Ben Faker

Christine Belgacem

Frédéric Bernardi
email: bernardi@ann.jussieu.fr

Stéphanie Hecht
email: hecht@ann.jussieu.fr

Salmon

Belgacem

Christine Bernardi

Frédéric Hecht

Stéphanie Salmon
email: stephanie.salmon@univ-reims.fr

Mixed finite

Keywords: Variational formulation, non-symmetric mixed problem, mixed finite elements, generalized saddle point theory, organic pollution

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Since the early (advection-)reaction formulation written by [START_REF] Streeter | A study of the pollution and natural purification of the Ohio river[END_REF] to study the Ohio river (see [START_REF] Streeter | A study of the pollution and natural purification of the Ohio river[END_REF]), sophisticated modeling of the organic pollution in stream-waters has been elaborated. Taylor's dispersion is particularly incorporated that bring substantial difficulties in the mathematical study. Readers interested in are referred to [START_REF] Okubo | Diffusion and Ecological Problems: Mathematical Models[END_REF][START_REF] Brown | The Enhanced Stream Water Quality Models QUAL2E and QUAL2E-UNCAS: Documentation and User Manual[END_REF][START_REF] Sawyer | Chemistry for Environmental Engineering and Science[END_REF] for the dispersion-reaction models so as more complete (non-linear) ones. The central element of such problems is the oxygen. The main tracers currently used are the density b of the Biochemical Oxygen Demand (BOD) and the concentration c of the Dissolved Oxygen (DO). In the steady state, they are solutions of the reaction-dispersion equations, in a bounded two or three-dimensional domain Ω,

-div (d ∇b) + r b = f in Ω, -div (d ∇c) + r * c + r b = g in Ω, c = α on ∂Ω, (1.1)
d ∂ n c = β on ∂Ω.
The symbol d is the dispersion coefficient and (r, r * ) are reaction parameters. They are all spacevarying. The BOD is the amount of oxygen per unit volume, necessary for the micro-organisms and aerobic bacteria to break down the organic matter contained in the water. The DO is the oxygen concentration housed in a unit volume of water. The right-hand side f in (1.1), describes the source of the pollution, while the datum g can be for instance the uptake oxygen from the atmosphere to reduce the deficit of oxygen caused by the biodegradation of the organic pollutants. Some authors favor the oxygen deficit density which is the gap between the saturation oxygen level and the actual dissolved oxygen content. The coupling term represented by r b in the second equation is the depletion of oxygen due to elevated BOD. The boundary data (α, β) are the measures of the DO and its flux at the boundary. Currently, no oxygen flux is generated by the environmental medium so that β is very near zero.

Differential system (1.1) is the steady form of the full time-dependent BOD/DO model, set in two-or three-dimensional body of water such as a lake, a lagoon or an estuary. It is actually obtained by adding to the elementary Streeter-Phelps differential equation the dispersion term. Solving the steady problem may have its own interest, the related problem aims the reconstruction of some polluting flux d ∂ n b, through the boundary ∂Ω, caused by many factor such as domestic, agricultural or breeding activities. The lack of boundary data on b is balanced by two conditions on the dissolved oxygen c. Indeed, measurements on c are easy and instantaneously obtained while those on b need conducting a strict chemical protocol and last five days. Note that we do not include the advection in our model. Nevertheless, the study conducted here extends as well to that case.

Deriving theoretical results for the steady model is important, in view of the unsteady problem which is ill-posed (see [START_REF] Ben Belgacem | Uniqueness in a Cauchy problem for reaction-diffusion system and inverse source problems in water pollution[END_REF][START_REF] Ben Belgacem | Uniqueness for an ill-posed reaction-dispersion model. Application to organic pollution in stream-waters[END_REF]). There is a great analogy between the steady dispersion-reaction BOD/DO model and the steady stream-function/vorticity formulation of the incompressible fluid flow in the two-dimensional case, due to the non-symmetry of the boundary conditions on b and c (see [START_REF] Bernardi | Mixed spectral element approximation of the Navier-Stokes equations in the stream-function and vorticity formulation[END_REF]). The main difference lies in the reaction terms in equations (1.1) which play a preponderous role. The fact that the related coefficients (r, r * ) are not equal arise tedious mathematical difficulties when they are non-constant. The stream-function/vorticity problem is a symmetric variational saddle point problem while the variational problem that follows from the BOD/DO model turns out to be non-symmetric. The corresponding theory originally developed by F. Brezzi (see [START_REF] Brezzi | Mixed and Hybrid Finite Element Methods[END_REF]) applies to the former system while for the latter one, the generalized theory proposed in [START_REF] Nicolaides | Existence, uniqueness and approximation for generalized saddle point problems[END_REF] (see also [START_REF] Bernardi | Generalized inf-sup condition for Chebyshev spectral approximation of the Stokes problem[END_REF]) is better suited and therefore necessary for our purpose. The inf-sup statements required in the saddle point analysis turn out to be complicated. The results proven here are restricted to a class of reaction parameters, roughly characterized by the fact that the oscillations of the ratio r * /r are bounded. Nevertheless, we do not consider this limitation as stringent since this assumption covers a wide part of the physical situations (see [START_REF] Jolánkai | Description of the CAL programme on Water Quality Modelling[END_REF]).

Once the structure of system (1.1) is understood, we construct a first finite element discrete problem by the Galerkin method. But the same difficulties as for the stream-function/vorticity problem lead to a lack of convergence of its solution. So, following the approach in [START_REF] Amara | Convergence of a finite element discretization of the Navier-Stokes equations in vorticity and stream function formulation[END_REF] and [START_REF] Amara | An optimal C 0 finite element algoruthm for the 2D biharmonic problem: theoretical analysis and numerical results[END_REF], we propose a modified problem where stabilization terms are added to overcome the previous difficulty. The a priori analysis is then performed for a wide class of reaction parameters (r, r * ). The mixed finite element method we propose is proved to be optimal. Some numerical experiments confirm the interest of the stabilization, in good coherence with the analysis.

The outline of the paper is as follows:

• In Section 2, we write the variational formulation of problem (1.1) and prove its well-posedness.

• Section 3 is devoted to the description of the first finite element discrete problem.

• In Section 4, we introduce a stabilized finite element discrete problem and prove its well-posedness together with optimal a priori estimates.

• Numerical experiments leading to a comparison of the two discrete problems are presented in Section 5.

• Some concluding remarks are stated in Section 6.

Functional notation. From now on, Ω is a bounded connected domain in R k , k = 2 or 3, and its boundary ∂Ω is Lipschitz-continuous. The Lebesgue space of square integrable functions over Ω is denoted by L 2 (Ω), and (•, •) is the associated scalar product. The Sobolev space H 1 (Ω) contains all the functions that belong to L 2 (Ω) so as their first-order derivatives. We also denote by H 1 0 (Ω), the subspace of H 1 (Ω) made of all functions whose traces on ∂Ω vanish (see [START_REF] Adams | Sobolev Spaces[END_REF]). The dual space of H 1 0 (Ω) is H -1 (Ω) and the duality pairing is represented by

•, • H -1 ,H 1 0 .
The space H 1/2 (∂Ω) is the range of H 1 (Ω) by the trace operator and H -1/2 (∂Ω) is its dual space. We refer to [START_REF] Adams | Sobolev Spaces[END_REF] for more details on these functional spaces.

The mixed variational framework

From now on, we assume that the reaction and diffusion coefficients r, r * and d are piecewise continuous on Ω and also that there exist positive real constants r , r , d and d such that

∀x ∈ Ω, r ≤ r(x), r * (x) ≤ r and d ≤ d(x) ≤ d . (2.1)
We also suppose the diffusion coefficient d to be piecewise continuously differentiable on Ω. Some further regularity of the boundary and these coefficients is needed for some specific properties of problem (1.1), we assume them only when needed.

To develop our analysis we make a simplification assumption about the boundary data, that is (α, β) = (0, 0), and discuss the case of non-homogeneous boundary data only at the end of this section. We take the data f in H -1 (Ω) and g in L 2 (Ω), respectively.

The variational formulation

The functional framework well fit to the system is the same as the one introduced in [START_REF] Bernardi | Mixed spectral element approximation of the Navier-Stokes equations in the stream-function and vorticity formulation[END_REF] (see also [START_REF] Bernardi | Incompressible Viscous Fluids and their Finite Element Discretization, Part I: The Stokes and Navier-Stokes equations[END_REF]Section II.4]) for the stream-function/vorticity formulation of the incompressible fluid flow. It has already been used in [START_REF] Ben Belgacem | Uniqueness in a Cauchy problem for reaction-diffusion system and inverse source problems in water pollution[END_REF] for the organic pollution with constant reaction parameters. Before providing details, we need some further notation. Consider the functional space

V = χ ∈ L 2 (Ω); div(d ∇χ) ∈ H -1 (Ω) .
which is a Hilbert space when endowed with the graph norm

χ V = div(d ∇χ) 2 H -1 (Ω) + χ 2 L 2 (Ω) 1/2
.

Observe that H 1 0 (Ω) is a closed subspace of V and that the norm

• V is equivalent to • H 1 (Ω) in H 1 0 (Ω).
To write down the mixed variational formulation of problem (1.1), we multiply the first line in (1.1) by a function ψ in H 1 0 (Ω). This leads to

-div(d ∇b) + r b, ψ H -1 ,H 1 0 = f, ψ H -1 ,H 1 0 .
Next, we multiply the second line in (1.1) by a function ϕ in V, and using the duality yields that

-div(d ∇ϕ) + r * ϕ, c H -1 ,H 1 0 + (r b, ϕ) = (g, ϕ).
To write the variational formulation of the system, let us introduce three bilinear forms,

∀(χ, ϕ) ∈ V × V, a(χ, ϕ) = (r χ, ϕ), ∀(ψ, ϕ) ∈ H 1 0 (Ω) × V, m(ψ, ϕ) = -div(d ∇ϕ) + r ϕ, ψ H -1 ,H 1 0 , ∀(ψ, ϕ) ∈ H 1 0 (Ω) × V, m * (ψ, ϕ) = -div(d ∇ϕ) + r * ϕ, ψ H -1 ,H 1 0 ,
and also the linear form ∀ϕ ∈ V, (ϕ) = (g, ϕ).

All these forms are continuous on the spaces where they are defined.

The mixed variational problem may then be expressed in terms of these bilinear forms as follows:

Find (b, c) in V × H 1 0 (Ω) fulfilling ∀ψ ∈ H 1 0 (Ω), m (ψ, b) = f, ψ H -1 ,H 1 0 , (2.2) ∀ϕ ∈ V, m * (c, ϕ) + a(b, ϕ) = (ϕ).
The following equivalence property is easily derived from the arguments in [4, Lemma 2.1].

Lemma 2.1 A pair (b, c) in V × H 1 0 (Ω) is a solution of the mixed variational problem (2.
2) if and only if it satisfies the boundary value system (1.1) with α = β = 0.

