, and c,ℓ X, will denote the 2 -orthogonal projector on X c,ℓ ( ).

DDR spaces

The discrete counterparts of the spaces appearing in the continuous de Rham complex are defined as follows:

grad,ℎ ≔ ℎ = ( ) ∈ T ℎ , ( ) ∈ F ℎ , ( ) ∈ E ℎ , ( ) ∈ V ℎ :
∈ P -1 ( ) for all ∈ T ℎ , ∈ P -1 ( ) for all ∈ F ℎ , ∈ P -1 ( ) for all ∈ E ℎ , and

∈ R for all ∈ V ℎ , curl,ℎ ≔ ℎ = ( R, , c R, ) ∈ T ℎ , ( R, , c R, ) ∈ F ℎ , ( ) ∈ E ℎ : R, ∈ R -1 ( ) and c R, ∈ R c, ( ) for all ∈ T ℎ , R, ∈ R -1 ( ) and c R, ∈ R c, ( ) for all ∈ F ℎ , and ∈ P ( ) for all ∈ E ℎ , div,ℎ ≔ ℎ = ( G, , c G, ) ∈ T ℎ , ( ) ∈ F ℎ :
G, ∈ G -1 ( ) and c G, ∈ G c, ( ) for all ∈ T ℎ , and ∈ P ( ) for all ∈ F ℎ , and P (T ℎ ) ≔ ℎ ∈ 2 (Ω) : ( ℎ ) | ∈ P ( ) for all ∈ T ℎ .

Local vector calculus operators and potentials

Gradient

For any ∈ E ℎ , the edge gradient : grad, → P ( ) is such that, for all ∈ grad, ,

∫ = - ∫ + , (1)
with derivative taken in the direction of and with • denoting the difference between vertex values on an edge such that, for any function ∈ 0 ( ) and any family { 1 , 2 } of vertex values such that points from 1 to 2 , ≔ 2 ( 2 ) -1 ( 1 ).

For any ∈ F ℎ , the face gradient : grad, → P ( ) and the scalar trace +1 : grad, → P +1 ( ) are such that, for all ∈ grad, ,

∫ • = - ∫ div + ∈ E ∫ ( • ) ∀ ∈ P ( ), (2)
∫ +1 div = - ∫ • + ∈ E ∫ ( • ) ∀ ∈ R c, +2 ( ).
Similarly, for all ∈ T ℎ , the element gradient : grad, → P ( ) is defined such that, for all ∈ grad, ,

∫ • = - ∫ div + ∈ F ∫ +1 ( • ) ∀ ∈ P ( ), (3)

Curl

For all ∈ F ℎ , the face curl : curl, → P ( ) and tangential trace t, : curl, → P ( ) are such that, for all ∈ curl, ,

∫ = ∫ R, • rot - ∈ E ∫ ∀ ∈ P ( ) (4)
and, for all ( ,

) ∈ P +1 0 ( ) × R c, ( ), ∫ t, • (rot + ) = ∫ + ∈ E ∫ + ∫ c R, • .
For all ∈ T ℎ , the element curl : curl, → P ( ) is defined such that, for all ∈ curl, ,

∫ • = ∫ R, • curl + ∈ F ∫ t,
• ( × ) ∀ ∈ P ( ).

(5)

Divergence

For all ∈ T ℎ , the element divergence : div, → P ( ) is defined by: For all ∈ div, ,

∫ = - ∫ G, • grad + ∈ F ∫ ∀ ∈ P ( ). (6)

DDR complex

The DDR complex reads:

0 grad,ℎ curl,ℎ div,ℎ P (T ℎ ) {0}, ℎ ℎ ℎ 0
where, for all

( ℎ , ℎ , ℎ ) ∈ grad,ℎ × curl,ℎ × div,ℎ , ℎ ℎ ≔ ( -1 R, , c, R, ) ∈ T ℎ , ( -1 R, , c, R, ) ∈ F ℎ , ( ) ∈ E ℎ , (7)
ℎ ℎ ≔ ( -1 G, , c, G, ) ∈ T ℎ , ( ) ∈ F ℎ , (8)
( ℎ ℎ ) | ≔ ∀ ∈ T ℎ .

Component norms

We endow the discrete spaces defined in Section 2.3 with the 2 -like norms defined as follows: For all

( ℎ , ℎ , ℎ ) ∈ grad,ℎ × curl,ℎ × div,ℎ , ||| ℎ ||| 2 grad,ℎ ≔ ∈ T ℎ ||| ||| 2 grad, with ||| ||| 2 grad, ≔ 2 2 ( ) + ℎ ∈ F ℎ ||| ||| 2 ∀ ∈ T ℎ , ||| ||| 2 grad, ≔ 2 2 ( ) + ℎ ∈ E ℎ ||| ||| 2 grad, ∀ ∈ F ℎ , ||| ||| 2 grad, ≔ 2 2 ( ) + ℎ ∈ V | | 2 ∀ ∈ E ℎ , ||| ℎ ||| 2 curl,ℎ ≔ ∈ T ℎ ||| ||| 2 curl, with ||| ||| 2 curl, ≔ R, 2 2 ( ;R 3 ) + c R, 2 2 ( ;R 3 ) + ℎ ∈ F ||| ||| 2 curl, ∀ ∈ T ℎ , ||| ||| 2 curl, ≔ R, 2 2 ( ;R 2 ) + c R, 2 2 ( ;R 2 ) + ℎ ∈ E 2 2 ( ) ∀ ∈ F ℎ , (9)
and

||| ℎ ||| 2 div,ℎ ≔ ∈ T ℎ ||| ||| 2 div, with ||| ||| 2 div, ≔ G, 2 2 ( ;R 3 ) + c G, 2 2 ( ;R 3 ) + ℎ ∈ F 2 2 ( ) ∀ ∈ T ℎ . (10)

Main results

Theorem 1 (Poincaré inequality for the gradient). For all ℎ ∈ grad,ℎ , it holds

inf ℎ ∈Ker ℎ ||| ℎ -ℎ ||| grad,ℎ ||| ℎ ℎ ||| curl,ℎ ,
with hidden constant only depending on Ω, the mesh regularity parameter, and .

