Igor Mozolevski
email: igor.mozolevski@ufsc.br

Serge Prudhomme
email: serge.prudhomme@polymtl.ca

Goal-oriented error estimation based on equilibrated-flux reconstruction for finite element approximations of elliptic problems

Keywords: Finite element method, goal-oriented estimates, quantity of interest, dual problem, asymptotically-exact error estimates

We propose an approach for goal-oriented error estimation in finite element approximations of second-order elliptic problems that combines the dualweighted residual method and equilibrated-flux reconstruction methods for the primal and dual problems. The objective is to be able to consider discretization schemes for the dual solution that may be different from those used for the primal solution. It is only assumed here that the discretization methods come with a priori error estimates and an equilibrated-flux reconstruction algorithm. A high-order discontinuous Galerkin (dG) method is actually the preferred choice for the approximation of the dual solution thanks to its flexibility and straightforward construction of equilibrated fluxes. One contribution of the paper is to show how the order of the dG method for asymptotic exactness of the proposed estimator can be chosen in the cases where a conforming finite element method, a dG method, or a mixed Raviart-Thomas method are used for the solution of the primal problem. Numerical experiments are also presented to illustrate the performance and convergence of the error estimates in quantities of interest with respect to the mesh size.

Introduction

A variety of finite element discretization schemes, such as the mixed finite element methods, the non-conforming finite element methods, or discontinuous Galerkin (dG) finite element methods, have been developed during the past decades in order to provide better approximation properties than those offered by the classical finite element method depending on the type of partial differential equations at hand. These methods have become increasingly popular and are now widely used for solving problems of various interest in engineering and sciences. At the same time, the necessity to obtain accurate finite element approximations to given boundaryvalue problems has stimulated the development of a posteriori error estimators that provide fully computable, reliable, and efficient error bounds in terms of the problem data and the finite element approximation. In the case of standard conforming finite element schemes, a posteriori error estimates with respect to global energy norms are indeed well-established, see e.g. [START_REF] Verfürth | A Review of A Posteriori Estimation and Adaptive Mesh -Refinement Techniques[END_REF][START_REF] Ainsworth | A posteriori error estimation in finite element analysis[END_REF][START_REF] Babuška | The Finite Element Method and its Reliability[END_REF][START_REF] Bangerth | Adaptive Finite Element Methods for Differential Equations[END_REF][START_REF] Neittaanmaki | Reliable methods for computer simulation[END_REF][START_REF] Han | A posteriori error analysis via duality theory[END_REF][START_REF] Repin | A Posteriori Estimates for Partial Differential Equations[END_REF][START_REF] Verfürth | A posteriori error estimation techniques for finite element methods[END_REF]. A posteriori error estimation for nonconforming finite element methods with application to second-order elliptic problems has recently seen significant progress and is still the subject of sustained research efforts, see e.g. [START_REF] Becker | Energy norm a posteriori error estimation for discontinuous Galerkin methods[END_REF][START_REF] Karakashian | A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems[END_REF][START_REF] Schneider | An analysis of discontinuous Galerkin methods for elliptic problems[END_REF][START_REF] Yang | A unified a posteriori error analysis for discontinuous Galerkin approximations of reactive transport equations[END_REF][START_REF] Houston | Energy norm a posteriori error estimation of hp-adaptive discontinuous Galerkin methods for elliptic problems[END_REF][START_REF] Ainsworth | A posteriori error estimation for discontinuous Galerkin finite element approximation[END_REF][START_REF] Carstensen | A unifying theory of a posteriori error control for discontinuous Galerkin FEM[END_REF][START_REF] Lovadina | A-posteriori error estimates for discontinuous Galerkin approximations of second order elliptic problems[END_REF][START_REF] Lazarov | Functional a posteriori error estimates for discontinuous Galerkin approximations of elliptic problems[END_REF][START_REF] Ern | Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusionreaction problems[END_REF][START_REF] Alaoui | Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems[END_REF] for dG methods and [START_REF] Carstensen | A posteriori error estimate for the mixed finite element method[END_REF][START_REF] Kim | A posteriori error analysis for locally conservative mixed methods[END_REF][START_REF] Ainsworth | A posteriori error estimation for lowest order Raviart-Thomas mixed finite elements[END_REF][START_REF] Carstensen | Lectures on adaptive mixed finite element methods[END_REF][START_REF] Kim | Guaranteed a posteriori error estimator for mixed finite element methods of elliptic problems[END_REF] for mixed finite element methods. We also refer the reader to [START_REF] Kim | A posteriori error estimators for locally conservative methods of nonlinear elliptic problems[END_REF][START_REF] Carstensen | A unifying theory of a posteriori error control for discontinuous Galerkin FEM[END_REF][START_REF] Ainsworth | A framework for obtaining guaranteed error bounds for finite element approximations[END_REF][START_REF] Ern | A unified framework for a posteriori error estimation in elliptic and parabolic problems with application to finite volumes[END_REF][START_REF] Dolejší | A Framework for Robust A Posteriori Error Control in Unsteady Nonlinear Advection-Diffusion Problems[END_REF] and references therein for the presentation of unifying frameworks on the topic.