We recall that, according to the analysis of the abstract problem achieved in [START_REF] Nicolaides | Existence, uniqueness and approximation for generalized saddle point problems[END_REF] and [START_REF] Bernardi | Incompressible Viscous Fluids and their Finite Element Discretization, Part I: The Stokes and Navier-Stokes equations[END_REF]Thm 1.3.14], necessary and sufficient conditions on the bilinear forms are required to ensure existence, uniqueness, and stability for the mixed problem (2.2). The bilinear form a(•, •) must satisfy an inf-sup condition and a positivity property on both null-spaces of the forms m(•, •) and m * (•, •); similar inf-sup conditions must also be satisfied by m(•,

•) and m * (•, •) in H 1 0 (Ω) × V.
∀ψ ∈ H 1 0 (Ω), sup ϕ∈V m(ψ, ϕ) ϕ V ≥ η ψ H 1 (Ω) . (2.3) Lemma 2.3
The bilinear form m * (•, •) satisfies the inf-sup condition on the space H 1 0 (Ω) × V, for a positive constant η * only depending on r and d ,

∀ψ ∈ H 1 0 (Ω), sup ϕ∈V m * (ψ, ϕ) ϕ V ≥ η * ψ H 1 (Ω) .
(2.4)

The bilinear form a(•, •)

The inf-sup conditions on a(•, •) use the kernels of the bilinear forms m(•, •) and m * (•, •) and require therefore their characterization. They are defined to be

N = ϕ ∈ V; ∀ψ ∈ H 1 0 (Ω), m (ψ, ϕ) = 0 N * = ϕ ∈ V; ∀ψ ∈ H 1 0 (Ω), m * (ψ, ϕ) = 0 .
The following statement is readily checked

N = ϕ ∈ V, -div(d ∇ϕ) + r ϕ = 0 in Ω , N * = ϕ ∈ V, -div(d ∇ϕ) + r * ϕ = 0 in Ω .
Note that N and N * are closed subspaces in V and are then Hilbert spaces, when endowed with • V . Furthermore, the norm • L 2 (Ω) is obviously equivalent to • V in both spaces N and N * , where the constants of equivalence depend only on r and r .

Remark 2.4 In the case f = 0, problem (2.2) turns out to be equivalent to the reduced one: Find b in N fulfilling ∀ϕ ∈ N * , a(b, ϕ) = (ϕ).

To solve this equation, one needs the couple of inf-sup conditions we look for.

The main purpose now is to bound from below the following inf-sup quantities

inf ψ∈N sup ϕ∈N * a(ϕ, ψ) ϕ V ψ V and inf ψ∈N * sup ϕ∈N a(ϕ, ψ) ϕ V ψ V .
This relies on the construction of a suitable isomorphism K between N and N * . To do this, for any χ in N , we consider the solution θ in

H 1 0 (Ω) of the Poisson problem -div (d ∇θ) + r * θ = (r -r * ) χ in Ω. (2.5)
This elliptic boundary value problem is obviously well-posed. Then, we set: Kχ = θ + χ. The function Kχ belongs to N * . The operator K is then well-defined from N into N * . Its properties are stated in the next lemma. For simplicity, we introduce the weighted norms defined by

χ L 2 r (Ω) = (χ, χ r) 1/2 and ϕ L 2 r * (Ω) = (ϕ, ϕ r * ) 1/2 .
Lemma 2.5 The operator K is an isomorphism between the spaces N and N * . Moreover, the following inequalities hold

∀χ ∈ N , σ χ L 2 (Ω) ≤ Kχ L 2 (Ω) ≤ σ χ L 2 (Ω) , (2.6)
for constants σ and σ only depending on r and r .

Proof. It is straightforward that equation (2.5) is equivalent to -div (d ∇θ) + r θ = (r -r * ) ϕ in Ω. (2.7)
With any function ϕ in N , we associate the function ϕ = Kχ in N * . We have that K -1 ϕ = ϕ -θ, so that the inverse K -1 of K is well-defined.

To prove (2.6), with any χ in N , we associate the function ϕ = Kχ = θ + χ in N * , where θ belongs to H 1 0 (Ω) and satisfies equation (2.5). It is therefore easily derived that

ϕ L 2 r * (Ω) = θ + χ L 2 r * (Ω) ≤ ς χ L 2 r (Ω) ,
for a positive constant ς only depending on the quantities r and r introduced in (2.1). Exactly the same arguments, now relying on equation (2.7), yield

χ L 2 r (Ω) = ϕ -θ L 2 r (Ω) ≤ 1 ς ϕ L 2 r * (Ω) ,
for another positive constant ς . By using the equivalence of the norms

• L 2 r (Ω) (resp. • L 2 r * (Ω) ) and • L 2 (Ω) , we conclude the proof.
This lemma is of great help for establishing the inf-sup conditions on a(•, •) for a class of reaction parameters. Let us introduce the assumption

osc r * r = max x∈Ω r * r (x) -min x∈Ω r * r (x) < 2. (2.8)
Before stating the inf-sup conditions, we need a further technical lemma. We skip its proof which is nearly obvious.

Lemma 2.6 Let ρ be a piecewise continuous and positive function in Ω. There exists a positive real number ξ such that

∀x ∈ Ω, 1 - (ρ 2 (x) -ξ) 2 4ρ 2 (x) ≥ ζ
for some ζ > 0, if and only if

osc ρ = max x∈Ω ρ(x) -min x∈Ω ρ(x) < 2.
Lemma 2.7 If assumption (2.8) holds, the bilinear form a(•, •) satisfies the two inf-sup conditions, for a positive constant τ , only depending on r and r ,

∀χ ∈ N , sup ϕ∈N * a(χ, ϕ) ϕ V ≥ τ χ V , ∀ϕ ∈ N * , sup χ∈N a(χ, ϕ) χ V ≥ τ ϕ V .
Proof. Let χ be given in N , we associate ϕ = Kχ in N * . The function θ = ϕ -χ belongs to H 1 0 (Ω) and satisfies (2.5), or equivalently

-div (d ∇θ) + r * ϕ -r χ = 0 in Ω.
Multiplying this equation by θ and using Green's formula results in

√ d∇θ 2 L 2 (Ω) d + ϕ 2 L 2 r * (Ω) + χ 2 L 2 r (Ω) = (r χ, ϕ) + (r * χ, ϕ) = a(χ, ϕ) + (r * χ, ϕ) = (1 + ξ)a(χ, ϕ) + ((r * -ξr) χ, ϕ),
where ξ is some positive real number to be appropriately fixed later on. Owing to Young's inequality ts ≤ t 2 + s 2 /4, this formula yields

(1 + ξ) a(χ, ϕ) + ϕ 2 L 2 r * (Ω) + ( (r * -ξr) 2 4rr * r χ, χ) ≥ ϕ 2 L 2 r * (Ω) + χ 2 L 2 r (Ω) whence (1 + ξ) a(χ, ϕ) ≥ Ω r χ 2 (x) 1 - (r * -ξr) 2 4rr * dx.
Now, in view of assumption (2.8) and Lemma 2.6, the real number ξ may be selected such that

min x∈Ω 1 - (r * -ξr) 2 4rr * = ζ > 0. (2.9)
Taking this bound into account yields that

a(χ, ϕ) χ L 2 (Ω) ≥ c ζ 1 + ξ χ L 2 (Ω) ≥ c ζ σ (1 + ξ) ϕ L 2 (Ω) = τ ϕ L 2 (Ω) .
The first inf-sup condition is therefore proved since K is an isomorphism and the norms

• L 2 (Ω)
and • V are equivalent on the kernels N and N * . The second one is checked following the same lines.

Remark 2.8 In the symmetric case where r = r * , the forms m(•, •) and m * (•, •) coincide, so that things are pretty easy. Indeed, the isomorphism K reduces to the identity. As in the previous proof,

choosing ϕ = Kχ = χ gives that a(χ, ϕ) = ϕ 2 L 2 r (Ω) . A direct consequence is the proof of the inf-sup conditions on a(•, •).
Remark 2.9 Despite the fact that we did not succeed in establishing both inf-sup conditions without assumption (2.8), we believe that this is only a technical problem and we suggest that it would be possible to prove that Lemma 2.7 holds without that assumption.

Existence and uniqueness

All the tools which are necessary and sufficient for the well-posedness of the mixed problem (2.2) are now available. We are hence in a position to state the main result of this section, which is straightforwardly derived from [6, Corollary 2.1] (see also [START_REF] Bernardi | Incompressible Viscous Fluids and their Finite Element Discretization, Part I: The Stokes and Navier-Stokes equations[END_REF]Thm 1.3.14]) and the generalized inf-sup conditions stated in Lemmas 2.2, 2.3 and 2.7.

Theorem 2.10 Assume that (2.8) holds true. Then, for any data

(f, g) in H -1 (Ω) × L 2 (Ω), the mixed problem (2.2) has a unique solution (b, c) in V × H 1 0 (Ω). Moreover this solution satisfies b V + c H 1 (Ω) ≤ C( f H -1 (Ω) + g L 2 (Ω) ).
(2.10)

Further regularity

We provide some indication about the "hidden" regularity of the solution (b, c) without investigating the issue in details. We then consider that the physical parameters d, r and r * are constant. We put them all to unity, only for simplicity. The conclusions we make here are also valid for smooth space-varying coefficients. We begin with a basic result. Proof. The second and third lines of problem (1.1) in the case α = 0 can be written as

-∆c + c = g -b in Ω, c = 0 on ∂Ω.
Due to the fact that g -b belongs to L 2 (Ω), we derive the desired result which follows from the elliptic regularity (see [14, Thms 2.4.2.5 & 3.2.1.2]).

Stating the regularity of the full solution (b, c) is more tricky and a further argument is needed.

Proposition 2.12 Assume that the domain Ω is a convex polygon or polyhedron or has a boundary of class C 1,1 . For any data (f, g) in

H -1 (Ω) × H 1 (Ω), the solution (b, c) of problem (1.1) belongs to H 1 (Ω) × H 3 (Ω).
Proof. Applying the Laplace operator to the second line in (1.1) yields the fourth order elliptic problem

-∆c + ∆ 2 c = -∆g -f + b in Ω, c = ∂ n c = 0 on ∂Ω.
The right-hand side of this equation belongs to H -1 (Ω). The regularity property of c is hence a consequence of [START_REF] Grisvard | Singularities in Boundary Value Problems[END_REF]Cor. 3.4.2], according to the geometry of Ω. The regularity of b is then a direct consequence of the second line of (1.1).

Nonhomogeneous boundary conditions

Enforcing the nonhomogeneous boundary conditions on the boundary ∂Ω is worth some comments.

In view of the functional framework used in our analysis, prescribing the Dirichlet condition c = α seems natural in H 1/2 (∂Ω) while imposing the Neumann one d ∂ n c = β arises some trouble. The main difficulty for this problem is to derive its variational formulation. It reads as follows:

Find (b, c) in V × H 1 0 (Ω) satisfying c |∂Ω = α and ∀ψ ∈ H 1 0 (Ω), m (ψ, b) = f, ψ H -1 ,H 1 0 , ∀ϕ ∈ V, m * (c, ϕ) + a(b, ϕ) = (ϕ) -β, ϕ ∂Ω .
The point is that the duality product β, ϕ ∂Ω is meaningless for all ϕ in V. To give a rigorous sense to it, we recall from [15, Section 1.5] that, if the boundary ∂Ω is smooth, any ϕ in V has a trace ϕ |∂Ω that belongs to H -1/2 (∂Ω). As a result for β, ϕ ∂Ω to make sense, it is necessary to assume that β lies in H 1/2 (∂Ω). This regularity seems too strong to consider on a Neumann data. This difficulty can be overcome by using an appropriate lifting and we have therefore the following result.