Proof. See Section 4.1.

Theorem 2 (Poincaré inequality for the curl). For all

ℎ ∈ curl,ℎ , it holds inf ℎ ∈Ker ℎ ||| ℎ - ℎ ||| curl,ℎ ||| ℎ ℎ ||| div,ℎ ,
with hidden constant only depending on Ω, the mesh regularity parameter, and .

Proof. See Section 4.2.

Theorem 3 (Poincaré inequality for the divergence). For all

ℎ ∈ div,ℎ , it holds inf ℎ ∈Ker ℎ ||| ℎ - ℎ ||| ℎ ,ℎ ℎ ℎ 2 (Ω) ,
with hidden constant only depending on Ω, the mesh regularity parameter, and .

Proof. See Section 4.3.

Mimetic Poincaré inequalities

This section contains Poincaré inequalities in mimetic spaces that are instrumental in proving the main results stated in the previous section. Their proofs rely on the use of a tetrahedral submesh ℎ = ℎ ∪ ℎ ∪ ℎ ∪ ℎ in the sense of [START_REF] Di Pietro | The Hybrid High-Order method for polytopal meshes[END_REF]Definition 1.8], with ℎ collecting the tetrahedral subelements and ℎ , ℎ , and ℎ their faces, edges, and vertices, respectively. We assume, for the sake of simplicity, that this submesh can be obtained adding as new vertices only centers of the faces and elements of M ℎ . As a result of the assumptions in [START_REF] Di Pietro | The Hybrid High-Order method for polytopal meshes[END_REF]Definition 1.8], the regularity parameter of the submesh only depends on that of M ℎ and, for a given element ∈ T ℎ , the diameters of the submesh entities contained in are comparable to ℎ uniformly in ℎ.

Mimetic Poincaré inequality for collections of vertex values Theorem 4 (Mimetic Poincaré inequality for collections of vertex values). Let

( ) ∈ V ℎ ∈ R V ℎ be a collection of values at vertices. Then, there is ∈ R such that ∈ T ℎ ℎ 3 ∈ V ( -) 2 ∈ T ℎ ℎ ∈ E | | 2 , ( 11
)
with hidden constant only depending on Ω and the mesh regularity parameter.

Proof. We extend the collection ( ) ∈ V ℎ to ℎ setting the values at face/element centers equal to the value taken at an arbitrary vertex of the face/element in question.

For any simplex ∈ ℎ and any vertex ∈ (with collecting the vertices of ), let , denote the restriction to of the piecewise affine "hat" function associated with given by (77), and let ℎ ∈ 1 (Ω) be the piecewise polynomial function defined by setting

( ℎ ) | ≔ ∈ ( -) , ∀ ∈ ℎ , (12)
where ∈ R is chosen so that the zero-average condition ∫ Ω ℎ = 0 is satisfied. We next prove the following norm equivalences:

ℎ 2 2 (Ω) ∈ T ℎ ℎ 3 ∈ V ( -) 2 , ( 13
)
grad ℎ 2 2 (Ω;R 3 ) ∈ T ℎ ℎ ∈ E | | 2 . ( 14
)
The conclusion follows from the above relations writing

∈ T ℎ ℎ 3 ∈ V ( -) 2 (13) ℎ 2 2 (Ω) grad ℎ 2 2 (Ω;R 3 ) (14)
∈ T ℎ ℎ ∈ E | | 2 ,
where the second inequality follows from the continuous Poincaré-Wirtinger inequality, which holds since [START_REF] Chave | A discrete Weber inequality on three-dimensional hybrid spaces with application to the HHO approximation of magnetostatics[END_REF]. For any ∈ T ℎ , by regularity of the submesh, we have ℎ ℎ for all ∈ (with collecting the subelements contained in ). It holds

∫ Ω ℎ = 0. (i) Proof of
ℎ 2 2 (Ω) (12) = ∈ T ℎ ∈ ∫ ∈ ( -) , 2 ∈ T ℎ ∈ ∈ ( -) 2 , 2 2 ( ) (82)
∈ T ℎ ℎ 3 ∈ ∈ ( -) 2 ∈ T ℎ ℎ 3 ∈ V ( -) 2 , (15)
where the second equivalence follows from the fact that card( ) 1, in the third one we have additionally used ℎ ℎ for all ∈ T ℎ and all ∈ , and the last one is justified by the choice we made at the beginning for , ∈ ℎ \ V ℎ . This readily gives [START_REF] Chave | A discrete Weber inequality on three-dimensional hybrid spaces with application to the HHO approximation of magnetostatics[END_REF].

(ii) Proof of [START_REF] Christiansen | Poincaré-Friedrichs inequalities of complexes of discrete distributional differential forms[END_REF]. The key argument to obtain ( 14) lies in the de Rham theorem: Let ( , ) ∈ (with collecting the edges of ) be the basis for the edge Nédélec space given by (78). Then, summing (86), we have

grad ∈ , = ∈ , . (16)
Starting from [START_REF] Di Pietro | An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness, Poincaré inequalities, and consistency[END_REF] and proceeding in a similar way as in ( 15) with ( 83) replacing (82), we have

∫ Ω grad ℎ 2 ∈ T ℎ ℎ ∈ ∈ | | 2 , (17)
with, for any ∈ R 3 , denoting the Euclidian norm of . Now, for any edge ∈ of any simplex ∈ , either ∈ E or, by the choice made at the beginning of this proof for with face or element center, can be computed as the sum of jumps along the boundary of (i.e.

= ∈ E for ∈ {-1, 0, 1}). Therefore, ∈ ∈ | | 2 ∈ E | | 2 ,
and we infer ( 14) from (17).

Mimetic Poincaré inequality for collections of edge values

If the topology of the domain is non-trivial, a suitable condition for each void must be satisfied in order to establish a mimetic Poincaré inequality for collections of edge values. Denote by 2 the second Betti number, i.e., the number of voids encapsulated by Ω. Let (F ) 1≤ ≤ 2 denote collections of boundary faces such that

≔ ∈ F
is the boundary if the th void. We start by proving a necessary and sufficient condition under which a function in the lowest-order Raviart-Thomas-Nédélec face space on the tetrahedral submesh is the curl of a function in the edge Nédélec space on the same mesh.

