T Abboud

M Salau

S Salmon
email: salmon@math.u-strasbg.fr

Coupling Harmonic Functions-Finite Elements for Solving the Stream Function-Vorticity Stokes Problem

Keywords: Stokes problem, stream function-vorticity formulation, harmonic functions, integral representation

We consider the bidimensional Stokes problem for incompressible fluids in stream function-vorticity form. The classical finite element method of degree one usually used does not allow the vorticity on the boundary of the domain to be computed satisfactorily when the meshes are unstructured and does not converge optimally. To better approach the vorticity along the boundary, we propose that harmonic functions obtained by integral representation be used. Numerical results are very satisfactory, and we prove that this new numerical scheme leads to an optimal convergence rate of order 1 for the natural norm of the vorticity and, under higher regularity assumptions, from 3/2 to 2 for the quadratic norm of the vorticity.

I. INTRODUCTION A. Motivations

Let ⍀ be a bounded connected open set in ޒ 2 . In order that error estimates may be obtained for our numerical schemes, we assume that ⍀ is polygonal. We denote by ⌫ ϭ Ѩ⍀ the boundary of ⍀. Modeling of the equilibrium of an incompressible and viscous fluid leads to the Navier-Stokes problem [START_REF] Landau | Me ´canique des fluides, volume VI of Physique the ´orique[END_REF]. If we neglect convection terms, when the viscosity is sufficiently dominant or the velocity of the fluid sufficiently small, we obtain the stationary Stokes problem which is (in primitive variables i.e., velocity u and pressure p):

Ϫ⌬u ϩ ٌp ϭ f in ⍀, div u ϭ 0 in ⍀, u ϭ 0 on ⌫,
where is the kinematic viscosity and f a field of given external forces.

If ⍀ is supposed to be simply connected and the velocity is divergence free, this twodimensional problem is often rewritten in terms of stream function and vorticity variables. Velocity is the curl of some stream function and vorticity is the curl of the velocity. A usual way of discretizing this new problem is to choose a finite element method and to use polynomial approximations of degree one for each variable. The stream function and the vorticity are though assumed to be in H 1 (⍀), but it is well known that first this problem is not mathematically well-posed (see for example, Girault-Raviart [START_REF] Girault | Finite element methods for Navier-Stokes equations, Theory and algorithms[END_REF]), and that second the vorticity is not satisfactorily approximated on the boundary of the domain when the meshes are unstructured (see e.g., Salmon [START_REF] Salmon | De ´veloppement nume ´rique de la formulation tourbillon-vitesse-pression pour le proble `me de Stokes[END_REF]). Nevertheless, it can be shown that, if ⍀ is convex, the scheme is convergent (Scholz [START_REF] Girault | Finite element methods for Navier-Stokes equations, Theory and algorithms[END_REF][START_REF] Scholz | A mixed method for 4th order problems using linear finite elements[END_REF]). The convergence for the quadratic norm of the vorticity is of order ͌ h , where h is the maximum diameter of the triangles in the mesh, and of order h 1Ϫ for the H 1 -norm of the stream function ( is an arbitrary strictly positive real). Notice that in the case of structured meshes, the usual scheme gives optimal numerical results (see e.g., [START_REF] Salmon | De ´veloppement nume ´rique de la formulation tourbillon-vitesse-pression pour le proble `me de Stokes[END_REF][START_REF] Dubois | First vorticity-velocity-pressure numerical scheme for the Stokes problem, Report IAT/CNAM 356/02[END_REF]) and moreover, superconvergence can be observed [START_REF] Girault | Finite element methods for Navier-Stokes equations, Theory and algorithms[END_REF].

However, as has been said, a major problem of the stream function-vorticity formulation arises in trying to obtain correct boundary values for the vorticity. Many articles deal with this aspect and propose new formulas for inclusion in the numerical scheme (see e.g., Napolitano et al. [START_REF] Napolitano | A review of vorticity conditions in the numerical solution of the Ϫ equations[END_REF] and references therein). Another idea due to Amara and Bernardi [START_REF] Amara | Convergence of a finite element discretization of the Navier-Stokes equations in vorticity and stream function formulation[END_REF] is to stabilize the usual formulation by adding jumps at interfaces of the triangles and thus improve the convergence.

However, from our point of view, the mathematically well-posed formulation of the problem should lead to a good numerical scheme. So, we work with the well-posed stream function-vorticity variational formulation which was introduced by Ruas [START_REF] Ruas | Variational approaches to the two-dimensional Stokes system in terms of the vorticity[END_REF] and Bernardi-Girault-Maday [START_REF] Bernardi | Mixed spectral element approximation of the Navier-Stokes equations in stream function and vorticity formulation[END_REF]. This formulation consists of looking for the vorticity in the space M(⍀) ϭ { ʦ L 2 (⍀), ⌬ ʦ H Ϫ1 (⍀)}, containing less regular functions than H 1 (⍀). We propose in the sequel to study a natural discretization of this space, which leads to a numerical scheme using harmonic functions to compute the vorticity on the boundary. Let us observe that the idea of using harmonic functions was first introduced by Glowinski-Pironneau [START_REF] Glowinski | Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem[END_REF] and also used by Quartapelle and Valz-Gris [START_REF] Quartapelle | Projection conditions on the vorticity in viscous incompressible flows[END_REF]. But these authors do not present any theoretical convergence results.

In the sequel, we prove that, in a polygonal domain, our numerical scheme is convergent of order at least ᏻ(h ) for the natural norm of the vorticity. Some of the authors have previously proposed a method based on discrete harmonic functions computed on refined meshes (Dubois et al. [START_REF] Dubois | Discrete harmonics for stream function-vorticity Stokes problem[END_REF]), which is quite time-consuming. Here, the novelty is to use integral representation for computing the real harmonic functions. Then, we prove both theoretically and numerically that, when ⍀ is moreover assumed to be convex, if the vorticity is assumed to belong to H 2 (⍀) [respectively to H 5/ 2 (⍀)] and if the stream function belongs to H 2 (⍀), their quadratic norms converge in fact as ᏻ(h 3/ 2 ) [respectively ᏻ(h 2 )]. The last part of the article is devoted to numerical experiments on different geometries and unstructured meshes, which are in agreement with theoretical results. In the whole article, ⍀ will be assumed to be at least connected and simply connected.

B. Notation

We shall consider the following spaces (see for example, Adams [START_REF] Adams | Sobolev spaces[END_REF]). Ᏸ(⍀) denotes the space of all indefinitely differentiable functions from ⍀ to ޒ with compact support and L 2 (⍀) the space of all classes of square integrable functions. For any integer m Ն 0 and any real p such that 1 Յ p Յ ϱ, W m,p (⍀) denotes the space of all functions v ʦ L p (⍀), whose partial derivatives in the sense of distributions, Ѩ ␣ v/Ѩ x i , ͉␣͉ Յ m, belong to L p (⍀). We define as usually H 1 (⍀) ϭ W 1,2 (⍀) and H 2 (⍀) ϭ W 2,2 (⍀). We denote by ʈ⅐ʈ m,p,⍀ (respectively, ͉⅐͉ m,p,⍀ ) the norms (respectively, the semi-norms) in the Sobolev spaces W m,p (⍀). We make the usual modification for p ϭ ϱ, and we drop the index 2 when p ϭ 2. The space H 0 1 (⍀) [respectively, H 0 2 (⍀)] is the closure of Ᏸ(⍀) with respect to the norm ʈ⅐ʈ 1,⍀ (respectively, ʈ⅐ʈ 2,⍀ ). Then, (⅐, ⅐) denotes the standard inner product in L 2 (⍀) and ͗⅐, ⅐͘ Ϫ1,1 the duality product between H 0 1 (⍀) and its topological dual space H Ϫ1 (⍀). Finally, ␥ denotes the trace operator from H 1 (⍀) onto H 1/ 2 (⌫), or from H 2 (⍀) onto H 3/ 2 (⌫) (see Lions-Magenes [START_REF] Lions | Proble `mes aux limites non homoge `nes et applications[END_REF]).

II. THE STREAM FUNCTION-VORTICITY FORM OF THE STOKES PROBLEM

Let f in (L 2 (⍀)) 2 be a field of given forces, we define curl f as (Ѩf 1 /Ѩ x 2 ) Ϫ (Ѩf 2 /Ѩ x 1 ). The steady-state Stokes problem consists of finding a stream function and a vorticity field solutions of

ϩ ⌬ ϭ 0 in ⍀ (2.1) Ϫ⌬ ϭ curl f in ⍀ (2.2) ϭ 0 on ⌫ (2.3) Ѩ Ѩn ϭ 0 on ⌫. (2.4)
Indeed, it consists of finding a velocity field u that is divergence free and can be written with the help of a stream function . We have

u ϭ curl ϭ ͩ Ϫ Ѩ Ѩx 2 , Ѩ Ѩx 1 ͪ T in ⍀.
Equation (2.1) means that the vorticity is the curl of the velocity. Equation (2.2) is the equilibrium equation for a viscous fluid, with kinematic viscosity equal to 1, and where convection terms are neglected. Boundary conditions (2.3) and (2.4) are consequences of u ϭ 0 on ⌫. For more details about this problem, we refer to [START_REF] Girault | Finite element methods for Navier-Stokes equations, Theory and algorithms[END_REF].

It is natural to discretize the problem (2.1)-(2.4) with a piecewise linear and continuous finite element method. As ⍀ is assumed to be polygonal, we can exactly cover it with a meshcomposed of triangular finite elements. The meshis assumed to be regular (in the sense defined in Ciarlet [START_REF] Ciarlet | The finite element methods for elliptic problems[END_REF]). The set H 1 denotes the space of continuous functions defined on ⍀ , which are polynomials of degree 1 in each triangle ofand H 0, 1 ϭ H 1 പ H 0 1 (⍀):

H 1 ϭ ͕ ʦ Ꮿ 0 ͑⍀ ͒, @K ʦ Ᏹ , ͉K ʦ ސ 1 ͑K͖͒, (2.5)
where Ᏹ is the set of triangles inand ސ 1 the space of all polynomials of total degree less or equal to 1. The discretization of the problem (2.1)-(2.4) consists of finding and such that:

ʦ H 0, 1 ͑boundary condition ͑2.3͒ is then satisfied͒ (2.6) ʦ H 1 (2.7)
in the following way. We first multiply (2.1) by a scalar function ʦ H 1 and integrate by parts (⌬ , ), we obtain, taking the boundary condition (2.4) into account:

͑ , ͒ Ϫ ٌ͑ , ٌ ͒ ϭ 0 @ ʦ H 1 .
(2.8)

Then we multiply (2.2) by a test function in H 0, 1 , integrate by parts Ϫ(⌬ , ) and (curl f, ) and, as vanishes on the boundary, we obtain:

ٌ͑ , ٌ͒ ϭ ͑f, curl ͒ @ ʦ H 0, 1 . (2.9)
This formulation (2.6)-(2.9) has been studied extensively (see Ciarlet-Raviart [START_REF] Ciarlet | A mixed finite element method for the biharmonic equation, Math Aspects Finite Elements Partial Differential Eq[END_REF], Glowinski-Pironneau [START_REF] Glowinski | Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem[END_REF] and [START_REF] Girault | Finite element methods for Navier-Stokes equations, Theory and algorithms[END_REF] among others) and presents some difficulties. First, the continuous formulation associated to it is not well-posed for any f in (L 2 (⍀)) 2 . Indeed, the Stokes problem can be seen as a biharmonic problem for the stream function:

ʦ H 0 2 (⍀) and ⌬ 2 ϭ curl f. In a variational form, this problem can be rewritten as follows:

͑⌬, ⌬͒ ϭ ͑f, curl ͒ @ ʦ H 0 2 ͑⍀͒.

(2.10) Problem (2.10) is well posed in H 0 2 (⍀) (see [START_REF] Ciarlet | The finite element methods for elliptic problems[END_REF]) and as ϭ Ϫ⌬, the vorticity cannot be more regular than square integrable. Second, error estimates derived for the scheme (2.6 -2.9) are not optimal. When ⍀ is assumed to be convex, the bound is in h 1/ 2 for the quadratic norm of the vorticity, where h is the maximum diameter of the elements of the triangulation [START_REF] Girault | Finite element methods for Navier-Stokes equations, Theory and algorithms[END_REF][START_REF] Scholz | A mixed method for 4th order problems using linear finite elements[END_REF] (it is numerically illustrated in Figure 3 at the end of this article).