Theorem 2.13 Assume that (2.8) holds true. Then, for any data (f, g) in H -1 (Ω) × L 2 (Ω) and (α, β) in H 1/2 (∂Ω) × H -1/2 (∂Ω), the boundary value system (1.1) has a unique solution (b, c) in

V × H 1 (Ω).
Proof. Since the problem is linear, the uniqueness of its solution follows from the fact that the only solution for zero data is zero, which is a consequence of Lemma 2.1 and Theorem 2.10. On the other hand, the problem

-div (d ∇c) + r * c = 0 in Ω d ∂ n c = β on ∂Ω,
has a unique solution c in H 1 (Ω). Then, the same arguments as for Lemma 2.1 yield that (b, c) is a solution of problem (1.1) if and only if (b, c 0 ), with

c 0 = c -c, is a solution in V × H 1 (Ω) of c 0 = α -c on ∂Ω, and
∀ψ ∈ H 1 0 (Ω), m (ψ, b) = f, ψ H -1 ,H 1 0 , ∀ϕ ∈ V, m * (c 0 , ϕ) + a(b, ϕ) = (ϕ).
Proving the existence of a solution to this problem is performed as for Theorem 2.10, whence the desired result. This is important, since even if these quantities do not appear in system (1.1), they have a physical meaning and computing them is of high interest.

A first finite element discrete problem

We aim to write a finite element discretization of the variational problem (2.2). The domain Ω is then assumed to be a polygon (k = 2) or a polyhedron (k = 3) to be splitted in a finite number of triangles or tetrahedra. We introduce a regular family (T h ) h of triangulations of Ω, in the sense that, for each h,

• Ω is the union of all elements of T h ;

• The intersection of two different elements of T h , if not empty, is a vertex or a whole edge or a whole face of both of them;

• The ratio of the diameter h K of any element K of T h to the diameter of its inscribed circle or sphere is smaller than a constant σ independent of h. As usual, h stands for the maximum of the diameters h K . We refer to [START_REF] Ciarlet | The Finite Element Method for Elliptic Problems[END_REF][START_REF] Brenner | The Mathematical Theory of Finite Element Methods[END_REF] for the basics of the finite element method.

The previous analysis of the continuous BOD-DO model illustrates the complication due by the lack of symmetry of the variational saddle point formulation. This is caused by the fact that the reaction parameters are different, and the same difficulty appears in the discrete case.

The discrete problem

We introduce the discrete spaces V h ⊂ V and H h ⊂ H 1 0 (Ω), defined by

V h = χ h ∈ H 1 (Ω); ∀K ∈ T h , (χ h ) |K ∈ P 1 (K) , H h = V h ∩ H 1 0 (Ω),
where P 1 (K) stands for the space of restrictions to K of affine functions on R k . Using finite elements of degree higher than unity is allowed and the analysis in this case can be carried out similarly. Then, the first discrete problem in the case α = β = 0 is constructed from (2.2) by the Ritz-Galerkin method. It reads as:

Find (b h , c h ) in V h × H h fulfilling ∀ψ h ∈ H h , m h (ψ h , b h ) = f, ψ h H -1 ,H 1 0 , (3.1)
∀ϕ h ∈ V h , m * ,h (c h , ϕ h ) + a(b h , ϕ h ) = (ϕ h ),
where the bilinear forms m h (•, •) and m * ,h (•, •) are defined by

∀(ψ h , ϕ h ) ∈ H h × V h , m h (ψ h , ϕ h ) = (d ∇ϕ h , ∇ψ h ) + (r ϕ h , ψ h ), ∀(ψ h , ϕ h ) ∈ H h × V h , m * ,h (ψ h , ϕ h ) = (d ∇ϕ h , ∇ψ h ) + (r * ϕ h , ψ h ).
Note that the new bilinear forms

m h (•, •) and m * ,h (•, •) coincide with m(•, •) and m * (•, •), respectively, on H 1 0 (Ω) × H 1 (Ω), hence on H h × V h .
Remark 3.1 Since H h is a closed subspace of H 1 0 (Ω), for reasons already stated in Section 2, the norm

• V is equivalent to • H 1 (Ω) in H h .
Remark 3.2 When the coefficients r and r * (and d) are too complex to be exactly taken into account in the practical computations, they are most often replaced by their Lagrange interpolates in V h . We do not consider this modification here because it only generates technicalities.

Existence and Uniqueness

Proving the well-posedness of the discrete problem (3.1) relies on the same arguments as for the continuous problem (2.2). Therefore, we need first some inf-sup conditions on the forms m h (•, •), and m * ,h (•, •). Lemma 3.3 There exist two positive constants η and η * independent of h such that the bilinear forms m h (•, •) and m * ,h satisfy the discrete inf-sup conditions

∀ψ h ∈ H h , sup ϕ h ∈V h m h (ψ h , ϕ h ) ϕ h V ≥ η ψ h H 1 (Ω) , ∀ψ h ∈ H h , sup ϕ h ∈V h m * ,h (ψ h , ϕ h ) ϕ h V ≥ η * ψ h H 1 (Ω) .
Proof. For any ψ h in H h , taking ϕ h equal to ψ h yields

m h (ψ h , ϕ h ) ≥ c ψ h 2 H 1 (Ω) = c ψ h H 1 (Ω) ϕ h H 1 (Ω) ≥ η ψ h H 1 (Ω) ϕ h V .
This and the same argument applied to m * ,h (•, •) yield the desired result.

Proving the required inf-sup conditions on the bilinear form a(•, •) requires more work. We first introduce the discrete kernels

N h = ϕ h ∈ V h ; ∀ψ h ∈ H h , m h (ψ h , ϕ h ) = 0 , N * ,h = ϕ h ∈ V h ; ∀ψ h ∈ H h , m * ,h (ψ h , ϕ h ) = 0 .
Note that N h is not a subspace of N nor N * ,h is included in N * . As for the continuous problem, the bilinear form a(•, •) should satisfy some inf-sup conditions on these subspaces and as for the continuous case we need the construction of an isomorphism K h between N h and N * ,h . To do this, with any χ h in N h , we associate the solution θ h in H h of the problem

∀ψ h ∈ H h , (d ∇θ h , ∇ψ h ) + (r * θ h , ψ h ) = (r -r * ) χ h , ψ h . (3.2)
This problem is well-posed, which enables us to set K h χ h = θ h + χ h . The operator K h is one-to-one from N h onto N * ,h and satisfies the following result.

Lemma 3.4 There exist postive constants σ and σ independent of h such that the following inequalities hold

∀χ h ∈ N h , σ χ h L 2 (Ω) ≤ K h χ h L 2 (Ω) ≤ σ χ h L 2 (Ω) .
Proof. The second inequality follows by taking ψ h equal to θ h in problem (3.2), noting that |∇θ h | H 1 (Ω) is nonnegative, and using the inequality

K h χ h L 2 (Ω) ≤ χ h L 2 (Ω) + θ h L 2 (Ω) .
The first inequality is derived by the same arguments when noting that problem (3.2) is equivalent to (with obvious notation)

∀ψ h ∈ H h , (d ∇θ h , ∇ψ h ) + (r θ h , ψ h ) = (r -r * ) ϕ h , ψ h .
The proof is complete.

The inf-sup conditions on a(•, •), restricted to V h × H h can be obtained following the same arguments as for the continuous problem. They require the Assumption (2.8) to be fulfilled. Lemma 3.5 Assume that (2.8) holds true. The bilinear form a(•, •) satisfies the two inf-sup conditions, for a positive constant τ independent of h,

∀χ h ∈ N h , sup ϕ h ∈N * ,h a(χ h , ϕ h ) ϕ h L 2 (Ω) ≥ τ χ h L 2 (Ω) , ∀ϕ h ∈ N * ,h , sup χ h ∈N h a(χ h , ϕ h ) χ h L 2 (Ω) ≥ τ ϕ h L 2 (Ω) .
These last inf-sup conditions do not involve the right norm of the functions, which should be • H 1 (Ω) . However, since all norms are equivalent on the finite-dimensional spaces N h and N * ,h , the well-posedness of problem (3.1) is a direct consequence of the previous results, see once more

h , c h ) in V h × H h .
Due to the weak norms in both inf-sup conditions of Lemma 3.5, a stability estimate analogous to (2.10) would involve constants depending on h. For the same reasons, the a priori error estimates that can be proved for this problem do not lead to the convergence of the solution for the discrete problem in all cases. [START_REF] Ben Belgacem | Uniqueness for an ill-posed reaction-dispersion model. Application to organic pollution in stream-waters[END_REF] The stabilized discrete problem and its a priori analysis A possible remedy to the difficulties encountered in the previous section, is to resort to the stabilized finite element discretization studied for the symmetric saddle point problems in [START_REF] Amara | An optimal C 0 finite element algoruthm for the 2D biharmonic problem: theoretical analysis and numerical results[END_REF]. The extension of this procedure to our non-symmetric problems arises some heavy technicalities. From now on and only for brevity of proofs, we take the coefficient d equal to 1.

The stabilized discrete problem

Let E h be the set of all edges (k = 2) or faces (k = 3) of elements of T h which are not contained in ∂Ω. For each e in E h , we denote by h e the diameter of e and by [•] e the jump through e. The stabilized discrete problem relies on the same discrete spaces as previously, it now reads:

Find (b h , c h ) in V h × H h fulfilling ∀ψ h ∈ H h , m h (ψ h , b h ) = f, ψ h H -1 ,H 1 0 , (4.1
)

∀ϕ h ∈ V h , m * ,h (c h , ϕ h ) + a ρ,h (b h , ϕ h ) = (ϕ h ),
where the new bilinear form a ρ,h (•, •) is defined by

a ρ,h (χ h , ϕ h ) = a(χ h , ϕ h ) + ρ e∈E h h e e [∂ n χ h ] e (τ )[∂ n ϕ h ] e (τ ) dτ.
ρ is a positive real number called regularization parameter. We refer to [START_REF] Amara | An optimal C 0 finite element algoruthm for the 2D biharmonic problem: theoretical analysis and numerical results[END_REF] (see also [START_REF] Amara | Convergence of a finite element discretization of the Navier-Stokes equations in vorticity and stream function formulation[END_REF]) for the introduction of this new form for the stream-function and vorticity formulation of the Stokes problem.

Remark 4.1 Observe that the bilinear form a ρ,h (•, •) is symmetric and positive-definite on V h . It is elliptic with respect to the L 2 -norm. Things are harder when we are involved in the natural norm of V. This issue is investigated in the subsequent.

a ρ,h (χ h , ϕ h ) = a(χ h , ϕ h ) + ρ e∈E h h e e [d∂ n χ h ] e (τ )[d∂ n ϕ h ] e (τ ) dτ + ρ K∈T h meas(K) K div (d ∇χ h ) div (d ∇ϕ h ) dx.
The overall analysis we undertake here is readily extended to this case, at the cost of more technical work, especially caused by the last term.

Well-posedness for the stabilized problem

Proving the existence, uniqueness and stability for the new discrete problem requires some preliminary lemmas which state several important properties of the augmented bilinear form a ρ,h (•, •), more precisely of the stabilizing form s h (•, •) defined by

s h (χ h , ϕ h ) = e∈E h h e e [∂ n χ h ] e (τ )[∂ n ϕ h ] e (τ ) dτ.
We begin with a result proven in [START_REF] Amara | An optimal C 0 finite element algoruthm for the 2D biharmonic problem: theoretical analysis and numerical results[END_REF]Proposition 6].