Lemma 5 (Condition on the cohomology). Denote by RT 1 ( ℎ ) and N 1 ( ℎ ) lowest-order face and edge finite element spaces on the submesh. Then, for all ℎ ∈ RT 1 ( ℎ ), there exists ℎ ∈ N 1 ( ℎ ) such that ℎ = curl ℎ if and only if

div ℎ = 0 and ∈ F ∫ ℎ • Ω = 0 for all integer such that 1 ≤ ≤ 2 , ( 18
)
with Ω denoting the unit normal vector to the boundary of Ω pointing out of Ω.

Proof. To check that ( 18) is necessary, notice that the first condition comes from the identity div curl = 0, while the second one follows from Green's theorem along with the fact that the flux of the curl of any function across a closed boundary is zero.

We next prove that condition ( 18) is sufficient using a counting argument. From the de Rham Theorem, we know that that the dimension of the space of harmonic forms is precisely the number of voids 2 . For all integer such that 1 ≤ ≤ 2 , we define the linear form such that, for any vector-valued function smooth enough,

( ) ≔ - ∈ F ∫ • Ω .
For any 1 ≤ ≤ 2 , let ∈ R 3 be any point inside the th void and consider the function :

R 3 ↦ → - - 3 ∈ R 3 .
Noticing that div = 0 and applying the divergence theorem inside the th void, we infer that ( ) = 0 if ≠ . Let us now show that ( ) > 0. Let > 0 be the distance between and Ω. Denoting by S the sphere of radius 2 centred in , we have that

∫ S • ( -) - = ∫ S 4 -2 = 4 > 0.
Applying once again the divergence theorem to on the volume V enclosed between S and , we have that 0

= ∫ V div = ( ) - ∫ S • ( -) - . Therefore, ( ) = ∫ S • ( -) - > 0. (19)
Denoting by 1 R T,ℎ the canonical interpolator onto R T 1 ( ℎ ), we know that, for any function , div( 1 R T,ℎ ) = ℎ P,0 (div ) and ( 1 R T,ℎ ) = ( ) by definition of the interpolator. Therefore, for any integer such that

1 ≤ ≤ 2 , 1 R T,ℎ
is a discrete harmonic form and, by a counting argument, the linearly independent family ( 1 R T,ℎ ) 1≤ ≤ 2 spans the space of discrete harmonic forms. Let now ℎ be such that div ℎ = 0. Then,

ℎ = curl ℎ + 2 =1 1 R T,ℎ
for some ℎ ∈ N 1 ( ℎ ) and ( ) 1≤ ≤ 2 ∈ R 2 . We prove that the condition ( ) = 0 for all 1 ≤ ≤ 2 is sufficient to ensure that ℎ is in the range of curl by contradiction. As a matter of fact, if this were not the case, then there would be 0 such that 0 ≠ 0. However, by [START_REF] Di Pietro | Cohomology of the discrete de Rham complex on domains of general topology[END_REF], this would also imply 0 ( ℎ ) = 0 0 ( 1 R T,ℎ 0 ) ≠ 0, which is the sought contradiction.

∈ F Ω = 0 for all integer such that 1 ≤ ≤ 2 ,
where Ω ∈ {-1, 1} is such that Ω points outside the domain Ω. Then, there is a collection

( ) ∈ E ℎ ∈ R E ℎ of values at edges such that, for all ∈ F ℎ , ∈ E = and ∈ T ℎ ℎ ∈ E 2 ∈ T ℎ ℎ -1 ∈ F 2 ( 21
)
with hidden constant only depending on Ω and the mesh regularity parameter.

Proof. Let ( , ) ∈ ℎ , ∈ and ( , ) ∈ ℎ , ∈ (with denoting the set of triangular faces of ) be the families of basis functions respectively given by (78) and (79) below. The main difficulty is to extend the family ( ) ∈ F ℎ to a family ( ) ∈ ℎ satisfying, for all ∈ ℎ , ∈ = 0.

We perform the construction locally on each element ∈ T ℎ . Let ∈ 2 ( ) be the piecewise constant function on such that

| ≔ - 1 | | ∈ ∩F ∀ ∈ . (22)
Recalling [START_REF] Di Pietro | A discrete three-dimensional divdiv complex on polyhedral meshes with application to a mixed formulation of the biharmonic problem[END_REF], by definition we have ∫ = 0. Hence, using Lion's Lemma [1, Theorem 3.1.e], we infer the existence of ∈ 1 0 ( ; R 3 ) such that

div = and | | 1 ( ;R 3 ) 2 ( ) ℎ -3 2 ∈ F | |, (23)
where the last inequality follows from the definition of after observing that | | ℎ 3 for all ∈ by mesh regularity. Setting ≔ ∫ • (with denoting the unit normal vector to , with orientation consistent with that of ), for all ∈ and all ∈ \ F . We infer that, for all ⊂ ℎ , ∈ = 0, noticing that

∈ = ∈ \F + ∈ ∩F = ∈ ∫ • + ∈ ∩F = ∫ div + ∈ ∩F (23), (22)
= -

∈ ∩F + ∈ ∩F = 0,
where we have used the fact that • = 0 on every face ∈ F lying on the boundary of to obtain the second equality. Let ≔ 1

| |

∫ and, for all integer such that 1 ≤ ≤ 3, denote by and the th components of and , respectively. It holds

| | = ∫ = ∫ • = ∫ • grad( - • ) = - ∫ div ( - • ) = - ∫ ( - • ) ℎ ∈ F | |, so that, recalling that | | ℎ 3 , 2 ( ;R 3 ) ℎ -1 2 ∈ F | |. (24)
Therefore, we can use the Poincaré inequality on the domain to write

2 ( ;R 3 ) ≤ - 2 ( ;R 3 ) + 2 ( ;R 3 ) ℎ | | 1 ( ;R 3 ) + 2 ( ;R 3 ) (23), (24) ℎ -1 2 ∈ F | |.
Combining this result with the continuous trace inequality we have, for all ∈ and all ∈ \ F ,

| | ℎ 2 ( ;R 3 ) ℎ 1 2 2 ( ;R 3 ) + ℎ 3 2 | | 1 ( ;R 3 ) ∈ F | |.
Therefore, summing over all tetrahedra inside and all tetrahedral faces, we obtain

