The Stokes complex for Virtual Elements in three dimensions

 The Authors c

DOI: 10.1142/S0218202520500128

The Stokes complex for Virtual Elements in three dimensions

L. Beir˜ ao da Veiga

, F. Dassi

and G. Vacca

Dipartimento di Matematica e Applicazioni, Universit` a di Milano Bicocca, Via Roberto Cozzi 55, 20126 Milano, Italy

∗

lourenco.beirao@unimib.it

†

franco.dassi@unimib.it

‡

giuseppe.vacca@unimib.it

Received 29 January 2019 Revised 12 November 2019 Accepted 14 November 2019

Published 23 March 2020 Communicated by F. Brezzi

This paper has two objectives. On one side, we develop and test numerically divergence- free Virtual Elements in three dimensions, for variable “polynomial” order. These are the natural extension of the two-dimensional divergence-free VEM elements, with some modification that allows for a better computational efficiency. We test the element’s performance both for the Stokes and (diffusion dominated) Navier–Stokes equation.

The second, and perhaps main, motivation is to show that our scheme, also in three dimensions, enjoys an underlying discrete Stokes complex structure. We build a pair of virtual discrete spaces based on general polytopal partitions, the first one being scalar and the second one being vector valued, such that when coupled with our velocity and pressure spaces, yield a discrete Stokes complex.

Keywords: Virtual elements; Stokes complex; Navier–Stokes equation; polygonal meshes.

AMS Subject Classification: 65N30, 76D05

1. Introduction

The Virtual Element Method (VEM) was introduced in Refs. 10 and 12 as a generalization of the Finite Element Method (FEM) allowing for general poly- topal meshes. Nowadays the VEM technology has reached a good level of suc- cess; among the many papers we here limit ourselves in citing a few sample works.

It was soon recognized that the flexibility of VEM allows to build elements that hold peculiar advantages also on more standard grids. One main example is that of “divergence-free” Virtual Elements for Stokes-type problems, ini- tiated in Refs. 5 and 17 and further developed in Refs. 18 and 57. An advantage of

‡

Corresponding author.

This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

477

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the proposed family of Virtual Elements is that, without the need of a high minimal polynomial degree as it happens in conforming FEM, it is able to yield a discrete divergence-free (conforming) velocity solution, which can be an interesting asset as explored for Finite Elements in Refs. 44, 45, 46, 48 and 53. For a wider look in the literature, other VEM for Stokes-type problems can be found in Refs. 28, 29, 31, 35, 42 and 50 while different polygonal methods for the same problem in Refs. 24, 33, 38 and 49.

This paper has two objectives. On one side, we develop and test numeri- cally for the first time the divergence-free Virtual Elements in three dimensions (for variable “polynomial” order k). These are the natural extension of the two- dimensional (2D) VEM elements of Refs. 17 and 18, with some modification that allows for a better computational efficiency. We first test the element’s performance for the Stokes and (diffusion dominated) Navier–Stokes equation for different kind of meshes (such as Voronoi, but also cubes and tetrahedra) and then show a spe- cific test that underlines the divergence-free property (in the spirit of Refs. 18 and 48).

The second, and perhaps main, motivation is to show that our scheme, also in three dimensions, enjoys an underlying discrete Stokes complex structure. That is, a discrete structure of the kind

R −−−−→ W ⁱ h −−−−−→ Σ ∇ h −−−−→ V

h −−−−→ Q

h −−−−→ 0,

where the image of each operator exactly corresponds to the kernel of the following one, thus mimicking the continuous complex

R −−−−→ H ⁱ

(Ω) −−−−−→ Σ(Ω) ^∇ −−−−→ [H

(Ω)]

−−−−→ L

(Ω) −−−−→ 0,

with Σ(Ω) denoting functions of L

(Ω) with curl in H

(Ω).

Discrete Stokes com- plexes have been extensively studied in the literature of Finite Elements since the presence of an underlying complex implies a series of interesting advantages (such as the divergence-free property), in addition to guaranteeing that the discrete scheme is able to correctly mimic the structure of the problem under study.

This motivation is therefore mainly theoretical in nature, but it serves the impor- tant purpose of giving a deeper foundation to our method. We therefore build a pair of virtual discrete spaces based on general polytopal partitions of Ω, the first one W h , conforming in H

(Ω) and the second one Σ _h conforming in Σ(Ω), such that, when coupled with our velocity and pressure spaces, yield a discrete Stokes com- plex. We also build a set of carefully chosen associated degrees of freedom (DoFs).

This construction was already developed in two dimensions in Ref. 19, but here

things are more involved due to the much more complex nature of the curl oper-

ator in 3D when compared to 2D. In this respect we must underline that, to the

best of the authors knowledge, no Stokes exact complex of the type above exists

for conforming Finite Elements in three dimensions. There exist FEM for different

(more regular) Stokes complexes, but at the price of developing cumbersome ele-

ments with a large minimal polynomial degree (we refer to Ref. 46 for an overview)

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or using a subdivision of the element.

We finally note that our construction holds for a general “polynomial” order k ≥ 2.

The paper is organized as follows. After introducing some notation and prelimi- naries in Sec. 2, the Virtual Element spaces and the associated DoFs are deployed in Sec. 3. In Sec. 4, we prove that the introduced spaces constitute an exact complex.

In Sec. 5, we describe the discrete problem, together with the associated projec- tors and bilinear forms. In Sec. 6, we provide the numerical tests. Finally, in the appendix we prove a useful lemma.