Lemma 4.3 There exists a positive constant µ independent of h such that

∀χ h ∈ V h , s h (χ h , χ h ) ≤ µ ∆χ h H -1 (Ω) .
The next lemmas provide the "inverse" inequality on the kernel spaces N h and N * ,h . The proof is an adaptation of the one made for Proposition 7 in [START_REF] Amara | An optimal C 0 finite element algoruthm for the 2D biharmonic problem: theoretical analysis and numerical results[END_REF]. Lemma 4.4 There exists a positive constant µ independent of h such that

∀χ h ∈ N h , ∆χ h H -1 (Ω) ≤ µ a 1,h (χ h , χ h ), ∀χ h ∈ N * ,h , ∆χ h H -1 (Ω) ≤ µ a 1,h (χ h , χ h ).
Proof. We handle the first estimate. The second one can established following the same lines. Let χ h ∈ N h be given. Then, for all ψ ∈ H 1 0 (Ω) and ψ h ∈ H h , we have that

(∇χ h , ∇ψ) = (∇χ h , ∇(ψ -ψ h )) -(χ h , ψ h r).
Making an integration by part and using the Cauchy-Schwarz inequality provide

(∇χ h , ∇ψ) = e∈E h e [∂ n χ h ] e (τ )(ψ -ψ h )(τ ) dτ -(χ h , ψ h r) ≤ s h (χ h , χ h ) e∈E h (h e ) -1 ψ -ψ h 2 L 2 (e) 1/2 -(χ h , ψ h r).
A suitable choice of ψ h , equal to the Clément interpolant of ψ, the Bramble-Hilbert argument together with the Poincaré inequality yield that

(∇χ h , ∇ψ) ≤ κ s h (χ h , χ h )|ψ| H 1 (Ω) + χ h L 2 r (Ω) ψ L 2 r (Ω) ≤ µ a 1,h (χ h , χ h )|ψ| H 1 (Ω) .
By taking into account the identity

∆χ h H -1 (Ω) = sup ψ∈H 1 0 (Ω) 1 |ψ| H 1 (Ω) (∇χ h , ∇ψ),
the proof is complete.

Corollary 4.5 The map χ h → a ρ,h (χ h , χ h )
is a norm on N h (resp. N * ,h ) that is equivalent to the norm of V, uniformly in h (in the sense that the equivalence constants are independent of h).

We will need also a third intermediary result, related to a bound of the bilinear form s h (•, •) by the H 1 -norm. Lemma 4.6 There exists a positive constant µ independent of h such that

∀χ h ∈ V h , s h (χ h , χ h ) ≤ µ |χ h | 2 H 1 (Ω) .
Proof. Let χ h belong to V h . Denoting by K and K the two elements of T h that share e, we have straightforwardly

h e e [∂ n χ h ] 2 e (τ ) dτ ≤ h e |(∇χ h ) |K | 2 L 2 (e) k + h e |(∇χ h ) |K | 2 L 2 (e) k .
Switching backward and forward to the reference element, provides the following bound

h e e [∂ n χ h ] 2 e (τ ) dτ ≤ κ |(χ h ) |K | 2 H 1 (K) + |(χ h ) |K | 2 H 1 (K ) ,
where the constant κ depends on the regularity parameter of the family of triangulations. Summing up over e, we derive that

e∈E h h e e [∂ n χ h ] 2 e (τ )) dτ ≤ µ |χ h | 2 H 1 (Ω) .
The proof is complete.

A first consequence of Lemma 4.3 is that the bilinear form a ρ,h (•, •) is continuous on V h × V h with respect to the natural norm • V . The continuity constant is bounded independently of h. Proposition 4.7 There exists a positive constant C ρ independent of h such that

∀(χ h , ϕ h ) ∈ V h × V h , a ρ,h (χ h , ϕ h ) ≤ C ρ χ h V ϕ h V .
Proof. Let χ h and ϕ h be in V h . By Cauchy-Schwarz inequality, there holds that

a ρ,h (χ h , ϕ h ) = a(χ h , ϕ h ) + ρs h (χ h , ϕ h ) ≤ χ h L 2 r (Ω) ϕ h L 2 r (Ω) + ρ s h (χ h , χ h ) s h (ϕ h , ϕ h ).
Invoking the result of Lemma 4.3 gives

a ρ,h (χ h , ϕ h ) ≤ χ h L 2 r (Ω) ϕ h L 2 r (Ω) + µ 2 ρ ∆χ h H -1 (Ω) ∆ϕ h H -1 (Ω) .
This yields that

a ρ,h (χ h , ϕ h ) ≤ C max(1, ρ) χ h V ϕ h V .
The proof is complete with

C ρ = C max(1, ρ).
Now, proving the desired inf-sup conditions for the augmented bilinear form a ρ,h (•, •) on the kernel spaces N h and N * ,h , with respect to the norm of V requires additional technical work. Let us first state the following lemma. Lemma 4.8 Assume that (2.8) holds true. There exists a positive constant ρ 0 such that, for any ρ ≤ ρ 0 , the following positivity property holds

∀χ h ∈ N h , a ρ,h (χ h , K h χ h ) ≥ λ a ρ,h (χ h , χ h ) a ρ,h (K h χ h , K h χ h ),
where the constant λ is positive and only depends on the reaction parameters (r, r * ).

Proof. Let χ h be given in N h . Thus, the function ϕ h = K h χ h belongs to N * ,h , while the function θ h = ϕ h -χ h is the solution of problem (3.2). Processing as in the proof of Lemma 2.7 we are allowed to assert that, for some ξ > 0, we have (2.8), it is possible to select ξ as in (2.9). This results in

|∇θ h | 2 H 1 (Ω) + ϕ h 2 L 2 r * (Ω) + χ h 2 L 2 r (Ω) = (1 + ξ)a(χ h , ϕ h ) + ((r * -ξr) χ h , ϕ h ), Then, given the identity -s h (θ h , θ h ) + s h (χ h , χ h ) + s h (ϕ h , ϕ h ) = 2s h (χ h , ϕ h ), we derive that |∇θ h | 2 H 1 (Ω) - 1 2 (1 + ξ)ρ s h (θ h , θ h ) + 1 2 (1 + ξ)ρ s h (χ h , χ h ) + 1 2 (1 + ξ)ρ s h (ϕ h , ϕ h ) + ϕ h 2 L 2 r * (Ω) + χ h 2 L 2 r (Ω) = (1 + ξ)a(χ h , ϕ h ) + (1 + ξ)ρ s h (χ h , ϕ h ) + ((r * -ξr) χ h , ϕ h ). Using Lemma 4.6 yields that 1 - 1 2 (1 + ξ)ρ µ |θ h | 2 H 1 (Ω) + 1 2 (1 + ξ)ρ s h (χ h , χ h ) + 1 2 (1 + ξ)ρ s h (ϕ h , ϕ h ) + ϕ h 2 L 2 r * (Ω) + χ h 2 L 2 r (Ω) ≤ (1 + ξ) a ρ,h (χ h , ϕ h ) + ((r * -ξr) χ h , ϕ h ). Recall that a ρ,h (•, •) = a(•, •) + ρ s h (•, •). Due to assumption
(1 + ξ) a ρ,h (χ h , ϕ h ) ≥ 1 - 1 2 (1 + ξ)ρ µ |θ h | 2 H 1 (Ω) + 1 2 (1 + ξ)ρ s h (χ h , χ h ) + 1 2 (1 + ξ)ρ s h (ϕ h , ϕ h ) + ζ χ h 2 L 2 r (Ω)
(note that ζ depends on r and r * ). Applying now Lemma 3.4, we get

(1 + ξ) a ρ,h (χ h , ϕ h ) ≥ 1 - 1 2 (1 + ξ)ρ µ |θ h | 2 H 1 (Ω) + 1 2 (1 + ξ)ρ s h (χ h , χ h ) + 1 2 (1 + ξ)ρ s h (ϕ h , ϕ h ) + ς χ h 2 L 2 r (Ω) + ς ϕ h 2 L 2
r * (Ω) . The constant ς is positive and does not depend neither on ρ nor on h. It is only sensitive to the reaction parameters (r, r * ). Next, if ρ is small enough, then the coefficient of |θ h | 2 H 1 (Ω) is nonnegative and can be dropped without changing the inequality. Hence, we obtain

(1 + ξ) a ρ,h (χ h , ϕ h ) ≥ 1 2 (1 + ξ)ρ s h (χ h , χ h ) + ς χ h 2 L 2 r (Ω) + 1 2 (1 + ξ)ρ s h (ϕ h , ϕ h ) + ς ϕ h 2 L 2 r * (Ω) .
This bound can be put under the following form

a ρ,h (χ h , ϕ h ) ≥ λ 2 a ρ,h (χ h , χ h ) + a ρ,h (ϕ h , ϕ h ) ,
for a constant λ > 0 that is dependent on (ξ, ς) (which are themselves dependent on (r, r * )). Next, thanks to the positive definiteness of a ρ,h (•, •), we finally obtain

a ρ,h (χ h , ϕ h ) ≥ λ a ρ,h (χ h , χ h ) a ρ,h (ϕ h , ϕ h ).
The proof is complete.

Remark 4.9 The following bound holds true

∀ϕ h ∈ N * ,h , a ρ,h ((K h ) -1 ϕ h , ϕ h ) ≥ λ a ρ,h ((K h ) -1 ϕ h , (K h ) -1 ϕ h ) a ρ,h (ϕ h , ϕ h ).
We are now in position to state the inf-sup conditions satisfied by a ρ,h (•, •) on the kernel spaces N h and N * ,h . Proposition 4.10 Assume that the regularization parameter ρ is sufficiently small. Then, when (2.8) holds true, the following inf-sup conditions on a ρ,h (•, •) are satisfied

∀χ h ∈ N h , sup ϕ h ∈N * ,h a ρ,h (χ h , ϕ h ) ϕ h V ≥ τ ρ χ h V , ∀ϕ h ∈ N * ,h , sup χ h ∈N h a ρ,h (χ h , ϕ h ) χ h V ≥ τ ρ ϕ h V .
The inf-sup constant τ ρ is expected to decay to zero when ρ decreases toward zero.

Proof. Let χ h be given in N h and denote ϕ h = K h χ h ∈ N * ,h . By Lemma 4.8, we derive

a ρ,h (χ h , ϕ h ) ϕ h V ≥ λ a ρ,h (χ h , χ h ) a ρ,h (ϕ h , ϕ h ) ϕ h V .
Now, applying Corollary 4.5 yields that

a ρ,h (χ h , ϕ h ) ϕ h V ≥ λ a ρ,h (χ h , χ h ) a ρ,h (ϕ h , ϕ h ) a 1,h (ϕ h , ϕ h ) ≥ λ(min(1, √ ρ)) 2 a 1,h (χ h , χ h ) ≥ c λ min(1, ρ) χ h V .
The constant τ ρ may be equal to λρ for small values of ρ. The second inf-sup condition can be established similarly using Remark 4.9 instead of Lemma 4.8. The proof is complete.

The following exisetence and uniqueness theorem is now a direct consequence of Lemma 3.3 and Propositions 4.7 and 4.10.

Theorem 4.11 Assume that (2.8) holds true and that ρ is small enough. For any data (f, g) in H -1 (Ω) × L 2 (Ω), the stabilized discrete problem (4.1) has a unique solution (b h , c h ) in V h × H h . The following stability holds

b h V + c h H 1 (Ω) ≤ C ρ ( f H -1 (Ω) + g L 2 (Ω) ). (4.2)
The constant C ρ is expected to blow up when ρ decays to zero.

A priori error estimates

From now on, we always assume that condition (2.8) holds true and that ρ is small enough, in order that the discrete problem (4.1) is well-posed. We now prove an a priori error estimate between the solutions (b, c) of problem (2.2) and (b h , c h ) of problem (4.1). As usual, we introduce the affine subspace

N h (f ) = ϕ h ∈ V h ; ∀ψ h ∈ H h , m h (ψ h , ϕ h ) = f, ψ h H -1 ,H 1 0 .
The following lemma relies on standard arguments, but takes into account the stabilization of the form a(•, •). It involves the norm • * defined by

v * = K∈T h h K v 2 H 3 2 (K) 1 2 .
Lemma 4.12 Assume that the part b of the solution (b, c) of problem (2.2) belongs to H 3/2 (Ω).