So, a different weak formulation was proposed by Ruas [START_REF] Ruas | Variational approaches to the two-dimensional Stokes system in terms of the vorticity[END_REF] and Bernardi et al. [START_REF] Bernardi | Mixed spectral element approximation of the Navier-Stokes equations in stream function and vorticity formulation[END_REF] who introduced the space

M(⍀) ϭ { ʦ L 2 (⍀), ⌬ ʦ H Ϫ1 (⍀)}, which is a Hilbert space for the norm ʈʈ M ϭ ͌ ʈʈ 0,⍀ 2 ϩ ʈ⌬ʈ Ϫ1,⍀ 2 . It consists of finding (, ) in M(⍀) ϫ H 0 1 (⍀)
with the following method. We test the first equation (2.1) with a function ʦ M(⍀) and the second one (2.2) with a function ʦ H 0 1 (⍀). It is then necessary to integrate twice by parts the term ͗⌬, ͘ Ϫ1,1 in order to include boundary conditions (2.3) and (2.4). We get

ͭ ͑, ͒ ϩ ͗⌬, ͘ Ϫ1,1 ϭ 0 @ ʦ M͑⍀͒ ͗Ϫ⌬, ͘ Ϫ1,1 ϭ ͑f, curl ͒ @ ʦ H 0 1 ͑⍀͒. (2.11)
This formulation (2.11) leads to a well-posed problem [START_REF] Bernardi | Mixed spectral element approximation of the Navier-Stokes equations in stream function and vorticity formulation[END_REF].

Proposition 2.1. The Sobolev space H 1 (⍀) is contained in M(⍀) with continuous imbedding. We have, ʈʈ M Յ ʈʈ 1,⍀ for all function in H 1 (⍀). Moreover if ʦ M(⍀) പ H 0 1 (⍀), its M-norm is equal to its H 0 1 -norm. Proof. First we notice that the laplacian of a function in H 1 (⍀) belongs to H Ϫ1 (⍀) and H 1 (⍀) is a subspace of M(⍀). Then to prove the continuity of the imbedding, we need the definition of the H Ϫ1 -norm. For all ʦ H 1 (⍀), we have

ʈ⌬ʈ Ϫ1,⍀ ϭ sup ʦH 0 1 ͑⍀͒ ͗⌬, ͘ Ϫ1,1 ͉͉ 1,⍀ ϭ sup ʦH 0 1 ͑⍀͒ Ϫٌ͑, ٌ͒ ʈٌʈ 0,⍀ because is in H 1 ͑⍀͒ and ͉⌫ ϭ 0, Յ ʈٌʈ 0,⍀ ͑using the Cauchy-Schwarz inequality͒. (2.12) An immediate consequence of this inequality is @ ʦ H 1 (⍀), ʈʈ M Յ ʈʈ 1,⍀ . Second, if ʦ M(⍀) പ H 0 1 (⍀)
, it suffices to take ϭ Ϫ in the inequality (2.12) to obtain ʈ⌬ʈ Ϫ1,⍀ ϭ ʈٌʈ 0,⍀ and then ʈʈ M ϭ ʈʈ 1,⍀ . y

Let us now introduce the kernel Ᏼ(⍀) of the bilinear form ͗⌬⅐, ⅐͘ Ϫ1,1 :

Ᏼ͑⍀͒ ϭ ͕ ʦ M͑⍀͒, ͗⌬, ͘ Ϫ1,1 ϭ 0, @ ʦ H 0 1 ͑⍀͖͒. Proposition 2.2. Characterization of the space Ᏼ(⍀): Ᏼ(⍀) ϭ { ʦ L 2 (⍀), ⌬ ϭ 0 in ᏰЈ(⍀)}.
There also exists a trace operator (still denoted by ␥) from Ᏼ(⍀) on H Ϫ1/ 2 (⌫).

The proof is quite classical and completely developed in [START_REF] Dubois | Discrete harmonics for stream function-vorticity Stokes problem[END_REF]. So, when we restrict the first equation of (2.11) to functions in Ᏼ(⍀), this new problem is well-posed according to Lax-Milgram's lemma [START_REF] Lax | Parabolic equations, in contributions to the theory of partial differential equations[END_REF]. Indeed Proposition 2.3. The L 2 -scalar product:

Ᏼ(⍀) ϫ Ᏼ(⍀) ʯ (, ) ۋ ͐ ⍀ ⅐ dx ʦ ޒ is elliptic on Ᏼ(⍀).
Proof. It is obvious that for in Ᏼ(⍀), ʈʈ M 2 ϭ ʈʈ 0,⍀ 2 . y Proposition 2.4. The space M(⍀) can be decomposed as follows: M(⍀) ϭ H 0 1 (⍀) ᮍ Ᏼ(⍀). Proof. The proof can also be found in [START_REF] Ruas | On the velocity-vorticity formulation of the Stokes system in two-and three-dimension spaces[END_REF]. We split ʦ M(⍀) into two parts: ϭ 0 ϩ ⌬ . On the one hand, since ⌬ ʦ H Ϫ1 (⍀), the component 0 is uniquely defined in H 0 1 (⍀) by the Dirichlet problem:

ͭ ⌬ 0 ϭ ⌬ in ⍀ ␥ 0 ϭ 0 on ⌫.
On the other hand, we define ⌬ by ⌬ ϭ Ϫ 0 . Then, it verifies ⌬ ⌬ ϭ 0 in ⍀, so ⌬ belongs to Ᏼ(⍀). y Moreover, we can observe that ␥ ⌬ ϭ ␥ on ⌫, with ␥ well defined in H Ϫ1/ 2 (⌫) (see Proposition 2.2).

We now rewrite the well-posed formulation of the Stokes problem (2.11) taking into account the decomposition given in Proposition 2.4.

Find

ϭ 0 ϩ ⌬ ʦ H 0 1 (⍀) ᮍ Ᏼ(⍀) and ʦ H 0 1 (⍀) such that Ά ͑a͒ ͑ 0 ϩ ⌬ , ͒ ϩ ͗⌬, ͘ Ϫ1,1 ϭ 0 @ ʦ H 0 1 ͑⍀͒. ͑b͒ ͑ 0 ϩ ⌬ , ͒ ϩ ͗⌬, ͘ Ϫ1,1 ϭ0 ϭ 0 @ ʦ Ᏼ͑⍀͒. ͑c͒ ͗Ϫ⌬ 0 , ͘ Ϫ1,1 Ϫ ͗⌬ ⌬ , ͘ Ϫ1,1 ϭ0 ϭ ͑f, curl ͒ @ ʦ H 0 1 ͑⍀͒.
Then it is obvious that the previous problem can be solved in the following way:

Ά ͑c͒ find 0 ʦ H 0 1 ͑⍀͒ such that ͗Ϫ⌬ 0 , ͘ Ϫ1,1 ϭ ٌ͑ 0 , ٌ͒ ϭ ͑f, curl ͒ @ ʦ H 0 1 ͑⍀͒. ͑b͒ find ⌬ ʦ Ᏼ͑⍀͒ such that ͑ ⌬ , ͒ ϭ Ϫ͑ 0 , ͒ @ ʦ Ᏼ͑⍀͒. ͑a͒ find ʦ H 0 1 ͑⍀͒ such that ͗Ϫ⌬, ͘ Ϫ1,1 ϭ ٌ͑, ٌ͒ ϭ ͑ 0 ϩ ⌬ , ͒ @ ʦ H 0 1 ͑⍀͒.
(2.13)

A previous work of the authors used with success homothetic mesh refinements to solve the second step of the above problem (Dubois et al. [START_REF] Salmon | De ´veloppement nume ´rique de la formulation tourbillon-vitesse-pression pour le proble `me de Stokes[END_REF][START_REF] Dubois | Discrete harmonics for stream function-vorticity Stokes problem[END_REF]). The following section gives an alternative way to solve this step by a direct use of harmonic functions, based on an integral representation.

III. DISCRETIZATION OF THE STOKES PROBLEM USING HARMONIC FUNCTIONS A. Discretization of the Space Ᏼ(⍀)

We recall that ⍀ being polygonal allows to entirely cover it with a mesh -. We denote by Ᏹ the set of triangles in -.

Definition. Family ᐁ of regular meshes [START_REF] Ciarlet | The finite element methods for elliptic problems[END_REF]. We assume thatbelongs to the set ᐁ of triangulations satisfying:

? Ͼ 0, @K ʦ Ᏹ , h K K Յ ,
where h K ϭ diam K and K is the diameter of the circle inscribed in K.

We introduce the trace of meshon the boundary ⌫. It is a set Ꮽ(-, ⌫) of edges of triangles of the mesh which are contained in ⌫. If N a (-, ⌫) is the number of these edges, we denote them by ,⌫) is also equal to the number of vertices of the meshon the boundary ⌫. Then we define the vector space Ꮿ generated by the characteristic functions of the edges ⌫ i ʦ Ꮽ(-, ⌫) of ⌫:

⌫ i , 1 Յ i Յ N a (-, ⌫). As ⌫ is closed, N a (-
Ꮿ ϭ Span͕q i ϭ މ ⌫i , ⌫ i ʦ Ꮽ͑-, ⌫͖͒, (3.1)
where މ ⌫ i is the characteristic function defined from ⌫ to ޒ by

މ ⌫i ͑x͒ ϭ ͭ 1 if x ʦ ⌫ i 0 if x ԫ ⌫ i .
The dimension of Ꮿ is exactly equal to the number N a (-, ⌫).

Remark 3.1. We can define the vector space ᏸ of continuous polynomial functions of degree one on the edges

⌫ i ʦ Ꮽ(-, ⌫) of ⌫. ᏸ ϭ ͕q ʦ Ꮿ 0 ͑⌫͒, q ͉⌫i ʦ ސ 1 ͑⌫ i ͒, ⌫ i ʦ Ꮽ͑-, ⌫͖͒. (3.2)
Then we denote by the simple layer operator applied to functions of Ꮿ .

Definition. Simple layer potential.

: Ꮿ ʯ q i ۋ i ʦ Ᏼ ,I
where i (x) ϭ q i ( x) ϭ ͐ ⌫ G( x, y)q i ( y) d␥ y @x ʦ ⍀ , and G(x, y) ϭ (1/2)log͉x Ϫ y͉ is the Green kernel.

We denote by Ᏼ ,I the discrete space spanned by functions i ϭ q i , for all q i ʦ Ꮿ . The space Ᏼ ,I is finite dimensional and, clearly, its dimension is equal to the dimension of Ꮿ . By construction, functions of Ᏼ ,I are harmonic. We shall denote by S ϭ ␥ the operator on the boundary. We introduce ␥ i (x) ϭ Sq i (x) ϭ ͐ ⌫ i G(x, y) d␥ y for all x on the boundary ⌫ and q i ϭ މ ⌫ i in Ꮿ . Remark 3.2. Our discretization will be conforming as Ᏼ ,I is contained in Ᏼ(⍀) ʚ M(⍀) and that functions in Ᏼ ,I have a trace on the boundary in H Ϫ1/ 2 (⌫) (see Proposition 2.2).

Theorem 3.3. The operator S is an isomorphism from the Sobolev space H s (⌫) onto H sϩ1 (⌫) for all real numbers (see e.g., Ne ´de ´lec [START_REF] Ne ´de ´lec | Approximation des e ´quations inte ´grales en me ´canique et en physique[END_REF], Dautray-Lions [START_REF] Dautray | Analyse mathe ´matique et calcul nume ´rique pour les sciences et les techniques[END_REF]).

Definition. We define the following subspace of H Ϫ3/ 2 (⌫):

Њ H Ϫ3/2 ͑⌫͒ ϭ ͕ ʦ H Ϫ3/2 ͑⌫͒, ͗, 1͘ Ϫ3/2,3/2 ϭ 0͖.
Proposition 3.4. For all q ʦ Њ H Ϫ3/ 2 (⌫), we can define the harmonic function q of L 2 (⍀):

q͑x͒ ϭ ͵ ⌫ G͑x, y͒q͑y͒ d␥ y @x ʦ ⍀ .
Then, there exists a constant C Ͼ 0 such that, for all q ʦ Њ H Ϫ3/ 2 (⌫):

ʈqʈ 0,⍀ Յ Cʈqʈ Ϫ3/2,⌫ . (3.3)
Proof. Let q be in Њ H Ϫ3/ 2 (⌫) and ⍀Ј be the interior of the complementary of ⍀. Notice that q is, by definition, the solution of the following problem [19 -21]:

Ά ⌬u ϭ 0 in ⍀ ഫ ⍀Ј ͓u͔ ϭ 0 on ⌫ ͫ Ѩu Ѩn ͬ ϭ q on ⌫,
where [] ϭ ͉⍀ Ϫ ͉⍀Ј is the jump of across ⌫. Vector n is the outer normal on ⌫. Notice that the condition ͗q, 1͘ Ϫ3/ 2,3/ 2 ϭ 0 is necessary to obtain the existence of a solution to the previous problem.

