Roland Becker

Daniela Capatina

Julie Joie

A DG METHOD FOR THE STOKES EQUATIONS RELATED TO NONCONFORMING APPROXIMATIONS

We study a discontinuous Galerkin method for the Stokes equations with a new stabilization of the viscous term. On the one hand, it allows us to recover, as the stabilization parameter tends towards innity, some stable and well-known nonconforming approximations of the Stokes problem. On the other hand, we can easily dene an a posteriori error indicator, based on the reconstruction of a locally conservative H(div)-tensor. An a priori error analysis is also carried out, yielding optimal convergence rates. Numerical tests illustrating the accuracy and the robustness of the scheme are presented.

Introduction

There is a large interest in discontinuous Galerkin nite element methods (dG) and the literature became quite impressive, that it would be impossible to mention here all the contributions. For a unied presentation of several existing dG methods for elliptic problems, we refer to [START_REF] Arnold | Unied analysis of discontinuous Galerkin methods for elliptic problems[END_REF].

Our approach for the Stokes equations follows the symmetric interior penalty method for elliptic problems of [START_REF] Arnold | An interior penalty nite element method with discontinuous elements[END_REF], which ensures well-posedness of the discrete formulation and optimal-order error estimates. It has been extended to the Stokes and Navier-Stokes equations by Girault, Rivière and Wheeler [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF]. The velocity is looked for in P k and the pressure belongs to P k-1 , with 1 ≤ k ≤ 3. The employed stabilization term is a penalization of the jumps of the velocities across the edges, penalizing the nonconformity of the discrete solution. Instead, we propose to penalize the L 2projection on P k-1 of the jumps, following a similar idea of Hansbo and Larson [START_REF] Hansbo | Discontinuous Galerkin and the Crouzeix-Raviart element: application to elasticity[END_REF] for the elasticity problem. This approach presents two main advantages.

First, we prove that the solution of the dG formulation tends, as the stabilization parameter γ goes to innity, towards the solution of the P k × P k-1 nonconforming approximation of the Stokes problem, which is known to be stable for 1 ≤ k ≤ 3. Moreover, the inf-sup constant with respect to the energy norm of our method is independent of γ whereas that of [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF] is O(1/ √ γ). Therefore, contrarily to [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF], our method is robust for large stabilization parameters; this phenomenon is highlighted by numerical experiments. Note that we limit ourselves to the case k ≤ 3, since for k ≥ 4 it is known (cf. [START_REF] Scott | Norm estimates for a maximal right inverse of teh divergence operator in spaces of piecewise polynomials[END_REF] ) that P cont k × P disc k-1 is a stable pair of spaces for the Stokes problem, provided that special mesh constructions are avoided.

Second, our choice of the stabilization allows us to construct a locally conservative vector approximation in the Raviart-Thomas space RT k-1 , which is further used to dene a simple a posteriori error estimator for the proposed numerical scheme. This feature is new as regards the dG approximations of the Stokes problem, though it has been previously mentioned by Kim [21] for the Laplace operator. Moreover, in the case k = 1 we show that the a posteriori dG error indicator tends, as γ → ∞, towards the error indicator for the nonconforming Crouzeix-Raviart approximation of the Stokes equations. Recent works on the H(div) ux reconstruction in view of a posteriori analysis include [START_REF] Luce | A local a posteriori error estimator based on equilibrated uxes[END_REF] for mixed nite element methods and [START_REF] Ern | An accurate H(div) ux reconstruction for discontinuous Galerkin approximations of elliptic problems[END_REF] for dG approximation of elliptic equations; note that in the latter, the reconstructed ux belongs to a larger space than ours, that is RT k .

We also discuss in this paper another variational formulation of the Stokes problem, written in terms of the strain rate tensor instead of the gradient of the velocity. The main advantage is the equivalence between its dG version and a three-elds formulation, allowing to recover the stress tensor in an obvious way. One may then be able to generalize it to non-Newtonian uids or to impose 1 other boundary conditions related to the normal stress. In order to retrieve a discrete Korn's type inequality for discontinuous velocities (cf. [START_REF] Brenner | Korn's inequalities for piecewise H 1 vector elds[END_REF] or [START_REF] Mardal | An observation on Korn's inequality for nonconforming nite element methods[END_REF]), we consider an additional stabilization term in the case k = 1. The dG method thus obtained seems to be new, and can be easily generalized to more complex constitutive laws involving nonlinear and/or convective terms. Another velocity-pressurestress formulations of the stationary Stokes equations is considered in [START_REF] Figueroa | Augmented mixed nite element methods for the stationary Stokes equations[END_REF], where an augmented mixed nite element method is studied. We show that the proposed discretizations are well-posed and yield optimal convergence rates, which are conrmed by numerical experiments.

Let us also cite a dierent approach for the discontinuous Galerkin approximation of the Stokes and Navier-Stokes equations, developed by Bassi and Rebay [START_REF] Bassi | A high-order accurate discontinuous nite element method for the numerical solution of the compressible Navier-Stokes equations[END_REF], [START_REF] Bassi | An articial compressibility ux for the discontinuous Galerkin solution of the incompressible Navier-Stokes equations[END_REF] and Cockburn et al. [START_REF] Cockburn | Local discontinuous Galerkin methods for the Stokes system[END_REF], [START_REF] Cockburn | A locally conservative LDG method for the incompressible Navier-Stokes equations[END_REF]. They all introduce the gradient of the velocity as a third variable but they discretize it by means of dierent numerical uxes. The link between the stabilization of [START_REF] Bassi | A high-order accurate discontinuous nite element method for the numerical solution of the compressible Navier-Stokes equations[END_REF], the usual one (cf. [START_REF] Arnold | An interior penalty nite element method with discontinuous elements[END_REF]) and ours will be discussed in Section 3.

A possible extension of this work consists in considering the instationary Stokes equations and developing a space-time dG algorithm. In [START_REF] Van Der Vegt | A space-time discontinuous Galerkin method for the time-dependent Oseen equations[END_REF], such an algorithm is proposed for the time-dependent Oseen equations, by using anisotropic Sobolev spaces on the space-time domain. Their dG discretization of the viscous term is dierent from ours and uses the approach of [START_REF] Bassi | An articial compressibility ux for the discontinuous Galerkin solution of the incompressible Navier-Stokes equations[END_REF], based on the introduction of an articial ux. One could then apply the dG time discretization of [START_REF] Van Der Vegt | A space-time discontinuous Galerkin method for the time-dependent Oseen equations[END_REF] together with our space discretization. Two other extensions, to more general boundary conditions and to the Navier-Stokes equations, are briey discussed in this article.

The outline of the paper is as follows. The model problem and three equivalent variational formulations are introduced in Section 2 and Section 3, respectively. Section 4 is devoted to the description of the numerical approximation, while in Section 5 we establish the well-posedness of the discrete problems. In Section 6, the relationship between our dG method and some wellknown nonconforming approximations of the Stokes problem is discussed. Section 7 is devoted to the derivation of optimal a priori error bounds. Next, a simple a posteriori error indicator is introduced and its eciency and reliability are proved in Section 8. Finally, some extensions are discussed in Section 9 and numerical tests are presented in the last section.

2. Notations and problem setting We agree to write the vectors in bold letters and the second-order tensors in underlined letters, τ = (τ ij ) 1≤i,j≤2 ; the product of two tensors will be denoted by :

τ : σ = τ ij σ ij .
For a given Hilbert space V , we put

V = {v = (v 1 , v 2 ); v i ∈ V, 1 ≤ i ≤ 2} = V × V , respectively V = {τ = (τ ij ); τ ij ∈ V, 1 ≤ i, j ≤ 2} .
We employ in this paper the summation convention of Einstein and we denote by the letter c any positive constant independent of the discretization parameter h. For any k ∈ N, we denote by P k the space of polynomials on R 2 of total degree ≤ k.

In what follows, Ω denotes a Lipschitz domain of R 2 . We use classical notations : H m (Ω) is the Sobolev space of order m ∈ N, • m,Ω and |•| m,Ω are its usual Hilbert norm, respectively semi-norm.

We are interested in the stationary Stokes equations, which describe the steady ow of an incompressible Newtonian uid at low Reynolds numbers. The governing equations are : the momentum conservation law

-∇ • (τ -pI) = f , the mass conservation law ∇ • u = 0
and the constitutive law of a Newtonian uid

τ = 2µD(u).
For the sake of simplicity, we consider a Dirichlet boundary condition u = g on ∂Ω.

Here above, τ denotes the viscous stress tensor, p the pressure, µ the uid's viscosity (supposed to be constant) and D(u) the deformation tensor, given by

D(u) = 1 2 ∇u + (∇u) t
with u the uid's velocity. The data of the problem are f ∈ L 2 (Ω) and g ∈ H 1/2 (∂Ω).

Three equivalent variational formulations

In what follows, we shall consider three mixed variational formulations of the previous boundary value problem, in view of the numerical approximation. For this purpose, let u g ∈ H 1 (Ω) be a continuous lifting of the boundary data g ∈ H 1/2 (∂Ω), such that :

u g = g on ∂Ω and u g 1,Ω ≤ c g 1/2,∂Ω . First, we consider the classical velocity-pressure formulation, obtained by substituting τ . The boundary value problem can then be written as follows :

   -µ∆u + ∇p = f inΩ ∇ • u = 0 inΩ u = g on∂Ω and its variational formulation is : (1)    (u, p) ∈ (u g + H 1 (Ω)) × L 2 0 (Ω) a(u, v) + b(p, v) = l(v) ∀v ∈ H 1 0 (Ω) b(q, u) = 0 ∀q ∈ L 2 0 (Ω),
where the bilinear and the linear forms are dened by

a(u, v) = µ Ω ∇u : ∇vdx, b(p, v) = - Ω p∇ • vdx, l(v) = Ω f • vdx.
Second, we use that ∆u = 2divD(u) and we get the following formulation :

(2)

   (u, p) ∈ (u g + H 1 (Ω)) × L 2 0 (Ω) c(u, v) + b(p, v) = l(v) ∀v ∈ H 1 0 (Ω) b(q, u) = 0 ∀q ∈ L 2 0 (Ω) with c(u, v) = 2µ Ω D(u) : D(v)dx.
We have used that the product between a symmetric and an anti-symmetric tensor vanishes, i.e.

Ω D(u) : ∇vdx = Ω D(u) : D(v)dx.
Finally, we consider a three-elds formulation in the unknowns (τ , u, p) :

       -divτ + ∇p = f in Ω ∇ • u = 0 in Ω τ -2µD(u) = 0 in Ω u = g on∂Ω.
Its weak form is given by

(3)        (u, p, τ ) ∈ (u g + H 1 (Ω)) × L 2 0 (Ω) × X b(p, v) + d(τ , v) = l(v) ∀v ∈ H 1 0 (Ω) b(q, u) = 0 ∀q ∈ L 2 0 (Ω) d(θ, u) -e(θ, τ ) = 0 ∀θ ∈ X,
where

d(τ , v) = Ω τ : D(v)dx e(θ, τ ) = 1 2µ Ω θ : τ dx
and where

X = {θ = (θ ij ) 1≤i,j≤2 ; θ ij = θ ji , θ ij ∈ L 2 (Ω), i, j = 1, 2} .
Note that the symmetry of the stress tensor is strongly imposed in the denition of the space X.

The proofs of the well-posedness of each of these mixed formulations are well known. We refer for instance to [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF] for the rst two formulations, whereas the formulation (3) is equivalent to [START_REF] Arnold | Unied analysis of discontinuous Galerkin methods for elliptic problems[END_REF]. Indeed, if (u, p) is the unique solution of (2) then (u, p, 2µD(u)) satises [START_REF] Bassi | An articial compressibility ux for the discontinuous Galerkin solution of the incompressible Navier-Stokes equations[END_REF], which shows the existence of a solution for (3); reciprocally, if (u, p, τ ) is a solution of (3) then (u, p) satises ( 2) and τ = 2µD(u), which ensures the uniqueness.

The three-elds formulation is useful in order to compute the ow of non-Newtonian uids, which is our further goal. Indeed, when considering such uids one cannot eliminate the stress tensor by means of the corresponding constitutive law, and hence one deals with formulations of at least three unknowns. Nevertheless, the numerical approximation which will be further introduced allows us to show the equivalence between the discrete versions of (3) and (2). This, in our opinion, justies the interest of the two-elds formulation [START_REF] Arnold | Unied analysis of discontinuous Galerkin methods for elliptic problems[END_REF].

Another justication is related to boundary conditions of Neumann's type : in (1), one has to impose (∇u)n on the boundary whereas in [START_REF] Arnold | Unied analysis of discontinuous Galerkin methods for elliptic problems[END_REF], one has to prescribe D(u)n (i.e., τ n in the Newtonian case) which is more meaningful from a physical point of view.