The following a priori error estimate holds

b -b h V ≤ (1 + C τ ρ ) inf β h ∈N h (f ) b -β h V + b -β h * + C τ ρ inf γ h ∈H h c -γ h H 1 (Ω) .
Proof. Let β h be any approximation of b in N h (f ). Thus, it follows from the first line in (4.1) that the function b h -β h belongs to N h . Applying the first inf-sup condition in Proposition 4.10 yields

b h -β h V ≤ τ -1 ρ sup ϕ h ∈N * ,h a ρ,h (b h -β h , ϕ h ) ϕ h V . (4.3)
To evaluate the quantity a ρ,h (b h -β h , ϕ h ), we observe by using problem (4.1) that

a ρ,h (b h -β h , ϕ h ) = (ϕ h ) -m * ,h (c h , ϕ h ) -a ρ,h (β h , ϕ h ).
Due to (2.2) together with the coincidence of the forms m * (•, •) and m * ,h on the discrete spaces, we derive that

a ρ,h (b h -β h , ϕ h ) = a(b -β h , ϕ h ) + m * (c -c h , ϕ h ) -ρ s h (β h , ϕ h ). Since ϕ h belongs to N * ,h we have for all γ h in H h m * (c -c h , ϕ h ) = m * (c -γ h , ϕ h ).
On the other hand, because of the regularity of the solution b inside Ω, we derive

-s h (β h , ϕ h ) = s h (b -β h , ϕ h ).
The fact that grad b belongs to L 2 (Ω) k and that its divergence belongs to L 2 (Ω) implies that each [∂ n b] e belongs to H -1/2 (e). Moreover, by an interpolation argument and since b belongs to H 3/2 (Ω), then the flux jump [∂ n b] e turns to be in L 2 (e). Combining this with a standard inequality on ϕ h give

-s h (β h , ϕ h ) ≤ e∈E h h e [∂ n (b -β h )] L 2 (e) [∂ n ϕ h ] L 2 (e) ≤ C b -β h * ϕ h V .
Remark 4.16 These estimates are fully optimal with respect to h. Furthermore, when combined with the stability property (4.2), they yield the convergence of (b h , c h ) towards (b, c) when h tends to zero, for any fixed parameter ρ.

Numerical experiments

Computations are realized by means of the code FreeFem++ developed by F. Hecht (see [START_REF] Hecht | New development in FreeFem++[END_REF]), where a script specifically dedicated to the pollution system is made. We wish to compare the two discrete problems, which we call in the following NS (for non stabilized) and S (for stabilized), in a basic case. The domain is the square

Ω =] - 1 2π , - 1 2π + 1[ 2 .
Both Neumann and Dirichlet conditions are enforced on c, and b is free of any boundary condition. The exact solution, represented in Figure 5.1, is given by b

(x 1 , x 2 ) = cos( 3π 2 x 1 ) cos( 3π 2 x 2 ), c(x 1 , x 2 ) = sin( 3π 2 x 1 ) sin( 3π 2 x 2 ).
The dispersion and reaction parameters are constant and fixed to d = 0.151 and (r, r * ) = (0.2, 0.4).

According to some specialized literature, they are close to real-life values. For reasons explained later on, we also consider this same solution in the disk with centre (0, 0) and radius 1/2, computed with the same coefficients. Figure 5.1 represents, from top to bottom, the exact solution and the solution issued from NS. For the non stabilized problem, we can see in this figure that the error on b is stronger than the error on c; this error is concentrated on the corners of the domain. To check that this is due to the discretization and not to the geometry of the domain, we now work with the domain Ω equal to the ball. In all the numerical simulations, the meshes are triangular and structured. Varying the mesh size on the square, we evaluate the errors on the BOD density b and the DO concentration c, with respect to the H 1 (Ω) and L 2 (Ω) norms, for both non-stabilized and stabilized approximations. The related convergence curves are plotted in Figure 5.3 in logarithmic scales.

• For the NS problem, there is no convergence of b h in the H 1 (Ω)-norm, and the slope in the L 2 (Ω)norm is evaluated to 1.09. This is an indication of the necessity to resort to stabilization. On the other hand, the slopes of the convergence curves for c in the (H 1 (Ω), L 2 (Ω)) norms are close to (0.99, 1.99).

• For the S problem, the slopes for c are close to the previous ones, equal to (0.99, 1.88). However, the slopes for b are now equal to (1.28, 1.67), so that the convergence for b is highly improved.

Finally, in order to illustrate the good convergence properties of the stabilized problem, Figure 5.4 represent the solutions b h and c h issued from S, first for a standard mesh (h = 1 48 ), second for a refined mesh (h = 1 96 ).

Conclusion

The steady pollution model studied here may be considered as a generalization of the well known saddle-point stream-function/vorticity ψ -ω problem used in the simulation of two-dimensional steady Stokes fluid flows (see [START_REF] Amara | An optimal C 0 finite element algoruthm for the 2D biharmonic problem: theoretical analysis and numerical results[END_REF][START_REF] Bernardi | Incompressible Viscous Fluids and their Finite Element Discretization, Part I: The Stokes and Navier-Stokes equations[END_REF]). The difference between the two models resides in facts that first the pollution system is valid in three dimensions and second that it is non-symmetric with an additional reaction term with space dependent kinetics. This, unexpectedly, brings tedious complications in the analysis of the exact and discretized versions of it. From the applied mathematics view, this stirs a renewed interest (we may even speak of a rehabilitation) in the technical and numerical work realized so-far on the mixed stream-function/vorticity formulation of the Stokes model. Somehow the road is pathed for studying the pollution model, the questions to solve, the methodology to follow so as the ideas already developed may be reinvested or recast for the pollution model with however substantially increased mathematical difficulties to handle. They are successfully solved in this work. The short-term continuation is to deal with the a posteriori issues. We are currently investigating residual based error estimations and the stated results together with a deeper numerical experimentation will be exposed in the second part of this paper. A more distant goal is to consider the full unsteady pollution model which differs tremendously of the unsteady ψ -ω system. They are opposite in nature, the time-dependent pollution model turns out to be ill-posed (see [START_REF] Ben Belgacem | Uniqueness in a Cauchy problem for reaction-diffusion system and inverse source problems in water pollution[END_REF]). Its space time discretization is expected to arise higher difficulties and requires some specific mathematical tools to pick up in the community inverse and ill-posed problems. We refer to the Ph.-D. thesis [START_REF] Khiari | [END_REF] to have some clues on these issues and on close ones but also to discover several applications of the unsteady pollution modes, especially in the inverse problem of (polluting) source detection and identification.

2. 2 Lemma 2 . 2

 222 The bilinear forms m(•, •) and m * (•, •) The next results easily follow from the imbedding of H 1 0 (Ω) into V, see [5, Lemma 3.2] or [7, Section 2], and are derived by taking ϕ equal to ψ. The bilinear form m(•, •) satisfies the inf-sup condition on the space H 1 0 (Ω) × V, for a positive constant η only depending on r and d ,

Lemma 2 . 11

 211 If the domain Ω is convex or has a boundary of class C 1,1 , the part c of the solution (b, c) of problem (1.1) belongs to H 2 (Ω).

Remark 2 . 14

 214 When the boundary ∂Ω and the data (f, g) are smooth enough, the unknown b belongs to H 1 (Ω), hence has a trace on ∂Ω. Moreover, we can write, for any function ψ in H 1 (Ω) d ∂ n b, ψ = (div(d ∇b), ψ) + (d ∇b, ∇ψ), which gives a sense to the BOD flux d ∂ n b.

[ 6 ,Theorem 3 . 6

 636 Corollary 2.1] or [8, Thm 1.3.14]. Assume that (2.8) holds true. For any data (f, g) in H -1 (Ω) × L 2 (Ω), the discrete problem (3.1) has a unique solution (b

Remark 4 . 2

 42 Handling high variations of the dispersion parameter d = d(x) is made by modifying the augmented part of the stabilized bilinear form a ρ,h (•, •). It is then transformed into

Figure 5 .

 5 2 represents the solution (b h , c h ) of the NS problem. The degradation of the accuracy at the vicinity of the boundary is obvious. Thus, the NS problem does not lead to the convergence of the solution, as appears in the next figure.

Figure 5 . 1 :

 51 Figure 5.1: Exact and NS computed solutions, b and b h in the left panels, c and c h in the right panels.

Figure 5 . 2 :

 52 Figure 5.2: NS Computed solutions in the disk, b h in the left panel, c h in the right panel.

Figure 5 . 3 :

 53 Figure 5.3: Convergence curves for the H 1 (Ω) and L 2 (Ω) norms, for both NS (left) and S (right) problems.

Figure 5 . 4 :

 54 Figure 5.4: S computed solutions, b h in the left panels, c h in the right panels, on different meshes.

Inserting all this into (4.3) gives

We conclude the proof by using a triangle inequality.

The next lemma is now just an extension of [13, Chap. II, eq. (1.16)].

Lemma 4.13 If the assumptions of Lemma 4.12 are satisfied, the following bound holds inf

Proof. For any function β h in V h , we deduce from the continuity and ellipticity of the form m h (•, •) on H h that there exists a function η h in H h which satisfies

Thus the function βh = β h + η h belongs to N h (f ) and estimating the norm of b -βh follows from a triangle inequality, the previous equation and a local inverse inequality appied to η h . Finally, we evaluate the error on the part c h of the solution.

Lemma 4.14 If the assumptions of Lemma 4.12 are satisfied, the following a priori error estimate holds

Proof. From the second inf-sup condition in Lemma 3.3, it suffices to evaluate the quantity, for any

Bounding the first two terms simply follows from Cauchy-Schwarz inequalities. To estimate the last one, we note that

use the same arguments as in the proof of Lemma 4.12 for the first quantity, Lemma 4.3 and a triangle inequality for the last one. This gives the desired result.

We are now in a position to state explicit error estimates. The proof follows from Lemmas 4.12, 4.13 and 4.14, together with the approximation properties of the spaces V h and H h and an interpolation argument for handling nonsmooth solutions (b, c). The constant C ρ is expected to blow up when ρ decays to zero.


Tobias Isenberg

Paul Asente

John Collomosse

T Metin Sezgin

Special Section on the 2011 Joint Symposium on Computational Aesthetics (CAe), Non-Photorealistic Animation and Rendering (NPAR), and Sketch-Based Interfaces and Modeling (SBIM)

The year 2011 marked the first year that the conferences CAe, NPAR, and SBIM all combined forces and in August organized a joint symposium in Vancouver, Canada, colocated with ACM SIGGRAPH. All fields share a strong interest in expressiveness-CAe with respect to the expressiveness of aesthetics, NPAR with respect to creating expressive imagery, and SBIM with respect to expressive interaction and modeling. The joint symposium including the associated art show thus was a tremendous success, attracting 161 participants from all over the world.