Following Ne ´de ´lec [START_REF] Ne ´de ´lec | Approximation des e ´quations inte ´grales en me ´canique et en physique[END_REF], we know that for all q ʦ H Ϫ1/ 2 (⌫), whose mean value is zero, q belongs to H 1 (⍀) and then to L 2 (⍀). So, we can consider the L 2 -norm of q when q is sufficiently regular. Then, if (3.3) is proven for regular functions, a classical density argument leads to the expected result.

Let us now prove this proposition for regular functions. We shall use methods similar to those used in the Aubin-Nitsche argument (see Aubin [START_REF] Aubin | Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by Galerkin's and finite difference methods[END_REF] and Nitsche [START_REF] Nitsche | Ein Kriterium fu ¨r die quasi-optimalitat des Ritzchen Verfahrens[END_REF]). Indeed,

ʈqʈ 0,⍀ ϭ sup gʦL 2 ͑⍀͒ ͑q, g͒ 0,⍀ ʈgʈ 0,⍀ .
We extend a function g ʦ L 2 (⍀) by zero outside ⍀ and we define g ˜:

g ˜ϭ ͭ g in ⍀ 0 in ⍀Ј.
Notice that, as g belongs to L 2 (⍀), g ˜belongs to L 2 ޒ( 2 ). We consider a function such that

Ϫ⌬ ϭ g ˜in ޒ 2 .
͑x͒ ϭ ᏻ͑log͉x͉͒ when ͉x͉ becomes large.

By local regularity of the Laplacian operator, the solution ͉⍀ belongs to H 2 (⍀) if ⍀ is bounded (Ne ´de ´lec [START_REF] Ne ´de ´lec | Re ´solution des e ´quations acoustiques et electromagne ´tiques[END_REF]) and verifies

ʈ ͉⍀ ʈ 2,⍀ Յ Cʈgʈ 0,⍀ . (3.4)
Because ⍀ is bounded, let B R be a ball, of radius R sufficiently large to contain ⍀ (we denote by S R the boundary of B R ). We have

͑q, g͒ 0,⍀ ϭ ͑q, g ޒ,0͒˜ 2 ϭ lim R3ϱ ͵ BR Ϫ⌬ ⅐ q dx.
Then, as ⌬q ϭ 0:

͵ BR Ϫ⌬ ⅐ q dx ϭ ͵ BR ͑⌬q ⅐ Ϫ ⌬ ⅐ q͒ dx ϭ ͵ ⍀ ͑⌬q ⅐ Ϫ ⌬ ⅐ q͒ dx ϩ ͵ BRگ⍀ ͑⌬q ⅐ Ϫ ⌬ ⅐ q͒ dx.
Notice that, on the one hand, ͉B R belongs to

H 2 (B R ) [respectively, ͉⍀ to H 2 (⍀)] as B R is bounded and so, ␥ ʦ H 3/ 2 ( R ) [respectively H 3/ 2 (⌫)] and Ѩ/Ѩn ʦ H 1/ 2 (S R ) [respectively H 1/ 2 (⌫)].
On the other hand, functions in L 2 (⍀) whose Laplacian is also in L 2 (⍀) have their normal trace in H Ϫ3/ 2 (⌫) (see [START_REF] Dautray | Analyse mathe ´matique et calcul nume ´rique pour les sciences et les techniques[END_REF]). So, we obtain by definition:

͵ ⍀ ͑⌬q ⅐ Ϫ ⌬ ⅐ q͒ dx ϩ ͵ BRگ⍀ ͑⌬q ⅐ Ϫ ⌬ ⅐ q͒ dx ϭ ͳ ͫ ѨSq Ѩn ͬ , ͓␥͔ʹ Ϫ3/2,3/2 ϩ ͳϪ ͫ Ѩ Ѩn ͬ , ͓Sq͔ʹ Ϫ1/2,1/2 ϩ ͵ SR ͩ ѨSq Ѩn ⅐ Ϫ Ѩ Ѩn ⅐ Sq ͪ d␥.
First, [␥] and [Ѩ/Ѩn] are continuous across ⌫ and S R since belongs to H 2 (B R ). Second, by construction: [ѨSq/Ѩn] ϭ q, [Sq] ϭ 0. Then, we obtain

͵ BR Ϫ⌬ ⅐ q dx ϭ ͗q, ␥͘ Ϫ3/2,3/2 ϩ ͵ SR ͩ ѨSq Ѩn ⅐ Ϫ Ѩ Ѩn ⅐ Sq ͪ d␥.
As acts as log R and Sq as 1/R (because the mean value of q is null) when R becomes large (see [START_REF] Ne ´de ´lec | Approximation des e ´quations inte ´grales en me ´canique et en physique[END_REF]) we obtain

ͯ͵ SR ѨSq Ѩn ⅐ d␥ ͯ Յ ͵ 0 2 1 R 2 ⅐ log R ⅐ R d ϳ log R R O ¡ R 3 ϩ ϱ 0,
and

ͯ͵ SR Ѩ Ѩn ⅐ Sq d␥ ͯ Յ ͵ 0 2 1 R ⅐ 1 R ⅐ R d ϳ 1 R O ¡ R 3 ϩ ϱ 0.
Finally, we deduce ͑q, g͒ 0,⍀ ϭ ͑q, g ޒ,0͒˜

H 1,I ϭ H 0, 1 ᮍ Ᏼ ,I . (3.5)
The dimension of H 1,I is the same as H 1 [see (2.5)] but near the boundary, we use harmonic functions instead of piecewise linear continuous functions.

B. Discrete Formulation

We propose the following discrete variational formulation of the Stokes problem based on problem (2.13):

ʦ H 0, 1 , ϭ 0 ϩ ⌬ ʦ H 1,I ϭ H 0, 1 ᮍ Ᏼ ,I (3.6)
ٌ͑ 0 , ٌ͒ ϭ ͑f, curl ͒ @ ʦ H 0, 1 (3.7)
͑ ⌬ , ͒ ϭ Ϫ͑ 0 , ͒ @ ʦ Ᏼ ,I (3.8)
ٌ͑ , ٌ͒ ϭ ͑ 0 ϩ ⌬ , ͒ @ ʦ H 0, 1 .

(3.9)

The above method is a conforming discretization of problem (2.13) since, as it was said in remark 3.2, H 1,I is a subset of M(⍀). Our approach for studying problem (3.6)-(3.9) follows ideas of [START_REF] Glowinski | Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem[END_REF] and Ruas [START_REF] Ruas | Approximating harmonic bases for a decoupled solution of viscous flow problems in Ϫ formulation[END_REF]. Proposition 3.5. Existence and uniqueness of a solution to problem (3.6)-(3.9). If f ʦ (L 2 (⍀)) 2 , problem (3.6)-(3.9) has a unique solution ( , ) ʦ H 0, 1 ϫ H 1,I , which depends continuously on the datum f. There exists a strictly positive constant C independent of the mesh such that ʈ ʈ M ϩ ʈٌ ʈ 0,⍀ Յ Cʈfʈ 0,⍀ .

(3.10)

Proof. As ⍀ is bounded, if C p denotes the Poincare ´constant, we have

@ ʦ H 0 1 ͑⍀͒, ʈʈ 0,⍀ Յ C p ͉͉ 1,⍀ .
Problem (3.7) is well posed according to Lax-Milgram's lemma, and Poincare ´'s inequality. There exists a unique 0 ʦ H 0, 1 satisfying [take ϭ 0 in (3.7)]

ʈٌ 0 ʈ 0,⍀ 2 ϭ ʈcurl 0 ʈ 0,⍀ 2 Յ ʈfʈ 0,⍀ ʈcurl 0 ʈ 0,⍀ .
As 0 ʦ H 0 1 (⍀), its M-norm is equal to its H 1 -norm (see Proposition 2.1) and

ʈ 0 ʈ M Յ ͱ1 ϩ C p 2 ʈfʈ 0,⍀ . (3.11)
We now study Equation (3.8). Function 0 is given and problem (3.8) has a unique solution ⌬ , as the L 2 -scalar product is M-elliptic on Ᏼ ,I (Proposition 2.3):

͑ ⌬ , ͒ ϭ Ϫ͑ 0 , ͒ @ ʦ Ᏼ ,I .
Taking ϭ ⌬ and thanks to (3.11), function

⌬ verifies ʈ ⌬ ʈ M ϭ ʈ ⌬ ʈ 0,⍀ Յ ʈ 0 ʈ 0,⍀ Յ ͱ1 ϩ C p 2 ʈfʈ 0,⍀ .
(3.12)

Finally, Equation (3.9) is formally identical to (3.7), so there exists (Lax-Milgram's lemma and Poincare ´'s inequality) a unique ʦ H 0, 1 such that

ٌ͑ , ٌ͒ ϭ ͑ 0 , ͒ ϩ ͑ ⌬ , ͒ @ ʦ H 0, 1 ,
and then

ʈٌ ʈ 0,⍀ Յ C p ͑ʈ 0 ʈ 0,⍀ ϩ ʈ ⌬ ʈ 0,⍀ ͒.

C. Interpolation Error

We recall the following result [START_REF] Lions | Proble `mes aux limites non homoge `nes et applications[END_REF]. Theorem 3.6. Interpolation between Sobolev spaces. Let s i and t i be two couples of positive reals for i ϭ 0 or i ϭ 1 and p ʦ ޒ such that 1 Յ p Յ ϱ. Let ⌸ be an operator of ᏸ(W s 0 ,p (⍀); W t 0 ,p (⍀)) പ ᏸ(W s 1 ,p (⍀); W t 1 ,p (⍀)) (ᏸ is the space of all linear and continuous functions). Then, for all ʦ ޒ പ [0; 1], operator ⌸ belongs to the interpolate space ᏸ ϭ ᏸ(W (1Ϫ )s 0 ϩs 1 ,p (⍀); W (1Ϫ )t 0 ϩt 1 ,p (⍀)) and we have ʈ⌸ʈ ᏸ Յ ʈ⌸ʈ ᏸ0 1Ϫ ʈ⌸ʈ ᏸ1 .

Proposition 3.7. Regularity of components of functions in M(⍀). Let be in M(⍀), we know

(see Proposition 2.4) that can be split: ϭ 0 ϩ ⌬ with 0 ʦ H 0 1 (⍀) and ⌬ ʦ Ᏼ(⍀) (i.e., harmonic). Then, if ⍀ is convex and if belongs to H 2 (⍀) പ M(⍀) ϭ H 2 (⍀), 0 and ⌬ belong also to H 2 (⍀). Moreover, there exists a constant C Ͼ 0 such that

ʈ 0 ʈ 2,⍀ Յ Cʈʈ 2,⍀ . (3.14) ʈ ⌬ ʈ 2,⍀ Յ Cʈʈ 2,⍀ .
(3.15)

Proof. Let us recall that 0 ʦ H 0 1 (⍀) is solution of

ͭ ⌬ 0 ϭ ⌬ in ⍀ ␥ 0 ϭ 0 on ⌫.
If belongs to H 2 (⍀), its Laplacian belongs to L 2 (⍀) and by the regularity of the Laplacian operator on a convex domain, 0 belongs to H 2 (⍀) (Agmon et al. [START_REF] Agmon | Estimates near the boundary for solutions in elliptic partial differential equations satisfying general boundary conditions I[END_REF]). Then, ⌬ ϭ Ϫ 0 belongs also to H 2 (⍀). Moreover, we have

ʈ 0 ʈ 2,⍀ Յ Cʈ⌬ʈ 0,⍀ Յ Cʈʈ 2,⍀ .
The inequality for ⌬ ϭ Ϫ 0 is deduced from the previous one by the triangular inequality.y Definition. Projection operator: Let p c be the L 2 -projection on the space of piecewise constants Ꮿ defined in (3.1).

p c : L 2 ͑⌫͒ 3 Ꮿ ۋ p c
such that ͵ ⌫ ͑p c Ϫ ͒ ⅐ q d␥ ϭ 0 @q ʦ Ꮿ .