2. Notations and Preliminaries

In this section, we introduce some basic tools and notations useful in the construc- tion and theoretical analysis of VEMs.

Throughout the paper, we will follow the usual notation for Sobolev spaces and norms.

Hence, for an open bounded domain ω, the norms in the spaces W _p ^s (ω) and L ^p (ω) are denoted by · W

and · _L

, respectively. Norm and seminorm in H ^s (ω) are denoted respectively by · s,ω and |·| s,ω , while ( ·, ·) ω and  ·  ω denote the L

-inner product and the L

-norm (the subscript ω may be omitted when ω is the whole computational domain Ω).

2.1. Basic notations and mesh assumptions

From now on, we will denote with P a general polyhedron having  V vertexes V ,

 e edges e and  f faces f .

For each polyhedron P , each face f of P and each edge e of f we denote with:

— n ^f _P (respectively, n _P ) the unit outward normal vector to f (respectively, to ∂P ),

— n ^e _f (respectively, n _f ) the unit vector in the plane of f that is normal to the edge e (respectively, to ∂f ) and outward with respect to f ,

— t ^e _f (respectively, t _f ) the unit vector tangent to e (respectively, to ∂f ) counter- clockwise with respect to n ^f _P ,

— τ ^f

and τ ^f

two orthogonal unit vectors lying on f and such that τ ^f

∧ τ ^f

= n ^f _P , i.e. τ

_f and τ

_f constitute the axis of a local coordinate system on f (see also Sec. 2.2),

— t _e a unit vector tangent to the edge e.

Notice that the vectors t ^e _f , t _f , τ ^f

and τ ^f

actually depend on P (we do not write such dependence explicitly for lightening the notations).

In the following O will denote a general geometrical entity (element, face, edge) having diameter h O .

Let Ω be the computational domain that we assume to be a contractible poly- hedron (i.e. simply connected polyhedron with boundary ∂Ω which consists of one connected component), with Lipschitz boundary. Let {Ω h } h be a sequence of decom- positions of Ω into general polyhedral elements P where h := sup P ∈Ω

h P .

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We suppose that for all h, each element P in Ω h is a contractible polyhedron that fulfills the following assumptions:

(A1) P is star-shaped with respect to a ball B P of radius ≥ h _P ,

(A2) every face f of P is star-shaped with respect to a disk B f of radius ≥ h P , (A3) every edge e in P satisfies h e ≥ h P ,

where  is a uniform positive constant. We remark that the hypotheses (A1)–

(A3), though not too restrictive in many practical cases, can be further relaxed, as investigated in Refs. 16, 25, 26 and 30.

The total number of vertexes, edges, faces and elements in the decomposition Ω _h are denoted respectively with L V , L e , L f , L P .

For any mesh object O and for n ∈ N we introduce the spaces:

• P n ( O) the polynomials on O of degree ≤ n (with the extended notation P ₋₁ ( O) = {0}),

• P n\m ( O) := P n ( O)\P m ( O) for m < n, denotes the polynomials in P n ( O) with monomials of degree strictly greater than m.

Moreover, for any mesh object O of dimension d we define

π n,d := dim( P n ( O)) = dim(P n ( R ^d )), (2.1) and thus dim( P _n\m ( O)) = π n,d − π m,d .

In the following the symbol  will denote a bound up to a generic positive constant, independent of the mesh size h, but which may depend on Ω, on the

“polynomial” order k and on the shape constant  in assumptions (A1)–(A3).

2.2. Vector calculus and de Rham complexes

Here below we fix some additional notation of the multivariable calculus.

Three-dimensional (3D) operators. In three dimensions we denote with x = (x

, x

, x

) the independent variable. With a usual notation the symbols ∇ and Δ denote the gradient and Laplacian for scalar functions, while Δ, ∇, ε, div and curl denote the vector Laplacian, the gradient and the symmetric gradient operator, the divergence and the curl operator for vector fields. Note that on each polyhedron P the following useful polynomial decompositions hold:

[ P n (P )]

= ∇(P n+1 (P )) ⊕ (x ∧ [P n−1 (P )]

), (2.2) [ P n (P )]

= curl( P n+1 (P )) ⊕ x P n−1 (P ). (2.3) Tangential operators. Let f be a face of a polyhedron P , we denote with x f :=

(x _{f 1} , x _{f 2} ) the independent variable (i.e. a local coordinate system on f associated

with the axes τ

_f and τ

_f introduced above). The tangential differential operators

are denoted by a subscript f . Therefore, the symbols ∇ f and Δ _f denote the gradient

and Laplacian for scalar functions. For instance if f lies on the plane x

= 0 and τ

_f

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and τ

_f are associated with the coordinate system (x

, x

) then ∇ f ϕ =  _∂ϕ

∂x

, _∂x ^∂ϕ

 and Δ _f ϕ = ^∂ _∂x

^ϕ

1

+ ^∂

^ϕ

∂x

, while ∇ϕ =  _∂ϕ

∂x

, _∂x ^∂ϕ

, _∂x ^∂ϕ



and Δϕ = ^∂ _∂x

^ϕ

+ ^∂

^ϕ

∂x

+ ^∂

^ϕ

∂x

. In the same way Δ _f , ∇ f , and div _f denote the vector Laplacian, the gradient operator and the divergence for vector fields on f (with respect to the coordinate x _f ). Furthermore for a scalar function ϕ and a vector field v := (v

, v

) we set

rot _f ϕ :=

 ∂ϕ

∂x _{f 2} , − ∂ϕ

∂x _{f 1}



and rot _f v := ∂v

∂x _{f 1} − ∂v

∂x _{f 2} . The following two-dimensional polynomial decompositions hold

[ P n (f )]

= ∇ f ( P n+1 (f )) ⊕ x ^⊥ _f P n−1 (f ), [ P n (f )]

= rot _f ( P n+1 (f )) ⊕ x f P n−1 (f ), where x ^⊥ _f := (x _{f 2} , −x _{f 1} ).