In practical applications, end users are however interested in error estimates in some specific features of the true solution. These so-called quantities of interest, either local or global, are represented as functionals defined on the vector space of trial solutions of the boundary-value problem. Error estimation with respect to such functionals is usually referred to as goal-oriented error estimation. The key ingredient for goal-oriented error estimation is the formulation of an auxiliary problem, the dual problem to the primal problem, whose solution provides necessary information for reliable estimates of the error in the goal functional. Several strategies for goal-oriented error estimation have been proposed in the case of elliptic problems: goal-oriented error estimates based on energy norm of the errors in the primal and dual solutions were introduced in [START_REF] Prudhomme | On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors[END_REF][START_REF] Oden | Goal-oriented error estimation and adaptivity for the finite element method[END_REF][START_REF] Oden | Estimation of modeling error in computational mechanics[END_REF][START_REF] Prudhomme | Computable error estimators and adaptive techniques for fluid flow problems[END_REF] and further developed by various authors, see for example [START_REF] Ainsworth | A posteriori error estimation in finite element analysis[END_REF][START_REF] Ainsworth | Guaranteed computable bounds on quan-tities of interest in finite element computations[END_REF], and references therein, error estimates using the dual-weighted residual method were proposed in [START_REF] Eriksson | Introduction to adaptive methods for differential equations[END_REF][START_REF] Becker | An optimal control approach to a posteriori error estimation in finite element methods[END_REF][START_REF] Bangerth | Adaptive Finite Element Methods for Differential Equations[END_REF]; functional a posteriori error estimates were developed in [START_REF] Neittaanmaki | Reliable methods for computer simulation[END_REF][START_REF] Repin | A Posteriori Estimates for Partial Differential Equations[END_REF]; estimates based on the gradient-recovery method were considered in [START_REF] Korotov | A posteriori error estimation of goal-oriented quantities by the superconvergence patch recovery[END_REF][START_REF] Korotov | A posteriori error estimation of goal-oriented quantities for elliptic type BVPs[END_REF][START_REF] Ovall | Asymptotically exact functional error estimators based on superconvergent gradient recovery[END_REF][START_REF] Neittaanmäki | New a posteriori error indicator in terms of linear functionals for linear elliptic problems[END_REF][START_REF] Kuzmin | Goal-oriented a posteriori error estimates for transport problems[END_REF]; finally, goal-oriented estimates for discontinuous Galerkin methods in the case of second-order elliptic problems were derived in [START_REF] Harriman | hp-version discontinuous Galerkin methods with interior penalty for partial differential equations with nonnegative characteristic form[END_REF].

The general approach to obtain goal-oriented error estimates consists, on one hand, in deriving an error representation involving the residual functional and the exact dual (adjoint) solution, and, on the other hand, in constructing a sufficiently accurate approximation of the adjoint solution in order to obtain a fully computable and reliable error estimate along with local refinement indicators. In the case of the classical conforming finite element method, such approximation to the adjoint solution is usually calculated on a refinement of the mesh used for the primal solution, with the same polynomial degree, or, preferably, on the same mesh, but with a higher polynomial degree.