Remark 3.8. It is also possible to define the L 2 -projection on the space of piecewise linear functions ᏸ defined in (3.2), p ᏸ : L 2 (⌫) 3 ᏸ , which is such that, for all ʦ L 2 (⌫), ͐ ⌫ ( p ᏸ Ϫ ) ⅐ q d␥ ϭ 0 @q ʦ ᏸ . Let h be the maximum diameter of triangles in -. The standard interpolation error for one-dimensional problems [START_REF] Ciarlet | The finite element methods for elliptic problems[END_REF] gives, for all in H 1 (⌫):

ʈ Ϫ p c ʈ 0,⌫ Յ Ch ͉͉ 1,⌫ .
(3.16) Lemma 3.9. For all ʦ H 1/ 2 (⌫), we have

ʈ Ϫ p c ʈ 0,⌫ Յ Ch 1/2 ʈʈ 1/2,⌫ .
(3.17)

Proof. Let I be the identity operator. As I Ϫ p c is a continuous operator from L 2 (⌫) into L 2 (⌫) with its norm bounded by a constant C 1 and is also continuous from L 2 (⌫) into H 1 (⌫) with its norm bounded by C 2 h [see (3.16)], we deduce that I Ϫ p c is continuous from L 2 (⌫) into H 1/ 2 (⌫) ( ϭ 1/ 2 in Theorem 3.6) with its norm bounded by a constant times h 1/ 2 which is inequality (3.17). y Proposition 3.10. For all ʦ H 1/ 2 (⌫), we have

ʈ Ϫ p c ʈ Ϫ3/2,⌫ Յ Ch 3/2 ʈʈ 1/2,⌫ . (3.18)
For all ʦ H 1 (⌫), we have

ʈ Ϫ p c ʈ Ϫ3/2,⌫ Յ Ch 2 ʈʈ 1,⌫ . (3.19)
Proof. By definition of the norm in H Ϫ3/ 2 (⌫), we have

ʈ Ϫ p c ʈ Ϫ3/2,⌫ ϭ sup ʦH 3/2 ͑⌫͒ ͗ Ϫ p c , ͘ Ϫ3/2,3/2 ʈʈ 3/2,⌫ .
As H 1/ 2 (⌫) and H 3/ 2 (⌫) are contained in L 2 (⌫), the duality product ͗ Ϫ p c , ͘ Ϫ3/ 2,3/ 2 can be rewritten as

͐ ⌫ ( Ϫ p c ) ⅐ d␥. By definition ͐ ⌫ ( Ϫ p c ) ⅐ d␥ ϭ 0 @ ʦ Ꮿ , so ͗ Ϫ p c , ͘ Ϫ3/2,3/2 ϭ ͵ ⌫ ͑ Ϫ p c ͒ ⅐ ͑ Ϫ ͒ d␥ @ ʦ Ꮿ Յ ʈ Ϫ p c ʈ 0,⌫ ʈ Ϫ ʈ 0,⌫ ᭙ ʦ Ꮿ ͑Cauchy-Schwarz inequality͒ Յ ʈ Ϫ p c ʈ 0,⌫ ʈ Ϫ p c ʈ 0,⌫ with ϭ p c ʦ Ꮿ Յ C 1 h 1/2 ʈʈ 1/2,⌫ C 2 h ʈʈ 1,⌫ using ͑3.17͒ and ͑3.16͒ Յ Ch 3/2 ʈʈ 1/2,⌫ ʈʈ 3/2,⌫ ,
which ends the proof of (3.18). Notice that if belongs to H 1 (⌫), using a classical interpolation estimate [see (3.16)], we have ʈ Ϫ p c ʈ 0,⌫ Յ Ch ʈʈ 1,⌫ , so we obtain

͗ Ϫ p c , ͘ Ϫ3/2,3/2 Յ ʈ Ϫ p c ʈ 0,⌫ ʈ Ϫ p c ʈ 0,⌫ with ϭ p c ʦ Ꮿ Յ C 1 h ʈʈ 1,⌫ C 2 h ʈʈ 1,⌫ Յ Ch 2 ʈʈ 1,⌫ ʈʈ 3/2,⌫ ,
which leads to formula (3.19). y Let ⌸ : H 2 (⍀) 3 H 1 be the classical Lagrange interpolation operator associated with mesh -.

Definition. The interpolation operator

: Ᏼ(⍀) പ H 2 (⍀) 3 Ᏼ ,I is defined by ⌬ ϭ
where is such that

͑x͒ ϭ p c ͑S Ϫ1 ⌬ ͒͑x) ϭ ⌫ G(x, y)⅐p c (S Ϫ1 ⌬ )(y) d␥ y @x ʦ ⍀ .
We define the interpolation operator ᏼ from M(⍀) പ H 2 (⍀) to H - 1,I ϭ H 0, 1 ᮍ Ᏼ ,I by the relations:

ᏼ : M͑⍀͒ പ H 2 ͑⍀͒ ʯ ϭ 0 ϩ ⌬ ۋ ᏼ ϭ ⌸ 0 ϩ ⌬ ʦ H 1,I .
Remark 3.11. We can also define the interpolation operator Ј :

Ᏼ(⍀) പ H 2 (⍀) 3 Ᏼ ,I by Ј ⌬ ϭ
, where is such that

͑x͒ ϭ p ᏸ ͑S Ϫ1 ⌬ ͒͑x) ϭ ͵ ⌫ G(x, y)⅐p ᏸ (S Ϫ1 ⌬ )(y) d␥ y @x ʦ ⍀ ,
and ᏼЈ from M(⍀) പ H 2 (⍀) to H - 1,I ϭ H 0, 1 ᮍ Ᏼ ,I by the relations:

ᏼЈ : M͑⍀͒ പ H 2 ͑⍀͒ ʯ ϭ 0 ϩ ⌬ ۋ ᏼЈ ϭ ⌸ 0 ϩ Ј ⌬ ʦ H 1,I .
Proposition 3.12. Error estimates: Forin a regular family of triangulations ᐁ ( Ͼ 0 fixed), for h small enough, and for given in M(⍀) decomposed into 0 and ⌬ , if we assume 0 ʦ H 2 (⍀) പ H 0 1 (⍀) and ʦ H 2 (⍀), then there exists some strictly positive constants, say C, only dependent on , such that

ʈ 0 Ϫ ⌸ 0 ʈ M Յ Ch ͉ 0 ͉ 2,⍀ .
(3.20)

ʈ ⌬ Ϫ ⌬ ʈ M Յ Ch 3/2 ʈʈ 2,⍀ . (3.21)
Moreover, if belongs to H 5/ 2 (⍀), we have

ʈ ⌬ Ϫ ⌬ ʈ M Յ Ch 2 ʈʈ 5/2,⍀ . (3.22)
Proof. As H 2 (⍀) ʚ Ꮿ 0 (⍀) when ⍀ is two-dimensional, we can use the classical interpolation operator, and we have the following interpolation error estimate (Ciarlet-Raviart [START_REF] Ciarlet | General Lagrange and Hermite interpolation in ޒ n with applications to finite element methods[END_REF]):

ʈ 0 Ϫ ⌸ 0 ʈ 1,⍀ Յ Ch ͉ 0 ͉ 2,⍀ . But ʈ 0 Ϫ ⌸ 0 ʈ M ϭ ʈ 0 Ϫ ⌸ 0 ʈ 1,⍀ because 0 Ϫ ⌸ 0 ʦ H 0 1 (⍀) (see Proposition 2.1), so relation (3.20) is established, if h is small enough.
We now interpolate the harmonic part ⌬ of ʦ M(⍀). By definition ⌬ verifies

ͭ ⌬ ⌬ ϭ 0 in ⍀ ␥ ⌬ ϭ ␥ on ⌫.
As is assumed to be in H 2 (⍀), its trace on ⌫, ␥, belongs to H 3/ 2 (⌫). As S is an isomorphism from H s (⌫) onto H sϩ1 (⌫) (see Theorem 3.3), there exists a unique q ʦ H 1/ 2 (⌫) such that Sq ϭ ␥q ϭ ␥ on ⌫. And because of uniqueness, q ϭ ⌬ on ⍀. Let now q ϭ p c q ʦ Ꮿ , we recall that ⌬ is

⌬ ͑x͒ ϭ q ͑x) ϭ ͵ ⌫ G(x, y)⅐q (y) d␥ y @x ʦ ⍀ . So, ʈ ⌬ Ϫ ⌬ ʈ M ϭ ʈ ⌬ Ϫ ⌬ ʈ 0,⍀ ϭ ʈq Ϫ q ʈ 0,⍀ .
As constants belong to Ꮿ , ͐ ⌫ (q Ϫ q ) d␥ ϭ 0, i.e. q Ϫ q ʦ Њ H Ϫ3/ 2 (⌫). Then proposition 3.4 gives

ʈq Ϫ q ʈ 0,⍀ Յ Cʈq Ϫ q ʈ Ϫ3/2,⌫ .
Then, from inequality (3.18), we obtain

ʈq Ϫ q ʈ Ϫ3/2,⌫ ϭ ʈq Ϫ p c qʈ Ϫ3/2,⌫ Յ Ch 3/2 ʈqʈ 1/2,⌫ .
And finally

ʈ ⌬ Ϫ ⌬ ʈ M Յ Ch 3/2 ʈqʈ 1/2,⌫ ϭ Ch 3/2 ʈS Ϫ1 ͑ ͒ʈ 1/2,⌫ Յ Ch 3/2 ʈ␥ʈ 3/2,⌫ because S is an isomorphism Յ Ch 3/2 ʈʈ 2,⍀
by continuity of the trace operator.

Notice that if is assumed to be in H 5/ 2 (⍀), its trace on ⌫, ␥, belongs to H 2 (⌫). So, there exists a unique q ʦ H 1 (⌫) such that Sq ϭ ␥q ϭ ␥ on ⌫. The same arguments as above lead to the inequality:

ʈq Ϫ q ʈ 0,⍀ Յ Cʈq Ϫ q ʈ Ϫ3/2,⌫ .
Then, from formula (3.19), we have

ʈq Ϫ q ʈ Ϫ3/2,⌫ ϭ ʈq Ϫ p c qʈ Ϫ3/2,⌫ Յ Ch 2 ʈqʈ 1,⌫ .
And finally

ʈ ⌬ Ϫ ⌬ ʈ M Յ Ch 2 ʈqʈ 1,⌫ ϭ Ch 2 ʈS Ϫ1 ͑ ͒ʈ 1,⌫ Յ Ch 2 ʈ␥ʈ 2,⌫ because S is an isomorphism Յ Ch 2 ʈʈ 5/2,⍀
by continuity of the trace operator. y Remark 3.13. Using linear interpolation on the boundary, we obtain

ʈ ⌬ Ϫ Ј ⌬ ʈ M Յ Ch 2 ʈʈ 2,⍀ . (3.23)
To prove the inequality with Ј , we use the same arguments as above. Projection is done on ᏸ : q ϭ p ᏸ q ʦ ᏸ , we define Ј ⌬ as the simple layer potential associated to q , i.e., Ј ⌬ ͑x͒ ϭ q ͑x) ϭ ͵ ⌫ G(x, y)⅐q (y) d␥ y @x ʦ ⍀ .

As we have proved for projection p c , using classical interpolation and interpolation between Sobolev spaces results, we have

ʈq Ϫ q ʈ Ϫ3/2,⌫ ϭ ʈq Ϫ p ᏸ qʈ Ϫ3/2,⌫ Յ Ch 2 ʈqʈ 1/2,⌫ .
The end of the proof, which leads to inequality (3.23), is the same as above.

D. Error Estimates

We define an auxiliary problem:

Find ʦ H 0, 1 , ϭ 0 ϩ ⌬ ʦ H 1,I such that ͭ ٌ͑ 0 , ٌ͒ ϭ ٌ͑g, ٌ͒ @ ʦ H 0, 1 . ͑ ⌬ , ͒ ϭ ͑m, ͒ @ ʦ Ᏼ ,I . ٌ͑ , ٌ͒ ϭ ٌ͑l, ٌ͒ ϩ ͑n, ͒ @ ʦ H 0, 1 . (3.24)
Proposition 3.14. Stability of discrete formulation (3.24).

Let g ʦ H 0 1 (⍀), m ʦ L 2 (⍀), l ʦ H 0 1 (⍀), n ʦ L 2 (⍀). Then problem (3.24) has a unique solution ( , ) ʦ H 0, 1 ϫ H 1,I
, which is stable in the following sense: there exists a constant C only dependent on the mesh family such that the following stability inequality holds: ʈ ʈ M ϩ ʈٌ ʈ 0,⍀ Յ C͕ʈٌgʈ 0,⍀ ϩ ʈmʈ 0,⍀ ϩ ʈٌlʈ 0,⍀ ϩ ʈnʈ 0,⍀ ͖.