Given a 3D vector-valued function v defined in P , the tangential component v f

of v with respect to the face f is defined by v _f := v − (v · n ^f _P )n ^f _P .

Noticing that v _f is a 3D vector field tangent to f , with a slight abuse of notations we define the 2D vector field v _τ on ∂P , such that on each face f its restriction to the face f satisfies

v _τ (x _f ) = v _f (x).

The 3D function v and its 2D tangential restriction v _τ are related by the curl-rot compatibility condition

curl v · n ^f _P = rot _f v _τ on any f ∈ ∂P. (2.4) Moreover, the Gauss theorem ensures the following rot-tangent component relation



f

rot _f v _τ df =



∂f

v · t f ds for any f ∈ ∂P . (2.5) Finally, for any scalar function v defined in P , we denote with v τ the scalar function defined in ∂P such that

v τ (x _f ) := v(x) _|f on each face f ∈ ∂P .

On a generic mesh object O with geometrical dimension d, on a face f and on a polyhedron P we define the following functional spaces:

L

( O) :=



v ∈ L

( O) s.t.



O v dO = 0

, Z(O) := {v ∈ [H

( O)]

s.t. div v = 0 in O}, H(div, O) := {v ∈ [L

( O)] ^d with div v ∈ L

( O)},

H(rot, f) := {v ∈ [L

(f )]

with rot _f v ∈ L

(f )},

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H(curl, P ) := {v ∈ [L

(P )]

with curl v ∈ [L

(P )]

}, Σ(P ) := {v ∈ [L

(P )]

with curl v ∈ [H

(P )]

},

Ψ(P ) := {v ∈ [L

(P )]

s.t. div v ∈ H

(P ), curl v ∈ [H

(P )]

} with the “homogeneous counterparts”

Z

( O) := {v ∈ Z(O) s.t. v = 0 on ∂O},

H

(div, O) := {v ∈ H(div, O) s.t. v · n O = 0 on ∂O}, H

(rot, f ) := {v ∈ H(rot, f ) s.t. v · t f = 0 on ∂f}, H

(curl, P ) := {v ∈ H(curl, P ) s.t. v τ = 0 on ∂P },

Σ

(P ) := {v ∈ Σ(P ) s.t. v τ = 0 and curl v = 0 on ∂P }, Ψ

(P ) :=



v ∈ Ψ(P ) s.t.



∂P

v · n P df = 0, v τ = 0 and curl v = 0 on ∂P

.

Remark 2.1. Notice that for each face f ∈ ∂P , the vector fields v f and v ∧ n ^f _P are different. In fact both lie in the plane of the face f , but v f is (π/2)-rotation in f (with respect to the axes τ

and τ

) of v ∧ n ^f _P . For instance let f lie on the plane x

= 0 with n ^f _P = (0, 0, 1), then for a vector field v = (v

, v

, v

) it holds that v _f = (v

, v

, 0) and v ∧ n ^f _P = (v

, −v

, 0). Therefore, both v f and v ∧ n ^f _P lie in the plane of f and v f is (π/2)-rotation of v ∧ n ^f _P (with respect to the orientation given by n ^f _P ). However, v _f = 0 if and only if v ∧ n ^f _P = 0. For that reason in the definition of Ψ

(P ), we consider a slightly different, but substantially equivalent, set of homogeneous boundary conditions to that considered in literature.

Recalling that a sequence is exact if the image of each operator coincides with the kernel of the following one, and that P is contractible, from (2.2) and (2.3) it is easy to check that the following sequence is exact:

R −−−→ P ⁱ n+2 (P ) −−−−→ [P ^∇ n+1 (P )]

−−−−→ [P

n (P )]

−−−−→ P

n−1 (P ) −−−→ 0,

(2.6) where i denotes the mapping that to every real number r associates the constant function identically equal to r and 0 is the mapping that to every function associates the number 0.

The 3D de Rham complex with minimal regularity (in a contractible domain Ω) is provided by (see for instance Refs. 8 and 37)

R −−−→ H ⁱ

(Ω) −−−−−→ H(curl, Ω) ^∇ −−−−→ H(div, Ω)

−−−−→ L

(Ω) −−−→ 0.

In this paper, we consider the de Rham sub-complex with enhanced smoothness

R −−−−→ H ⁱ

(Ω) −−−−−→ Σ(Ω) ^∇ −−−−→ [H

(Ω)]

−−−−→ L

(Ω) −−−−→ 0,

(2.7)

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that is suitable for the Stokes (Navier–Stokes) problem. Therefore, our goal is to construct conforming virtual element spaces

W h ⊆ H

(Ω), Σ _h ⊆ Σ(Ω), V h ⊆ [H

(Ω)]

, Q h ⊆ L

(Ω) (2.8) that mimic the complex (2.7), i.e. are such that

R −−−−→ W ⁱ h −−−−−→ Σ ∇ h −−−−→ V

h −−−−→ Q

h −−−−→ 0

(2.9) is an exact sub-complex of (2.7).