4. Discretization by means of DG methods 4.1. Preliminary notations and results. From now on, we assume that Ω is a polygonal domain. Let (T h ) h>0 be a regular family of triangulations of Ω consisting of triangles : Ω = T ∈T h T. We agree to denote by ε int h the set of internal edges of T h , by ε ∂ h the set of edges situated on the boundary ∂Ω and by ε h the set of all edges of T h ,

ε h = ε int h ∪ ε ∂ h .
As usually, let h T be the diameter of the triangle T and let h = max

T ∈T h h T .
On every edge e belonging to ε int h , such that {e} = ∂T 1 ∩ ∂T 2 , we dene once for all the unit normal n e oriented from T 1 towards T 2 . Then, for a given function ϕ, we dene the average and the jump across the edge e as follows :

[ϕ] = ϕ /T 1 -ϕ /T 2 , {ϕ} = 1 2 ϕ /T 1 + ϕ /T 2 .
If e is situated on the boundary ∂Ω, we agree to take as n e the outward unit normal n; in this case, the jump and the average of ϕ are equal to the trace of ϕ on e. We agree to denote the L 2 (e)-orthogonal projection of a given function ϕ ∈ L 2 (e) on the polynomial space

P k (k ∈ N) by π k ϕ.
In what follows, we take k = 1, 2 or 3 and we introduce the nite dimensional spaces :

V h = v h ∈ L 2 (Ω); (v h ) /T ∈ P k , ∀T ∈ T h , Q h = q h ∈ L 2 0 (Ω); (q h ) /T ∈ P k-1 , ∀T ∈ T h , X h = θ h ∈ X; (θ h ) /T ∈ P k-1 , ∀T ∈ T h .
Let us next recall some approximation results for the spaces V h and Q h (see also [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF]). For each k = 1, 2, 3, it is known that there exist two interpolation operators i h ∈ L(L 2 0 (Ω); Q h ) and I h ∈ L(H 1 (Ω); V h ) such that, for any T ∈ T h and any e ∈ ε h , one has :

T r(i h q -q)dx = 0, ∀r ∈ P k-1 , ∀q ∈ L 2 0 (Ω) (4) and
T r∇ • (I h v -v)dx = 0, ∀r ∈ P k-1 , ∀v ∈ H 1 (Ω) e r • [I h v]ds = 0, ∀r ∈ P k-1 , ∀v ∈ H 1 0 (Ω).
(5) Moreover, for s ∈ [0, k] the following interpolation estimates hold :

∀q ∈ L 2 0 (Ω) ∩ H s (Ω), q -i h q 0,T ≤ Ch s T |q| s,T ∀v ∈ H s+1 (Ω), |v -I h v| 1,T ≤ Ch s T |v| s+1,∆ T (6)
where ∆ T is a suitable macro-element containing T . The case k = 1 follows from [START_REF] Crouzeix | Conforming and nonconforming nite element methods for solving the stationary Stokes equations[END_REF] (with ∆ T = T ), k = 2 from [START_REF] Fortin | A nonconforming piecewise quadratic nite element on triangles[END_REF] and k = 3 from [START_REF] Crouzeix | Nonconforming nite elements for the Stokes problem[END_REF].

Let us now introduce a bilinear form on (H 1 (Ω) + V h ) × (H 1 (Ω) + V h ) representing our new stabilization term : (7)

J(u, v) = µ e∈ε h 1 |e| e [π k-1 u] • [π k-1 v]ds.
Note that J(u, v) = 0 for any u ∈ H 1 0 (Ω).

Finally, let | • | 1,h denote the H 1 -broken semi-norm, dened as follows :

| v | 1,h =   T ∈T h ∇v 2 0,T   1/2 , ∀v ∈ H 1 (Ω) + V h
and let us also introduce the following semi-norm on

H 1 (Ω) + V h : |v| = µ|v| 2 1,h + γJ(v, v) 1/2
where γ > 0 is a stabilization parameter. Then we can prove :

Lemma 4.1. The application v → | v | is a norm on V h . Proof. Let v ∈ V h such that |v| 1,h = J(v, v) = 0.
On the one hand, it follows that v is piecewise constant on every triangle T ∈ T h and on the other hand, one gets that [π k-1 v] = 0 on every edge e ∈ ε h (k = 1, 2, 3). The last assertion translates into :

e [v] • rds = e [π k-1 v] • rds = 0, ∀r ∈ P k-1 and ∀e ∈ ε h .
Hence, v is continuous, respectively zero at the k Gauss points of any edge e ∈ ε int h , respectively e ∈ ε ∂ h . Together with the property v piecewise constant, it implies that v = 0 on Ω, which concludes the proof. 4.2. First two-elds formulation. We consider the next discrete dG formulation of [START_REF] Arnold | An interior penalty nite element method with discontinuous elements[END_REF], where Nitsche's method (cf. [START_REF] Nitsche | Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind[END_REF]) is employed in order to treat the nonhomogeneous boundary condition :

(8)    (u h , p h ) ∈ V h × Q h a h (u h , v h ) + b h (p h , v h ) = l h (v h ) ∀v h ∈ V h b h (q h , u h ) = g h (q h ) ∀q h ∈ Q h .
The bilinear, respectively linear forms are dened as follows :

a h (•, •) = A 0 (•, •) + A 1 (•, •) + γJ(•, •) A 0 (u h , v h ) = µ T ∈T h T ∇u h : ∇v h dx A 1 (u h , v h ) = -µ e∈ε h e ∂u h ∂n e • [v h ] ds + e ∂v h ∂n e • [u h ] ds b h (q h , v h ) = - T ∈T h T q h ∇ • v h dx + e∈ε h e {q h }[v h • n e ]ds l h (v h ) = T ∈T h T f • v h dx -µ e∈ε ∂ h e ∂v h ∂n e • gds + µ e∈ε ∂ h γ |e| e π k-1 g • π k-1 v h ds g h (q h ) = e∈ε ∂ h e q h g • n e ds.
The rst part of A 1 (•, •) comes from the integration by part of -∆u, whereas the last part is added in order to obtain a symmetric form a h (•, •).

In [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF], the authors study a dierent discontinuous Galerkin method for the Stokes and Navier-Stokes equations, where the stabilization term is given by :

J * (u h , v h ) = µ e∈ε h 1 | e | e [u h ] • [v h ]ds.
We shall prove in what follows that, contrarily to the dG method of [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF], ours is robust with respect to the stabilization parameter, i.e. the solution of (8) tends, as γ tends to innity, towards the unique solution of the Stokes problem discretized by P k × P k-1 nonconforming nite elements, for all k = 1, 2, 3. Moreover, we can compute the corresponding edge integrals by a lower degree Gauss formula than in [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF]. Finally, our choice of the stabilization allows us to construct a locally conservative vector approximation in the Raviart-Thomas space, which is further used to dene a simple a posteriori error indicator for the dG method.

Another stabilization term Ω R([u h ]) • R([v h ]
)dx was proposed by Bassi and Rebay in [START_REF] Bassi | A high-order accurate discontinuous nite element method for the numerical solution of the compressible Navier-Stokes equations[END_REF], where R is a lifting of the jumps across the edges in P disc k . In order to improve the computational eciency and memory use, they replace the contributions from the global lifting operator R with a local lifting operator R e , dened by

K⊃e K R e (w) • v h dx = e w • {v h } ds, ∀v h ∈ P disc k .
So nally the stabilization term is approximated by

J # (u h , v h ) = e∈ε h K⊃e γ K K R e ([u h ]) • R e ([v h ])dx.
By means of a simple computation, one can notice that for k = 1, one actually has that R e (w) /e = |e| |K| π 1 w + 1 2 π 0 w which gives :

K R e ([u h ]) • R e ([v h ])dx = 1 2 e [u h ] • R e ([v h ])ds = |e| 2 |K| e [u h ] • [v h ]ds + |e| 4 |K| e [π 0 u h ] • [π 0 v h ]ds.
In conclusion, by choosing γ K = γ one gets on regular meshes that

J # (•, •) = J * (•, •) + J(•, •).
If we change the denition of the lifting operator and we look for it in P k-1 instead of P k , then it follows for k = 1 that R e (w) /e = |e| 2 |K| π 0 w and therefore

J # (•, •) = J(•, •), since K R e ([u h ]) • R e ([v h ])dx = |e| 4 |K| e [π 0 u h ] • [π 0 v h ]ds.
Similar computations could be carried out for k = 2 or 3, allowing to express J # (•, •) in terms of J * (•, •) and J(•, •). 4.3. Second two-elds formulation. We consider the following discrete version of ( 2) :

   (U h , P h ) ∈ V h × Q h c h (U h , v h ) + b h (P h , v h ) = f h (v h ) ∀v h ∈ V h b h (q h , U h ) = g h (q h ) ∀q h ∈ Q h (9)
where :

c h (•, •) = C 0 (•, •) + C 1 (•, •) + γJ(•, •) + γ 1 J 1 (•, •) C 0 (u h , v h ) = 2µ T ∈T h T D(u h ) : D(v h )dx C 1 (u h , v h ) = -2µ e∈ε h e {D(u h )n e } • [v h ] ds + e {D(v h )n e } • [u h ] ds J 1 (u h , v h ) = e∈ε int h 1 |e| e [π 1 (u h • n e )][π 1 (v h • n e )]ds f h (v h ) = T ∈T h T f • v h dx -2µ e∈ε ∂ h e (D(v h )n e ) • gds +µγ e∈ε ∂ h 1 |e| e π k-1 g • π k-1 v h ds
and where γ > 0 and γ 1 > 0 are now two stabilization parameters (which can be chosen independent of h). A second stabilization term J 1 (•, •) is added in order to retrieve the coercivity of the bilinear form c h (•, •), thanks to a discrete Korn's type inequality.

In view of the analysis of (9), we introduce the following semi-norm on H 1 (Ω) + V h :

[[v]] =   2µ T ∈T h D(v) 2 0,T + γJ(v, v) + γ 1 J 1 (v, v)   1/2
and we show the next result :

Lemma 4.2. The application v → [[v]] is a norm on V h , for all k = 1, 2, 3. Proof. Let [[v]] = 0. Then [π k-1 v] = 0 on every edge e ∈ ε h , [π 1 v • n e ] = 0 on every internal edge e ∈ ε int h and D(v) = 0 on every triangle T ∈ T h . The last relation implies that v is a rigid motion on every triangle T ∈ T h (i.e. v = a T + b T x ⊥ with a T ∈ R 2 , b T ∈ R and x ⊥ = (x 2 , -x 1 )
), so v is piecewise linear.

For k = 2 or 3, it follows that π k-1 v = v and therefore we can deduce similarly to the previous lemma that v is continuous across the edges and null on the boundary. So v ∈ H 1 0 (Ω) and the classical Korn inequality implies that v = 0 on Ω.

For k = 1, we have that v • n e is continuous across the internal edges but v is continuous only at the midpoints of the edges. Let us note (cf. [START_REF] Mardal | An observation on Korn's inequality for nonconforming nite element methods[END_REF]) that the tangential trace on a segment of a rigid motion is a constant (and not a fully linear function) : indeed,

(b T x ⊥ ) • t e = b T x • n e is constant along the edge e of equation x • n e = const.
Then we can immediately deduce that v • t e is also continuous across the internal edges,

hence v ∈ H 1 (Ω). So now v = a + bx ⊥ on Ω with π 0 v = 0 on every boundary edge e ∈ ε ∂ h , which implies v = 0. Remark. For k = 2 or 3, it is not necessary to add the new stabilization term J 1 (•, •) in order to enhance the coercivity of c h (•, •) since KerJ ⊂ KerJ 1 .
We only add it in order to obtain a unied presentation of the method for all k. 4.4. Three-elds formulation. We consider the following dG approximation of ( 3) :

(10)          (U h , P h , τ h ) ∈ V h × Q h × X h k h (U h , v h ) + b h (P h , v h ) + d h (τ h , v h ) = f h (v h ) ∀v h ∈ V h b h (q h , U h ) = g h (q h ) ∀q h ∈ Q h d h (θ h , U h ) -e(θ h , τ h ) = 0 ∀θ h ∈ X h ,
where

k h (•, •) = C 1 (•, •) + γJ(•, •) + γ 1 J 1 (•, •) d h (θ h , v h ) = T ∈T h T θ h : D(v h )dx.

5.