Proof. In the following, C will denote various constants. In the first equation of problem (3.24) we take ϭ 0 , and using Poincare ´'s inequality, we obtain

ʈ 0 ʈ M Յ Cʈٌ 0 ʈ 0,⍀ Յ Cʈٌgʈ 0,⍀ ,
as 0 belongs to H 0 1 (⍀), its H 1 -norm is equal to its M-norm. In the second equation of problem (3.24), we take ϭ ⌬ , as ⌬ ⌬ ϭ 0, we obtain

ʈ ⌬ ʈ M ϭ ʈ ⌬ ʈ 0,⍀ Յ ʈmʈ 0,⍀ .
Finally, in the last equation of problem (3.24), we take ϭ . As belongs to H 0 1 (⍀), using Poincare ´'s inequality, we have ʈٌ ʈ 0,⍀ Յ C͑ʈٌlʈ 0,⍀ ϩ ʈnʈ 0,⍀ ͒.

By combining these three inequalities, Proposition 3.14 is proven. y Proposition 3.15. The error is bounded by the interpolation error. Let (, ) be the solution of the continuous stream function-vorticity formulation (2.13) and ( , ) solution of the associated discrete problem (3.6)-(3.9). There exists a constant C Ͼ 0 independent ofsuch that

ʈ Ϫ ʈ M ϩ ʈ Ϫ ʈ 1,⍀ Յ C͑ʈ 0 Ϫ ⌸ 0 ʈ M ϩ ʈ ⌬ Ϫ ⌬ ʈ M ϩ ʈ Ϫ ⌸ ʈ 1,⍀ ͒.
Proof. The continuous problem (2.13) is written with a discrete test function ϭ 0 ϩ ⌬ ʦ H 1,I since H 1,I ʚ M(⍀) and with ʦ H 0, 1 ʚ H 0 1 (⍀):

ͭ ٌ͑ 0 , ٌ͒ ϭ ͑f, curl ͒ @ ʦ H 0, 1 . ͑ ⌬ , ͒ ϭ Ϫ͑ 0 , ͒ @ ʦ Ᏼ ,I . ٌ͑, ٌ͒ ϭ ͑ 0 ϩ ⌬ , ͒ @ ʦ H 0, 1 .
(3.25)

The discrete problem is

ͭ ٌ͑ 0 , ٌ͒ ϭ ͑f, curl ͒ @ ʦ H 0, 1 ͑ ⌬ , ͒ ϭ Ϫ͑ 0 , ͒ @ ʦ Ᏼ ,I . ٌ͑ , ٌ͒ ϭ ͑ 0 ϩ ⌬ , ͒ @ ʦ H 0, 1 . (3.26)
Subtracting (3.25) from (3.26), we obtain

ͭ ٌ͑͑ 0 Ϫ 0 ͒, ٌ͒ ϭ 0 @ ʦ H 0, 1 . ͑ ⌬ Ϫ ⌬ , ͒ ϭ Ϫ͑ 0 Ϫ 0 , ͒ @ ʦ Ᏼ ,I . ٌ͑͑ Ϫ ͒, ٌ͒ ϭ ͑͑ 0 Ϫ 0 ͒ ϩ ͑ ⌬ Ϫ ⌬ ͒, ͒ @ ʦ H 0, 1 .
(3.27)

We now introduce the interpolants of and ϭ 0 ϩ ⌬ on the mesh -. In the first equation of problem (3.27), we add and subtract ⌸ 0 and obtain

ٌ͑͑ 0 Ϫ ⌸ 0 ͒, ٌ͒ ϭ ٌ͑͑ 0 Ϫ ⌸ 0 ͒, ٌ͒ @ ʦ H 0, 1 . (3.28)
Adding and subtracting ⌬ in the second equation, we obtain

͑ ⌬ Ϫ ⌬ , ͒ ϭ ͑ ⌬ Ϫ ⌬ , ͒ ϩ ͑ 0 Ϫ 0 , ͒ @ ʦ Ᏼ ,I . (3.29)
And finally using same techniques for the last equation, we have

ٌ͑͑ Ϫ ⌸ ͒, ٌ͒ ϭ ٌ͑͑ Ϫ ⌸ ͒, ٌ͒ Ϫ ͑ 0 Ϫ 0 , ͒ Ϫ ͑ ⌬ Ϫ ⌬ , ͒ @ ʦ H 0, 1 .
(3.30) Equations (3.28), (3.29), and (3.30) lead to the following problem:

Ά ٌ͑͑ 0 Ϫ ⌸ 0 ͒, ٌ͒ ϭ ٌ͑g, ٌ͒ @ ʦ H 0, 1 . ͑ ⌬ Ϫ ⌬ , ͒ ϭ ͑m, ͒ @ ʦ Ᏼ ,I ٌ͑͑ Ϫ ⌸ ͒, ٌ͒ ϭ ٌ͑l, ٌ͒ ϩ ͑n, ͒ @ ʦ H 0, 1 ,
which is the auxiliary problem (3.24) previously defined with g

ϭ 0 Ϫ ⌸ 0 in H 0 1 (⍀), m ϭ ( ⌬ Ϫ ⌬ ) ϩ ( 0 Ϫ 0 ) in L 2 (⍀), l ϭ Ϫ ⌸ in H 0 1 (⍀), n ϭ Ϫ( 0 Ϫ 0 ) Ϫ ( ⌬ Ϫ ⌬ ) in L 2 (⍀)
. By applying the triangular inequality and Proposition 3.14, we have

ʈ Ϫ ʈ M ϩ ʈ Ϫ ʈ 1,⍀ Յ ʈ Ϫ ᏼ ʈ M ϩ ʈ Ϫ ⌸ ʈ 1,⍀ interpolation error ϩ ʈᏼ Ϫ ʈ M ϩ ʈ⌸ Ϫ ʈ 1,⍀ Յ ʈ Ϫ ᏼ ʈ M ϩ ʈ Ϫ ⌸ ʈ 1,⍀ ϩ C͑ʈٌ͑ 0 Ϫ ⌸ 0 ͒ʈ 0,⍀ ϩ ʈ ⌬ Ϫ ⌬ ϩ 0 Ϫ 0 ʈ 0,⍀ ϩ ʈٌ͑ Ϫ ⌸ ͒ʈ 0,⍀ ϩ ʈ 0 Ϫ 0 ϩ ⌬ Ϫ ⌬ ʈ 0,⍀ ) Յ ʈ Ϫ ᏼ ʈ M ϩ ʈ Ϫ ⌸ ʈ 1,⍀ ϩ C͑ʈ 0 Ϫ ⌸ 0 ʈ M ϩ ʈ ⌬ Ϫ ⌬ ʈ M ϩ ʈ 0 Ϫ 0 ʈ 0,⍀ ϩ ʈٌ͑ Ϫ ⌸ ͒ʈ 0,⍀ ϩ ʈ 0 Ϫ 0 ʈ 0,⍀ ϩ ʈ ⌬ Ϫ ⌬ ʈ M ),
which leads to the announced result. y The inequality obtained in Proposition 3.15 and the interpolation error lead to the following. Theorem 3.16. First convergence result. Ifbelongs to a regular family of triangulation ᐁ ( Ͼ 0) and h is small enough, the discrete problem (3.6)-(3.9) has a unique solution ( , )

ʦ H 0, 1 ϫ H 1,I
, associated with a stable discretization of the Stokes problem (2.1)-(2.4). If ʦ H 2 (⍀) പ H 0 1 (⍀) and ϭ 0 ϩ ⌬ such that ʦ H 2 (⍀) and 0 ʦ H 2 (⍀), then

? C͑ ͒ Ͼ 0, @-ʦ ᐁ , ʈ Ϫ ʈ M ϩ ʈ Ϫ ʈ 1,⍀ Յ C͑͒h ͉͑ 0 ͉ 2,⍀ ϩ ʈʈ 2,⍀ ϩ ͉͉ 2,⍀ ͒.
Proof. With the help of the inequality in Proposition 3.15 (C denotes various constants),

ʈ Ϫ ʈ M ϩ ʈ Ϫ ʈ 1,⍀ Յ C͑ʈ 0 Ϫ ⌸ 0 ʈ M ϩ ʈ ⌬ Ϫ ⌬ ʈ M ϩ ʈ Ϫ ⌸ ʈ 1,⍀ ͒ Յ C͑h ͉ 0 ͉ 2,⍀ ϩ h 3/2 ʈʈ 2,⍀ ϩ h ͉͉ 2,⍀ ͒ by Proposition 3.12 Յ Ch ͉͑ 0 ͉ 2,⍀ ϩ ʈʈ 2,⍀ ϩ ͉͉ 2,⍀ ͒
as h can be assumed to be less or equal to 1. y Remark 3.17. If ⍀ is assumed to be convex, proposition 3.7 leads to

? C͑ ͒ Ͼ 0, @-ʦ ᐁ , ʈ Ϫ ʈ M ϩ ʈ Ϫ ʈ 1,⍀ Յ C͑͒h ͑ʈʈ 2,⍀ ϩ ͉͉ 2,⍀ ͒.
Theorem 3.16 is important because it shows that using a space of harmonic functions along the boundary gives an error of order ᏻ(h ) when ʦ H 2 (⍀), which improves previous known results and is equivalent to those proved in [START_REF] Dubois | Discrete harmonics for stream function-vorticity Stokes problem[END_REF]. Theorem 3.18. Second convergence result. If ⍀ is assumed to be convex and under the same assumptions as in Theorem 3.16, we have

? C͑͒ Ͼ 0, @-ʦ ᐁ , ʈ Ϫ ʈ 0,⍀ ϩ ʈ Ϫ ʈ 0,⍀ Յ C͑ ͒h 3/2 ͑ʈʈ 2,⍀ ϩ ʈʈ 2,⍀ ͒.
Under the same assumptions, if moreover ʦ H 5/ 2 (⍀), we have

? C͑͒ Ͼ 0, @-ʦ ᐁ , ʈ Ϫ ʈ 0,⍀ ϩ ʈ Ϫ ʈ 0,⍀ Յ C͑ ͒h 2 ͑ʈʈ 5/2,⍀ ϩ ʈʈ 2,⍀ ͒.
Proof. If belongs to H 2 (⍀), from Proposition 3.12, formula (3.21), we know that

ʈ ⌬ Ϫ ⌬ ʈ 0,⍀ Յ C͑ ͒h 3/2 ʈʈ 2,⍀ .
If belongs to H 5/ 2 (⍀), from Proposition 3.12, formula (3.22), we know that

ʈ ⌬ Ϫ ⌬ ʈ 0,⍀ Յ C͑ ͒h 2 ʈʈ 5/2,⍀ .
We just have to use

ʈ 0 Ϫ 0 ʈ 0,⍀ Յ C͑ ͒h 2 ʈʈ 2,⍀ Յ C͑ ͒h 2 ʈʈ 5/2,⍀ , and
ʈ Ϫ ʈ 0,⍀ Յ C͑ ͒h 2 ͑ʈʈ 2,⍀ ϩ ʈʈ 5/2,⍀ ͒.
These inequalities are proven using the classical Aubin-Nitsche argument when the domain ⍀ is convex so that regularity on the adjoint problem is obtained ( [START_REF] Aubin | Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by Galerkin's and finite difference methods[END_REF][START_REF] Nitsche | Ein Kriterium fu ¨r die quasi-optimalitat des Ritzchen Verfahrens[END_REF]). y The last part of the previous theorem says that if the solution is more regular than usual, the convergence is of order 2. This is illustrated in the numerical examples in the next section where the solutions are very regular.

ʈ ⌬ Ϫ ⌬ ʈ 0,⍀ Յ C͑ ͒h 2 ʈʈ 2,⍀ ,
from which we deduce the following theorem.

Theorem 3.20. We assume that ⍀ is convex, thatbelongs to a regular family of triangulations, that ʦ H 2 (⍀) പ H 0 1 (⍀) and that ʦ H 2 (⍀). If we use space ᏸ , we have

? C͑͒ Ͼ 0, @-ʦ ᐁ , ʈ Ϫ ʈ 0,⍀ ϩ ʈ Ϫ ʈ 0,⍀ Յ C͑ ͒h 2 ͑ʈʈ 2,⍀ ϩ ʈʈ 2,⍀ ͒.