3. The Virtual Element Spaces

This section is devoted to the construction of conforming virtual element spaces (2.8) that compose the virtual sub-complex (2.9). As we will see, the space W h

consists of the lowest degree 3D nodal VEM space,

whereas the spaces V _h and Q h (that are the spaces actually used in the discretization of the problem) are the 3D counterparts of the inf-sup stable couple of spaces introduced in Refs. 18 and 57. Therefore, the main novelty of this section is in the construction of the Σ-conforming space Σ _h .

In order to facilitate the reading, we present the spaces in the reverse order, from right to left in the sequence (2.9). In particular, in accordance with (2.9), the space Σ _h will be carefully designed to fit curl Σ _h ⊆ V h .

We stress that the readers mainly interested on the virtual element approxima- tion of the 3D Navier–Stokes equation (and not on the virtual de Rham sequence) can skip Secs. 3.3, 3.4 and 4.

One essential idea in the VEM construction is to define suitable (computable) polynomial projections. For any n ∈ N and each polyhedron/face O we introduce the following polynomial projections:

• the L

-projection Π

_n : L

( O) → P n ( O), defined for any v ∈ L

( O) by



O q n (v − Π

n v)dO = 0 for all q n ∈ P n ( O), (3.1) with obvious extension for vector functions Π

_n : [L

( O)]

→ [P n ( O)]

, and tensor functions Π

_n : [L

( O)]

→ [P n ( O)]

,

• the H

-seminorm projection Π ^∇,O _n : H

( O) → P _n ( O), defined for any v ∈ H

( O) by

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩



O ∇ q n · ∇(v − Π ^∇,O _n v)dO = 0 for all q n ∈ P n ( O),



∂O (v − Π ^∇,O _n v)dσ = 0,

(3.2)

with obvious extension for vector functions Π ^∇,O _n : [H

( O)]

→ [P n ( O)]

. Let k ≥ 2 be the polynomial degree of accuracy of the method. We recall that, in standard finite element fashion, the VEM spaces are first defined elementwise and then assembled globally.

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3.1. Scalar L

-conforming space

We start our construction with the rightmost discrete space Q h in (2.9). Since we are not requiring any smoothness on Q h , the local space Q h (P ) is simply defined by

Q h (P ) := P k−1 (P ),

having dimension (cf. (2.1)) dim(Q h (P )) = π k−1,3 . The corresponding DoFs are chosen, defining for each q ∈ Q h (P ) the following linear operators

• D Q : the moments up to order k − 1 of q, i.e.



P

qp k−1 dP for any p k−1 ∈ P k−1 (P ).

The global space is given by

Q h := {q ∈ L

(Ω) s.t. q |P ∈ Q h (P ) for all P ∈ Ω h }. (3.3) It is straightforward to see that the dimension of Q h is

dim(Q h ) = π k−1,3 L P . (3.4)

3.2. Vector H

-conforming VEM space

The subsequent space in the de Rham complex (2.9) is the vector-valued H

- conforming virtual element space V _h . The space V _h is a 3D version of the space

(see also the guidelines in the appendix of Ref. 17). We make an extensive use of the enhanced technique

in order to achieve the computability of the polynomial projections stated in Proposition 5.1.

We first consider on each face f of the element P , the face space

B k (f ) := {v ∈ H

(f ) s.t. (i) v |∂f ∈ C

(∂f ), v |e ∈ P _k (e) for all e ∈ ∂f , (ii) Δ _f v ∈ P k+1 (f ),

(iii) (v − Π ^∇,f _k v,  p k+1 ) _f = 0 for all p k+1 ∈ P _k+1\k−2 (f )} (3.5) and the boundary space

B k (∂P ) := {v ∈ C

(∂P ) such that v |f ∈ B k (f ) for any f ∈ ∂P }

that is a modification of the standard boundary nodal VEM.

Indeed the “super- enhanced” constraints (condition iii) in the definition (3.5)) are needed to exactly compute the polynomial projection Π

_k+1 (see Proposition 5.1).

On the polyhedron P we first define the virtual element space  V _h (P ) V  _h (P ) := {v ∈ [H

(P )]

s.t. (i) v _|∂P ∈ [B k (∂P )]

,

(ii) Δv + ∇s ∈ x ∧ [P k−1 (P )]

for some s ∈ L

(P ),

(iii) div v ∈ P k−1 (P )}, (3.6)

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and afterwards our velocity space

V _h (P ) := {v ∈  V _h (P ) s.t. (v − Π ^∇,P _k v, x ∧ p _k−1 ) _P = 0

for all p _k−1 ∈ [P k−1\k−3 (P )]

}. (3.7) The definition above is the 3D counterpart of the virtual elements,

in particular we remark that the enhancing constraints in definition (3.7) are necessary to achieve the computability of the L

-projection Π

_k (see Proposition 5.1). Moreover, notice that the space V _h (P ) contains [P k (P )]

and this will guarantee the good approximation property of the space (cf. Theorem 5.1).

Proposition 3.1. The dimension of V _h (P ) is given by

dim(V _h (P )) = 3 V + 3(k − 1) e + 3π k−2,2  f + 3π k−2,3 .