Well-posedness of the approximated problems The aim of this section is to prove the well-posedness of the previous discrete formulations. For this purpose, we shall apply the Babǔska-Brezzi theorem for the two-elds formulations ( 8) and ( 9), and we shall directly prove the well-posedness of the three-elds formulation [START_REF] Crouzeix | Nonconforming nite elements for the Stokes problem[END_REF] by showing its equivalence with [START_REF] Cockburn | Local discontinuous Galerkin methods for the Stokes system[END_REF]. Let us begin by checking the coercivity of the bilinear forms

a h (•, •), respectively c h (•, •) on the discrete kernel Ker h b h , as well as the inf-sup condition for b h (•, •) with respect to the norms |•| , respectively [[•]].
Lemma 5.1. For γ large enough, there exists a constant α 1 > 0 independent of h and µ such that

∀v ∈ V h , a h (v, v) ≥ α 1 | v | 2 .
Proof. One has by denition that

a h (v, v) = | v | 2 +A 1 (v, v)
so one only has to control the term (11)

A 1 (v, v) = -2µ e∈ε h e ∂v ∂n e • [v] ds = -2µ e∈ε h e ∂v ∂n e • [π k-1 v] ds.
Thanks to the Cauchy-Schwarz inequality, it follows that

A 1 (v, v) ≥ -2µ e∈ε h |e| ∂v ∂n e 2 0,e 1/2 e∈ε h 1 |e| [π k-1 v] 2 0,e 1/2
.

A classical scaling argument together with the equivalence of norms in nite dimensional spaces yields, for any polynomial function w, that :

|e| w 0,e ≤ c w 0,T where c is a constant independent of the discretization. Now let e ∈ ε int h such that {e} = ∂T 1 ∩ ∂T 2 with T 1 , T 2 ∈ T h . Obviously, |e| ∂v ∂n e 0,e ≤ |e| 2 2 i=1 (∇v) /T i 0,e ≤ c 2 (|v| 1,T 1 + |v| 1,T 2 ).
A similar argument holds on a boundary edge e ∈ ε ∂ h . So by summing upon all edges it follows that

A 1 (v, v) ≥ -2c √ µ|v| 1,h J(v, v)
and nally,

a h (v, v) ≥ µ|v| 2 1,h + γJ(v, v) - 2c √ γ ( √ µ|v| 1,h ) γJ(v, v) .
In conclusion, a h (•, •) is positive denite for γ > c 2 so the statement is established.

Lemma 5.2. For γ large enough, there exists a constant α 2 > 0 independent of h and µ such that

∀v ∈ V h , c h (v, v) ≥ α 2 [[v]] 2 .
Proof. The proof is completely similar to the one of Lemma 5.1. Noting that

c h (v, v) = [[v]] 2 -4µ e∈ε h e {D(v)n e } • [v] ds,
one immediately gets

c h (v, v) ≥ [[v]] 2 -4 µ e∈ε h |e| {D(v)n e } 2 0,e 1/2 J(v, v).
Using that Lemma 5.3. There exists a constant β 1 > 0 independent of h, µ and γ such that :

|e| {D(v)n e } 0,e ≤ c D(v) 0,T 1 + D(v) 0,T 2 on {e} = ∂T 1 ∩ ∂T 2 nally
inf q∈Q h sup v∈V h b h (q, v) q 0,Ω | v | ≥ β 1 √ µ .
Proof. The proof is rather classical. With any q ∈ Q h , we shall associate w ∈ V h satisfying :

b h (q, w) = q 2 0,Ω and | w | ≤ c q 0,Ω . For this purpose, we make use of the continuous inf-sup condition for the Stokes problem (see for instance [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF]). So let q ∈ Q h ⊂ L 2 0 (Ω) and let z ∈ H 1 0 (Ω) such that :

∇ • z = q z 1,Ω ≤ c q 0,Ω .
Then we put w = I h z ∈ V h . By construction, we have according to (5) on every T ∈ T h and every e ∈ ε h :

T q∇ • I h zdx = T q∇ • zdx = q 2 0,Ω , e {q} [I h z • n e ]ds = 0 so b h (q, w) = q 2 0,Ω . Next, let us point out that J(w, w) = µ e∈ε h 1 |e| e [π k-1 w] • [π k-1 w]ds = µ e∈ε h 1 |e| e [π k-1 w] • [I h z]ds = 0,
thanks to [START_REF] Brenner | Poincaré-Friedrichs inequalities for piecewise H 1 functions[END_REF]. Therefore, using the interpolation estimate (6) we get

| w | = √ µ |I h z| 1,h ≤ c √ µ |z| 1,h ≤ c √ µ q 0,Ω which nally yields sup v∈V h b h (q, v) | v | ≥ b h (q, w) | w | ≥ q 2 0,Ω c √ µ q 0,Ω ≥ β 1 √ µ q 0,Ω .
Lemma 5.4. There exists a constant β 2 > 0 independent of h, µ and γ such that :

inf q∈Q h sup v∈V h b h (q, v) q 0,Ω [[v]] ≥ β 2 √ µ .
Proof. We closely follow the proof of Lemma 5.3. With an arbitrary q ∈ Q h , we associate the same function w = I h z ∈ V h as previously and we next prove that

[[w]] =   2µ T ∈T h D(w) 2 0,T + γ 1 J 1 (w, w)   1/2 ≤ c √ µ q 0,Ω .
For k = 2 or 3, one has that J 1 (w, w) = 0, since J(w, w) = 0. So the result is obvious, since by ( 6)

T ∈T h D(w) 2 0,T ≤ c |z| 1,h ≤ c q 0,Ω .
For k = 1, we still have to bound the term

J 1 (w, w) = µ e∈ε int h 1 |e| [w • n e ] 2 0,e .
This is achieved by combining the following ingredients : the fact that [z • n e ] = 0, the trace inequality on {e} = ∂T 1 ∩ ∂T 2 :

1 √ |e| [(I h z -z) • n e ] 0,e ≤ c 1 h T 1 I h z -z 0,T 1 + 1 h T 2 I h z -z 0,T 2 + |I h z -z| 1,T 1 + |I h z -z| 1,T 2 (12)
and the interpolation properties of the Crouzeix-Raviart operator I h . Thus, we get

J 1 (w, w) ≤ cµ |z| 2 1,h
which allows us to conclude. Remark. Note that the constants β 1 and β 2 are independent of γ, which is not the case when considering the stabilization term J * (•, •) of [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF]. We obtain the same coercivity constant α 1 as in [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF], but with dierent energy norms (we shall see in Lemma 7.2 that

J * (v h , v h ) ≤ c |v h | on V h ).
In order to establish the continuity of the bilinear form a h (•, •), it is sucient to notice that the following bounds hold true :

µ e∈ε h e ∂u h ∂n e • [v h ] ds = µ e∈ε h e ∂u h ∂n e • [π k-1 v h ] ds ≤ µ e∈ε h |e| ∇u h 2 0,e 1/2 J(v h , v h ) 1/2 ≤ µ |u h | 2 1,h 1/2 J(v h , v h ) 1/2 ≤ |u h | |v h | .
Here above, we have applied to ϕ = ∇u h the next trace inequality on e ⊂ ∂T combined with an inverse inequality :

1 |e| ϕ 0,e ≤ c( 1 h T ϕ 0,T + |ϕ| 1,T ) ≤ c h T ϕ 0,T .
Similar arguments yield the continuity of c h (•, •) :

µ e∈ε h e {D(u h )n e } • [v h ] ds ≤   µ T ∈T h D(u h ) 2 0,T   1/2 J(v h , v h ) 1/2 ≤ [[u h ]] [[v h ]]
and of b h (•, •):

e∈ε h e {q h } [v h • n e ] ds = e∈ε h e {q h } [π k-1 v h • n e ] ds ≤ 1 √ µ e∈ε h |e| q h 2 0,e 1/2 J(v h , v h ) 1/2 ≤ 1 √ µ q h 0,T J(v h , v h ) 1/2 .
The continuity of the linear forms l h (•), f h (•) and g h (•) follows with the same arguments.

We are now able to state the main results of this section.

Theorem 5.5. For γ suciently large, each of the mixed problems ( 8) and ( 9) has a unique solution.

Proof. According to Lemmas 5.1 and 5.3 for problem [START_REF] Cockburn | A locally conservative LDG method for the incompressible Navier-Stokes equations[END_REF], respectively Lemmas 5.2 and 5.4 for (9), the hypotheses of the Babǔska-Brezzi theorem are satised (cf. [START_REF] Brezzi | Mixed and hybrid nite element methods[END_REF]). Therefore, these mixed variational formulations are well-posed. Theorem 5.6. For γ suciently large, problem [START_REF] Crouzeix | Nonconforming nite elements for the Stokes problem[END_REF] has a unique solution. Moreover, its solution

is (U h , P h , 2µD(U h )) where (U h , P h ) is the unique solution of (9). Proof. It is obvious that (U h , P h , 2µD(U h )) belongs to V h × Q h × X h
and satises the variational problem [START_REF] Crouzeix | Nonconforming nite elements for the Stokes problem[END_REF]. Since we are dealing with nite dimensional spaces, this equally ensures the uniqueness of the solution.

6. Robustness with respect to the stabilization parameter For the sake of simplicity, we suppose in what follows that g = 0. We study here the behaviour of our dG method when γ → ∞ and we prove its robustness with respect to large stabilization parameters. More precisely, we prove that our dG method is robust with respect to large stabilization parameters since its solution converges, when γ → ∞, towards the solution of the P k × P k-1 nonconforming nite element approximation of the Stokes problem.

Let us rst consider the formulation [START_REF] Arnold | An interior penalty nite element method with discontinuous elements[END_REF]. Its nonconforming approximation is given by ( 13)

   (u * h , p * h ) ∈ H h × Q h , A 0 (u * h , v h ) + b h (p * h , v h ) = l h (v h ) ∀v h ∈ H h b h (q h , u * h ) = 0 ∀q h ∈ Q h where H h = v ∈ L 2 (Ω); (v h ) /T ∈ P k , ∀T ∈ T h , v h continuous (resp. null) at the k Gauss points of e ∈ ε int h (resp. ε ∂ h ) .
For k = 1, H h is the well-known P 1 nonconforming space of Crouzeix-Raviart [START_REF] Crouzeix | Conforming and nonconforming nite element methods for solving the stationary Stokes equations[END_REF]; k = 2 corresponds to the P 2 nonconforming nite elements of Fortin-Soulie [START_REF] Fortin | A nonconforming piecewise quadratic nite element on triangles[END_REF], whereas for k = 3 we retrieve the P 3 nonconforming nite elements of Crouzeix-Falk [START_REF] Crouzeix | Nonconforming nite elements for the Stokes problem[END_REF]. It is a well-known result that ( 13) is well-posed for k = 1, 2, 3, thanks to a discrete Poincaré inequality on H h .

It is important to notice that our choice of the stabilization yields ( 14)

Ker h J = v h ∈ V h ; [π k-1 v h ] /e = 0, ∀e ∈ ε h = H h .
Theorem 6.1. Let (u h , p h ) be the solution of ( 8) and (u * h , p * h ) the solution of [START_REF] Ern | An accurate H(div) ux reconstruction for discontinuous Galerkin approximations of elliptic problems[END_REF]. Then one has :

lim γ →∞ ( | u h -u * h | + p h -p * h 0,Ω ) = 0.
Proof. Let us rst show that the sequence (u h , p h ) γ is bounded with respect to γ.

By taking v h = u h as test-function in [START_REF] Cockburn | A locally conservative LDG method for the incompressible Navier-Stokes equations[END_REF] and by using the second variational equation, one classically gets that

a h (u h , u h ) = l h (u h ) ≤ f 0,Ω u h 0,Ω .
Following the proof of Lemma 5.1, we have that

a h (u h , u h ) ≥ µ|u h | 2 1,h + γJ(u h , u h ) -2c √ µ|u h | 1,h J(u h , u h )
with c a constant independent of h, µ and γ. We next recall a Poincaré-Friedrichs inequality for discontinuous nite element spaces (see Brenner [START_REF] Brenner | Poincaré-Friedrichs inequalities for piecewise H 1 functions[END_REF]) :

(15)

v 0,Ω ≤ c   |v| 2 1,h + e∈ε int h 1 |e| [π 0 v] 2 0,e + φ(v)   1/2 , ∀v ∈ V h
where φ : H 1 (Ω) → R is a continuous semi-norm such that for a constant function c, φ(c) = 0 if and only if c = 0. We choose ( 16)

φ(v) = e∈ε ∂ h π 0 v 2 0,e
and we use that π 0 v 0,e ≤ π k-1 v 0,e , since

π 0 v 2 0,e = e π 0 v • vds = e π 0 v • π k-1 vds ≤ π k-1 v 0,e π 0 v 0,e .
Then we can deduce a slightly dierent Poincaré-Friedrichs inequality, which will be employed in the rest of the paper :

v 0,Ω ≤ c |v| 2 1,h + 1 µ J(v, v) 1/2 , ∀v ∈ V h . (17)
Gathering together the last inequalities yields, for γ large enough, that | u h | ≤ C √ µ , with C independent of h, µ and γ.