In this special section we are pleased to present extended versions of the best research papers from this 2011 conference. We therefore continue the tradition started a year ago with the special section in this journal on Non-Photorealistic Animation and Rendering [START_REF] Collomosse | Special Section on Non-Photorealistic Animation and Rendering (NPAR)[END_REF] and include 10 papers, covering a broad range of topics associated with expressive aesthetics, depiction, interaction, and modeling. These papers were selected from the best papers accepted to the conference and their authors submitted a revised journal version. These manuscripts were extended from the conference versions with additional material in the form of new research results, additional study results, new and extended techniques, and/or additional detail and examples on the original approach. The journal submissions underwent at least one thorough journal review cycle, in addition to the previous conference reviewing. Where it was possible, we asked the reviewers from the previous conference review cycle to also review the journal version, but we also asked additional expert reviewers to ensure the high quality of this special section.

From the proceedings of CAe 2011, four extended papers are included in this special issue. The first of them is authored by Alexa and Matusik [START_REF] Alexa | Irregular Pit Placement for Dithering Images by Self-Occlusion[END_REF] and addresses the topic of creating dithered images by physically placing small drill holes into a carrier material. Through self-occlusion these cylindrical holes create the appearance of different gray values, an observation that the authors use to com-Email addresses: isenberg@cs.rug.nl (Tobias Isenberg), asente@adobe.com (Paul Asente), j.collomosse@surrey.ac.uk (John Collomosse), mtsezgin@ku.edu.tr (T. Metin Sezgin) pute an optimal placement of holes to represent a given input image while specifically taking image features into account. The topic of creating images by manipulating a certain carrier structure is also the topic of the paper by Inglis and Kaplan [START_REF] Inglis | Op Art Rendering with Lines and Curves[END_REF] who present an approach that allows users to produce images in the style of Op Art which create a very interesting aesthetic. In this technique the bending of a set of parallel lines in the background of the image is used to encode virtual features, which in turn can be used to represent two-color source images. The authors describe a technique that minimizes potential artifacts and extend the approach to also encode 3-color images, create the illusion of 3D shapes, and thus to use curves instead of straight lines as a carrier. The paper by Lynch et al. [START_REF] Lynch | ColourVis: Exploring Colour in Digital Images[END_REF] analyzes the use of color by artists over time. They sample the histogram of colors used in different paintings, classify them into different color ranges, and then create a visualization from this data as inspired by parallel coordinates and stacked line graphs. The resulting images facilitate the analysis not only of how color usage changed over time for a specific artist or a whole artistic direction but also allow to compare different artists/art directions with each other. Finally, the extended version of the best paper of the CAe 2011 conference was authored by Lockyer et al. [START_REF] Lockyer | Affective Motion Textures[END_REF] and approaches the topic of communicating emotion or creating affect through motion textures. The authors present a tool for motion brushing to influence affective impressions and report on on-going qualitative evaluations with visual effects design professionals about how such a tool can enhance their every-day work. Four papers were selected from the NPAR 2011 for extension and presentation in this special issue. The first paper "XDoG: An eXtended Difference-of-Gaussians Compendium including Advanced Image Stylization" [START_REF] Winnemöller | XDoG: An eXtended Difference-of-Gaussians Compendium including Advanced Image Stylization[END_REF] responds to the growing volume of the NPR literature adopting edge enhancing filtering operators to emphasize and stylize edges to create a variety of artistic effects in images. For the first time, the paper draws together the many works in this area to present both a comprehensive overview of these techniques, and a new formulation of edge enhancing filtering-the extended Difference of Gaussian (XDoG). The XDoG formulation both envelopes many of the previously presented edge stylization effects, and enables a gamut of new stylization effects. "Towards Automatic and Flexible Concept Transfer" [START_REF] Murray | Towards Automatic and Flexible Concept Transfer[END_REF] contrasts the low-level operators of XDog with a high level semantic analysis of color distribution in images. Images may be automatically classified into concepts like "romantic," "luscious," "spicy," or concepts may be transferred to new images. The parameterization of the stylization process at a high conceptual level contrasts many earlier approaches that require many low-level user parameters to be set to control a rendering, and so may improve usability and adoption of NPR techniques. The third paper in the issue, "Emotional Response and Visual Attention to Non-Photorealistic Images" [START_REF] Mould | Emotional Response and Visual Attention to Non-Photorealistic Images[END_REF] complements this automated technique with a 42 participant evaluation of the perceived emotional context attached by users particular stylization effects. The final extended paper within the NPAR section of this issue is unusual among those commonly appearing at these symposia, given the focus on maintaining photorealism in rendering. "Ray Prioritization Using Stylization and Visual Saliency" [START_REF] Steinberger | Ray Prioritization Using Stylization and Visual Saliency[END_REF] presents a new approach for maintaining photorealistic detail in scene rendering, to achieve a guaranteed frame rate when rendering complex scenes. This is achievable through their exploitation of non-photorealistic cues in the rendering process. This final paper won the best paper prize at NPAR 2011-which, as with the selection all four extended papers presented here, was determined through the double-blind review process of NPAR 2011.

This year, four papers from the SBIM track were invited to this special section. Of these papers, two made it past a rigorous review process and are included here. These papers not only introduce sketch-based interfaces, but also spend considerable space discussing potential real-use scenarios. The first article by Applegate et al. [START_REF] Applegate | A Sketch-Based System for Highway Design with User-Specified Regions of Influence[END_REF] describes a sketch-based system for highway design. The highway design process is little known to the non-practitioners, but it is a good example of the kinds of processes that can benefit from free-hand sketch-based interaction. In this paper, the authors describe automated techniques for generating road networks efficiently. Their tool allows the designer to specify a number of task dependent constraints efficiently through a sketch-based interface. The second paper from the SBIM track introduces Sketch Expressa sketching interface for creating facial animations [START_REF] Miranda | Sketch Express: A Sketching Interface for Facial Animation[END_REF]. Here, Miranda et al. describe a system that allows animators to construct facial animations by sketching on either a 3D mesh, a 2.5D canvas or a 2D canvas. In each case, the pen input collected from the animators is used to modify the underlying animation rig. The authors also report two applications of their system-one in an artistic design context, and the other in an educational game context for people with autism spectrum disorders.

This special section would, of course, not have been possible without the support of many people. Aside from the paper authors we would like to thank, in particular, Joaquim Jorge for offering the special section to the three conferences and for helping with the process, as well as to all staff at Elsevier for the efficient production process.

We also thank all involved in organizing CAe/NPAR/SBIM 2011 in Vancouver and, specifically, our general, arts, and program co-chairs Allen Bevans, Douglas Cunningham, Ellen Do, Tracy Hammond, Jean-Claude Leon, Andy Nealen, and Andres Wanner; the poster/demo chairs Jan Eric Kyprianidis, Beryl Plimmer, Bernhard Riecke, and Holger Winnemöller; our publicity chair Christian Richardt; and our ACM liaison person Brian Wyvill. Tobias Isenberg, University of Groningen (for CAe) Paul Asente, Adobe Systems (for NPAR) John Collomosse, University of Surrey (for NPAR) T. Metin Sezgin, Koc ¸University (for SBIM) Guest Editors of the Special Section on CAe/NPAR/SBIM 2011. April 2012.

Tobias Isenberg is an assistant professor for computer graphics and interactive systems at the University of Groningen, the Netherlands. Since September 2010, he also holds a Digiteo Chair of Excellence in collaboration with the French research institutions CNRS and INRIA in Orsay. He works on topics in computational aesthetics as well as non-photorealistic and illustrative rendering and explores applications, for example, in scientific visualization. He also investigates interaction approaches for nonphotorealistic and illustrative visualization, in particular, using large, touch-sensitive displays.

Paul Asente is a member of the Visual Computing Lab at Adobe Systems. His major interests include vector graphics and stylization, and many of his research results have appeared as features in Adobe's creative applications. He received his PhD from Stanford University in 1987.

John


Vincent Bretagnolle

Elsa Berthet

Managing grasslands biodiversity at a landscape level to foster ecosystem services in intensive cereal systems: from ecological knowledge to collective action

Keywords: Agro-ecosystem, ecosystem services, grasslands, cereal crops, biodiversity, design, collective action, commons

Effective solutions for integrating agricultural development and conservation of biodiversity at the landscape scale remain to be identified. We present a case study in an intensively farmed French cereal plain, where the reintroduction of grasslands has been proposed first for conservation purposes in order to protect the Little Bustard, a highly threatened bird species. Monitoring the effects of grassland "experimental" implementation revealed other beneficial effects on virtually all components of the trophic web in these agroecosystems, particularly at the landscape level. Indeed, in intensive cereal systems, perennial habitats such as grasslands are radically different from annual crops in terms of level and frequency of disturbance (plowing, planting, spraying etc.). In these highly fragmented and disturbed habitats, the presence, abundance and distribution of grasslands therefore have a critical role in ecological and environmental regulatory processes. We provide evidence that grasslands, particularly alfalfa, are the support of many ecosystem services, such as pollination, biological control, in addition to forage production. To maximize their provision, it is critical to rationalize the inclusion of grasslands in the cropping system (in time, space and according to management practices). However, currently, grasslands are severely depleted by farmers who privilege cereal crops for economic reasons (including CAP subsidies). We therefore raise the issue of whether crop allocation at the landscape scale can be changed without public funding, in order to increase the proportion of grasslands. In other words, how to overcome the reluctance of cereal farmers to produce forage crop? A solution explored here is to identify the interdependencies between farmers related to the ecosystem services grasslands provide at the landscape scale. The recognition of grassland emergent functions when considered at the landscape scale gives them a status of common good: a good that should be collectively managed to maximize ecosystem services. This consideration leads to involve new stakeholders such as citizens, scientists, government bodies or NGOs in the collective management of grasslands and opens an innovative way to reconcile agriculture and conservation at the landscape scale.

Introduction

In response to the growing concerns about environmental crisis, ecosystem based approaches are increasingly being called upon [START_REF] Grumbine | What Is Ecosystem Management?[END_REF]. Protection and management of whole ecosystem structure and functions however needs integrating ecological, social, economic and institutional factors, and therefore call for holistic approaches in contrast to targeted actions in order to protect species of conservation concern. Ecosystem-based approaches require improving our understanding of the underlying biophysical processes in order to take decisions at the right spatial and temporal scales [START_REF] Anderies | A Framework to Analyze the Robustness of Social-ecological Systems from an Institutional Perspective[END_REF]). Given that individual management actions can affect multiple interacting ecological processes, a critical aspect is the coordination of decisions among interacting stakeholders [START_REF] Granek | Ecosystem services as a common language for coastal ecosystem-based management[END_REF][START_REF] White | The value of coordinated management of interacting ecosystem services[END_REF]. Indeed, coordinated management is supposed to optimize ecosystem state, the provision of multiple ecosystem services and consequently the joint value of an ecosystem to society [START_REF] Slocombe | IMPLEMENTING ECOSYSTEM-BASED MANAGEMENT[END_REF][START_REF] Stallman | Ecosystem services in agriculture: Determining suitability for provision by collective management[END_REF][START_REF] White | The value of coordinated management of interacting ecosystem services[END_REF]). However, coordinating decisions among ecosystem managers and stakeholders is a difficult task, which raises particular and complex governance issues. Two main approaches have been proposed for the provision of ecosystem services as public goods: governmental or market regulations [START_REF] Swinton | Ecosystem services and agriculture: Cultivating agricultural ecosystems for diverse benefits[END_REF]). However a growing number of studies advocate for the potential role of a "cooperative" approach [START_REF] Ostrom | Governing the Commons. The Evolution of Institutions for Collective Action[END_REF][START_REF] Stallman | Ecosystem services in agriculture: Determining suitability for provision by collective management[END_REF]. In studies of socio-ecological systems, many variables have been identified as affecting the patterns of interactions and outcomes observed in empirical studies [START_REF] Anderies | A Framework to Analyze the Robustness of Social-ecological Systems from an Institutional Perspective[END_REF][START_REF] Olsson | Adaptive Comanagement for Building Resilience in Social-Ecological Systems[END_REF]), and frameworks have been developed to enhance our understanding of the conditions under which cooperation is maintained or will evolve [START_REF] Ostrom | A diagnostic approach for going beyond panaceas[END_REF].