Moreover, the following linear operators D

split into five subsets constitute a set of DoFs for V _h (P ):

• D

: the values of v at the vertexes of P ,

• D

: the values of v at k − 1 distinct points of every edge e of P ,

• D

: the face moments of v (split into normal and tangential components)



f

(v · n ^f _P )p k−2 df,



f

v _τ · p _k−2 df, for all p k−2 ∈ P k−2 (f ) and p _k−2 ∈ [P k−2 (f )]

,

• D

: the volume moments of v



P

v · (x ∧ p _k−3 )dP for all p _k−3 ∈ [P k−3 (P )]

,

• D

: the volume moments of div v



P

(div v) p k−1 dP for all p k−1 ∈ P k−1\0 (P ).

Proof. We only sketch the proof since it follows the guidelines of Proposition 3.1 in Ref. 57 for the analogous 2D space. First of all, recalling (2.1) and polynomial decomposition (2.2), simple computations yield

#D

= 3 V , #D

= 3(k − 1) e , #D

= 3π k−2,2  f ,

#D

= 3π k−2,3 − π k−1,3 + 1, #D

= π k−1,3 − 1, (3.8) and therefore

#D

= 3 V + 3(k − 1) e + 3π k−2,2  f + 3π k−2,3 .

Now employing Proposition 2 and Remark 5 in Ref. 2, it can be shown that the DoFs D

, D

, D

are unisolvent for the space [ B k (∂P )]

. Therefore, it holds that

dim([ B k (∂P )]

) = 3 V + 3(k − 1) e + 3π k−2,2  f , (3.9)

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which in turn implies (recalling (3.6) and (3.7)) dim(V _h (P )) ≥ #D

.

Now the result follows by proving that D

(v) = 0 implies that v is identically zero, which can be easily shown first working on ∂P and then inside P . As a consequence the linear operators D

are unisolvent for V _h and in particular dim(V _h (P )) =

#D

.

The global space V _h is defined by gluing the local spaces with the obvious associated sets of global DoFs:

V _h := {v ∈ [H

(Ω)]

s.t. v _|P ∈ V h (P ) for all P ∈ Ω h }. (3.10) The dimension of V _h is given by

dim(V _h ) = 3L V + 3(k − 1)L e + 3π k−2,2 L f + 3π k−2,3 L P . (3.11) We also consider the discrete kernel

Z _h :=



v ∈ V h s.t.



Ω

div v q dΩ = 0 for all q ∈ Q h

, (3.12)

and the local version Z _h (P ) :=



v ∈ V h (P ) s.t.



P div v q dP = 0 for all q ∈ Q h (P )

.

A crucial observation is that, extending to the 3D case the result in Ref. 17, the proposed discrete pressure space (3.3) and the H

-conforming velocity space (3.10) are such that div V _h ⊆ Q h . As a consequence the crucial kernel inclusion holds

Z _h ⊆ Z. (3.13)

The inclusion here above and explicit computations (cf. (3.8)) yield that dim(Z _h ) = 3L V + 3(k − 1)L e + 3π k−2,2 L f + (3π k−2,3 − π k−1,3 )N P .

Property (3.13) has several important advantages, as explored in Refs. 18, 46 and 48.

Remark 3.1. In definition (3.7) the H

-seminorm projection Π ^∇,P _k can be actu- ally replaced by any polynomial projection Π ^P _k that is computable on the basis of the DoFs D

(in the sense of Proposition 5.1). This change clearly propagates throughout the rest of the analysis (see definitions (3.17) and (3.27)). An analogous observation holds also for the operator Π ^∇,f _k in condition (iii) of definition (3.5).

This remark allows to make use of computationally cheaper projections, as done in the numerical tests of Sec. 6.

3.3. Vector Σ-conforming VEM space

In this subsection, we consider the construction of the Σ-conforming virtual space

Σ _h in (2.9). As mentioned before, this brick constitutes the main novelty in the

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construction of the virtual de Rham sequence (2.9), that is carefully designed to satisfy curl Σ _h = Z _h .

We start by introducing on each face f ∈ ∂P the face space

S _k (f ) := {σ ∈ H(div f , f ) ∩ H(rot f , f ) s.t. (i) (σ · t e ) _|e ∈ P

(e), ∀ e ∈ ∂f, (ii) div _f σ = 0, (iii) rot f σ ∈ B k (f )}, (3.14) and the boundary space

S _k (∂P ) := {σ ∈ [L

(∂P )]

s.t. σ _τ ∈ S k (f ) for any f ∈ ∂P ,

(σ _f

· t e ) _|e = (σ _f

· t e ) _|e , ∀ e ⊆ ∂f

∩ ∂f

, f

, f

∈ ∂P }. (3.15) We recall that the differential problem in definition (3.14) posed on the simply connected face f is well posed if and only if the prescribed problem data satisfy the compatibility condition (2.5).