The inf-sup condition of Lemma 5.3 together with the continuity of a h (•, •) now leads to :

p h 0,Ω ≤ √ µ β 1 sup v h ∈V h b h (p h , v h ) | v h | = √ µ β 1 sup v h ∈V h l h (v h ) -a h (u h , v h ) | v h | ≤ √ µ β 1 f 0,Ω sup v h ∈V h v h 0,Ω | v h | + 2 | u h |
so according to [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF], one gets for γ large enough that p h 0,Ω ≤ C with C independent of h , µ and γ. Therefore, there exist two subsequences of (u h ) γ and (p h ) γ which converge as γ → ∞ towards

u ∞ h ∈ V h , respectively p ∞ h ∈ Q h . From the variational problem (8) one next deduces that u ∞ h ∈ Ker h J = H h .
Using that

A 1 (u h , v h ) = -µ e∈ε h e ∂u h ∂n e • [π k-1 v h ] ds + e ∂v h ∂n e • [π k-1 u h ] ds = 0
for any u h , v h ∈ H h , it follows by passing to the limit in ( 8) that (u ∞ h , p ∞ h ) satises the limit problem [START_REF] Ern | An accurate H(div) ux reconstruction for discontinuous Galerkin approximations of elliptic problems[END_REF]. The well-posedness of ( 13) implies on the one hand, that

(u ∞ h , p ∞ h ) = (u * h , p * h )
and on the other hand, that the whole sequences (u h ) γ and (p h ) γ are convergent.

Remark. If the stabilization term J(•, •) is replaced by J * (•, •) of [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF], then the limit (u ∞ h , p ∞ h ) belongs to Ker h J * × Q h , that is to the (P k -continuous)×(P k-1 -discontinuous) nite element spaces. This is not a stable pair of spaces for the Stokes problem, hence the dG method developed in [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF] is not robust for any k as γ tends towards innity.

We are now interested in the robustness of the formulation [START_REF] Cockburn | Local discontinuous Galerkin methods for the Stokes system[END_REF]. Let us consider the discretization of the variational problem (2) by means of P k × P k-1 nonconforming nite elements, which reads as follows : (18)

   (U * h , P * h ) ∈ H h × Q h C 0 (U * h , v h ) + γ 1 J 1 (U * h , v h ) + b h (P * h , v h ) = f h (v h ) ∀v h ∈ H h b h (q h , U * h ) = 0 ∀q h ∈ Q h .
Remark. In the case k = 2 or 3, one actually has that

J 1 (u h , v h ) = 0 for all u h , v h ∈ H h . Since for any u h , v h ∈ H h one has that c h (u h , v h ) = C 0 (u h , v h ) + γ 1 J 1 (u h , v h ),
one can immediately deduce the well-posedness of the mixed formulation [START_REF] Grisvard | Elliptic problems is nonsmooth domains[END_REF], thanks to the Babǔska-Brezzi theorem and to Lemmas 5.2 and 5.4.

Theorem 6.2. Let (U h , P h ) be the solution of ( 9) and (U * h , P * h ) the solution of [START_REF] Grisvard | Elliptic problems is nonsmooth domains[END_REF]. Then, for γ 1 xed, one has that :

lim γ →∞ ([[U h -U * h ]] + P h -P * h 0,Ω ) = 0
Proof. The proof is very similar to the one of the previous theorem. The only dierence concerns the discrete Poincaré inequality (17) on V h , which has to be replaced here by a Korn's type inequality. For this purpose, we recall the folowing result, rst established by Brenner in [START_REF] Brenner | Korn's inequalities for piecewise H 1 vector elds[END_REF] in a stronger form and then improved by Mardal and Winther in [START_REF] Mardal | An observation on Korn's inequality for nonconforming nite element methods[END_REF] :

(19) |v| 1,h ≤ c   T ∈T h D(v) 2 0,T + 1 µ J 1 (v, v) + φ(v)   1/2 , ∀v ∈ V h
where now φ : H 1 (Ω) → R is a continuous semi-norm such that if φ(v) = 0 for a rigid motion v, then v is a constant vector. With the same choice of φ as in [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF], we can now deduce the following Korn inequalities on V h :

|v| 1,h ≤ c   T ∈T h D(v) 2 0,T + e∈ε ∂ h 1 |e| π k-1 v 2 0,e + 1 µ J 1 (v, v)   1/2 , ( 20
)
v 0,Ω ≤ c   T ∈T h D(v) 2 0,T + 1 µ J(v, v) + 1 µ J 1 (v, v)   1/2 . ( 21
)
The last estimate allows us to conclude as in Theorem 6.1.

As regards the three-elds formulation [START_REF] Crouzeix | Nonconforming nite elements for the Stokes problem[END_REF], it is obvious that for xed γ 1 ,

lim γ →∞ τ h -τ * h 0,Ω = 0
where

τ h = 2µD(U h ) and τ * h = 2µD(U * h ). Note that (U * h , P * h , τ * h )
is the unique solution of the P k × P k-1 × P k-1 nonconforming approximation of the three-elds formulation (3), namely :

       (U * h , P * h , τ * h ) ∈ H h × Q h × X h γ 1 J 1 (U * h , v h ) + b h (P * h , v h ) + d h (τ * h , v h ) = f h (v h ) ∀v h ∈ H h b h (q h , U * h ) = 0 ∀q h ∈ Q h d h (θ h , U * h ) -e(θ h , τ * h ) = 0 ∀θ h ∈ X h .

7.

A priori error estimates This section is devoted to the derivation of optimal a priori error estimates for both the velocity and the pressure, for the two dG formulations (8) and ( 9). We will rst consider the energy norm of the error and then the L 2 -norm of the velocity error, in order to prove a O(h k+1 ) convergence rate for the latter. Let us rst establish some auxiliary results.

a h (u, v h ) + b h (p, v h ) = l h (v h ), ∀v h ∈ V h b h (q h , u) = 0, ∀q h ∈ Q h .
Proof. The proof is classical (see also [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF]), therefore we only give here a sketch of the proof, for the sake of clarity. The second relation is obvious, since ∇•u = 0 on any triangle and [u • n e ] = 0 across any internal edge e. The rst consistency property is obtained after integrating by parts on each element and using the following regularity of the Stokes problem (cf. [START_REF] Grisvard | Elliptic problems is nonsmooth domains[END_REF]) with data f ∈ L 4/3 (Ω) on a Lipschitz polygon : (u, p) ∈ W 2,4/3 (Ω) × W 1,4/3 (Ω). This ensures that both ∇u and p have a trace on each line segment e, which moreover belongs to L 2 (e). Since µ∇u -pI ∈ H(div, Ω) and u ∈ H 1 (Ω), it follows that µ ∂u ∂ne -pn and u are continuous across any internal edge, so the desired result holds true. Lemma 7.2. There exists a constant c > 0 independent of h and µ such that :

e∈ε h 1 |e| [v h ] 2 0,e 1/2 ≤ c √ µ |v h | , ∀v h ∈ V h .
Proof. We obviously have, on any e ∈ E h :

[v h ] 0,e ≤ [v h -π k-1 v h ] 0,e + [π k-1 v h ] 0,e .
Let T be a triangle such that e ⊂ ∂T . Then

1 |e| v h -π k-1 v h 0,e ≤ 1 |e| v h -π T k-1 v h 0,e ≤ c h T v h -π T k-1 v h 0,T ≤ c|v h | 1,T
where π T k-1 denotes the L 2 (T )-orthogonal projection on P k-1 on the element T . These inequalities yield the desired statement, for γ ≥ 1 .

Lemma 7.3. Suppose u ∈ H k+1 (Ω). Then there exists a constant c > 0 independent of h and µ such that

∀v h ∈ V h , |a h (u -I h u, v h )| ≤ c √ µh k |v h | |u| k+1,Ω .
Proof. We recall that a h (•,

•) = A 0 (•, •) + A 1 (•, •) + γJ(•, •).
The Cauchy-Schwarz inequality immediately gives that

A 0 (u -I h u, v h ) ≤ µ |v h | 1,h |u -I h u h | 1,h ≤ c √ µh k |v h | |u| k+1,Ω .
The property (5) of the interpolation operator I h gives on the one hand, that

J(u -I h u, v h ) = µ e∈ε h 1 |e| e [π k-1 I h u] • [π k-1 v h ]ds = 0
and on the other hand, that

e∈ε h e ∂v h ∂n e • [u -I h u] ds = 0.

So we only have to bound the remaining term

µ e∈ε h e ∂(u-I h u) ∂ne • [v h ] ds.
For this purpose, let us introduce the classical Lagrange interpolation operator of polynomial degree k, denoted by L h . Then we can write that e ∂(u -

I h u) ∂n e • [v h ]ds = e ∂(u -L h u) ∂n e • [v h ]ds + e ∂(L h u -I h u) ∂n e • [π k-1 v h ]ds ≤ [v h ] 0,e {∇(u -L h u)} 0,e + [π k-1 v h ] 0,e {∇(L h u -I h u)} 0,e .
Let e be an internal edge common to the triangles T 1 and T 2 ; the proof is completely similar for a boundary edge. Thanks to the trace inequality [START_REF] Dari | Error estimators for nonconforming nite element approximations of the Stokes problem[END_REF] and to classical interpolation estimates for L h , one obtains :

1 |e| {∇(u -L h u)} 0,e ≤ c 1 h T 1 ∇(u -L h u) 0,T 1 + 1 h T 2 ∇(u -L h u) 0,T 2 + |∇(u -L h u)| 1,T 1 ∪T 2 ≤ ch k-1 |u| k+1,T 1 ∪T 2 .
It follows, thanks to Lemma 7.2, that

e∈ε h e ∂(u -L h u) ∂n e • [v h ]ds ≤ ch k e∈ε h 1 |e| [v h ] 2 0,e 1/2 |u| k+1,Ω ≤ c √ µ h k |v h | |u| k+1,Ω .
Next, using that L h u -I h u is piecewise polynomial, we obtain by means of a scaling argument that :

1 |e| {∇(L h u -I h u)} 0,e ≤ c 1 h T 1 ∇(L h u -I h u) 0,T 1 + 1 h T 2 ∇(L h u -I h u) 0,T 2 .
The triangle inequality implies, on any triangle T ,

|L h u -I h u| 1,T ≤ |L h u -u| 1,T + |u -I h u| 1,T ≤ ch k T |u| k+1,∆ T so nally, e∈ε h e ∂(L h u -I h u) ∂n e • [π k-1 v h ]ds ≤ c √ µ h k J(v h , v h )|u| k+1,Ω .
It is now sucient to gather together the previous estimates in order to end the proof.

Lemma 7.4. Suppose p ∈ H k (Ω). Then there exists a constant c > 0 independent of h and µ such that

∀v h ∈ V h , |b h (p -i h p, v h )| ≤ c √ µ h k |v h | |p| k,Ω .
Proof. We recall that

b h (p -i h p, v h ) = - T ∈T h T (p -i h p)∇ • v h dx + e∈ε h e {p -i h p}[v h • n e ]ds = e∈ε h e {p -i h p}[v h • n e ]ds.
Taking into account Lemma 7.2, we get for γ ≥ 1 that

|b h (p -i h p, v h )| ≤ c √ µ e∈ε h |e| {p -i h p} 2 0,e 1/2 |v h | .
On each edge e ∈ ε h such that e ⊂ ∂T , we bound the term p -i h p 0,e as in the proof of Lemma 7.3. Denoting by l h the Lagrange interpolation operator on P k-1 for k = 2 or 3, we obtain :

|e| p -i h p 0,e ≤ |e| ( p -l h p 0,e + l h p -i h p 0,e ) ≤ c p -l h p 0,T + h T |p -l h p| 1,T + l h p -i h p 0,T ≤ c h k |p| k,T .
For k = 1, we directly have :

|e| p -i h p 0,e ≤ c p -i h p 0,T + h T |p -i h p| 1,T ≤ ch|p| 1,T .
So the announced result holds. Remark. For γ < 1, the two previous lemmas also hold true, but with constants dependent of γ.

Theorem 7.5. Let (u, p) ∈ H k+1 (Ω) × H k (Ω) be the solution of the continuous Stokes problem and let γ be suciently large (as in Lemma 5.1). Then the solution (u h , p h ) of the discrete problem (8) satises the following a priori error bounds :

| u -u h | ≤ ch k ( √ µ|u| k+1,Ω + 1 √ µ |p| k,Ω ) (22)
p -p h 0,Ω ≤ ch k (µ|u| k+1,Ω + |p| k,Ω ) (23)
with a constant c independent of h and µ.