∈ ∈ 2 ∈ F 2 . ( 25
)
We next define the following piecewise polynomial function:

ℎ ≔ ∈ ℎ ∈ , ∈ (div; Ω). (26)
Since div ℎ = 0 and, for all 1 ≤ ≤ 2 , ∈ F Ω ∫ ℎ • = 0, we can use Lemma 5 to infer from the uniform Poincaré inequality on the simplicial de Rham complex [START_REF] Arnold | Finite Element Exterior Calculus[END_REF] the existence of

ℎ ≔ ∈ ℎ ∈ , ∈ (curl; Ω) such that curl ℎ = ℎ and ℎ 2 (Ω;R 3 ) ℎ 2 (Ω;R 3 ) . ( 27
)
Summing (87), we have

curl ℎ = ∈ ℎ ∈ ∈ E , . (28)
Hence, equating ( 26) and (28), we infer that, for all ∈ F ℎ ⊂ ℎ , ∈ E = . Moreover, noticing that both , and , are only supported in , we have

ℎ 2 2 (Ω;R 3 ) = ∈ T ℎ ∈ ∫ ∈ , 2 ∈ T ℎ ∈ ∈ 2 , 2 2 ( ;R 3 ) (83) ∈ T ℎ ∈ ℎ ∈ 2 ∈ T ℎ ℎ ∈ E 2 (29) and ℎ 2 2 (Ω;R 3 ) = ∈ T ℎ ∈ ∫ ∈ , 2 ∈ T ℎ ∈ ∈ 2 , 2 2 ( ;R 3 ) (84) ∈ T ℎ ∈ ℎ -1 ∈ 2 (25) ∈ T ℎ ℎ -1 ∈ F 2 , (30)
where, in the last passage, we have additionally used the fact that ℎ -1 ℎ -1 for all ∈ by mesh regularity. Finally, to prove [START_REF] Lemaire | Discrete Weber inequalities and related Maxwell compactness for hybrid spaces over polyhedral partitions of domains with general topology[END_REF], it suffices to write

∈ T ℎ ℎ ∈ E 2 (29) ℎ 2 2 (Ω;R 3 ) (27) ℎ 2 2 (Ω;R 3 ) (30) ∈ T ℎ ℎ -1 ∈ F 2 .

Mimetic Poincaré inequality for collections of face values

Theorem 7 (Mimetic Poincaré inequality for collections of face values). Let ( ) ∈ T ℎ ∈ R T ℎ be a collection of values at elements. Then, there is a collection ( ) ∈ F ℎ ∈ R F ℎ of values at faces such that, for all ∈ T ℎ , ∈ F

= and

∈ T ℎ ℎ -1 ∈ F 2 ∈ T ℎ ℎ -3 2 (31)
with hidden constant only depending on Ω and the mesh regularity parameter.

Proof. Let ( ) ∈ ℎ and ( ) ∈ ℎ be the basis functions of the face Raviart-Thomas-Nédélec and of the fully discontinuous piecewise affine spaces on the tetrahedral submesh, respectively given by ( 79) and (80) below. Define the following piecewise polynomial function:

ℎ ≔ ∈ T ℎ ∈ | | | | . ( 32
)
We infer from the uniform Poincaré inequality on the simplicial de Rham complex [START_REF] Arnold | Finite Element Exterior Calculus[END_REF] the existence of

ℎ ≔ ∈ ℎ ∈ RT 1 ( ℎ ) ⊂ (div; Ω) such that div ℎ = ℎ and ℎ 2 (Ω;R 3 ) ℎ 2 (Ω) . (33)
Summing (88), on the other hand, we obtain

div ℎ = ∈ T ℎ ∈ ∈ , (34)
where is the orientation of relative to . Since each is supported in , we infer equating (32) and (34) that, for all ∈ T ℎ and all ∈ ,

| | | | = ∈ .
For all ∈ T ℎ , summing this relation over ∈ , we have

∈ | | | | = ∈ ∈ =⇒ ¨¨¨¨B 1 ∈ | | | | = ∈ F
, where we have used the fact that the contributions from the simplicial faces internal to cancel out in the right-hand side. It remains to check that (31) holds. Noticing that the only supported in are those associated with its faces collected in the set , we have

ℎ 2 2 (Ω;R 3 ) = ∈ T ℎ ∈ ∫ ∈ 2 ∈ T ℎ ∈ ∈ 2 2 2 ( ;R 3 ) (84) ∈ T ℎ ℎ -1 ∈ ∈ 2 ∈ T ℎ ℎ -1 ∈ F 2 , ( 35
)
where we have used ℎ -1 ℎ -1 (consequence of mesh regularity) in the third line. Likewise, we have

ℎ 2 2 (Ω) = ∈ T ℎ ∈ 2 | | 2 | | 2 2 2 ( ) (85) ∈ T ℎ ℎ -3 2 , ( 36
)
where we have used ℎ ℎ and | | | |. Finally, to prove (31), we write

∈ T ℎ ℎ -1 ∈ F 2 (35) ℎ 2 2 (Ω;R 3 ) (33) ℎ 2 2 (Ω) (36) ∈ T ℎ ℎ -3 2 .

Proofs of Poincaré inequalities in DDR spaces

Poincaré inequality for the gradient

We start with the following preliminary lemma. (37)

Proof. We provide an explicit definition of ℎ and check that (37) holds. Specifically, we let ℎ ∈ grad,ℎ be such that, for all ∈ E ℎ ,

= ∫ ∀ ∈ E ℎ , (38)
∫ = - ∫ + ∀ ∈ P 0 ( ), (39)
for all ∈ F ℎ ,

∫ div = - ∫ • + ∈ E ∫ ( • ) ∀ ∈ R c, ( ), (40)
and, for all ∈ T ℎ ,

∫ div + ∈ F ∫ ( • ) ∀ ∈ R c, ( ). (41)
Notice that the vertex values ( ) ∈ V ℎ are only defined up to a global constant and that (resp., ) is well-defined by condition (40) (resp., (41)) since div : R c, ( ) → P -1 ( ) (resp., div : R c, ( ) → P -1 ( )) is an isomorphism. The equality of the components associated with an element ∈ T ℎ is proved in a similar way. First, using again [7, Lemma 14], this time with ( , ) = (0, 3) (which corresponds to [16, Proposition 1]), we infer that

Equality of the discrete gradient. Let us first briefly show that

∫ • curl = - ∈ F ∫ • ( × ) for all ∈ G c, +1 ( ).
Accounting for (44), this yields R, = R, . Then, plugging the definition (41) of into (3), we get

c, R, = c, R,
. These equalities give

= ∀ ∈ T ℎ . (45)
Gathering ( 43), (44), and (45), and recalling the definition ( 7) of the discrete gradient (42) follows.