On the polyhedron P we first define the enlarged virtual space:

Σ  _h (P ) :=



ϕ ∈ Ψ(P ) s.t. (i) ϕ _|∂P ∈ S k (∂P ),



∂P

ϕ · n P df = 0,

(ii) (curl ϕ) _|∂P ∈ [B k (∂P )]

, (iii)



P

Δ ϕ · Δ ψ dP =



P

p _k−1 · ψ dP, ∀ ψ ∈ Ψ

(P ),

for some p _k−1 ∈ [P k−1 (P )]

∩ Z(P )

, (3.16)

and afterwards the final space

Σ _h (P ) := {ϕ ∈  Σ _h (P ) s.t. (curl ϕ − Π ^∇,P _k curl ϕ, x ∧ p _k−1 ) _P = 0

for all p _k−1 ∈ [P _k−1\k−3 (P )]

}. (3.17) We address the well-posedness of the biharmonic problem in definition (3.16) in Appendix A. Note that, in accordance with the target curl Σ _h ⊆ V h (respectively, curl  Σ _h ⊆  V _h ) the enhanced constraints in (3.17) are the curl version of those in (3.7) and (ii) in definition (3.16) corresponds to (i) in (3.6). Whereas we will see that curl of the solutions of the biharmonic problem (iii) in (3.16) are solutions to the Stokes problem (ii)–(iii) in (3.6) (see Proposition 4.2).

Proposition 3.2. The dimension of Σ _h (P ) is given by

dim(Σ _h (P )) = 3 V + (3k − 2) e + (3π k−2,2 − 1) f + 3π k−2,3 − π k−1,3 + 1.

Moreover, the following linear operators D

constitute a set of DoFs for Σ _h (P ) namely:

• D

: the values of curl ϕ at the vertexes of P ,

• D

: the values of curl ϕ at k − 1 distinct points of every edge e of P ,

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• D

: the face moments of curl ϕ (split into normal and tangential components)



f

(curl ϕ · n ^f _P ) p k−2 df for all p k−2 ∈ P _k−2\0 (f ),



f

(curl ϕ) _τ · p _k−2 df for all p _k−2 ∈ [P _k−2 (f )]

,

• D

: the volume moments of curl ϕ



P

curl ϕ · (x ∧ p _k−3 )dP for all p _k−3 ∈ [P _k−3 (P )]

,

• D

: the edge mean value of ϕ · t e , i.e.

1

|e|



e

ϕ · t e ds.

Proof. We prove that the linear operators D

plus the additional moments D 

:



P

curl ϕ · (x ∧ p _k−1 )dP for all p _k−1 ∈ [P k−1\k−3 (P )]

,

constitute a set of DoFs for the enlarged space  Σ _h (P ) in (3.16). Once the proof for Σ  _h (P ) is given, the extension to the smaller space Σ h (P ) easily follows by employ- ing standard techniques for VEM enhanced spaces (see Ref. 2 and Proposition 5.1 in Ref. 19). We preliminary observe that D

+  D

are equivalent to prescribe the moments 

P curl ϕ · (x ∧ p _k−1 )dP for all p _k−1 ∈ [P k−1 (P )]

.

We start the proof counting the number of the linear operators  D

= D

+  D

. Using similar computations as in (3.8) we have

#D

= 3 V , #D

= 3(k − 1) e , #D

= (3π k−2,2 − 1) f ,

#D

+ #  D

= 3π k,3 − π k+1,3 + 1, #D

=  e , and thus

#  D

= 3 V + (3k − 2) e + (3π k−2,2 − 1) f + 3π k,3 − π k+1,3 + 1. (3.18) Employing Theorem A.1, given p _k−1 ∈ [P k−1 (P )]

∩ Z(P ), g ∈ [B k (∂P )]

and h ∈ S k (∂P ) satisfying the compatibility condition (cf. (2.4)) g · n ^f _P = rot _f h _τ on any f ∈ ∂P , there exists a unique function ϕ ∈ Ψ(P ) such that

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩



P

Δ ϕ · Δ ψ dP =



P

p _k−1 · ψ dP for all ψ ∈ Ψ

(P ),



∂P

ϕ · n P df = 0,

ϕ _τ = h _τ on ∂P ,

curl ϕ = g on ∂P .

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Therefore,

dim(  Σ _h (P )) = dim([P k−1 (P )]

∩ Z(P )) + dim([B k (∂P )]

)

+ dim(S _k (∂P )) − dim( B k (f )) f , (3.19) where the last term ( −dim(B k (f )) f ) ensues from the compatibility condition among h and g mentioned above. We calculate the addenda in the right-hand side of (3.19). Regarding the first term in (3.19), we preliminary note that the following characterization ensues from the exact sequence (2.6) and polynomial decomposi- tion (2.2)

[ P k−1 (P )]

∩ Z(P ) = curl([P k (P )]

) = curl(x ∧ [P k−1 (P )]

). (3.20) Employing again the exact sequence (2.6), curl restricted to (x ∧ [P k−1 (P )]

) is actually an isomorphism, therefore from (3.20) and (3.8) it follows that

dim([ P k−1 (P )]

∩ Z(P )) = dim(x ∧ [P k−1 (P )]

) = 3π k,3 − π k+1,3 + 1. (3.21) From definitions (3.14) and (3.15), direct computations yield

dim(S _k (∂P )) =  e + (dim( B _k (f )) − 1) f , (3.22) where the −1 in the formula above is due to the compatibility condition (2.5).

Collecting (3.21), (3.9) and (3.22) in (3.19) (compare with (3.18)) we get dim(  Σ _h (P )) = #  D

.

Having proved that #  D

is equal to dim(  Σ _h (P )), in order to validate that the linear operators  D

constitute a set of DoFs for  Σ _h (P ) we have to check that they are unisolvent. Let ϕ ∈  Σ _h (P ) such that  D

(ϕ) = 0, we need to show that ϕ is identically zero. It is straightforward that D

(ϕ) = D

(ϕ) = 0 implies

(curl ϕ) _|∂f = 0 for any f ∈ ∂P . (3.23) Recalling the well-known results for nodal boundary spaces,

it is quite obvious to check that (3.23) D

(ϕ) = 0 implies

(curl ϕ) _f = 0 for any f ∈ ∂P .