IV. NUMERICAL APPLICATIONS

The first numerical experiments have been performed on a unit square with an analytical solution (Bercovier-Engelman test [START_REF] Bercovier | A finite element for the numerical solution of viscous incompressible flows[END_REF]):

f 1 ͑x, y͒ ϭ 256͑x 2 ͑x Ϫ 1͒ 2 ͑12y Ϫ 6͒ ϩ y͑y Ϫ 1͒͑2y Ϫ 1͒͑12x 2 Ϫ 12x ϩ 2͒͒ f 2 ͑x, y͒ ϭ Ϫf 1 ͑y, x͒,
for which we obtain (x, y) ϭ Ϫ128( y 2 ( y Ϫ 1) 2 x 2 ( x Ϫ 1) 2 ) and (x, y) ϭ 256( y 2 ( y Ϫ 1) 2 (6x 2 Ϫ 6x ϩ 1) ϩ x 2 ( x Ϫ 1) 2 (6y 2 Ϫ 6y ϩ 1)). For the second numerical experiments, we have extended the tests to circles. They have been performed on a circle of radius 2 with an analytical solution (test suggested by Ruas [START_REF] Ruas | Me ´thodes vitesse-tourbillon pour les e ´quations de Navier-Stokes[END_REF]):

f 1 ͑x, y͒ ϭ Ϫ32y, f 2 ͑x, y͒ ϭ 32x, which gives (x, y) ϭ (4 Ϫ x 2 Ϫ y 2 ) 2 and (x, y) ϭ 32 Ϫ 16x 2 Ϫ 16y 2 .
Remark 4.1. Notice that for error estimates on the circle, we have to add a boundary approximation error. The boundary is approximated by polynomials of degree 1 and it seems that this error is of the same order than the approximation error with polynomials of degree zero.

We have worked with unstructured meshes obtained with EMC2, mesh generator of Modulef (Bernadou et al. [START_REF] Bernadou | MODULEF: Une bibliothe `que modulaire d'e ´le ´ments finis[END_REF]), see Fig. 1. Notice that structured meshes give good results without any stabilization, see [START_REF] Girault | Finite element methods for Navier-Stokes equations, Theory and algorithms[END_REF] and for numerical results see [START_REF] Salmon | De ´veloppement nume ´rique de la formulation tourbillon-vitesse-pression pour le proble `me de Stokes[END_REF] and [START_REF] Dubois | First vorticity-velocity-pressure numerical scheme for the Stokes problem, Report IAT/CNAM 356/02[END_REF]. For the sake of simplicity, we use the space Ꮿ of functions which are constant on each edge for the integral representation. It would be interesting to test the linear integral representation but the results were so satisfying with the constant one that we did not even try. Remark 4.2. All the integrals for assembling the mass matrix in Equation (3.8) were computed with the help of a Gauss formula using 7 quadrature points and we obtain results in accordance with the theory. The cost of such an integration could be reduced by using less points far from the boundary. Although errors due to numerical integration were not studied here, they seem to be dominated by other errors and do not pollute results. For the first test, the analytical vorticity attains its extremum on the middle of each edge of the square and its value is then ϩ16.00. And for the second one the extremum (Ϫ32) is attained on the whole boundary. We recall that the number of discrete harmonic functions is equal to the number of vertices on the boundary and that the classical method uses piecewise linear functions for approximating the vorticity and the stream function. Figure 2 gives the values of the vorticity on the boundary obtained by the classical and the integral method on the same mesh. In fact with the classical method, extrema of the vorticity blow up on the boundary (see Fig. 2).

We recall that problem (2.6)-(2.9) is not stable and error bounds are in h 1/ 2 for the L 2 -norm of the vorticity-as numerically illustrated on Figs. 3 and4-andh 1Ϫ for the H 1 -norm of the stream function [START_REF] Girault | Finite element methods for Navier-Stokes equations, Theory and algorithms[END_REF]. Using only constants on edges with regular solutions, we have obtained, as expected by Theorem 3.18, convergence of order 2 for the L 2 -norm of the vorticity and for the L 2 -norm of the stream function (see Figs. 3 and4).

V. CONCLUSION

We have studied the well-posed Stokes problem in stream function and vorticity form. We have shown that using a space of real harmonic functions is sufficient to obtain on the one hand a better solution and on the other hand better estimations on the convergence than those obtained previously. We have proposed a way of approaching numerically the space of real harmonic functions with the help of an integral representation which yields a large gain of time compared to previous results obtained in [START_REF] Dubois | Discrete harmonics for stream function-vorticity Stokes problem[END_REF]. We have shown theoretically and numerically that by this way we obtain convergence with optimal rate in some cases on the quadratic norm of the vorticity.

We use the same method for a vorticity-velocity-pressure formulation of the Stokes problem that allows more general boundary conditions [START_REF] Salmon | De ´veloppement nume ´rique de la formulation tourbillon-vitesse-pression pour le proble `me de Stokes[END_REF][START_REF] Dubois | First vorticity-velocity-pressure numerical scheme for the Stokes problem, Report IAT/CNAM 356/02[END_REF]. The results are not published yet but they are really satisfying as, once again, the solution and the convergence of the method are improved. We insist on the fact that the additional work of the method (the computation of the matrix) needs to be done only once. So, the additional cost of our scheme shall be less important for a more realistic problem like the time dependent Stokes or Navier-Stokes equations.

Using ( 3 .

 3 [START_REF] Quartapelle | Projection conditions on the vorticity in viscous incompressible flows[END_REF]) and (3.12), we obtain ʈٌ ʈ 0,⍀ Յ CЈʈfʈ 0,⍀ .(3.13)Combining (3.11), (3.12), and (3.13), we obtain the announced result (3.10). y To obtain error estimates and thus convergence, we need a stability result (Proposition 3.14) and an interpolation error (Proposition 3.12).

Remark 3 . 19 .

 319 If we use the linear interpolation for the boundary values of the vorticity instead of piecewise constant one, we know from formula (3.23) that

FIG. 1 .

 1 FIG. 1. Two unstructured meshes obtained by EMC2.

FIG. 2 .

 2 FIG. 2. Comparison: vorticity on the boundary-Bercovier-Engelman test.

  2 ϭ ͗q, ␥͘ Ϫ3/2,3/2 Յ ʈqʈ Ϫ3/2,⌫ ʈ␥ʈ 3/2,⌫

	Using relation (3.4), it leads to
	ʈqʈ 0,⍀ ϭ sup gʦL 2 ͑⍀͒	͑q, g͒ ʈgʈ 0,⍀	Յ C sup gʦL 2 ͑⍀͒	ʈqʈ Ϫ3/2,⌫ ʈgʈ 0,⍀ ʈgʈ 0,⍀	Յ Cʈqʈ Ϫ3/2,⌫ .
					y
	Definition. For the discretization of M(⍀), we set
			Յ Cʈqʈ Ϫ3/2,⌫ ʈʈ 2,⍀	by continuity of the trace.


Rosalie Vandromme
email: r.vandromme@brgm.fr

Olivier Cerdan

Aurore Gay

Anthony Foucher

Sébastien Salvador- Blanes

Valentin Landemaine

Marc Desmet

The VERSEAU -TRACKSED Project: origin of Loire River basin sediments

Introduction:

In France, since the beginning of 20 th century, rural landscapes have been completely modified by human activities. These practices have resulted in profound sedimentary and morphological alterations (channel bed incision, deposition of fine sediment, bank erosion, etc.), detrimental to the achievement of good water status [1]. Several research efforts have already investigated either global budgets at the river basin or continental scale or local detailed budget at the plot to the field scale. However, very few studies have tried to analyse the connectivity between fluxes and storages and to draw the links between the different scales. In this broad context, the purpose of this study is to examine source-to-sink dynamic of the sediment cycle for the Loire River Basin. This project is broken down into two steps: the first step's aim is to understand poorly studied processes such as sediment production by agricultural drainage or bank erosion by catchment monitoring. The second step is to elaborate a distributed model of sediment connectivity from hillslopes to basin outlet.

Methods:.for the first step, two catchment sites are studied, using historical data or monitoring: the linear (21 km) of two small streams ("La Ligoire") and the Louroux lake catchment. Most of those two catchments are intensively cultivated and have been extensively submitted to subsurface drainage using drain tiles. The objectives of this part are threefold: 1) quantify incision and deposition processes since the channelization of the streams (1970), (2) quantify in-channel deposition rates of fine sediments, and, (3) explain the spatial distribution of these deposits. For the second step, the modeling approach is based on the use of indicators to describe hillslope processes, potential downstream retention, attempting to link river basin characteristics to a prediction of sediment exports in rivers. It provides insight in the identification of the most influent sediment redistribution processes on the total sediment fluxes and on the differences between various basin typologies [2] [3].

Results and discussion:

The Ligoire study shows an important stream incision (around 30 cm in 40 years) and the influence of water obstacles on sediment distribution. The Louroux monitoring should allow a better understanding of the origin and dynamics of sediment transport within small intensively cultivated plain catchments. The first results from the second step on mean annual suspended sediment loads, show that catchments contribute from 4.8x10 2 to 3.7.10 5 t.yr -1 to the overall Loire river sediment exports (which equals 8.6x10 6 t.yr -1 ) and area-specific suspended sediment yields have been calculated (Fig. 1). Investigations on catchments global characteristics should then allow the identification of dominant processes on sediment redistribution and help to draw local then regional distributed sediment budgets to bridge the gap between the different spatial scales. Contribution of hillslopes to the overall catchment budget should finally help to assess in-stream contributions and redistribution processes.

Fig. 1 :

 1 Fig. 1: Specific sediment yields in the Loire River Basin.


Quentin Bernical

Xavier Joulia

Isabelle Noirot-Le Borgne

Pascal Floquet

Pierre Baurens

Guillaume Boissonnet

Sustainability assessment of an integrated High Temperature Steam Electrolysis -enhanced Biomass to Liquid Fuel process

This work addresses the integrated design of the High Temperature Steam Electrolysis (HTSE) and Biomass to Liquid (BtL) hybrid process. The comprehensive gate-to-gate analysis includes BtL and hydrogen production on-site operations. Simulations are carried out using commercial process simulation software -ProSimPlus ® -to allow physical modeling as well as mass and energy balances; modeling is based on standard elementary and user modules and supported by various CEA (French Atomic Energy and Alternative Energies Commission) previous works.

The framework for assessment is proposed. Considering productivity, efficiency, cost and environmental issues, five sustainability criteria are chosen: carbon matter yield; energy efficiency; greenhouse gas emissions; electrolysis water use; levelized biofuels production cost.

Simulations results verify that hydrogen input is almost doubling the biogenic carbon conversion into liquid biofuels. They bring out significant secondary energy saving for HTSE compared to standard process; and clear advantages over alkaline electrolysis considering technical and environmental criteria. HTSE coupling allows a sharp increase in productivity (biofuels/biomass) with a limited increase in cost and environmental impacts at almost constant primary efficiency.

Introduction

Transport undergoes an increase in energy consumption, consisting almost exclusively in petroleum products. [START_REF]IEA Scoreboard 2009: 35 Key Energy Trends over 35 Years[END_REF] This has a negative impact on major challenges of the energy system, such as GreenHouse Gas Emissions (GHGE), energy insecurity and depletion of resources. Biomass could address these issues and is the raw energy most considered regarding the substitution of oil. [START_REF] Boissonnet | Working Group III. IPCC Special Report on Renewable Energy Sources and Climate Change Mitigation[END_REF][START_REF]Directive 2009/28/EC[END_REF] However the first-generation biofuels (using agricultural biomass) are criticized for their environmental impacts (mainly change of land use). They also compete with the agriculture food chain (land use, pressure on prices) and generate significant non-recoverable by-product. The second generation liquid biofuels (using lignocellulosic biomass from forest resources, and agriculture and forestry residues) are promising regarding substitution of petroleum: they offer both interesting energy efficiency and potential for GHGE cut. [START_REF]Directive 2009/28/EC[END_REF][START_REF] Edwards | Well-to-wheels analysis of future automotive fuels and powertrains in the european context[END_REF] Additionally, the Biomass to Liquid (BtL) thermochemical route can produce fuels similar to conventional fuels with higher quality [START_REF] Gill | Combustion characteristics and emissions of Fischer-Tropsch diesel fuels in IC engines[END_REF] ; this allows to keep the current distribution network and internal combustion engines and thus to benefit from their maturity. This route is based on the gasification of the biomass followed by Fischer-Tropsch Synthesis (FTS), producing naphtha, gasoline, kerosene and diesel; so BtL fuels are also called FT biofuels.