Agro-ecosystems represent a very interesting, though under studied, socio-ecological systems. Most studies suppose that the stakeholders in charge of management of ecosystem services are identified, that cooperation among them already exists, and that the resources used are considered as stocks to maintain [START_REF] Anderies | A Framework to Analyze the Robustness of Social-ecological Systems from an Institutional Perspective[END_REF]. Therefore management actions generally consist in regulating resource exploitation and solutions can be identified by problem-solving approaches. In contrast, in the case of agricultural ecosystems, collective action to maintain their sustainability is more complex: first, there is a spatial scalemismatch between ecosystem services typically managed at territory or watershed levels and agriculture practice managed at the farm scale [START_REF] Goldman | Institutional incentives for managing the landscape: Inducing cooperation for the production of ecosystem services[END_REF]). Second, while in theory, ecological knowledge in such systems is probably enough to propose effective solutions in order to maintain ecosystem services [START_REF] Pelosi | The spatial scale mismatch between ecological processes and agricultural management: Do difficulties come from underlying theoretical frameworks?[END_REF], the practical way to implement these solutions and involve stakeholders in the collective action is far from being achieved. Third, in agro-ecosystems, economic value of crop productions is a highly rewarded target, therefore in potential conflict with other services (particularly those in relation to environmental impacts). Last but not least, many people live within these agroecosystems, and therefore, there are many stakeholders, both quantitatively and qualitatively.

In this paper, we focus our analysis on such situations where solutions are not known initially, i.e. where potential services provided by agricultural ecosystems are multiple, interacting and dynamic. Management solutions to enhance their provision are generally unknown and require a complex design process, being all the more difficult since performance criteria are not known initially. In particular, solutions must be shared and viewed as effective and feasible by heterogeneous stakeholders with sometimes conflicting interests. In addition, in such systems, there is no legitimate designer or expert in charge of the design of such solutions: people in charge of designing and managing agricultural ecosystems (farmers, researchers, extension services, local authorities, naturalist organizations, etc.) are scattered and their actions are generally not coordinated. Hence the issue of designing relevant solutions to enhance the sustainability of an agricultural ecosystem raises in itself a governance problem.

In order to highlight the difficulties raised when looking for solutions to design sustainable agro-ecosystems, we propose to draw upon an empirical case, which to a certain extent can be considered as an ideal type. The agro-ecosystem considered is an intensive cereal cropped farming system in the west of France. Cereal farmland covers more than 50% of French farmed areas. In these systems, biodiversity loss, soil fertility degradation and water pollution have reached critical stages. The area under study is a habitat for various threatened bird species, which makes it a hotspot for conservation actions. We first describe a design process carried out by a research center in ecology (Centre d'Etudes Biologiques de Chizé, CEBC) that combines a "classical" scientific production of knowledge in ecology with the design of conservation actions. Then we analyze a second design process, which is in the continuity of the previous one, but which involves a wider group of stakeholders. It is carried out for the implementation of a short-scale fodder crop supply chain conducted by an agricultural cooperative (Coopérative Entente Agricole, CEA). Analyzing these two design phases under the framework of the most recent design theories, we highlight their main steps and originality. We reveal the importance and stakes of such design processes to develop locally adapted and accepted collective ecosystem-based management strategies. Finally we draw upon this analysis to discuss the literature on governance of socio-ecological systems.

Study area

The study area, "Zone Atelier Plaine & Val de Sèvre" (South of Département des Deux-Sèvres, in the arable plain of Niort-Brioux (46°15'N, 0°30'W) is situated in Région Poitou-Charentes, western France. It is a 430 km 2 farmland intensively cultivated, mainly with wheat and winter barley, sunflower, maize, oilseed rape, alfalfa and grasslands. Hedgerows and small forest fragments are still present but irregularly distributed). Fields are however characterised by relatively small size (of about 4 ha on average). In 2010, cereals accounted for 44 % of the land use, grassland (including hay meadows, alfalfa, pastures, setasides and fallows) c.15 %, and spring-sown crops c. 24%. The presence of 17 Annex 1 species of the EU Bird Directive (notably the Little Bustard) has led to the designation of half of the study area (207.6 km 2 ) as a Special Protection Area in 2004 (ZPS Niort Sud-Est, FR5412007). Mixed farming systems have been decreasing for the last 30 years, but cattle and goat farming remain, though a global shift toward industrial husbandry has occurred. The number of farmers has halved over the past 30 years, while average farm size has doubled (French agricultural general census, 2000).

Method and conceptual framework

Our analysis was twofold. An ex-post analysis of the first design phase was carried out between May and August 2010. It was based on a literature review in addition to scientific reports or research projects produced by CEBC researchers. We also conducted 23 interviews with researchers of CEBC, government bodies, environmental associations, farming organizations and local authorities. These first interviews were aimed at reconstructing the design of conservation actions for the Little Bustard Tetrax tetrax [START_REF] Berthet | Analyzing the Design Process of Farming Practices Ensuring Little Bustard Conservation: Lessons for Collective Landscape Management[END_REF].

The analysis of the second design phase is based on an on-going intervention research study [START_REF] Hatchuel | Collaborating for Management Research: from Action Research to Intervention Research in Management. Handbook of Collaborative Management Research[END_REF]. Indeed, we accompany the design process led by local stakeholders while analyzing it. We started this study by carrying out 18 interviews between January and February 2011 with members of the cooperative CEA (President, board members, manager, technicians and farmers), CEBC researchers and local authorities. We then contributed to the organization of a designed workshop in May 2011, the method of which will be presented below. We are currently following the implementation of the alfalfa local supply chain and the governance issues it raises.

Our theoretical framework is based on the Concept-Knowledge (C-K) theory [START_REF] Hatchuel | A new approach of innovative design: an introduction to C-K theory[END_REF][START_REF] Hatchuel | C-K design theory: an advanced formulation[END_REF]. C-K theory is one of the most recent and general design theories. It focuses on the specificities of innovative design reasoning. The central proposition of this theory is a formal distinction between "Concepts" (C), i.e. proposals still partly unknown and requiring a design process (whether the proposal is achievable or not remains uncertain in the current state of knowledge); and "Knowledge" (K), i.e. proposals having a logical status (proposals can be assessed by anyone as being true or false).

Knowledge is what designers already know or what they learn progressively during the design process. A central finding of the C-K theory is that a concept is the necessary departure point of a design process [START_REF] Hatchuel | A new approach of innovative design: an introduction to C-K theory[END_REF]. The formalism proposed by the theory is a kind of map of design reasoning. We used the theory to analyze ex-post the design reasoning that was developed by scientists in the case studied, but that today is implicit. The first flow diagram we display below highlights only the key steps of this reasoning and does not intend to reflect in an exhaustive way all the reasoning of stakeholders over the past 15 years of the project, or to be chronological. Its construction was iterative and took into account feedback from the people involved in the design process.

Results and discussion

From a flagship species natural history knowledge to the role of grasslands in ecosystem function

a) Improving the knowledge about Little Bustard biology to target efficient conservation actions

In order to stop the biodiversity loss in the agro-ecosystem of the Région Poitou-Charentes, naturalists targeted the conservation of a key stone species which was an apex predator, the Little Bustard, assuming that it was a mean to have the strongest impact on the ecosystem preservation as a whole [START_REF] Andelman | Umbrellas and flagships: Efficient conservation surrogates or expensive mistakes?[END_REF][START_REF] Mace | Biodiversity and ecosystem services: a multilayered relationship[END_REF]. The conservationists could have promoted a traditional approach such as excluding agricultural activities and establishing a nature reserve such as in National parks [START_REF] Grumbine | What Is Ecosystem Management?[END_REF]. However economic interests were such that it was impossible (see also [START_REF] Phalan | Reconciling Food Production and Biodiversity Conservation: Land Sharing and Land Sparing Compared[END_REF]for the classical land sharing/land sparing debate). As a consequence, the starting point of the design process was to develop new agricultural practices that would make it possible to conserve the Little Bustard. No obvious or satisfactory solutions existed a priori. Therefore, "developing new agricultural practices to conserve Little Bustards in the cereal plain" is what in the framework of the C-K theory we qualify as an initial concept: an unknown proposal which is the starting point of a design process.

To begin with, CEBC researchers sought to understand the main causes of Little Bustard population decrease. They identified a parameter as being the most significant in terms of conservation strategy: female productivity [START_REF] Bretagnolle | Modelling population reinforcement at a large spatial scale as a conservation strategy for the declining little bustard (Tetrax tetrax) in agricultural habitats[END_REF]. Given the rapid decline of the Little Bustard in agricultural areas (> -13% per year between 1997 and 2002: [START_REF] Bretagnolle | The role of grassland areas within arable cropping systems for the conservation of biodiversity at the regional level. Grassland Productivity and Ecosystem Services[END_REF], two complementary strategies were set: reproduction in captivity, and increase in habitat quality in the agro-ecosystem. The latter was the most important, and consisted in "improving reproduction capacity in the agro-ecosystem", that can be seen as a concept. This concept required the production of new knowledge regarding Little Bustard reproduction, as very little information was available about where and when this species bred, laid eggs, how chicks were fed, etc. Based on studies carried out between 1997 and 2001, researchers found that half of the clutches could not hatch, 40% of which was due to the destruction of nests by farming practices. In addition, nearly 40% of clutches arriving at hatching failed because of food shortage during the early chick rearing period, when chicks rely solely on insects [START_REF] Inchausti | Predicting short-term extinction risk for the declining Little Bustard () in intensive agricultural habitats[END_REF]. This problem of chick food shortage was targeted as a priority by conservationists as it was a major limiting factor for Little Bustard fecundity. At this stage we can see in the flow diagram (Fig. 1) that various solutions had been proposed based on this knowledge progress, but this one has been particularly explored as it appeared to have a greater potential in terms of efficiency and feasibility.

b) The use of metapopulation theory: toward a new representation of the agroecosystem ecological functioning

Aiming at maintaining a sufficient quantity of grasshoppers favored a new concept that eventually emerged from the previous one: "increasing the carrying capacity of the entire agro-ecosystem". Monitoring insects on the study site showed that the repeated application of herbicides that deprive insects of food and to a lesser extent of insecticides was a major problem and that plowing was found to destroy egg-laying habitats for grasshoppers [START_REF] Badenhausser | Acridid (Orthoptera: Acrididae) abundance in Western European Grasslands: sampling methodology and temporal fluctuations[END_REF]. In order to improve grasshopper populations, disturbance regime had to be lower, pesticides use had to be banned and breeding habitat for grasshoppers should be distributed optimally in space. Metapopulation theory was thus used, which actually changed the representation of the agro-ecosystem. This theory predicts that a local extinction of a population can be compensated for by colonization, provided that an adequate regional supply of the population exists and that individuals can disperse to new local habitat when population has gone extinct [START_REF] Levins | Some demographic and genetic consequences of environmental heterogeneity for biological control[END_REF][START_REF] Hanski | Habitat connectivity, habitat continuity, and metapopulations in dynamic landscapes[END_REF]). The agroecosystem was thus modeled as a matrix, composed of semi-natural areas (high quality habitat) and cropped areas (low quality habitat). This matrix is considered of high-quality if it allows for migration rates balancing the rate of local extinctions [START_REF] Perfecto | Nature's Matrix: Linking Agriculture, Conservation and Food Sovereignty[END_REF])(see Fig. 1: Path 7 & Path 8). Various options to regenerate grasshopper populations were thus explored: patches of land area purchased from farmers and excluded from production, fallows or grasslands. As grasslands are productive areas and have value for farmers, this solution was preferred since it was thought to be more acceptable for farmers. Moreover, grasslands were given priority compared to linear infrastructures (hedges, grass strips) because they contribute to diversify the landscape mosaic required for Little Bustards reproduction, being also nesting areas.

c) Grasslands as a targeted object in terms of conservation management Grasslands were then considered as a lever to reintroduce insect population regulations in a highly disturbed agro-ecosystem and regenerate the trophic web (an important target for Little Bustards that can be considered as an apex predator in such systems). However a new bundle of knowledge was needed: how to distribute such grasslands among annual crops in order to optimize grasshopper population size? Based on studies taking into account their dispersal distance and on empirical observation, grassland percentage in the landscape was estimated to lie between 10% and 20%, and the scientists set a target of 15% of agricultural land as grasslands, preferably randomly scattered across the landscape.