2. Continuity. Using the fact that, for all ∈ T ℎ , ℎ ℎ for all ∈ F ∪ E and that the number of faces of each element and of edges of each face is 1 by mesh regularity, we have

∈ T ℎ ℎ ∈ F ℎ ∈ E ℎ ∈ V | | 2 ∈ T ℎ ℎ 3 ∈ V | | 2 (11) ∈ T ℎ ℎ ∈ E | | 2 ,
where, to apply Theorem 4, we have used the fact that vertex values are defined up to a constant. Taking absolute values in (38) and using a Cauchy-Schwarz inequality in the right-hand side, on the other hand,

we obtain | | ℎ 1 2 2 ( ) ≤ ℎ 1 2
2 ( ) for all ∈ T ℎ such that ∈ E . Plugging this bound into the previous expression, we obtain

∈ T ℎ ℎ ∈ F ℎ ∈ E ℎ ∈ V | | 2 ∈ T ℎ ℎ 2 ∈ E 2 2 ( ) ||| ℎ ℎ ||| 2 curl,ℎ , (46)
where the last inequality is obtained recalling the definition (9) of |||•||| curl,ℎ with ℎ = ℎ ℎ and using mesh regularity.

Taking in (39) such that = , using Cauchy-Schwarz and trace inequalities in the right-hand side, simplifying, and squaring, we obtain

2 2 ( ) ℎ 2 2 2 ( ) + ℎ ∈ V | | 2 ,
so that, using the fact that ℎ ≤ ℎ for all ∈ E ℎ in the first term,

∈ T ℎ ℎ ∈ F ℎ ∈ E 2 2 ( ) ℎ 2 ∈ T ℎ ℎ ∈ F ℎ ∈ E 2 2 ( ) + ∈ T ℎ ℎ ∈ F ℎ ∈ E ℎ ∈ V | | 2 (9), ( 46
) ||| ℎ ℎ ||| 2 curl,ℎ , (47)
where, in the last inequality, we have additionally used the fact that ℎ ≤ diam(Ω) 1.

To estimate the face components, we first take in (40) such that div = (this is possible since div : R c, ( ) → P -1 ( ) is an isomorphism by [2, Corollary 7.3]) use Cauchy-Schwarz and trace inequalities in the right-hand side to obtain

2 2 ( ) ℎ 2 2 2 ( ;R 2 ) + ℎ ∈ E 2 2 ( ) ,
so that, using the fact that ℎ ≤ ℎ for all ∈ F ℎ in the first term,

∈ T ℎ ℎ ∈ F 2 2 ( ) ℎ 2 ∈ T ℎ ℎ ∈ F 2 2 ( ;R 2 ) + ∈ T ℎ ℎ ∈ F ℎ ∈ E 2 2 ( ) (9), (47)
||| ℎ ℎ ||| 2 curl,ℎ , (48)
where we have again used the fact that ℎ 1 for the first term. The estimate of the element components is entirely similar: taking in (41) ∈ R c, ( ) such that div = (which is possible since div : R c, ( ) → P -1 ( ) is an isomorphism again by [2, Corollary 7.3]), we obtain

2 2 ( ) ℎ 2 2 2 ( ;R 3 ) + ℎ ∈ F 2 2 ( ) .
Summing over ∈ T ℎ , using the fact that ℎ ≤ ℎ 1 for all ∈ T ℎ for the first term and (48) for the second, and recalling the definition (9) of |||•||| curl,ℎ , we arrive at

∈ T ℎ 2 2 ( ) ||| ℎ ℎ ||| 2 curl,ℎ . (49)
Summing ( 46), ( 47), (48), and (49), the continuity bound in (37) follows.

Proof of Theorem 1. Let ℎ ∈ grad,ℎ be given by Lemma 8. Noticing that

ℎ - ℎ ∈ Ker ℎ , we write inf ℎ ∈Ker ℎ ||| ℎ -ℎ ||| grad,ℎ ≤ ||| ℎ -( ℎ - ℎ )||| grad,ℎ = ||| ℎ ||| grad,ℎ (37) ||| ℎ ℎ
||| curl,ℎ .

Poincaré inequality for the curl

The proof of Theorem 2 is analogous to that of Theorem 1 provided we establish the following result.

Lemma 9 (Continuous inverse of the discrete curl). For any ℎ ∈ curl,ℎ , there is ℎ ∈ curl,ℎ such that

ℎ ℎ = ℎ ℎ and ||| ℎ ||| curl,ℎ ||| ℎ ℎ ||| div,ℎ . ( 50
) Proof. Let ≔ ∫ ∀ ∈ F ℎ . (51)
Recalling that ℎ ℎ ℎ = 0 and using the definition (6) of it holds, for all ∈ T ℎ ,

0 = ∫ = ∈ F ∫ = ∈ F ,
showing that (20) is met. If the domain encapsulates 2 > 0 voids, we infer using (4) for = 1, that, for all integers

1 ≤ ≤ 2 , ∈ F Ω = ∈ F Ω ∫ = - ∈ F Ω ∈ E ∫ = 0,
where we have used the fact that each edge in the summation appears exactly twice with opposite coefficients, and therefore cancels out. We can thus invoke Theorem 6 to infer the existence of a collection of values at edges ( ) ∈ R E ℎ satisfying [START_REF] Lemaire | Discrete Weber inequalities and related Maxwell compactness for hybrid spaces over polyhedral partitions of domains with general topology[END_REF]. We then take ∈ P 0 ( ) such that