In order to get also the normal component of (curl ϕ) _|f equal to zero, based on D

(ϕ) = 0, it is sufficient to observe that the compatibility conditions (2.4) and (2.5) give



f

curl ϕ · n ^f _P df =



f

rot _f ϕ _τ df =



∂f

ϕ · t f ds

= 

e∈∂f



e

ϕ · t ^e _f ds = 

e∈∂f

|e| D

(ϕ)t _e · t ^e _f for any f ∈ ∂P

(3.24)

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that is equal to 0 since D

(ϕ) = 0. Therefore, we have proved that

curl ϕ = 0 on ∂P . (3.25)

Moreover, being D

(ϕ) = 0, from (3.25) and (2.4) and definition (3.14) for any f ∈ ∂P we infer

(ϕ _f · t _e ) _|e = 0 for any e ∈ ∂f , div f ϕ _τ = 0 and rot _f ϕ _τ = 0 on f , and thus we obtain

ϕ _τ = 0 for any f ∈ ∂P . (3.26)

Finally, by definition of  Σ _h (P ), there exists  p _k−1 ∈ [P k−1 (P )]

∩ Z(P ) such that



P

Δ ϕ · Δ ψ dP =



P

p _k−1 · ψ dP for all ψ ∈ Ψ

(P ).

Therefore, being ϕ ∈ Ψ

(P ) (cf. (3.25) and (3.26)) we infer

Δ ϕ

=



P

Δ ϕ · Δ ϕ dP =



P

p _k−1 · ϕ dP

=



P

curl (x ∧ q _k−1 ) · ϕ dP (characterization (3.20))

=



P

x ∧ q _k−1 · curl ϕ dP (integration by parts + (3.26)) and thus, since D

(ϕ) and  D

(ϕ) are 0, we obtain Δ ϕ

= 0. Now the proof follows by the fact that Δ · 

is a norm on Ψ

(P ) (see Lemma 5.2 in Ref. 43 and (A.1)).

Notice that the DoFs D

are conveniently chosen in order to have a direct correspondence between the curl of the Lagrange-type basis functions of Σ _h and the Lagrange basis functions of V _h .

Remark 3.2. A careful inspection of Theorem A.1 (see also Remark 5.1 in Ref. 43 and Ref. 20) reveals that the space (3.17) admits the equivalent formulation

Σ _h (P ) := {ϕ ∈ Ψ(P ) s.t. (i) ϕ _|∂P ∈ S k (∂P ), (ii) (curl ϕ) |∂P ∈ [B k (∂P )]

, (iii) Δ

ϕ ∈ [P k−1 (P )]

∩ Z(P ), (iv) div ϕ = 0,

(v) (curl ϕ − Π ^∇,P _k curl ϕ, x ∧ p _k−1 ) _P = 0, ∀  p _k−1 ∈ [P _k−1\k−3 (P )]

}.

(3.27) The global space Σ _h is defined by collecting the local spaces Σ _h (P ), i.e.

Σ _h := {ϕ ∈ Σ(Ω) s.t. ϕ _|P ∈ Σ h (P ) for all P ∈ Ω h }. (3.28) The global set of DoFs is the global counterpart of D

, in particular the choice of DoFs D

establishes the conforming property curl Σ _h ⊆ [H

(Ω)]

. The dimension of Σ _h is given by

dim(Σ _h ) = 3L V + (3k − 2)L e + (3π k−2,2 − 1)L f + (3π k−2,3 − π k−1,3 + 1)L P .

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3.4. Scalar H

-conforming VEM space

In this section, we briefly define the H

-conforming space W h in the virtual complex (2.9). The space W h consists of low-order nodal VEM.

We first introduce the low-order boundary space

B

(f ) := {v ∈ H

(f ) s.t. v |∂f ∈ C(∂f), v _|e ∈ P

(e) for all e ∈ ∂f, Δ f v = 0}, (3.29) and then we consider the VEM space on the polyhedron P

W h (P ) := {v ∈ H

(P ) s.t. v |∂P ∈ C

(∂P ), v |f ∈ B

(f ) ∀ f ∈ ∂P , Δv = 0}, (3.30) with the associated set of DoFs:

• D W : the values of v at the vertexes of the polyhedron P .

It is straightforward to see that the dimension of W h (P ) is dim(W h (P )) =  V . The global space is obtained by collecting the local spaces

W h := {v ∈ H

(Ω) s.t. v _|P ∈ W h (P ) for all P ∈ Ω h } (3.31) with the obviously associated DoFs. The dimension of W h is given by dim(W h ) = L V .

4. The Virtual Elements de Rham Sequence

The aim of this section is to show that the set of virtual spaces introduced in Sec. 3 realizes the exact sequence (2.9).

Theorem 4.1. The sequence (2.9) constitutes an exact complex.

The theorem follows by Propositions 4.1–4.3, here below, stating that the image of each operator in (2.9) coincides with the kernel of the following one.

Proposition 4.1. Let W h and Σ _h be the spaces defined in (3.31) and (3.28), respectively. Then

∇W _h = {ϕ ∈ Σ _h s.t. curl ϕ = 0 in Ω }.