In this paper, we propose an alternative approach to goal-oriented estimation by considering an error representation that does not uses the orthogonality property and is amenable to different types of discretization of the primal and dual problems. We only suppose here that the method used to discretize the primal problem produces piecewise polynomial solutions from which equilibrated fluxes can be reconstructed (this is not a restrictive property) and satisfies standard a priori error estimates in the L 2 and energy norms. Using an error representation similar to the one used in the dual-weighted residual method, we use the flux-equilibration technique to decompose the error into a computable error estimator and a higher-order remainder. It is well known, see e.g. [START_REF] Ainsworth | Guaranteed computable bounds on quan-tities of interest in finite element computations[END_REF], that in order to obtain an efficient error estimator (effectivity indices remain close to unity), the dual problem must be approximated using a higher-order approximation than the one used in the finite element approximation of the primary problem. We propose in this paper, in order to approximate the dual solution, that a high-order discontinuous Galerkin method be used and applied on the same mesh as that used for the discretization of the primal problem. The choice of the dG method seems natural owing to its flexibility in using non-uniform high-order polynomials. Furthermore, the dG method benefits from the fact that it is locally conservative; it implies that the construction of equilibrated fluxes, needed for the evaluation of the proposed error estimator, is rather straightforward, see e.g. [START_REF] Kim | A posteriori error estimators for locally conservative methods of nonlinear elliptic problems[END_REF][START_REF] Ern | An accurate H(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems[END_REF][START_REF] Carstensen | Effective postprocessing for equilibration a posteriori error estimators[END_REF][START_REF] Braess | An equilibrated a posteriori error estimator fort he interior penalty discontinuous Galerkin method[END_REF]. Finally, we show that, depending on the approximation properties of the primal method and problem data, the order of the dG method, used to solve the dual problem, can be chosen in such a way that the respective error estimator is asymptotically exact.

The paper is organized as follows. Section 2 introduces the model (primal) problem, the corresponding dual problem, and preliminary notation for goal-oriented error estimation. Section 3 presents a suitable error representation for the goal functional based on the use of equilibrated fluxes in terms of the finite element solutions to the primal and dual problems. We also show how the error representation can be decomposed into a fully computable error estimator and a higher-order term, which can be evaluated using 1) a priori estimates with respect to the primal and dual discretization schemes and 2) methods for reconstruction of equilibrated fluxes. We briefly introduce in Section 4 the symmetric version of the dG method. We then show how the order of the dG method, used for the discretization of the dual problem, can be chosen to guarantee asymptotical exactness of the error estimator when discretizing the primal problem by high-order dG methods, conforming finite element methods, and mixed finite element methods, as described in Sections 5, 6, and 7, respectively. Finally, we present some numerical examples to demonstrate the performance of the estimators in Section 8 before concluding in Section 9.

Model problem

In this work, we are primarily interested in general linear boundary-value problems defined in terms of elliptic second-order partial differential equations, but the results could be extended to more general situations. For the sake of clarity in the presentation, we shall restrict ourselves to a simple model problem. Let Ω ∈ R 2 be a polygonal domain with boundary ∂Ω. We consider the homogeneous Dirichlet boundary-value problem whose solution u satisfies

-∇ • (D∇u) = f in Ω, u = 0 on ∂Ω. (1)
where coefficient D = D(x), x ∈ Ω, is a piecewise constant strictly positive scalar function and f ∈ L 2 (Ω). The weak formulation of the problem reads:

Find u ∈ H 1 0 (Ω) such that B(u, v) = F (v), ∀v ∈ H 1 0 (Ω) (2)
where B and F are the bilinear and linear forms on H 1 0 (Ω), respectively, defined as:

B(u, v) = Ω D∇u • ∇v, (3)
F (v) = Ω f v. (4)
We shall suppose here that the elliptic regularity property holds true for the model problem; i.e. there exist a unique weak solution u ∈ H 2 (Ω) to (2) and a constant C > 0 that only depends on Ω, such that

u H 2 (Ω) ≤ C f L 2 (Ω) . (5)
Note that the conditions for elliptic regularity are ensured if the polygonal domain Ω is assumed convex; in more general cases, the regularity of the solution usually depends on the internal angles between the boundary edges of Ω, see [START_REF] Grisvard | Singularities in Boundary Value Problems[END_REF]. In the remainder of the paper, we will denote by C a generic constant that may depend on the problem data and other model parameters. This dependence will be made explicit in the text when necessary.