∫ = ℎ = ∀ ∈ E ℎ . (52)
For all ∈ F ℎ , the face components R, ∈ R -1 ( ) and c R, ∈ R c, ( ) are selected such that

∫ R, • rot = ∫ + ∈ E ∫ ∀ ∈ P 0 ( ) (53) and c
R, = 0. (54)
Similarly, for any ∈ T ℎ , the element components R, ∈ R -1 ( ) and c R, ∈ R c, ( ) satisfy The equality of the projections on R c, ( ) results plugging (55) into the definition (5) of with test function ∈ G c, ( ). Gathering the above results, we obtain = for all ∈ T ℎ , from which ℎ ℎ = ℎ ℎ follows recalling the definition (8) of the discrete curl. 2. Continuity. Let us now show the bound in (50). Concerning edge components, we write

∫ R, • curl = ∫ • + ∈ F ∫ t, • ( × ) ∀ ∈ G c, ( ) (55)
∈ T ℎ ℎ ∈ F ℎ ∈ E 2 2 ( ) ∈ T ℎ ℎ ∈ E 2 ∈ T ℎ ℎ -1 ∈ F ∫ 2 ||| ℎ ℎ ||| 2 div,ℎ , (57)
where we have used (52) along with the fact that each edge ∈ E is shared by exactly two faces in F , the mimetic Poincaré inequality [START_REF] Lemaire | Discrete Weber inequalities and related Maxwell compactness for hybrid spaces over polyhedral partitions of domains with general topology[END_REF] together with the definition (51) of in the second passage, and

∫ | | 1 2 2 ( ) ℎ 2 ( ) ≤ ℎ 2 (
) for all ∈ F ℎ and all ∈ T ℎ to which belongs together with the definition [START_REF] Botti | A Hybrid High-Order method for nonlinear elasticity[END_REF] of |||•||| div,ℎ to conclude.

To estimate the face component, we let in (53) be such that rot = R, and use Cauchy-Schwarz, inverse, and trace inequalities in the right-hand side to infer R,

2 ( ;R 2 ) ℎ 2 ( ) + ℎ 1 2 ∈ E 2 ( ) .
Squaring the above relation and using standard inequalities for the square of a finite sum of terms, we obtain, after noticing that ℎ ≤ ℎ for all ∈ F ℎ ,

∈ T ℎ ℎ ∈ F R, 2 2 ( ;R 2 ) ℎ 2 ∈ T ℎ ℎ ∈ F 2 2 ( ) + ∈ T ℎ ℎ ∈ F ℎ ∈ E 2 2 ( ) ||| ℎ ℎ ||| 2 div,ℎ , (58)
where the conclusion follows noticing that ℎ ≤ diam(Ω) 1, recalling the definition [START_REF] Botti | A Hybrid High-Order method for nonlinear elasticity[END_REF] of |||•||| div,ℎ for the first term, and invoking (57) for the second one.

The estimate of the element component is obtained in a similar way, starting from (55) with such that curl = R, , leading to

∈ T ℎ R, 2 2 ( ;R 3 ) ||| ℎ ℎ ||| 2 div,ℎ . (59)
Summing ( 57), (58), and (59), recalling the definition (9) of ||| ℎ ||| curl,ℎ as well as ( 54) and ( 56), the bound in (50) follows.

Poincaré inequality for the divergence

Theorem 3 is established in the same way as Theorem 1 (see the end of Section 4.1) starting from the following result.

Lemma 10 (Continuous inverse of the discrete divergence). For any ℎ ∈ div,ℎ , there is ℎ ∈ div,ℎ such that

ℎ ℎ = ℎ ℎ and ||| ℎ ||| div,ℎ ℎ ℎ 2 (Ω) . ( 60
)
Proof. The face components of ℎ are obtained applying Theorem 7 with = ∫ for all ∈ T ℎ and letting ∈ P 0 ( ) be such that

∫ = | | = ∀ ∈ F ℎ . (61)
For all ∈ T ℎ , the element component G, ∈ G -1 ( ) is defined by the following relation:

∫ G, • grad = - ∫ + ∈ F ∫ ∀ ∈ P 0 ( ). (62)
Finally, we set

c G, = 0 ∀ ∈ T ℎ . (63)
1. Equality of the discrete divergence. To check the first condition in (60), it suffices to show that = for a generic ∈ T ℎ . To this end, we start by noticing that

∫ = = ∈ F = ∈ F ∫ = ∫ ,
showing that 0 P, ( ) = 0 P, ( ). To show the equivalence of the higher-order components, it suffices to use (62) in ( 6) written for to infer ∫ = ∫ for all ∈ P 0 ( ).

Continuity.

Let us now show the continuity bound in (60). Observing that, by (61), for all ∈ F ℎ it holds

2 2 ( ) = | | 2 = | | -1 2 ℎ -2 2
for all ∈ T ℎ such that ∈ F (the last inequality being a consequence of mesh regularity), we have

∈ T ℎ ℎ ∈ F 2 2 ( ) ∈ T ℎ ℎ -1 ∈ F 2 (31) ∈ T ℎ ℎ -3 2 ∈ T ℎ ℎ -3 | | 0 P, ( ) 2 2 ( ) ℎ ℎ 2 2 (Ω) , (64)
where, to pass to the second line, we have used the mesh regularity assumption to infer ℎ -3 | | 1. To estimate the element components, we take in (62) such that grad = G, , use Cauchy-Schwarz, trace, and inverse inequalities in the right-hand side, pass to the square and use standard inequalities for the square of a finite sum of terms to obtain

G, 2 2 ( ;R 3 ) ℎ 2 2 2 ( ) + ℎ ∈ F 2 2 ( ) .
Summing the above relation over ∈ T ℎ , using the fact that ℎ ≤ diam(Ω) 1 for the first term in the right-hand side and (64) for the second, we obtain

∈ T ℎ G, 2 2 ( ;R 3 ) ℎ ℎ 2 2 (Ω) . (65)
Summing ( 64) to (65) and recalling the definition [START_REF] Botti | A Hybrid High-Order method for nonlinear elasticity[END_REF] of |||•||| div,ℎ along with (63) yields the inequality in (60).