Proof. Essentially we need to prove that

( i1) for every w ∈ W h , ∇w ∈ Σ h and curl( ∇w) = 0,

( i2) for every ϕ ∈ Σ _h with curl ϕ = 0, there exists w ∈ W h such that ∇w = ϕ.

For what concerns the inclusion (i1), every w ∈ W h clearly satisfies curl( ∇w) = 0 ∈ [H

(Ω)]

, therefore we need to verify that ( ∇w) _|P ∈ Σ h (P ) for any P ∈ Ω h . Notice that the tangential component of ∇w satisfies

( ∇ w) τ = ∇ f w τ on each face f ∈ ∂P . (4.1)

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From definition (3.29) and (3.14), for any f ∈ ∂P we infer (i) ( ∇w · t e ) _|e = ∂w

∂t e

∈ P

(e), ∀ e ∈ ∂f, (w |e ∈ P

(e)),

(ii) div _f ( ∇ f w τ ) = 0 in f (Δ _f w τ = 0),

(iii) rot _f ( ∇ f w τ ) = 0 in f (vector calculus identity) that, recalling (4.1), implies ( ∇w) τ ∈ S k (f ). Moreover, w ∈ C

(∂P ) entails

[( ∇w) f

· t e ] _|e = [( ∇w) f

· t e ] _|e for any e ⊆ ∂f

∩ f

, and thus (cf. definition (3.15))

( ∇w) _|∂P ∈ S k (∂P ). (4.2)

Furthermore, definition (3.30) implies (i)



∂P

∇w · n P df =



P

Δw dP = 0, (div. theorem + Δw = 0), (ii) curl( ∇w) = 0 in P (vector calculus identity),

(iii) Δ( ∇w) = ∇(Δw) = 0 in P (Δ w = 0).

(4.3) Collecting (4.2) and (4.3) in definition (3.17), we easily obtain (i1).

We prove now the property (i2). Consider ϕ ∈ Σ h such that curl ϕ = 0. Since (2.7) is an exact sequence, there exists unique (up to constant) w ∈ H 

(Ω) such that ∇  w = ϕ. Therefore, for any face f in the decomposition Ω h , the tangential component of ∇  w satisfies (cf. definition (3.14))

( ∇  w) τ = ∇ _f w  τ = ϕ _τ ∈ [L

(f )]

on f . Hence on each face f the function  w fulfills

 ( ∇  w · t e ) _|e = (ϕ · t _e ) _|e ∈ P

(e) on any e ∈ ∂f ,

w  τ ∈ H

(f ) in f . (4.4)

From (4.4) it follows that w restricted to the mesh skeleton is continuous and piece-  wise linear. Thus the function w is well defined (single valued) on the vertexes of  the decomposition Ω _h and D

( w) makes sense. Let now w ∈ W  h be the interpolant function of w in the sense of DoFs, i.e. the function uniquely determined by 

D

(w) = D

( w).  (4.5)

Inclusion (i1) guarantees that ∇w ∈ Σ h . Hence, by Proposition 3.2, w realizes (i2) if and only if D

( ∇w) = D

(ϕ). Being curl( ∇w) = curl ϕ = 0, this reduce to verify that

D

( ∇w) = D

(ϕ).

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For any edge e in the decomposition Ω h , we denote with ν

and ν

the two endpoints of e, with t e pointing from ν

to ν

. Therefore, from (4.5) and (4.4), we infer

D

( ∇w) = 1

|e|



e

∇w · t e ds = D

(w) − D

(w) = D

( w) − D 

( w) 

= 1

|e|



e

∇  w · t e ds = 1

|e|



e

ϕ · t _e ds = D

(ϕ),

that concludes the proof.

Proposition 4.2. Let Σ _h and Z _h be the spaces defined in (3.28) and (3.12), respec- tively. Then

curl Σ _h = Z _h .

Proof. The proof follows by showing the following points:

( i1) for every ϕ ∈ Σ h , curl ϕ ∈ Z h ,

( i2) for every v ∈ Z h , there exists ϕ ∈ Σ h such that curl ϕ = v.

Let us analyze the inclusion (i1). Let ϕ ∈ Σ h , clearly curl ϕ ∈ Z(Ω). Therefore, we need to verify that (curl ϕ) _|P ∈ V h (P ) for any P ∈ Ω h . It is evident that the constraints in definition (3.17) are the curl version of the constraints in definition (3.7), and (ii) in (3.16) corresponds to (i) in (3.6). Hence it remains to show that curl ϕ is the velocity solution of the Stokes problem associated with definition (3.7) on each element P .

By definition (3.17) the function ϕ satisfies



P

Δ ϕ · Δ ψ dP =



P

p _k−1 · ψ dP, ∀ ψ ∈ Ψ

(P ),

which, recalling (3.20) and by an integration by parts, yields



P

Δ ϕ · Δ ψ dP =



P

(x ∧ p _k−1 ) · curl ψ dP, ∀ ψ ∈ Ψ

(P ). (4.6)

In particular, the last equation is still valid restricting to all ψ ∈ Ψ

(P ) ∩ Z(P ).

Therefore, using the identity Δ = −curl curl + ∇ div and an integration by parts (coupled with the homogeneous boundary condition curl ψ = 0 on ∂P ), it is easy to show that (4.6) implies



P

−Δ(curl ϕ) · curl ψ dP =



P

(x ∧ p _k−1 ) · curl ψ dP, ∀ ψ ∈ Ψ

(P ) ∩ Z(P ).

(4.7)

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