Proof. According to Lemma 5.1, one has for γ large enough that

α 1 | u h -I h u | 2 ≤ a h (u h -I h u, u h -I h u) = a h (u h , u h -I h u) -a h (u, u h -I h u) + a h (u -I h u, u h -I h u) = l h (u h -I h u) -b h (p h , u h -I h u) -a h (u, u h -I h u) + a h (u -I h u, u h -I h u).
Thanks to the consistency property stated in Lemma 7.1, one next gets that ( 24)

α 1 | u h -I h u | 2 ≤ b h (p -p h , u h -I h u) + a h (u -I h u, u h -I h u).
On the one hand, the continuity property proved in Lemma 7.3 implies that

a h (u -I h u, u h -I h u) ≤ c √ µh k |u| k+1,Ω |u h -I h u| .
On the other hand, we write that

b h (p -p h , u h -I h u) = b h (p -i h p, u h -I h u) + b h (i h p -p h , u h -I h u)
and we notice that, thanks to the second equation of ( 8) and to the interpolation properties of

I h , b h (i h p -p h , u h -I h u) = -b h (i h p -p h , I h u) = -b h (i h p -p h , u) = 0.
Using now Lemma 7.4, it follows that

b h (p -p h , u h -I h u) ≤ c √ µ h k |p| k,Ω |u h -I h u| .
From [START_REF] Mardal | An observation on Korn's inequality for nonconforming nite element methods[END_REF], we obtain that ( 25)

| u h -I h u | ≤ ch k ( √ µ|u| k+1,Ω + 1 √ µ |p| k,Ω ).
Finally, by means of the triangle inequality and thanks to the relation ( 26)

J(u -I h u, u -I h u) = J(I h u, I h u) = 0,
we deduce (22). In order to establish the error estimate for the pressure, we write that

p -p h 0,Ω ≤ p -i h p 0,Ω + i h p -p h 0,Ω .
According to the discrete inf-sup condition (see Lemma 5.3), one has that

i h p -p h 0,Ω ≤ √ µ β 1 sup v h ∈V h b h (i h p -p h , v h ) |v h | = √ µ β 1 sup v h ∈V h b h (i h p -p, v h ) + b h (p -p h , v h ) |v h | .
The continuity property of Lemma 7.4 gives :

b h (i h p -p, v h ) ≤ c √ µ h k |v h | |p| k,Ω
whereas Lemma 7.1 together with the rst variational equation of ( 8) yield

b h (p -p h , v h ) = -a h (u -u h , v h ), ∀v h ∈ V h .
One can then show, using Lemma 7.3 and the continuity of

a h (•, •) on V h , that a h (u -u h , v h ) = a h (u -I h u, v h ) + a h (I h u -u h , v h ) ≤ c √ µh k |u| k+1,Ω |v h | + 2 | I h u -u h | |v h | .
In conclusion, we obtain that

i h p -p h 0,Ω ≤ ch k (µ|u| k+1,Ω + |p| k,Ω ) + c √ µ | I h u -u h | ,
which together with (25) imply the desired estimate ( 23).

Theorem 7.6. Suppose Ω is convex, (u, p) ∈ H k+1 (Ω) × H k (Ω) and γ is suciently large. Then there exists a constant c independent of h and µ such that

u -u h 0,Ω ≤ ch k+1 (|u| k+1,Ω + 1 µ |p| k,Ω ).
Proof. We adapt Nitsche's argument to our dG discretization. We follow the same steps as in [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF], but with a dierent norm |•| resulting from a dierent stabilization.

Let us rst recall that, due to the convexity of Ω, the Stokes problem

   -µ∆φ + ∇ξ = ψ inΩ ∇ • φ = 0 inΩ φ = 0 on∂Ω admits a unique solution (φ, ξ) ∈ H 2 (Ω) × H 1 (Ω) which moreover satises : (27) µ φ 2,Ω + ξ 1,Ω ≤ c ψ 0,Ω .
We next consider the dual problem with ψ = u h -u and we write, thanks to an integration by parts on each triangle T ∈ T h and to the regularity of φ and ξ, that

u -u h 2 0,Ω = Ω (u h -u) • (-µ∆φ + ∇ξ)dx = a h (φ, u h -u) + b h (ξ, u h -u).
By using the orthogonality equations (cf. Lemma 7.1)

a h (u h -u, v h ) + b h (p h -p, v h ) = 0, ∀v h ∈ V h b h (q h , u h -u) = 0, ∀q h ∈ Q h ,
we obtain, by choosing v h = I h φ and q h = i h ξ : (28)

u -u h 2 0,Ω = a h (φ -I h φ, u h -u) + b h (ξ -i h ξ, u h -u) -b h (p h -p, I h φ).
In what follows, we estimate each of the three righthand side terms.

The last one can be bounded exactly as in [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF]. Let us give some details, for the sake of clarity. First, thanks to the properties of φ and of the interpolation operators I h and i h , we write that

b h (p h -p, I h φ) = b h (i h p -p, I h φ -φ) = e∈ε h e {i h p -p}[(I h φ -φ) • n e ]ds.
Then we employ interpolation estimates (see also Lemma 7.4) and the regularity stated in [START_REF] Scott | Norm estimates for a maximal right inverse of teh divergence operator in spaces of piecewise polynomials[END_REF] in order to conclude that (29

) b h (p h -p, I h φ) ≤ c µ h k+1 |p| k,Ω u -u h 0,Ω .
Concerning the second righthand side term of ( 28), we can write thanks to (4) and to the properties of u that

b h (ξ -i h ξ, u h -u) = e∈ε h e {ξ -i h ξ}[(u h -L h u) • n e ]ds ≤ ch u -u h 0,Ω e∈ε h 1 |e| [u h -L h u] 2 0,e 1/2
.

By using Lemma 7.2, the fact that [u] = [L h u] = 0 as well as the error estimate ( 22), it follows that

e∈ε h 1 |e| [u h -L h u] 2 0,e 1/2 ≤ c |u -L h u| 1,Ω + 1 √ µ |u -u h | ≤ ch k (|u| k+1,Ω + 1 µ |p| k,Ω ) so nally, ( 30
) b h (ξ -i h ξ, u h -u) ≤ ch k+1 (|u| k+1,Ω + 1 µ |p| k,Ω ) u -u h 0,Ω .
We still have to estimate

a h (φ -I h φ, u h -u) = A 0 (φ -I h φ, u h -u) + A 1 (φ -I h φ, u h -u) + γJ(φ -I h φ, u h -u).
The following bound is straightforward :

A 0 (φ -I h φ, u h -u) + γJ(φ -I h φ, u h -u) ≤ |φ -I h φ| |u h -u| ≤ ch k+1 (|u| k+1,Ω + 1 µ |p| k,Ω ) u -u h 0,Ω ,
so let us next consider

A 1 (φ-I h φ, u h -u) = -µ e∈ε h e ∂(φ -I h φ) ∂n e • [u h -u] ds + e ∂(u h -u) ∂n e • [φ -I h φ] ds .
Similarly to the proof of Lemma 7.3, we obtain that

µ e∈ε h e ∂(φ -I h φ) ∂n e • [u h -u] ds ≤ cµh|φ| 2,Ω e∈ε h 1 |e| [u h -L h u] 2 0,e 1/2 ≤ ch k+1 (|u| k+1,Ω + 1 µ |p| k,Ω ) u -u h 0,Ω ,
whereas we can write, thanks to [START_REF] Brenner | Poincaré-Friedrichs inequalities for piecewise H 1 functions[END_REF] and to the fact that ∂(u h -L h u) ∂ne ∈ P k-1 on every edge e ∈ ε h , that :

µ e∈ε h e ∂(u h -u) ∂n e • [φ -I h φ] ds = µ e∈ε h e ∂(L h u -u) ∂n e • [φ -I h φ] ds ≤ cµh k+1 |φ| 2,Ω |u| k+1,Ω ≤ ch k+1 |u| k+1,Ω u -u h 0,Ω .
The previous estimates yield that (31)

a h (φ -I h φ, u h -u) ≤ ch k+1 (|u| k+1,Ω + 1 µ |p| k,Ω ) u -u h 0,Ω .
By gathering together ( 28), ( 29), ( 30) and (31), one now obtains the announced result. One can equally establish optimal error bounds for the other two-elds formulation (9).

Theorem 7.7. Let (u, p) ∈ H k+1 (Ω) × H k (Ω) be the solution of the continuous Stokes problem and let γ be suciently large (as in Lemma 5.1). Then the solution (U h , P h ) of ( 9) satises :

[[u -U h ]] ≤ ch k ( √ µ|u| k+1,Ω + 1 √ µ |p| k,Ω ) p -P h 0,Ω ≤ ch k (µ|u| k+1,Ω + |p| k,Ω )
with a constant c independent of h and µ. Moreover, if Ω is convex then

u -U h 0,Ω ≤ ch k+1 (|u| k+1,Ω + 1 µ |p| k,Ω ).
Proof. The proof is similar to those of Theorem 7.5 and Theorem 7.6.

Concerning the error in the energy norm, one only needs to note the following additional estimates, for any v h ∈ V h :

J 1 (u -I h u, v h ) = 0 for k = 2, 3 J 1 (u -I h u, v h ) ≤ c √ µ   e∈ε int h 1 |e| [u -I h u] 2 0,e   1/2 [[v h ]] ≤ c √ µh k |u| k+1,Ω [[v h ]] for k = 1 |v h | ≤ c [[v h ]] ,
with c independent of h and µ. The rst one results from the trace inequality ( 12) and the interpolation error ( 6) while the second one is an immediate consequence of [START_REF] Karakashian | A posteriori error estimates for a discontinuous Galerkin approximation of a second-order elliptic problem[END_REF].

As regards the L 2 -norm of the velocity error, the main change is that one now has to bound the term

e∈ε h 1 √ |e| [U h -L h u] 0,e .
Thanks to Lemma 7.2 and to the relations [L h u] = [u] = 0, this can be done as follows :

e∈ε h 1 |e| [U h -L h u] 0,e ≤ c √ µ |U h -L h u| ≤ c √ µ [[U h -L h u]] ≤ c D(u) -D(L h u) 0,Ω + 1 √ µ [[u -U h ]] ≤ ch k (|u| k+1,Ω + 1 µ |p| k,Ω ).
8. A posteriori error estimates Our analysis follows the idea of Kim [START_REF] Kim | A posteriori error analysis for locally conservative mixed methods[END_REF], who proposed an a posteriori error indicator for the Laplace equation based on the reconstruction of a locally conservative H(div, Ω)-conforming vector approximation.

We perform here only the a posteriori analysis of problem [START_REF] Cockburn | A locally conservative LDG method for the incompressible Navier-Stokes equations[END_REF]; analoguous results can be established for the formulation [START_REF] Cockburn | Local discontinuous Galerkin methods for the Stokes system[END_REF]. For this purpose, we put

H(div, Ω) = θ ∈ L 2 (Ω); divθ ∈ L 2 (Ω)
and we introduce the Raviart-Thomas nite element space (cf. [START_REF] Roberts | Mixed and Hybrid Methods, volume II of Handbook of Numerical Analysis[END_REF])

Σ h = θ h ∈ H(div, Ω); (θ h ) /T ∈ RT k-1 , ∀T ∈ T h where RT k-1 = P k-1 + P k-1 ⊗ x.
Then we construct a tensor σ h ∈ Σ h from the solution (u h , p h ) of ( 8) by specifying its degrees of freedom as follows :

σ h n e = µ ∂u h ∂n e - µγ |e| [π k-1 u h ] -{p h } n e , ∀e ∈ ε h and for k = 2 or 3, T σ h : rdx = T (µ∇u h -p h I) : rdx, ∀T ∈ T h and ∀r ∈ P k-2 .
By taking as test-function v h in the dG formulation (8) a piecewise polynomial of degree k -1, it follows that (32)

T (divσ h + f ) • rdx = 0, ∀T ∈ T h and ∀r ∈ P k-1 so one has (divσ h ) /T = -π T k-1 f on every triangle T ∈ T h . Obviously, σ h is locally conservative. It is useful to introduce σ = µ∇u -pI which clearly belongs to H(div, Ω).
We next dene, following [START_REF] Kim | A posteriori error analysis for locally conservative mixed methods[END_REF], a residual-type error estimator by

η 2 h = 1 µ T ∈T h σ h -µ∇u h + p h I 2 0,T + J * (u h , u h ).
In order to establish the reliability of η h , let us denote by S(•, •) the bilinear form of the continuous Stokes problem (1), which we extend on (H 1 0 (Ω) + V h ) × L 2 0 (Ω) as follows :

S((u, p), (v, q)) = µ T ∈T h T ∇u : ∇vdx - T ∈T h T p∇ • vdx + T ∈T h T q∇ • udx and let (φ, ξ) ∈ H 1 0 (Ω) × L 2 0 (Ω)
be the unique solution of (33)

S((φ, ξ), (v, q)) = S((u h , p h ), (v, q)), ∀(v, q) ∈ H 1 0 (Ω) × L 2 0 (Ω).
Then we split the error by means of the triangle inequality :

and in what follows, we bound each righthand-side term with respect to η h . Lemme 1. There exists c > 0 depending on Ω and k such that

√ µ |u -φ| 1,Ω + 1 √ µ p -ζ 0,Ω ≤ c 1 √ µ σ h -µ∇u h + p h I 0,Ω + T ∈T h h T √ µ f -π T k-1 f 0,T + J * (u h , u h ) .
Proof. The well-posedness of the continuous Stokes problem implies that :