Stability analysis of a DDR scheme for the magnetostatics problem

We apply the Poincaré inequalities stated in Section 2.7 to the stability analysis of a DDR scheme for the magnetostatics problem which generalises the one presented in [START_REF] Di Pietro | An arbitrary-order method for magnetostatics on polyhedral meshes based on a discrete de Rham sequence[END_REF] to domains with non-trivial topology. We introduce the space of discrete harmonic forms div,ℎ ≔ ℎ ∈ div,ℎ : ℎ ℎ = 0 and ( ℎ , ℎ ℎ ) div,ℎ = 0 for all ℎ ∈ curl,ℎ .

For a given source term ∈ 1 (Ω; R 3 ), we consider the following DDR approximation of the magnetostatics problem: Find (

ℎ , ℎ , ℎ ) ∈ curl,ℎ × div,ℎ × div,ℎ such that, for all ( ℎ , ℎ , ℎ ) ∈ curl,ℎ × div,ℎ × div,ℎ , ℎ (( ℎ , ℎ , ℎ ), ( ℎ , ℎ , ℎ )) = ( div,ℎ , ℎ ) div,ℎ ,
where the bilinear form ℎ : curl,ℎ × div,ℎ × div,ℎ 2 → R is given by:

ℎ (( ℎ , ℎ , ℎ ), ( ℎ , ℎ , ℎ )) ≔ ( ℎ , ℎ ) curl,ℎ -( ℎ , ℎ ℎ ) div,ℎ + ( ℎ ℎ , ℎ ) div,ℎ + ( ℎ ℎ , ℎ ℎ ) 2 (Ω) + ( ℎ , ℎ ) div,ℎ + ( ℎ , ℎ ) div,ℎ , (66)
with discrete 2 -like products (•, •) curl,ℎ and (•, •) div,ℎ defined as in [START_REF] Di Pietro | An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness, Poincaré inequalities, and consistency[END_REF]Section 4.4]. We define the graph norm on this product space as

( ℎ , ℎ , ℎ ) ℎ ≔ ℎ 2 curl,ℎ + ℎ ℎ 2 div,ℎ + ℎ 2 div,ℎ + ℎ ℎ 2 2 (Ω) + ℎ 2 div,ℎ 1 2
, (67)

with norms • curl,ℎ ≔ (•, •) 1 2 curl,ℎ on curl,ℎ and • div,ℎ ≔ (•, •) 1 2
div,ℎ on div,ℎ induced by the corresponding discrete 2 -products. ) be given, and let P > 0 be the maximum of the hidden constants in the continuity estimates (60) for the divergence and (50) for the curl. Let ℎ ∈ div,ℎ be given by (60), i.e., such that

We define

ℎ ≔ 2 2 P ℎ - ℎ , ℎ ≔ ℎ ℎ + ℎ + 2 2 P ℎ , ℎ ≔ ℎ -2 2 P ℎ . ( 72
)
The following bound is readily inferred using triangle inequalities: Simplifying, the conclusion follows.

1 | | ℎ -3 , (85)
where, for { , , , } = {0, 1, 2, 3}, = • is the dihedral angle associated to the edge .

Theorem 6 (

 6 Mimetic Poincaré inequality for collections of edge values). Let ( ) ∈ F ℎ ∈ R F ℎ be a collection of values at faces satisfying ∈ F = 0 for all ∈ T ℎ and (20)

Lemma 8 (

 8 Continuous inverse of the discrete gradient). For all ℎ ∈ grad,ℎ , there is ℎ

  now ∈ F ℎ . By[START_REF] Bonaldi | An exterior calculus framework for polytopal methods[END_REF] Lemma 14] for ( , ) = (0, 2), ∫ • rot = -∈ E∫ for all ∈ P +1 0 ( ), so that, by (43), R, = R, . On the other hand, plugging the definition (40) of into (2), we readily infer that c,

1 .

 1 that (55) defines R, uniquely since curl : G c, ( ) ↦ → R -1 ( ) is an isomorphism by [2, Corollary 7.3]. Equality of the discrete curl. By definition of , it holds 0 P, = 0 P, for all ∈ F ℎ . The equality of the higher-order components is obtained plugging (53) into the definition (4) of , which leads to ∫ = ∫ for all ∈ P 0 ( ). By [16, Proposition 4] (which corresponds to [7, Lemma 14] with ( , ) = (3, 1)), the equality of the face curls implies G, ( ) = G, ( ).

ℎ ℎ = ℎ ℎ and ℎ div,ℎ ≤ P ℎ ℎ 2 (

 2 applied to ℎ , we infer the existence of ℎ such that ℎ ℎ = ℎ ℎ and ℎ curl,ℎ ≤ P ℎ ℎ div,ℎ .

  Theorem 11 (Stability of the discrete bilinear form). The discrete bilinear form (66) is inf-sup stable for the graph norm • ℎ , i.e., for all (

ℎ , ℎ , ℎ ) ∈ curl,ℎ × div,ℎ × div,ℎ , it holds |||( ℎ , ℎ , ℎ )||| ℎ sup ( ℎ , ℎ , ℎ ) ∈ curl,ℎ × div,ℎ × div,ℎ \{0} ℎ (( ℎ , ℎ , ℎ ), ( ℎ , ℎ , ℎ )) |||( ℎ , ℎ , ℎ )||| ℎ . (

68

)

Proof. Let ( ℎ , ℎ , ℎ

  ℎ ℎ ) 2 ( T ℎ ) -( ℎ ℎ ,Denoting by $ the supremum in (68), we then use the previous bound to write