However, lignocellulosic resources are significant but not plentiful: in Europe, the primary energy feedstock from forestry [START_REF] Eea | How much biomass can Europe use without harming the environment?[END_REF] is about the third of the final energy consumption for transportation [START_REF]IEA Scoreboard 2009: 35 Key Energy Trends over 35 Years[END_REF] . As a consequence it is necessary to use the biogenic carbon to the utmost. This increase in productivity is possible using optimized energy efficiency of the BtL processes, but also by providing extra hydrogen input. Hydrogen production coupled with BtL process is therefore studied. [START_REF] Hawkes | Liquid Fuel Production from Biomass via High Temperature Steam Electrolysis[END_REF][START_REF] Seiler | Technical and economical evaluation of enhanced biomass to liquid fuel processes[END_REF] This interest is obvious referring to chemical equations: biomass gasification is roughly described by eq. ( 1) (1) while desired FTS (producing mainly saturates [START_REF] Gill | Combustion characteristics and emissions of Fischer-Tropsch diesel fuels in IC engines[END_REF] ) is described by eq. ( 2) for alkanes, (2) with n belonging to [5, 20], which leads to eq. ( 3) for the BtL process.

(3)

Only one-third of the biogenic carbon is valorized in biofuels. With extra hydrogen input, the process equation can be described by eq. (4) instead. (4) Then, by adding extra hydrogen, two-third of the biogenic carbon can theoretically be valorized in biofuels.

For consistency, primary energy to produce hydrogen has to be in line with energy issues such as GHGE, this is possible considering electrical power due to numerous alternatives in raw energy. Water electrolysis produces both hydrogen for extra input and oxygen for gasification, in the required one half ratio shown above (eq. ( 5)).

(5) Furthermore, water reuse from BtL could theoretically lead to full matter integration for the hybrid process (eq. ( 6)). Note that, in this case, there is still no net extra matter input to the process, only extra energy as electrical power is required. (6) Contrary to standard alkaline electrolysis, High Temperature Steam Electrolysis (HTSE) can use heat at different temperature levels, both for water vaporization and for the high temperature electrochemical reaction, so it allows the recovery of free heat from BtL [START_REF] Hawkes | Liquid Fuel Production from Biomass via High Temperature Steam Electrolysis[END_REF] . HTSE also allows to reuse easily the FTS coproduced water (and so to get close to eq. ( 6)) due to inherent handling of hydrocarbons and additionally it offers very high secondary efficiency [START_REF] Manage | A technoeconomic appraisal of hydrogen generation and the case for solid oxide electrolyser cells[END_REF] . Figure 1 sums-up these relevant synergies of HTSE coupling to BtL process. This paper aims at defining the framework for assessment, validation and quantification of the relevance of the HTSE and BtL hybrid process regarding sustainability, suitable for instance for multicriteria optimization. Both integrated design and further optimization require a detailed modeling: selected technologies and method used are first presented through a brief description of the modeling, in the next section. Sustainability criteria and matching performance data of the base case integrated design are then discussed. Finally the proposed framework and a comparison with the standard BtL process are summed-up.

Process description

Integrated process

The BtL process is based on two main steps: gasification of the biomass into syngas then FTS of desired fuels: naphtha, gasoline, kerosene and diesel. Cobalt catalyst is chosen for FTS in order to favor the diesel range, most used in Europe 1 . In addition, the complete process includes preparation or recovery operations that impact process performance data. Operations involving heat or matter are simulated, from raw biomass to liquid fuels. Simulations are carried out using a commercial process simulation software (ProSimPlus ® ) to allow physical modeling, including mass and energy balances as well as thermodynamic computing. Concerning reactions modeling, equilibrium is considered when possible (i.e. temperature (EFR and combustions) or catalyst (reforming, WGS) provides near-equilibrium output), accounting for potential changes in the reaction conditions; in case of non-standard kinetics (torrefaction, FTS, hydrocraking), phenomenological modeling at fixed conditions and conversions is used; HTSE is managed by user-built module. The process model is built from standard elementary and user modules, supported by various CEA (French Atomic Energy and Alternative Energies Commission) previous works, including modeling work and selection of relevant elementary processes and technologies [START_REF] Seiler | Technical and economical evaluation of enhanced biomass to liquid fuel processes[END_REF][START_REF] Boissonnet | Process simulation, thermal and economic assessment of several technical options for btl production[END_REF][START_REF] Boissonnet | BtL process development: simulation and techno-economic assessment of several technical options[END_REF] . Modeling is briefly presented in the next section.

BtL process

Fixed characteristics of the raw material is a major simplification, commonly assumed for process analysis. Wood is considered as biomass; it is 51% C, 6% H, 41% O, 2% N and 35% water (mass, dry basis) with a Lower Heating Value (LHV) of 18 MJ/kg. First, raw biomass is dried by direct contact with flue gas or heated air. This type of drying is the most widely used industrially, due to its maturity and low cost. [START_REF] Fagernäs | Drying of biomass for second generation synfuel production[END_REF] Drying is modeled by the mixing of process exhaust gases and wet biomass followed by separation of steam and flue gas from the biomass. For this first order modeling, output residual moisture of the biomass, output steam and flue gas temperature (over temperature from dew point), and input hot gas temperature (through the use additional fresh air) are set to match standard values. Secondly, torrefaction [START_REF] Prins | More efficient biomass gasification via torrefaction[END_REF] is used to lower the energy consumption of biomass grinding. Mechanical grinding is necessary to achieve the required particle size for fluidization of biomass for feeding into the EFR. A fraction of the biomass is converted into a poor gas mixture, usually burned to provide energy to the reaction, resulting in a slight loss of biogenic carbon. Modeling includes fixed reaction at 550 K, torrefied wood moisture uptake and gas combustion.

Adiabatic pressurized oxygen-blown EFR is used for gasification. The thermodynamic equilibrium and the operating temperature (1600-1900 K) avoid the formation of methane and thus maximize the conversion of biomass into useful compounds for the FTS. It also avoids the formation of particulates, all this in a single step. In addition, working under pressure is particularly useful for large installations. [START_REF] Boissonnet | BtL process development: simulation and techno-economic assessment of several technical options[END_REF][START_REF] Tijmensen | Exploration of the possibilities for production of Fischer Tropsch liquids and power via biomass gasification[END_REF] These benefits can be found on the production cost, and this reactor benefits from development for the coal-based integrated gasification combined cycle. Oxygen is recovered from electrolysis if available, or supposed to be otherwise produced by an Air Separation Unit (ASU). Modeling includes oxygen compression (if produced from ASU), oxygen demand computing (to meet the temperature), CO 2 compression, steam generation and equilibrium reaction at 1673 K, 3 MPa.

Between gasification and synthesis, it is necessary to process the gas in order to comply with the requirements of the FTS. [START_REF] Hamelinck | Production of FT transportation fuels from biomass; technical options, process analysis and optimisation, and development potential[END_REF] First gas quenching aims at fixing syngas composition. Modeling includes water consumption computing (water quench) to reach 1170 K and fixed sudden WGS reaction. Radiant heat exchanger or recirculating cold raw syngas can also be considered as dry quench in order to minimize sudden shift reaction. Considering the gas cleaning step, only CO 2 (diluent) removal is modeled. Both chemical (amines) and physical (methanol) washings are considered. Chemical washing generates a high heat demand (set at 420 K [START_REF] Tock | Thermochemical production of liquid fuels from biomass: Thermo-economic modeling, process design and process integration analysis[END_REF] ) for solvent regeneration while physical washing consumes more electrical power. Modeling includes separation of CO 2 and water from the syngas as well as heat and electrical power consumption.

Cobalt catalyzed FTS requires near stoichiometric H 2 /CO feed and works at around 500 K, temperature range for efficient liquid production is narrow. Modeling of the reaction is performed with a fixed α parameter (the probability of propagation instead of termination in the polymerization process, typically alkanes' growth), and includes short-loop tail gas recycling as well as products separation. Some works [START_REF] Hamelinck | Production of FT transportation fuels from biomass; technical options, process analysis and optimisation, and development potential[END_REF] offer correlations for α, but the validity range is narrow and values obtained are not in agreement with state-of-art FTS regarding liquid production. Therefore α is set in this work at 0.92, value which is both currently achievable [START_REF] Van Vliet | Fischer-Tropsch diesel production in a well-to-wheel perspective: A carbon, energy flow and cost analysis[END_REF] and ensuring a good conversion into liquids [START_REF] Tijmensen | Exploration of the possibilities for production of Fischer Tropsch liquids and power via biomass gasification[END_REF] . Products are modeled as alkanes up to C 20 H 42 , while a single component is used for wax (C 20+ ).

Finally too light (tail gas) or too heavy (wax) products from the FTS are valued given that they can contain up to a quarter of the introduced carbon each. Wax upgrading is performed by hydrocracking long-chains molecules into liquids and gas. Modeling includes a fixed reaction (alkanes' distribution) and hydrogen consumption. The tail gas recycling improves performance on the system by reinjecting the biogenic carbon upstream. [START_REF] Prins | Exergetic optimisation of a production process of Fischer-Tropsch fuels from biomass[END_REF] The tail gas is reformed using conventional steam reforming. This reaction requires high temperature heat (1100 K); modeling includes steam generation and equilibrium reaction. As with any recycling, a part must be purged to prevent the accumulation of inerts (mainly nitrogen from biomass itself) and it is burned to provide high temperature heat.

Hydrogen adjustment

An adjustment of hydrogen is required to bring the H 2 /CO ratio of the synthesis gas (lower than 1 out of gasification) to stoichiometric conditions (higher than 2) for the cobalt catalyzed FTS. This adjustment can be either a conversion of a part of the syngas into H 2 and CO 2 using WGS reaction or an extra hydrogen input to the process in order to increase the carbon yield.

Using steam reforming of natural gas to produce hydrogen creates GHGE. With low-carbon electricity (e.g. renewable or nuclear power), the Alkaline Electrolysis (AE) (consuming only electricity) can reduce these GHGE, the downside is an increase in production cost and a poor energy yield with generation of waste (80°C) heat [START_REF] Manage | A technoeconomic appraisal of hydrogen generation and the case for solid oxide electrolyser cells[END_REF] . Finally, HTSE (using heat both to vaporize the water and to the electrochemical reaction at 1000 K) brings the advantage of better energy efficiency of the process, but with an increased cost. Modeling of HTSE includes steam generation, water consumption and excess oxygen production computing, and HTSE reaction with a user-build module, at 3 MPa and with hydrogen production matching the consumption for syngas adjustment. WGS, NG reforming (not presented here) and AE can be considered as touchstones for HTSE performance and suitability.

Energy calculations

Electrical power consumption is obtained from consumption of pumps and compressors as well as electrolysis and ASU if used. Concerning heat, heat integration [START_REF] March | Introduction to pinch technology[END_REF] is performed as part of the simulation from the results obtained simulating the complete flowsheet, which are inputoutput temperatures and enthalpy changes along the operations (steam generations, reactions, …) involving heat in the process. Thus, contrary to matter computing, heat integration is computed for the hybrid process, as shown in Table 1. 4 shows the importance of heat integration for the whole process while major sources and HTSE thermal coupling is studied later (Figure 5). Hot utility is considered to be produced using natural gas; cold utility is valued by producing electrical power (with 40% exergetic efficiency) if it is higher than 373 K, modeling a steam turbine (slight power loop in Figure 4).

As the standard pinch analysis, this integration is a techno-economic sub-optimization based on the parameter (minimum temperature difference in exchangers), supposed to offer a tradeoff between investment cost and energy savings: temperatures of sinks and sources are corrected using half matching typical fixed values [START_REF] March | Introduction to pinch technology[END_REF] . This method allows targeting both utilities and capital cost without explicit heat exchange network.

Criteria and preliminary analysis

In order to study the relevance of the HTSE and BtL hybrid process, issues of secondgeneration biofuels bring us to select several criteria as performance indices or compliance constraints. Matching key points of the process are discussed using the base case results, in order to provide a relevant framework and analysis for further multicriteria optimization, not considered in this study.

Energy

The first criterion needed for an energy transformation process is the energy efficiency of such a process. This technical criterion is used in all publications on the BtL process, expressed in energy (lower or higher heating value for fuels) or, in some papers, exergy (work equivalent) [START_REF] Prins | Exergetic optimisation of a production process of Fischer-Tropsch fuels from biomass[END_REF] .