√ µ |u -φ| 1,Ω + 1 √ µ p -ζ 0,Ω ≤ c sup (v,q)∈H 1 0 (Ω)×L 2 0 (Ω) S((u -φ, p -ξ), (v, q)) √ µ |v| 1,Ω + 1 √ µ q 0,Ω
where the constant depends on Ω. Using now that

S((u -φ, p -ξ), (v, q)) = S((u -u h , p -p h ), (v, q)) = Ω (σ -σ h ) : ∇vdx + T ∈T h T (σ h -µ∇u h + p h I) : ∇vdx + T ∈T h T q∇ • (u -u h )dx
it follows, after integrating by parts and after using (32), that

S((u -φ, p -ξ), (v, q)) = T ∈T h T (σ h -µ∇u h + p h I) : ∇vdx - T q∇ • u h dx + T ∈T h T (f -π T k-1 f ) • vdx which yields √ µ |u -φ| 1,Ω + 1 √ µ p -ζ 0,Ω ≤ c √ µ T ∈T h h T f -π T k-1 f 0,T + σ h -µ∇u h + p h I 0,T + µ ∇ • u h 0,T
In order to bound ∇ • u h 0,T , let us take as test-function q h in (8) the function given by ∇ • u h on T and 0 elsewhere. Then

∇ • u h 2 0,T = 1 2 e⊂∂T e (∇ • u h )[u h • n e ]ds ≤ c ∇ • u h 0,T e⊂∂T 1 |e| [u h ] 2 0,e 1/2
, where c depends on the polynomial degree k. So it follows that

  T ∈T h µ ∇ • u h 2 0,T   1/2 ≤ c J * (u h , u h )
which nally leads to the announced estimate. Lemme 2. There exists c > 0 depending on k and Ω such that :

√ µ |u h -φ| 1,h + 1 √ µ p h -ζ 0,Ω ≤ c J * (u h , u h ).
Proof. The continuous inf-sup condition on b(•, •) implies that

c p h -ζ 0,Ω ≤ sup v∈H 1 0 (Ω) b(p h -ξ, v) |v| 1,Ω = sup v∈H 1 0 (Ω) a(u h -φ, v) |v| 1,Ω ≤ µ |u h -φ| 1,h therefore it is sucient to bound √ µ |u h -φ| 1,h .
A simple calculation together with (33) yield, for any v ∈ H 1 0 (Ω) :

µ |u h -φ| 2 1,h -µ |u h -v| 2 1,h = S((u h -φ, p h -ξ), (u h -φ, p h -ξ)) -S((u h -v, p h -ξ), (u h -v, p h -ξ)) = S((u h -φ, p h -ξ), (v -φ, 0)) + S((v -φ, 0), (u h -φ, p h -ξ)) -S((φ -v, 0), (φ -v, 0)) = 2 T ∈T h T (p h -ξ)∇ • (v -φ)dx -µ |φ -v| 2 1,Ω ≤ 2 T ∈T h T (p h -ξ)∇ • (v -u h )dx.
Then it follows, with C = 1/c, that

|u h -φ| 2 1,h ≤ |u h -v| 2 1,h + 2C |u h -φ| 1,h |u h -v| 1,Ω
and hence,

|u h -φ| 1,h ≤ (C + 1 + C 2 ) |u h -v| 1,h , ∀v ∈ H 1 0 (Ω).
We proceed as in [START_REF] Kim | A posteriori error analysis for locally conservative mixed methods[END_REF] (see also [START_REF] Karakashian | A posteriori error estimates for a discontinuous Galerkin approximation of a second-order elliptic problem[END_REF]) and we take for v ∈ C 0 (Ω) the piecewise P k function dened by its values at the Lagrangian nodes z as follows :

v(z) = (u h ) /T (z),
where T ∈ T h is such that z ∈ T . Then one can establish (cf. [START_REF] Kim | A posteriori error analysis for locally conservative mixed methods[END_REF]) that :

|u h -v| 1,h ≤ c e∈ε h 1 |e| [u h ] 0,e
where c depends on k. This ends the lemma's proof. The two previous lemmas together with the obvious inequality :

J(u h , u h ) ≤ J * (u h , u h )
allow us to conclude to the reliability of η h , stated in the next theorem.

Theorem 8.1. There exists a constant c depending on k and Ω such that :

|u -u h | + 1 √ µ p -p h 0,Ω ≤ (c + √ γ)η h + c T ∈T h h T √ µ f -π T k-1 f 0,T .
We are now interested in the eciency of the a posteriori error indicator. Let us introduce for any triangle T ∈ T h the local contribution

η 2 T = 1 µ σ h -µ∇u h + p h I 2 0,T + e⊂∂T µ |e| [u h ] 2 0,e .
It is useful to recall some results of Verfürth [START_REF] Verfürth | A review of a posteriori error estimation and adaptive mesh-renement techniques[END_REF]. Let b T denote the cubic bubble function on T ∈ T h and b e the quadratic bubble function on e ∈ ε h , satisfying 0 ≤ b T ≤ 1 = max b T and 0 ≤ b e ≤ 1 = max b e . Then there exists a constant C depending only on the minimum angle of T h and on the polynomial degree r such that, for any v ∈ P r (T ) : (34)

v 0,T ≤ C b 1/2 T v 0,T , v 0,e ≤ C b 1/2 e v 0,e .
Moreover, there exists an operator P e which extends any function dened on e ∈ ε h to the triangle T and satises :

C 1 |e| v 0,e ≤ b e P e v 0,T ≤ C 2 |e| v 0,e , ∀v ∈ P r (e).

Then we have :

Theorem 8.2. There exists a constant c depending on the minimum angle of T h and on k such that, for any T ∈ T h , one has

η 2 T ≤ c   µ( S∈ω T |u -u h | 2 1,S ) + 1 µ p -p h 2 0,ω T + e⊂∂T (1 + γ) 2 µ |e| [π k-1 u h ] 2 0,e  
where ω T is the set of all elements sharing an edge with T . Consequently,

η h ≤ c √ µ |u -u h | 1,Ω + (1 + γ) J(u h , u h ) + 1 √ µ p -p h 0,Ω .
Proof. In order to bound the term 1 √ µ σ h -µ∇u h + p h I 0,T , we point out that σ h -µ∇u h + p h I belongs to RT k-1 (T ) and for k ≥ 2, its L 2 (T )-orthogonal projection on P k-2 is zero. Therefore σ h -µ∇u h + p h I 0,T and (σ h -µ∇u h + p h I)n e 0,∂T are equivalent norms. A scaling argument yields

σ h -µ∇u h + p h I 0,T ≤ c e⊂∂T |e| (σ h -µ∇u h + p h I)n e 0,e
with a constant c depending on k. For a given edge e ⊂ ∂T , one next has

(σ h -µ∇u h + p h I)n e = ± [µ∇u h -p h I] n e - µγ |e| [π k-1 u h ]
the sign depending on the orientation of n e with respect to T . So

σ h -µ∇u h + p h I 0,T ≤ c e⊂∂T |e| [µ∇u h -p h I] n e 0,e + µγ √ |e| [π k-1 u h ] 0,e .
Let us remark that an integration by parts gives, for any w ∈ H 1 0 (Ω), that

T ∈T h T (σ -µ∇u h + p h I) : ∇wdx + T ∈T h T div (σ -µ∇u h + p h I) • wdx = T ∈T h ∂T (σ -µ∇u h + p h I)n • wds = -e∈ε h e [µ∇u h -p h I] n e • wds.
We now employ the argument used by Verfüsrth in [START_REF] Verfürth | A review of a posteriori error estimation and adaptive mesh-renement techniques[END_REF], which is based on the weighted norms by the bubble functions and on inverse inequalities. By taking w = b T div(σ -µ∇u h + p h I) and using (34), we rst get that

h T div(σ -µ∇u h + p h I) 0,T ≤ c σ -µ∇u h + p h I 0,T .
We next take w = b e P e ([µ∇u h -p h I] n e ) and we obtain, thanks to (34) and (35),

|e| [µ∇u h -p h I] n e 0,e ≤ c σ -µ∇u h + p h I 0,T 1 ∪T 2 ,
with c now depending on the minimum angle of T h and on k. This nally implies that :

1 √ µ σ h -µ∇u h + p h I 0,T ≤ c √ µ S∈ω T |u -u h | 1,S + 1 √ µ p -p h 0,ω T + γ e⊂∂T √ µ √ |e| [π k-1 u h ] 0,e .
We still have to bound the remaining term

√ µ √ |e| [u h ]
0,e of η T , on every edge e ⊂ ∂T . For this purpose, we proceed similarly to the proof of Lemma 7.2 and, using that [u] = 0, we write that:

[u h ] 0,e ≤ [(u h -u) -π k-1 (u h -u)] 0,e + [π k-1 u h ] 0,e .
Let S a triangle of T h such that e ⊂ ∂S. The trace inequality together with the interpolation properties of π S k-1 next give that

1 √ |e| (u h -u) -π k-1 (u h -u) 0,e ≤ 1 √ |e| (u h -u) -π S k-1 (u h -u) 0,e ≤ c 1 h S (u h -u) -π S k-1 (u h -u) 0,S + |(u h -u) -π S k-1 (u h -u)| 1,S ≤ c|u h -u| 1,S with c depending on k. So it follows that e⊂∂T √ µ |e| [u h ] 0,e ≤ c √ µ( S∈ω T |u h -u| 1,S ) + e⊂∂T √ µ |e| [π k-1 u h ] 0,e
which concludes the proof. Finally, we show in what follows that the a posteriori error indicator η h tends, as γ → ∞, towards an a posteriori error estimator of the nonconforming discretization (13) similar to the one developed in [START_REF] Dari | Error estimators for nonconforming nite element approximations of the Stokes problem[END_REF] . For simplicity, we restrict ourselves to the case k = 1 and piecewise constant right-hand side.

Theorem 8.3. Let k = 1 and the right-hand side f be piecewise constant with respect to T h . Then we have with u * h the nonconforming nite element solution of ( 13):

(36)

lim γ→∞ η 2 h = 1 4µ T ∈T h f ⊗ (x -x T ) 2 0,T + J * (u * h , u * h ).
Proof. Following the idea of Marini, we dene on each triangle T an element of RT 0 by

σ * h = µ∇u * h -p * h I - 1 2 f ⊗ (x -x T ).
Let any v h ∈ V h . Then obviously divσ * h = -f = divσ h , σ * h ∈ Σ h and also, using the fact that

T (x -x T )dx = 0, e∈ε h e σ * h n e • [v h ] ds = Ω divσ * h • v h dx + Ω σ * h : ∇v h dx = T ∈T h (- T f • v h dx + µ T ∇u * h : ∇v h dx - T p * h ∇ • v h dx).
Meanwhile, a simple computation yields, thanks to problem [START_REF] Cockburn | A locally conservative LDG method for the incompressible Navier-Stokes equations[END_REF] :

e∈ε h e σ h n e • [v h ] ds = - Ω f • v h dx + T ∈T h (µ T ∇u h ∇v h dx - T p h ∇ • v h dx) -µ e∈ε h e ∂v h ∂n e • [u h ] ds.
Theorem 6.1 next gives that

lim γ→∞ e∈ε h e σ h n e • [v h ] ds = - Ω f • v h dx + µ Ω ∇u * h : ∇v h dx - Ω p * h ∇ • v h dx = e∈ε h e σ * h n e • [v h ] ds.
By taking as v h the restriction on T of the basis function of the Crouzeix-Raviart space associated with e ⊂ ∂T and zero elsewhere, it follows that

lim γ→∞ (σ h -σ * h )n e 0,e = 0, ∀e ∈ ε h so lim γ→∞ σ h -σ * h 0,Ω = 0.
Passing to the limit in the expression of η h yields (36). Remarque 1. Thanks to the continuity condition of the nonconforming space, the term J * (u * h , u * h ) can be bounded by the term µ e∈ε h |e| e [ ∂u h ∂te ] 2 ds where t e denotes the tangent of edge e. This yields the form of the nonconformity error known from the literature, see for example [START_REF] Dari | Error estimators for nonconforming nite element approximations of the Stokes problem[END_REF].

In the general case f = π 0 f , we obtain an additionnal higher order term. 9. Extensions Several extensions of the proposed dG method for the steady Stokes equations can be envisaged. In what follows, we briey discuss two of them.