	Plugging (75) and (76) into (74), we have
											))
										ℎ
	≥	5 2 P 4		ℎ		2 curl,ℎ +	ℎ ℎ	2 div,ℎ +		ℎ	2 div,ℎ +	5 2 P 4	ℎ ℎ	2	2 (Ω) +	2 3 ℎ	2 div,ℎ -	1 3 ℎ ℎ	2 div,ℎ
	≥	5 2 P 4		ℎ		2 curl,ℎ +	ℎ ℎ	2 div,ℎ +		ℎ	2 div,ℎ +	11 2 P 12	ℎ ℎ	2	2 (Ω) +	1 3 ℎ	2 div,ℎ
		|||(	ℎ ,	ℎ ,	ℎ	)||| 2 ℎ .
	|||(	ℎ ,	ℎ ,	ℎ	)||| 2 ℎ		ℎ ((								ℎ	)||| ℎ	(73)	$ |||(
																	(73)
	Plugging the test functions (72) into the expression (66) of ℎ gives
				= 2 2 P	ℎ	2 curl,ℎ -(	ℎ ,		ℎ ) curl,ℎ
						-	h h h h h h h h h h 2 2 P ( ℎ , ℎ ℎ ) div,ℎ + (	ℎ ,	ℎ ℎ ) div,ℎ
						+	ℎ ℎ		2 div,ℎ +	$ $ $ $ $ $ $ $ ( ℎ ℎ ,	h h h h h h h h h h 2 2 P ( ℎ ℎ , ℎ ) div,ℎ
						-( ℎ ℎ ,	$ $ $ $ $			¨¨¨ ℎ ℎ ) 2 ( T ℎ ) + 2 2 P	ℎ ℎ	2	2 (Ω)	(74)
						+	$ $ $ $ $ $ $ $ ( ℎ , ℎ ℎ ) div,ℎ +		ℎ	2 div,ℎ +	2 2 P (
														2 2 P (	ℎ ,	ℎ	) div,ℎ
				= 2 2 P	ℎ	2 curl,ℎ -(			ℎ ℎ	2 div,ℎ +	ℎ	2 div,ℎ
						+ 2 2 P		ℎ ℎ	2	2 (Ω) + (	ℎ ,	ℎ ℎ +
	Using Cauchy-Schwarz and generalised Young inequalities, we have
			(	ℎ ,		ℎ ) curl,ℎ	(70) ≤ P ℎ curl,ℎ ℎ ℎ div,ℎ ≤	3 2 P 4	ℎ	2 curl,ℎ +	1 3 ℎ ℎ	2 div,ℎ .	(75)
	Moreover, the decomposition	ℎ =		ℎ +	ℎ ℎ +
											ℎ div,ℎ P		ℎ ℎ 2 (Ω) ≤	1 3 ℎ	2 div,ℎ +	3 2 P 4	ℎ ℎ

, (71), (

67

)

|||( ℎ , ℎ , ℎ )||| 2 ℎ . ℎ (( ℎ , ℎ , ℎ ), ( ℎ , ℎ , ℎ )) ℎ ) div,ℎ + ℎ ℎ , ℎ ) div,ℎ + ( ℎ , ℎ ) div,ℎℎ , ℎ ) curl,ℎ + ℎ ) div,ℎ . ℎ gives ( ℎ , ℎ ℎ + ℎ ) div,ℎ = ℎ 2 div,ℎ -( ℎ , ℎ ) div,ℎ with ( ℎ , ℎ ) div,ℎ ≤ ℎ (( ℎ , ℎ , ℎ ), ( ℎ , ℎ , ℎ , ℎ , ℎ ), ( ℎ , ℎ , ℎ )) ≤ $ |||( ℎ , ℎ , ℎ , ℎ , ℎ )||| ℎ .

(Ω) . (76)

2 ( ) =

Acknowledgements

Daniele Di Pietro acknowledges the partial support of Agence Nationale de la Recherche through the grant "HIPOTHEC". Both authors acknowledge the partial support of Agence Nationale de la Recherche through the grant ANR-16-IDEX-0006 "RHAMNUS".

A Results on the trimmed finite element sequence on tetrahedral meshes

In this section we provide the explicit expression of the polynomial basis functions used in Section 3. These bases can be easily described on a reference element (see [START_REF] Arnold | Periodic table of the Finite Elements[END_REF]). However, in order the compute their norms, we need to know their expression on the physical element. The transformation from the reference to the physical element is given by the pullback of the mapping between the two.

We consider a simplex with vertices 0 , 1 , 2 , and 3 ordered so that, denoting by the coordinate vector of , 1 -0 , 2 -0 , 3 -0 forms a direct basis of R 3 . The basis for the local affine Lagrange space P - 1 Λ 0 ( ) P 1 ( ) spanned by "hat" functions is

(77) The basis of the lowest-order local face Raviart-Thomas-Nédélec space P -

where the function is associated to the edge with vertices and . The basis of the lowest-order local edge Nédélec space

with function associated to the simplicial face with vertices , , and . Finally, the basis of P -

Lemma 12 (Dual basis). The following identities hold:

where

Proof. The proof of (81a) readily follows from the orthogonality of the cross product.

Let us check (81b) for 23 , the other being similar. For ( ) ∈ {23, 13, 12, 03, 02, 01}, we have

=0 @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ [( , we have

Lemma 13 (Norm of the basis function). The functions given by (77)-( 80) have the following 2 -norms: For all ∈ {0, 1, 2, 3}, all ( ) ∈ {23, 13, 12, 03, 02, 01}, and all ( ) ∈ {123, 023, 013, 012},

Proof. We will only show the computation for one function of each space, the others being similar. In order to integrate over the simplex S, we consider the change of variable induced by : ( 1 , 2 , 3 ) ↦ →

Let us first consider the family given by (77). Using the orthogonality of the cross product, and the identity ( × ) • = det( , , ), we notice that

Hence, we have ∫

.

Then, we proceed with the family (78). We have

Expanding the product, we obtain

Finally, we prove that (84) holds for 123 given by (79). We have

Lemma 14 (Link with the differential operators). For all face of , we define ∈ {-1, 1} such that is outward pointing. Then, the followings identities hold; Then, to prove (87), we use the identity curl( × ) = div( ⊗ -⊗ ) in (78) to write

where ( ) is such that { , , , } = {0, 1, 2, 3} and < . Noticing that is an inward pointing to the face ˆ , and lies outward pointing in the plane of ˆ , we have

where we inserted in the second argument of the determinant to get the second inequality. Insertingin (90) and replacing the denominator according to (91), we obtain curl = 6| | ˆ 2( -) + ˆ 2( -) We infer (87) recalling (89). Finally, let us prove (86). We only prove the equality for 0 , the other three being similar. By the assumption on the basis, we have