Exergy is particularly useful to point out irreversibilities, so it is relevant for a given process analysis. It also allows to value heat, that is why it is used in Figure 4. Given that no steam credit is considered [START_REF] Hinderink | On the efficiency and sustainability of the process industry[END_REF] for the processes under study, only high-exergy inputs/outputs are involved, so energy does not distort the efficiencies. Additionally, exergy of fuels is higher than LHV (biomass: 113%, liquid fuels: 107% [START_REF] Szargut | Exergy analysis of thermal, chemical, and metallurgical processes[END_REF] ) but exergy equals power for electricity. Thus, exergy inputs/outputs data favor the use of electrical power over biomass; we choose not to use exergy in studying this electrical power hybrid process relevance. Finally, we must account for upstream electrical power production and inferred technical energy efficiency of this prior conversion [START_REF] Hinderink | On the efficiency and sustainability of the process industry[END_REF] . The energy criterion is then expressed as the yield of equivalent primary energy: the primary energy of a fuel (biomass, produced liquid biofuel) is taken as its LHV; electrical power is affected by a conversion efficiency [START_REF] Hawkes | Liquid Fuel Production from Biomass via High Temperature Steam Electrolysis[END_REF][START_REF] Tock | Thermochemical production of liquid fuels from biomass: Thermo-economic modeling, process design and process integration analysis[END_REF] , fixed here at 1/2.6 or 39%. Considering terminology, primary energy is found in nature (e.g. wind, biomass or natural gas); on the contrary secondary energy has been converted (e.g. electricity) and thus it has been affected at least once by a conversion efficiency; primary energy is sometimes referred as thermal value [START_REF] Hawkes | Liquid Fuel Production from Biomass via High Temperature Steam Electrolysis[END_REF] .

Primary efficiencies are then computed according to eq. ( 7) using net costing inputs and outputs (Figure 4): biofuels are outputs; and biomass, electrical power and natural gas are inputs.

Regarding heat integration, the composites curves (Figure 5) plot cumulative heat demand and heat excess versus availability temperature for process operations. They show two pinch points:

heat integration is limited in two different points; further integration would require acting on several operations. Contrary to expectations, the heat of FTS is not required to HTSE steam generation: in the studied case, the steam can be generated by the heat of syngas cooling from gasification. Otherwise, i.e. to achieve heat integration between these two operations, adjusting FTS temperature or HTSE pressure is needed. For example, a synthesis at 500 K allows producing hydrogen by HTSE at 1.8 MPa maximum (taking into account pressure losses and difference temperature in exchangers); to avoid further compression of hydrogen, EFR should operate at 2.3 MPa. This analysis addresses the first (500 K) pinch point. Regarding the second pinch point, high temperature heat consumption due to HTSE reaction can be supplied either by heat or by electrical power (Joule effect), given that from a thermodynamic point of view only free Gibbs energy is required as electrical power for electrolysis. The results presented here use high temperature heat for the HTSE reaction, that is why heat consumption appears in the cold composite curve. On the contrary, by fully supplying reaction enthalpy by electrical power, the surface is reduced (and so capital cost) but it would not be possible to value high temperature heat from a process source (purge combustion). Thus the flexibility of HTSE regarding pressure and heat behavior allows more advanced heat integration. Regarding HTSE efficiency, standard process secondary (i.e. gate-to-gate) efficiency is 46% while hybrid process efficiency is 59%. The specific or marginal 23 efficiency of HTSE, calculated as additional energy in produced biofuels against additional input energy compared to the standard process, is 89% (using exergy, it would reach 95%). This point illustrates how efficient the HTSE is. However, accounting for upstream electrical power production efficiency (39%), primary specific efficiency of HTSE is cut down to 36%. Recalculated breakeven electrical power production efficiency is 47%: for more efficient production (e.g. renewable is conventionally 100%) the hybrid process is more efficient than the standard one, but for less efficient electrical power generation (as in this study) the HTSE and BtL integrated process is considered less efficient than the standard process.

Biogenic carbon

The main function of the coupling is to enhance the conversion of biogenic carbon into liquid biofuels, so a yield is required as a technical performance criterion, biomass as input and fuels as output. The process basically removes oxygen from the input, leading to limited mass yield: eq. ( 6) mass yield is 40%. Biogenic carbon yield (also called carbon utilization [START_REF] Hawkes | Liquid Fuel Production from Biomass via High Temperature Steam Electrolysis[END_REF] ) is therefore selected as criterion, maximum biogenic carbon yield being 100% (in this case the process indeed requires net additional matter input as hydrogen element). Flow diagrams are presented for the standard and HTSE coupling case in Figure 6. CO 2 formations translate in loss of energetic biogenic carbon; roughly one-third of input biogenic carbon remains as material for the produced biofuel in the standard process. The loss avoidable using extra hydrogen input instead of WGS adjustment is roughly an additional third of the input biomass. Water quench also induces an important shift, that is why dry quench is proposed as alternative in Figure 3 in order to save a sixth of biogenic carbon as support for fuel (not studied here). Biogenic carbon losses at pretreatment (torrefaction) and gasification are due to the chosen EFR technology; on the other hand this technology offers to perform CO 2 reuse for the injection of biomass, and to recycle 18 tail gas thanks to minimum inert (nitrogen) input in the process. Due to remaining CO 2 in the washed syngas in the standard case there is almost no net CO 2 formation for tail gas purge and reforming.

GHGE

The impact of biofuel in GHGE is the subject of much research, the change of land use is particularly discussed and impacting. Additionally it is questionable whether the biomass is a renewable energy or not. These variables are outside the scope of this work: it is assumed that the biomass available at the entrance of the process and biogenic CO 2 emitted during combustions (in the process or as biofuel) are not responsible for any GHGE. Additionally, avoided emissions are directly related to productivity (or in this study, the conversion of biogenic carbon), therefore raw emissions are preferred. GHGE are calculated for the process from natural gas consumption for hot utility and for electrical power production from indirect emissions.

GHGE due to electrical power highly impact the relevance of the hybrid process: if they are low (nuclear or renewable), the HTSE-coupled BtL process offer biofuels suitable with European Union requirements [START_REF]Directive 2009/28/EC[END_REF] ; for moderate GHGE due to electrical power (> 150 g eq fossil CO2 / kWh) the produced biofuels can't meet European Union 2018 requirements; and finally for high GHGE (> 400 g eq fossil CO2 / kWh) the hybrid process leads to higher GHGE than fossil fuels (Figure 7). This is another example of how the relevance of the HTSE and BtL hybrid process depends on electrical power characteristics.

Water use

Water consumption of energy processes is under recent and growing concern. Fresh water consumption is used in life cycle analyses and water consumption due to cooling (cold utility of heat integration) is presented in recent studies [START_REF] Martín | Process Optimization of FT-Diesel Production from Lignocellulosic Switchgrass[END_REF] using a process engineering approach. As electrolysis uses water as raw material, additional fresh water consumption due to electrolysis is accounted here in order to compare the hybrid process from the standard BtL process. Focusing on matter, water consumption due to upstream electrical power production or to process cooling is not accounted here. The process is always globally a net producer of water due to moisture of raw biomass; the additional water consumption is calculated then without waste water recycling facilities.

Consumption of fresh water by HTSE can be notably reduced with simple reuse of the water byproduced by the FTS. This water meets specifications for HTSE catalysts due to harsh specifications for FTS [START_REF] Hamelinck | Production of FT transportation fuels from biomass; technical options, process analysis and optimisation, and development potential[END_REF] and organics are harmless for HTSE.

Cost

The cost criterion is included in almost all studies of the process. Indeed the production cost of the fuel must be given special attention, as an indicator of economic suitability of the process; even though the confidence on the calculated production cost is narrow due to sensitivity to cost assessment parameters. As standard economic evaluation, the levelized cost is calculated from annualized investment and operating costs.

Results

Simulations have been carried out for the proposed base case (Figure 3) for standard BtL process and electrolysis. Two technical criteria (yield or efficiency; to maximize) are used for the two main functions of the process and three other criteria (expressed per produced liquid biofuels) are presented (the standard economic criterion and two environmental criteria).

Computing issues are summarized in Figure 8. Base case results (Table 2) are consistent with expectations: the use of electrolysis can nearly double the conversion of biogenic carbon, but this leads to lower performance in terms of environmental and economic criterion; the HTSE use can limit the drop in performance compared to the AE. Concerning energy, the secondary energy efficiency increases with HTSE but the equivalent primary energy efficiency is lower (39%) than the standard BtL process (42%). This is due to poor efficiency of electrical power production: the criterion in equivalent primary energy is indeed necessary. Concerning GHGE emitted by the process, they are strongly linked to power consumption in case of electrolysis, given that little natural gas is used compared to electrical power. The biogenic carbon yield and additional water use are the most accurate results, directly based on flowsheet simulation. However they still suffer from major assumptions, such as fixed moisture (35%) and chosen LHV (18 MJ/kg) of the input biomass. Primary efficiency, GHGE and levelized cost strongly depend on the parameters used for their calculation. Regarding GHGE and primary efficiency these parameters are electrical power characteristics, chosen here to account for French industrial electricity mix.

Conclusion

This paper presents the comprehensive framework for the assessment of HTSE-enhanced BtL process regarding sustainability. The challenges of the energy system and particularly secondgeneration biofuels lead to select five criteria: carbon matter yield; energy efficiency; greenhouse gas emissions; water use; levelized biofuels production cost. Results are presented for base case flowsheets; they allow discussing key points of the process.

The first concluding elements in studying integrated HTSE-BtL process verify that hydrogen input is almost doubling the biogenic carbon conversion compared to the standard BtL process.

They bring out significant secondary energy saving for HTSE; and clear advantages over AE considering technical and environmental criteria.

HTSE is therefore suitable and promising, although its performance data is strongly linked to electrical power characterization. With a clean (< 150 g eq fossil CO2 / kWh) and efficient (> 50%) production, HTSE coupling offer a more efficient biofuel production with limited environmental impact (< 100 g eq fossil CO2 / kWh, +12 kg / GJ liquid biofuels ) and over cost (0.1 € 2011, France / L liquid biofuels ) while almost doubling the productivity (biofuels/biomass). On the contrary GHGE intensive (> 400 g eq fossil CO2 / kWh) electrical power production would lead to more GHGE than fossil fuels.

Finally the results presented suffer from major assumptions for estimated or average parameters, such as input biomass or electricity characteristics. For a specific case study, these parameters have to be chosen matching situations and sensitivity analysis have to be carried out matching inputs data accuracy; statistical evaluation would also be appropriate. Regarding the process modeling, multicriteria optimization would make the proposed comparison more accurate by providing adapted operating points for the integrated design under study; this is

Figure 1 .

 1 Figure 1. Principle and interest of BtL and HTSE hybrid process to produce biofuels.

  Selected technologies are shown in the block diagram Figure 2 (EFR: Entrained Flow Reactor, WGS: Water Gas Shift). Regarding extra-hydrogen input and alternate technologies shown in Figure 3, base case is used for the results presented here with three options for hydrogen ratio adaptation: HTSE, Alkaline Electrolysis (AE) and WGS.

Figure 2 .

 2 Figure 2. Block diagram of the standard BtL process. Material streams in italics, operations in

Figure 3 .

 3 Figure 3. Block diagram of the integrated HTSE-enhanced BtL process. Material streams in

Figure 4 .

 4 Figure 4. Heat integration for the BtL and HTSE hybrid process: Grassmann (exergy flow)

Figure 5 .

 5 Figure 5. Heat integration for the BtL and HTSE hybrid process: composite curves with main

Figure 6 .

 6 Figure 6. Biogenic carbon flows for the standard and HTSE coupled BtL process, CO 2

Figure 7 .

 7 Figure 7. Raw GHGE for the standard and HTSE coupled BtL process, versus GHGE of electrical power. French and German electricity mix 1,24,25 , and European threshold values are

Figure 8 .

 8 Figure 8. Framework for BtL and HTSE hybrid process evaluation.

Table 1 .

 1 Heat and matter integration of BtL and HTSE hybrid process: computing issues.

		HTSE	BtL	Note
		Produces required H 2	Requires H 2
	H 2 , O 2			Excess O 2
		and additional O 2	aand also O 2
		internal reuse	Requires and produces H 2 O	HTSE additional
	H 2 O	and reuse from FTS	internal recycling (treatment required, not simulated)	water use computed
				Net heat
	Heat	Complete heat integration performed	exchange is from BtL to HTSE
	Figure

  [START_REF] Boissonnet | BtL process development: simulation and techno-economic assessment of several technical options[END_REF][START_REF] Chauvel | Manuel d'évaluation économique des procédés[END_REF] Capacity based correlations are applied for a 100 t anhydrous biomass / h or 500 MW biomass heat power plant; as a consequence capital expenditure (capex) cost of the integrated plant is almost twice as high as for the standard plant, matching the increase in productivity. Biomass price is 70 €/t, power 70 €/MWh and natural gas 35 €/MWh.

Table 2 .

 2 Comparison of standard and HTSE hybrid BtL process.

	Standard BtL	BtL + HTSE

currently under study. Further work would take into account multi-period issues (both characterization and modeling) in order to study the suitability of the integrated process with inconstant electrical power.

Acknowledgment

Quentin Bernical acknowledges the CEA (French Atomic Energy and Alternative Energies Commission) for providing a CFR PhD funding.

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