We are rst interested in the treatement of more general boundary conditions and, in view of the generalization to non-Newtonian uids, we focus on the second variational formulation (2). We recall that τ = 2µD(u). Besides the usual Dirichlet boundary condition, one may want to prescribe a Neumann condition τ n -pn = χ. Nevertheless, for certain applications it is important to dispose of a larger panel of boundary conditions. For instance, the exact solution of the Poiseuille ow does not satisfy the previous Neumann condition on the outlet boundary. So we consider a partition of the boundary ∂Ω = Γ 1 ∪ Γ 2 ∪ Γ 3 ∪ Γ 4 such that Γ 2 = ∂Ω and we impose :

u • t = g t , u • n = g n on Γ 1 (pure Dirichlet) τ n • n -p = χ n , τ n • t = χ t on Γ 2 (pure Neumann) u • t = g t , τ n • n -p = χ n on Γ 3 u • n = g n , τ n • t = χ t on Γ 4 .
Then, since the boundary terms are treated by means of Nitsche's method, the bilinear forms J(•, •), C 1 (•, •) and b h (•, •) have to be changed as follows :

J(u h , v h ) = µ e∈ε int h ∪ε 1 h ∪ε 3 h 1 |e| e [π k-1 u h • t e ][π k-1 v h • t e ]ds +µ e∈ε int h ∪ε 1 h ∪ε 4 h 1 |e| e [π k-1 u h • n e ][π k-1 v h • n e ]ds C 1 (u h , v h ) = -2µ e∈ε int h ∪ε 1 h ∪ε 3 h e {D(v h )n e • t e } [u h • t e ]ds + e {D(u h )n e • t e } [v h • t e ]ds -2µ e∈ε int h ∪ε 1 h ∪ε 4 h e {D(v h )n e • n e } [u h • n e ]ds + e {D(u h )n e • n e } [v h • n e ]ds b h (q h , v h ) = - T ∈T h T q h ∇ • v h dx + e∈ε int h ∪ε 1 h ∪ε 4 h e {q h } [u h • n e ]ds.
The linear forms are modied accordingly :

f h (v h ) = T ∈T h T f • v h dx + e∈ε 2 h ∪ε 3 h e χ 1 v h • n e ds + e∈ε 2 h ∪ε 4 h e χ 2 v h • t e ds -2µ( e∈ε 1 h ∪ε 3 h e D(v h )n e • t e g 1 ds + e∈ε 1 h ∪ε 4 h e D(v h )n e • n e g 2 ds) +µγ( e∈ε 1 h ∪ε 3 h 1 |e| e π k-1 g 1 π k-1 v h • t e ds + e∈ε 1 h ∪ε 4 h 1 |e| e π k-1 g 2 π k-1 v h • n e ds) g h (q h ) = e∈ε 1 h ∪ε 4 h e q h g 2 ds.
Next, we propose to extend our discretization to the Navier-Stokes equations by following the approach of Girault et al. [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF], which yields a priori error estimates. For the sake of completeness, we recall here below the discretization of the nonlinear convective term given in [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF]. The authors adapted to the case of discontinuous velocities the upwind scheme introduced by Lesaint and Raviart [22] and considered :

c N S (u, v) = T ∈T h T u • ∇uvdx + ∂T - |{u} • n T | (u int -u ext ) • v int ds + 1 2 T ∈T h T (∇ • u)u • vdx - 1 2 e∈ε h e [u] • n e {u • v}ds
where ∂T -= {x ∈ ∂T ; {x} • n T < 0}. It is important, for the mathematical analysis, to note that c N S (u, u) ≥ 0.

10. Numerical tests In this section, we present several numerical experiments in order to conrm the theoretical results. We are interested in the convergence rate and in the inuence of the stabilization parameter. Comparisons with the dG method proposed in [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF] are carried out, illustrating the robustness of our method. The developed codes are written in C++ and use the in-house C++ library CONCHA (http://uppa-inria.univ-pau.fr/concha). The triangulation is obtained by rst meshing the domain into quadrilaterals and then decomposing each quadrilateral into triangles as in the gure below. One thus ends up with a "criss-cross" mesh, cf. Fig. 1. At each renement step, the discretization parameter h is divided by 2; we denote by ne the total number of triangles. First formulation : convergence rates for dierent k For k = 3, we have rst taken γ = 10 as in the previous tests, but then the method didn't converge. This is not in contradiction with the theoretical results since according to Lemma 5.1, γ has to be large enough and it depends on the polynomial degree. The inuence of the stabilization parameter will be studied more extensively in the next subsection; for the moment, we have simply performed similar tests with a larger value of γ (γ = 100). Behavior of the errors with respect to γ for k = 2 (µ = 10) One can notice that, contrarily to the stabilization of [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF], ours yields a stable scheme independently of γ. The method of [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF] leads to bigger errors, which increase with γ . 10.2.2. Comparison of solutions for a Poiseuille ow. We now consider a Poiseuille ow in the domain Ω = [0; 0.06] × [-0.01; 0.01]. On the inow, we set u • t = 0 and u • n, whereas on the outow we impose a homogeneous Neumann condition : µ(∇u)n -pn = 0.

We rst set a parabolic velocity on the inow u • n = a(0.01 2 -y 2 ), which yields the following exact solution of the Stokes problem: u = (a(0.01 2 -y 2 ), 0), p = bx + c . For k ≥ 2 both dG codes give the exact solution, as expected. We now let the stabilization parameter γ vary and we compare the two solutions obtained for k = 1. The numerical tests are carried out on a unstructured mesh consisting of 10954 triangles. We have obtained similar results for the velocity eld, for γ between 10 and 10000, even though the method of [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF] presents some instabilities at large γ (see Fig. 9 and 10). Nevertheless, signicant dierences between the two methods appear as regards the computation of the pressure, therefore we have performed a more detailed study with respect to γ. One may see in Fig. 11 that the method of [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF] is clearly less accurate and less stable than ours.

We have equally tested the two dG methods on non-smooth solutions, by imposing in the previous test-case u • n = 1 on the inow boundary. Note that the exact velocity does not belong to H 1 (Ω). In order to dispose of a reference solution, we have computed it by means of nonconforming nite elements of Crouzeix-Raviart (see Fig. 12). We have now employed a criss-cross mesh consisting of 18432 triangles. One can notice again a lack of accuracy of the method of [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF], which becomes visible at rather small values of the stabilization parameter, such as γ = 20. As γ increases, the pressure computed with the method of [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF] gets worse whereas our method is robust. (e) Our method for γ = 10000 (f) Method of [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF] for γ = 10000 (g) Our method for γ = 100000 (h) Method of [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF] for γ = 100000 (c) Our method for γ = 100 (d) Method of [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF] for γ = 100 (e) Our method for γ = 1000 (f) Method of [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF] for γ = 1000 (g) Our method for γ = 10000 (h) Method of [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF] for γ = 10000

  leads to the desired result, for γ suciently large. Let us now focus on the inf-sup condition for b h (•, •), with respect to both norms |•| and [[•]].

Lemma 7 . 1 .

 71 The solution (u, p) of the continuous Stokes problem satises the consistency properties :

10. 1 .

 1 Mesh convergence.[START_REF] Crouzeix | Nonconforming nite elements for the Stokes problem[END_REF].1.1. First two-elds formulation. We rst study the behavior of the numerical scheme (8) with respect to mesh renement, for k = 1, 2 and 3. We consider the exact solution of the Stokes problem with non-homogeneous Dirichlet conditions : (37) u(x, y) = π cos(πx) sin(πy) -π sin(πx) cos(πy) , p(x, y) = sin(πx) sin(πy) on the square Ω = [-1, 1] × [-1, 1].

Figure 1 .

 1 Figure 1. Triangular mesh from quadrilateral one We have represented in Fig. 2 the logarithm of the errors in terms of the logarithm of ne, for all k. As expected, we numerically obtain a convergence rate O(h k ) for the pressure and for the velocity in the energy norm, and an improved order of convergence O(h k+1 ) for the velocity in the L 2 -norm. The results below are obtained for γ = 10 and µ = 1.

Figure 2 .

 2 Figure 2.First formulation : convergence rates for dierent k For k = 3, we have rst taken γ = 10 as in the previous tests, but then the method didn't converge. This is not in contradiction with the theoretical results since according to Lemma 5.1, γ has to be large enough and it depends on the polynomial degree. The inuence of the stabilization parameter will be studied more extensively in the next subsection; for the moment, we have simply performed similar tests with a larger value of γ (γ = 100).

Figure 5 .

 5 Figure 5. Behavior of the errors with respect to γ for k = 1 (µ = 1)

Figure 7 .Figure 8 .

 78 Figure 7. Behavior of the errors with respect to γ for k = 3 (µ = 100)

Figure 9 .

 9 Figure 9. Comparison of U 0 for a Poiseuille ow at γ = 100000

  (a) Our method for γ = 100 (b) Method of[START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF] for γ = 100 (c) Our method for γ = 1000 (d) Method of[START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF] for γ = 1000

Figure 11 .

 11 Figure 11. Poiseuille ow : comparison of pressures for dierent γ

  (a) Our method for γ = 10(b) Method of[START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF] for γ = 10

Figure 13 .

 13 Figure 13. Comparison of pressures for a non-smooth solution

√ µ |u -u h | 1,Ω + 1 √ µ p -p h 0,Ω ≤ √ µ |u -φ| 1,Ω + 1 √ µ p -ζ 0,Ω + √ µ |u h -φ| 1,Ω + 1 √ µ p h -ζ 0,Ω

We present in Tables 1, 2 and 3 the values of the errors on dierent meshes for k equal to 1, 2 and 3 respectively and, for each error, the ratio between the value computed on the previous (coarse) mesh and on the actual (rened) one. Table 3 : Ratio of the errors on succesive meshes for k = 3 (γ = 100) A similar behavior of the error was observed for dierent values of the viscosity. To illustrate this point, we present the results obtained for the same test-case, but with µ = 100. As predicted by the theory, the viscosity has no inuence on the convergence rate (see Fig. 3 ). Figure 3. Convergence rate of the errors for k = 1 and µ = 100 10.1.2. Second two-elds formulation. We now perform the same test as in the previous paragraph, but we employ the variational formulation [START_REF] Cockburn | Local discontinuous Galerkin methods for the Stokes system[END_REF]. We recall that the additionnal stabilization term J 1 (•, •), whose role is to ensure the discrete Korn inequality, is necessary only in the case k = 1. For this reason, we have chosen to illustrate the convergence rate of the dG method only for k = 1.

We have represented in Fig. 4 the corresponding error curves in log scale, for the velocity (in the energy norm and in the L 2 -norm) and for the pressure. They are in agreement with the theoretical results, that is : 4 : Ratio of the errors on succesive meshes for k = 1 (γ = 10, µ = 1)

Figure 4. Second formulation : convergence rate of the errors for k = 1 10.2. Behavior with respect to the stabilization parameter. In this subsection, we let γ vary and compare the results given by our numerical method for [START_REF] Cockburn | A locally conservative LDG method for the incompressible Navier-Stokes equations[END_REF] with those given by the scheme of [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF], which we call GRW in what follows. We rst consider the previous test-case for which the exact solution is known, and then we treat the Poiseuille ow. We are interested in the computed errors and solutions for large γ, on a xed mesh. 10.2.1. Comparison of errors. Let the exact solution be given by (37). We employ a mesh consisting of 4096 elements. We next compare the velocity errors in energy norm and the pressure errors computed by the two methods (ours in continuous lines, the one of [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF] in dotted lines), for dierent values of γ. Fig. 5 corresponds to the case k = 1, Fig. 6 to k = 2 while Fig. 7 corresponds to k = 3. Dierent values of the viscosity have been chosen, in order to emphasize the independence of the schemes on this parameter. We recall that the energy norm of [START_REF] Girault | A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems[END_REF] associated with the velocity is dierent from ours, and is given by : µ |v| 2 1,h + γµ


domain: info spi

text:
Daniel Fiuza
email: danielfiuzacm@gmail.com

Francisco Junior

Fabio Gomes

Ivana Cristina

Odorico Monteiro
email: odorico@ufc.br

Mauro Oliveira
email: amauroboliveira@gmail.com

A SmartNextGISSA for Monitoring and Predicting of Comorbidities

A casual or late diagnosis given the lack of knowledge of the severity of a problem or asymptomatology of patients with comorbidity can cause serious consequences for the patient. Therefore, there is an urgent need for decision-making systems based on continuous monitoring in real time. This research presents SmartNextGISSA, an intelligent mechanism used in NextGISSA, an architecture developed with the objective of assisting decision-making in a situation room in monitoring patients with comorbidity, based on the crossing of data collected in real time from this patient and his health history. NextGISSA predictively and continuously monitors in real time patients with comorbidities (Pregnant Women, Hypertensives, Diabetics and Bedridden) of GISSA, an intelligent governance platform for decision-making in Digital Health, a product already operational in several municipalities by the startup "Avicenna Intelligent Governance". With the intention of validating the machine learning module of the architecture proposed in NextGISSA, the SmartNextGISSA module was implemented, as an emphasis of this article, which is characterized by a composition of the NextGISSA architecture responsible for the inference about the data of an individual classifying it with different comorbidity risk levels through machine learning algorithms. Thus, we present a development pipeline that consist of a comparative analysis between binary classification algorithms, such as Naive Bayes, Decision Tree and Random Forest, using the CRISP-DM methodology, a concept applied to data mining processes and, later, the model was implemented in a web application.

Introduction

Scientific and technological development has provided means of using digital information and communication technologies (TDIC), widely disseminated in the most diverse instances of society. Thus, many areas, such as health, have taken advantage of the benefits of these technologies, such as Machine Learning, applied to management, evaluation and analysis systems.