The design flow diagram, the method used was not pro process using various represen made it possible to formulate first part of the design process use of metapopulation theor ecological engineering initiati may restore regulation process finished yet and the question o was still an open question.

Restoring grasslands

CEBC ecologists proposed environmental scheme (AES). NATURA 2000 SPA; local, n CEBC was given the role of became landscape matrix des 1995, the researchers have com of the location of Little Busta with regard to Little Bustard lo However this situation of c AES are contracts with farme friendly agricultural practices m, applied to knowledge-based conservation a problem-solving (i.e. trial and error), but a entations of the agro-ecosystem. This innovativ te new concepts, which then guided knowledg ess was a quite classical conservationist reaso ory introduced a spatial dimension and op iatives, considering grasslands as ecological i esses in the agro-ecosystem. However the desig n of how to develop grassland areas and manag design reasoning led by CEBC researchers using C-K ds in the cereal plain: from theory to implem sed the conversion of cereal crops into grasslan S). Such AES was made feasible because the , national and European authorities validated th of AES operator in this SPA. Hence researc esigners, by the way of reintroducing grassla ompiled a GIS database at the scale of their stu stards as well as crop use, allowing to develop location and also to monitor their effectiveness f centralized management strategy presents var ers intended to promote the implementation in return for an annual subsidy to offset the n actions, shows that a gradually targeted tive design reasoning edge production. The soning. However the opened the door to l infrastructures that esign process was not age their distribution K Theory formalism lementation lands as a local agrithe study site was a the scheme, and the archers progressively slands. Indeed, since study area (450 km²) lop targeted schemes ess. various shortcomings.

of environmentally he costs involved and possible income reductions. This system is very costly (compensations can reach 500€/ha/year), limited in time (5 years), and cannot be extended outside the SPA delineation (with a buffer of 2 km allowed in this precise case). Above all, as farmers have only slightly participated to the design of such practices, and perceive it as a constraint, their main motivation to adopt this solution is the financial compensation.

Therefore, other attempts aimed at restoring grassland areas were searched for, as an alternative to the government-funded economic compensation. These new solutions required the cooperation of ecologists with agronomists and social-scientists. A first solution considered was to make cattle breeders increase their fodder production. However many breeders are unable or unwilling to put more land into grassland as they prefer diversifying their sources of income, and actually maximize wheat income. A more innovative alternative is to make cereal farmers produce forage crops. However to a certain extent cereal farmers consider grasslands as non-productive areas, as there is no market for hay or silage. Exchanges exist only on an informal ad hoc basis, as cattle breeders generally grow their own fodder or buy compound feeds. Hence a critical condition to conduce cereal farmers to grow fodder crops was to create a market for fodder.

Pursuing the objective to find both efficient and sustainable solutions, the researchers looked for the fodder crop that would be the most easily adopted by regional farmers. Alfalfa was chosen first because it was historically grown in the region: it is pretty well adapted to the pedo-climatic conditions and suitable for goat cattle feeding, one of the main husbandry sectors in the region. Second, as a legume, alfalfa is also known for its agronomic interests such as atmospheric nitrogen fixation. Thus in addition to launch a large research program to identify the ecosystem services provided by alfalfa in a context of an intensive cereal cropped agro-ecosystem, the researchers explored the possibilities to create a market for fodder crops. However such solutions are difficult to set up by an actor external to the farming sector.

From scientific-based to multi-stakeholder design process

a) A new partnership between ecology researchers and an agricultural organization to develop alfalfa production in the cereal plain

While ecology researchers enlarged progressively their knowledge base about agriculture, farmers and related organizations also initiated reflections to renew their practices in order to make them more environmentally friendly. This is the case of the agricultural cooperative CEA (Cooperative Entente Agricole) who decided in 2005 to initiate a shift in its strategy toward sustainable development. As the cooperative had very few skills with regard to environmental preservation, it started off with looking for partners. With the CEBC, the cooperative was proposed to set up a short-scale alfalfa supply chain. They agreed on this proposition not only as it seemed to be a systemic solution to local environmental problems, but also as it was in the scope of their activity to contribute to set up a supply chain. For the CEBC researchers, this was an opportunity to develop alfalfa production without public funding, and with the support of actors from the agricultural sector.

However although the project of alfalfa supply chain was agreed by both stakeholders, many questions remained open. At a first glance, this project was an opportunity for actors with rather conflicting interests to collaborate in the framework of common objectives. In reality the initial motivations and constraints of both parties were significantly different. Indeed the cooperative aimed to maximize alfalfa production while ecologists aimed to develop areas with as less disturbance as possible. "Grasslands" actually referred to different objects for each stakeholder.

Indeed the proposal of developing alfalfa crops in the cereal plain was the result of a long design process led by researchers. This design process had given alfalfa multiple properties that farmers were not aware of, such as insect population regulations or weed regulation [START_REF] Meiss | Perennial lucerne affects weed community trajectories in grain crop rotations[END_REF]. Other solutions had been left aside by ecology researchers as they were considered as less efficient. As a consequence, a crucial preliminary step to the project was to make explicit all the expected ecological properties, but also to express the properties expected by other stakeholders than conservationists. Moreover, there was a need to explore the variety of alfalfa production modalities and to decide collectively on their selection.

b) Initiating a collective design process to set up common objectives

To tackle these issues, the project leaders decided to launch a collective design workshop in order to specify the objectives of the project and agree on priorities. It also aimed at looking for innovative solutions for the implementation of the project. Lastly, it was a first step in developing ad-hoc governance rules, in particular identifying the role that the cooperative should play. Before this workshop, the cooperative had planned to implement the project with a classical bilateral contract for alfalfa production with conventional practices. An expected outcome of the workshop was to re-open the alternatives for the implementation of the project. The design workshop was facilitated by a researcher in management sciences using the KCP method [START_REF] Hatchuel | Design theory and collective creativity: a theoretical framework to evaluate KCP process[END_REF], which was developed on the basis of C-K theory to foster innovative design. The method structures collective exploration into three steps: a Knowledge-phase (K-phase) that make it possible to share internal and external knowledge; a Concept-phase (C-phase) grounded by surprising and strongly contrasting propositions ("concept projectors") that orient creativity; and a Proposal-phase (P-phase) aimed to synthesize the results and elaborate a design strategy.

The organization committee agreed on a list of 30 participants: CEA board members, technicians and member farmers; researchers in ecology and agronomy; local authorities' representatives and extension services. The majority of participants were internal to the cooperative. However this workshop was also an occasion to gather stakeholders who did not know each other before. The two first phases were organized on a single day. The K-phase aimed to review the cutting-edge knowledge about alfalfa production and related environmental issues. Four contrasting concepts projectors were identified by the facilitators to help the participants explore a large array of alfalfa properties: for instance, "premium quality alfalfa", or "alfalfa that farmers like to produce". This phase drove participants to identify the need for external knowledge useful for the project. Then for the Proposal-phase, the facilitator carried out a thorough analysis of the results. The aim was to analyze the knowledge capitalized and to identify some frequently recurring and innovative propositions. The results were presented to the organization committee then to all workshop participants.

c) Toward a creative resolution of conflicts

At the beginning of the process, naturalists and farmers had very different views and expectations about alfalfa: for naturalists, it was an ecological habitat whereas for farmers it was a productive area. The design workshop made it possible to extend the attributes of alfalfa, i.e. its design parameters and expected functions to build a common ground for all stakeholders. Antagonisms between key parameters were pointed out, such as harvest date or herbicide use, and flexibility possibilities were explored to overcome them. The alfalfa production modalities proposed were not focused on trade-offs between fodder production and nature conservation, but rather on the identification of new values that could be collectively created by farmers, such as enhancing pollination by staggering harvest dates or improving water quality by concentrating grasslands around drinking water catchments. The identification of such ecosystem services and of their potential related values highlighted strong interdependences between agro-ecosystem stakeholders, especially when considering the regulation functions of grasslands that most stakeholders were not aware of, e.g. biogeochemical cycles.

The spatial distribution of alfalfa plots can be controlled to enhance Little Bustard conservation, and they would then be dispersed. But such control would also be relevant to protect drinking water catchments or to optimize biological control, in which case other spatial distribution would be required. Thus, by increasing the number of targeted objectives, some have become antagonists. The design phase of the alfalfa supply chain project is thus an opportunity to explore feasible options and their related ecosystem service bundles. This makes it possible then to hierarchize and select collectively the ecosystem services to produce in priority. This exploration revealed the importance of coordinating grassland management at the landscape scale. According to the configurations of grasslands chosen by the stakeholders, the ecological and productive properties of the ecosystem will not be the same. This actually revealed the status of "common good" of the agro-ecosystem. Common goods refer to a field of the economy for which it is difficult to develop physical or institutional means for excluding beneficiaries although these goods are in limited quantity. The case of grasslands is interesting because it leads to thinking in terms of "design of common goods." So far, several strategies proposed to safeguard common goods have been identified, including those developed by local communities, and analyzed by Ostrom [START_REF] Ostrom | Governing the Commons. The Evolution of Institutions for Collective Action[END_REF][START_REF] Ostrom | Collective action and the evolution of social norms[END_REF]. She identified several design principles that help communities sustain and build their cooperation over long periods of time [START_REF] Ostrom | Collective action and the evolution of social norms[END_REF]. However, she considers common goods as an existing collectively-owned stock to manage. In our case, grasslands are private goods, but as soon as they are managed as common goods they have to be designed, as do also new farming practices and new modes of collective action. Here, on the basis of our observations, it now appears likely that the action of designating private goods as commons can help achieve a collective interest, and that the collective characteristic of these goods is not a problem, but rather a solution. It opens up the way to collective design processes of new modes of governance to manage sustainably agricultural landscapes. Thanks to the knowledge production in ecology, grasslands have been identified as a potential infrastructure to sustain the provision of agro-ecosystem services. However the stakeholders have then to agree on how to manage this infrastructure to provide the ecosystem services they select. The legitimacy of actors who will take over the assessment of the agro-ecosystem trajectory remains however an open question. Are an agricultural cooperative, or a research center, legitimate to determine the agro-ecosystem trajectory? This research-action program aims to generate new knowledge to identify relevant parameters values, but also tools and methods for the design and multi-criteria assessment of actions to implement: shared evaluation grids, simulation tools, ad-hoc method of collective design.

Figure 1 :

 1 Figure 1: A summary of the de