In the context of the Covid pandemic, the term comorbidity has become commonplace on a daily basis. The name is given to a set of causes that aggravate a disease and, in this way, can worsen a clinical condition. This study is justified, since a casual or late diagnosis given the lack of knowledge of the seriousness of the problem or even asymptomatology of patients with comorbidity urges the taking of measures with a fruitful and continuous follow-up in real time.

Machine learning can target different users with some comorbidity. With the intention of validating the machine learning module of the architecture proposed in NextGISSA, the SmartNextGISSA module was implemented, as an emphasis of this article, which is characterized by a composition of the NextGISSA architecture responsible for the inference about the data of an individual classifying it with different comorbidity risk levels through machine learning algorithms.

As for the choice of the research universe -population with comorbidity, it is known that this public is one of the priorities in care within the Primary Care policy of the public health network.

In view of this, the object of study is problematized through the following guiding question of this research: How to validate the machine learning module of the architecture proposed in NextGISSA from a scenario to monitor predictively and continuously in real time and in a way smart the patients with comorbidity found in the report of enrolled people (Pregnant Women, Hypertensive, Diabetic and Bedridden) of GISSA?

Thus, this article details the machine learning process applied to a dataset of individuals with hypertension to evaluate the application of supervised machine learning methods, specifically, binary classification algorithms, and implement them in the SmartNextGISSA module in order to to validate the architecture proposed in NextGISSA.

Therefore, the main contribution of this article includes validating the proposed implementation of the SmartNextGISSA module embedded in the NextGISSA architecture in order to provide a predictive model capable of inferring an individual's risk of comorbidity.

Thus, this article is structured as follows: Section two presents related works that use artificial intelligence are presented. In section three, the methodology used in the data mining process for the elaboration of the model. Section four presents an application scenario for SmartNextGISSA. In section five, the results obtained are presented and discussed and, finally, in section six, the conclusion and future aspirations of this article.

As for the choice of the research universe -population with comorbidity, it is known that this public is one of the priorities in care within the Primary Care policy of the public health network.

In view of this, the object of study is problematized through the following guiding question of this research: How to validate the machine learning module of the architecture proposed in NextGISSA from a scenario to monitor predictively and continuously in real time and in a way smart the patients with comorbidity found in the report of enrolled people (Pregnant Women, Hypertensive, Diabetic and Bedridden) of GISSA?

Therefore, the main contribution of this article includes validating the proposed implementation of the SmartNextGISSA module embedded in the NextGISSA architecture in order to provide a predictive model capable of inferring an individual's risk of comorbidity.

Thus, this article is structured as follows: Section two presents the methodology used in the data mining process for the elaboration of the model. In section three, related works that use artificial intelligence are presented. Section four presents an application scenario for SmartNextGISSA. In section five, the results obtained are presented and discussed and, finally, in section six, the conclusion and future aspirations of this article.

2 Related Works BRAGA, Oton Crispim (2017) proposed an intelligent solution based on classifiers as an inference mechanism, capable of helping health professionals during the process of clinical management of diseases transmitted by the Aedes Aegypti mosquito, identifying the diagnosis based on symptoms and test results. The work was divided into two steps: one focused on pre-diagnosis, considering symptoms and clinical history, anamnesis; and another focused on the final diagnosis, also considering the results of specific tests, such as serology tests. The study uses a methodology based on Data Mining to guarantee knowledge based on examples. After several tests and adjustments in machine learning algorithms, it was possible to define two learning models capable of inferring the probability of a patient being infected with a certain disease, with an accuracy of up to 91.6%. From these models, they were able to build an intelligent API to support decision-making during the clinical management of dengue and chikungunya. The solution allows several applications to access the learning models. Among them, a popular consultation mobile application for the identification of dengue and chikungunya, MARCIA, an interoperable system for the clinical management of chikungunya.

Costa [START_REF] Filho | Um Sistema Para Governança em Saúde Digital Baseado em Aprendizado de Máquina[END_REF] proposes Smart GISSA, which is an architecture built from the Intelligent Governance System in Health Systems -GISSA, an innovative solution in the health area already highlighted in this work. This solution provides managers, servers and users with intelligent governance by promoting the systemic integration of information. Smart GISSA's architecture, in addition to extracting, transforming and loading data into dashboards, includes machine learning models to specialize decision-making in public health, in which two new Data Mining methodologies are proposed, focusing on risk analysis of death and in epidemiological surveillance to predict epidemics. It was possible to verify that this architecture already proposed to GISSA by Costa [START_REF] Filho | Um Sistema Para Governança em Saúde Digital Baseado em Aprendizado de Máquina[END_REF] does not use IOT and is not aimed at the hypertensive public.

In this way, the GISSA platform has undergone continuous evolution since its conception in 2009, as a result of research, development and innovation. Thus, research has been incorporated into GISSA, notably in the recent master's research (Francisco Junior, 2022).

From the study carried out on the related works and the proposed solutions, it was observed that almost all of the works directly related to the subject under study, Machine Learning, are aimed at different situations of case studies. Thus, this study arose from the need and possibility of converging Machine Learning and any comorbidity (hypertensive, diabetic, cardiac) on an existing platform, adding value to the architecture, in this case, of GISSA, in an expanded architecture.

Methodology

The research takes place from the performance of GISSA (Gissa, 2022), having as spatial delimitation the city of Fortaleza. The GISSA platform operates in a datacenter, in the cloud, provided by the company Avicena Software e Serviços Ltda. It can be accessed through the web, by computer terminals, smartphones and tablets and proactively communicating through the use of the system, accessing panels (dashboards), with access to summarized and analytical information, filterable, weighted, classified according to risks and compliance with standards quality for services provided, reporting and sending alerts and categorized messages. 2. Understanding the Data: Considering the definition of the problem and the compatibility of the data with the NextGISSA platform, the dataset was composed with health data from individuals characterized with 11 predictive variables (age, gender, height, weight, blood pressure, cholesterol, glucose level, smoker, alcoholic and active) and are classified into two groups of individuals (hypertensive and non-hypertensive). Data were obtained from resources available on the Kaggle platform and then exploratory data analysis was applied in order to describe and qualify them through statistical measurements. Among the analyses, the quality of the data was verified (presence of missing values, outliers, amount of data, among others) and the balance of data, indicating that they are approximately equally distributed between the two groups, with 34,979 cases of cardiac and 35,021 non-cardiac cases, totaling a sample of 70,000 cases.

3. Data preparation: This step consist of pre-processing the data, summarized by the steps of selecting columns and rows of interest (filtering), cleaning the data (removing missing values and outliers) and formatting the data (conversion of values categorical values into numerical values). After the pre-processing step, the data were normalized in order to transform the attributes into a similar scale.

Modeling:

The process of defining the model is based on the pipeline prepared by Costa Filho (2021), since binary classification algorithms can be applied to the data set used in this article. Thus, some steps were performed to identify and evaluate the model to be defined, among them, the data loading and pre-processing, exploratory data analysis, hyperparameter optimization. The developed flow is presented in Figure 2. In the training stage, the binary classification algorithms, Logistic Regression (LR), Random Forest (RF) and Decision Tree (DT) were applied using the Scikit-learn library. After training, the model was evaluated based on the main metrics, accuracy, precision, recall, f1 and the AUROC curve (Area Under ROC Curve). The features of the dataset are (F1) age in days of the individual, (F2) gender, (F3) height, (F4) weight, (F5) blood pressure, (F6) cholesterol level, (F7) glucose level, (F8) smoker, (F9) alcoholic and (F10) physically active.

5. Evaluation: In this stage, the performance of the supervised classifiers used in the training stage was evaluated. Then, the GridSearch strategy associated with the K-Fold Cross Validation (CV) method was applied, thus allowing the identification of the best parameters from combinations of hyperparameters. For the RF and DT classifiers, hyperparameters, number of estimators and maximum number of features were used, as shown in Table 1, while in LR penalty, C and solver from the Scikit-learn library were used, according to . Implementation: After evaluating the model from the AUROC curves and performance metrics, its parameters were serialized and made available in a Rest API destined to a web application for validating the SmartNextGISSA module. In this context, this module's web service consumes the API that performs the inference based on the trained model. For this, an endpoint was made available that receives a POST request according to the format:

POST https://<host>/predict { "input": "[ 40, 140, 289, 172, 0, 0, 1, 0, 1, 0, 0,1, 0, 0, 1, 0, 1,0,0,0,1]" }

The API returns, through the POST response, the value of the class (0 for non-hypertensive and 1 for hypertensive).

Response: [{"result": 1}]

Intelligent mechanisms are used to monitor vital data (weight, blood pressure, heart rate) and check these data against the patient's history through artificial intelligence algorithms.

4 SmartNextGISSA Architecture This work concerns the use of Intelligent Systems in dealing with comorbidities, expanding the functionality of GISSA, a Public Health Governance product already in operation in several municipalities in Northeast Brazil (GISSA Manual, 2021). GISSA is a computing platform that automatically collects the databases of the Ministry of Health systems: E-SUS, CNES, SIM, SINASC, SI-PNI, SINAN through data extractor robots, analyzes this information using various intelligent and transforming them into integrated technologies information , useful for decision-making at different levels of health management.

Named SmartNextGISSA, the proposal presented here includes an intelligent module on the GISSA platform for monitoring and predicting cases of comorbidities, notably patients enrolled in the GISSA system (diabetic, hypertensive), in the city of Fortaleza-Ceará, as shown in figure 3. The SmartNextGISSA is proposed as an additional component to GISSA. In this context, the service will operate in the Machine Learning module of the NextGISSA scenario, receiving the history of patients from the databases monitored by the GISSA platform in order to filter individuals at potential risk through the inference of the trained model, thus allowing, that these cases move to the next stage of this scenario.

Results

Several experiments were carried out following the CRISP-DM methodology in order to optimize the model's performance. In the comparative performance analysis between the algorithms used, the Logistic Regression (RL) classifier obtained better accuracy in relation to the others, as shown in Figure 4. In the evaluation, the AUROCC curve was also used, which represents the general performance an estimator, since this metric considers all computed values of sensitivity and specificity. The greater the capacity of the estimator to discriminate individuals with and without hypertension, the more the curve approaches the upper left corner and the AUROCC will approach 1. In the application of the Grid Search optimization strategy associated with the Cross Validation (CV) technique, the Regression Logistic (RL) classifier was obtained as the best performance with the following parameters: C= 1.0, penalty = l1 e solver = liblinear.

After training with the mentioned values, improvements were obtained in the following metrics: accuracy = 0.720381, precision: 0.678812

The area under the ROC curve for this parameterization is shown in Figure 5 and presented as a value of 0.7220. AI-based technologies using machine learning have the potential to transform healthcare. In practice, applications are used for more accurate diagnosis, identification of physiological patterns, decision support for processes in personalized medicine, In the case of study, it is proposed to use it for patients with comorbidities, such as hypertension. This article presented SmartNextGISSA, an intelligent system for the prediction of comorbidities, via monitoring of the patient's vital health signs, from their clinical history. SmartNextGISSA offers a monitoring environment by automating some services such as obtaining historical patient data, in addition to expanding the range of GISSA functionalities by including an intelligent module for monitoring and predicting comorbidities. To carry out the proof of concept of the proposed solution, a prototype will be built to carry out the tests in order to evaluate the effectiveness of SmartNextGISSA in carrying out online monitoring, integrating solutions that complement each other to signal alerts to the patient about their health status in case of any intercurrence.

As future work, we will assess monitoring for other comorbidities in addition to the report of people enrolled in GISSA, aiming to expand its scope and also identify improvements in the proposed solution.

Figure 1 :

 1 Figure 1: Proposal of the SmartNextGISSA System in the GISSA architecture Considering that the pipeline was developed based on the CRISP-DM methodology (SCHRÖER, Christoph; KRUSE, Felix; GÓMEZ, Jorge M., 2021), the process was segmented into phases as proposed by the concept: 1. Understanding the Business: This step was characterized by understanding the problem based on the research carried out associated with the business domain of the NextGISSA platform, leading to the search for a dataset that corresponds to the scenario covered by NextGISSA.

Figure 2 :

 2 Figure 2: General flow for defining the model.

Figure 3 :

 3 Figure 3: SmartNextGISSA Architecture

Figure 4 :

 4 Figure 4: Accuracy and Precision

Figure 5 :

 5 Figure 5: The area under the ROC curve

Table 2 .

 2

	Parameters	Description

Table 1 -

 1 Random Forest and Decision Tree classifier evaluation parameters

	Parameters Description	Tested Values
	penalty	Specify the norm of the penalty	l1, l2
	C	Inverse of regularization strength	0.0001, 0.01, 1, 100, 10000
	solver	Algorithm to use in the optimization problem	liblinear

Table 2 -

 2 Logistic Regression classifier evaluation parameters 6