We assume that we shall be interested in the linear goal functional Q defined on L 2 (Ω) such that:

Q(v) = Ω qv, (6)
where q ∈ L 2 (Ω) denotes the Riesz representer of Q. We introduce the corresponding dual problem, in strong form as:

-∇ • (D∇p) = q in Ω, p = 0 on ∂Ω, (7)
and in weak form as:

Find p ∈ H 1 0 (Ω) such that B(v, p) = Q(v), ∀v ∈ H 1 0 (Ω). ( 8
)
We again emphasize that, owing to the elliptic regularity and the fact that q ∈ L 2 (Ω), we have that p ∈ H 2 (Ω). It is also straightforward to show from (2) and (8) that

F (p) = B(u, p) = Q(u). (9)
Let T h , h > 0, be a family of shape-regular triangular meshes on Ω, see e.g. [START_REF] Brenner | The Mathematical Theory of Finite Element Methods[END_REF], where h = max T∈T h h(T ) denotes the mesh size and h(T ) is the diameter of mesh element T . We assume here that the edges of two neighboring elements will perfectly match with each other, in the sense that the meshes will be free of so-called hanging nodes. For a triangular mesh T h , we denote by E h the set of all edges in the mesh; E h can be further decomposed into the set of interior edges E i h and the set of boundary edges

E ∂ h , that is E h = E i h ∪ E ∂ h .
v along E by [[v]] := v T -| E -v T + | E , E ∈ E i h , [[v]] := v| E , E ∈ E ∂ h , (10)
where v ± | E denotes the trace on the respective side of E. For E ∈ E i h , we denote by n E the unit normal vector to E pointing from T -toward T + , whereas for E ∈ E ∂ h , we set n E equal to the unit external normal n to ∂Ω. The orientation of n E for interior faces is chosen in accordance with the definition of the jump; in such a case, the arbitrariness in the choice of T - and T + in the definition of the interior edge is irrelevant. We also introduce the standard arithmetic average of v along E as

{{v}} = 1 2 v T -| E + v T + | E , E ∈ E i h , {{v}} = [[v]] = v| E , E ∈ E ∂ h . (11)
Finally, let P k (T ) denote the vector space of polynomials on T of degree k or less. We then introduce the finite element space V k h as:

V k h := {v h ∈ L 2 (Ω) : v h | T ∈ P k (T ), ∀T ∈ T h }. ( 12
)
For a given mesh T h , let us denote by u h ∈ V k h an arbitrary approximation of the exact solution u to the primal problem [START_REF] Ainsworth | A posteriori error estimation for lowest order Raviart-Thomas mixed finite elements[END_REF]. The objective in this paper is to study the approximation error in the goal functional

E = Q(u) -Q(u h ). ( 13
)
At this stage, it is not necessary to specify the numerical scheme with which the approximate solution is obtained. Since the goal functional considered here is assumed linear and continuous on L 2 (Ω), it is sufficient to suppose that u h converges to the exact solution u in L 2 (Ω) in order for Q(u h ) to be a reasonable approximation of Q(u).

We will make use of the following definition in the remainder of the paper:

Definition 1. The error E = Q(u) -Q(u h )
in the goal functional is said to be properly of order l > 0 with respect to the discrete solution u h if there exist constants C > 0 and C > 0, that depend on u and Q only, such that

Ch l ≤ |Q(u) -Q(u h )| ≤ Ch l (14)
for sufficiently small h > 0.

We propose below an asymptotically exact estimator of the error in the goal functional [START_REF] Braess | Equilibrated residual error estimator for edge elements[END_REF] based on the condition that the error be properly of order l > 0 with respect to a discrete solution family {u h } h>0 .

Goal-oriented error representation

In this section, we briefly review the notion of equilibrated fluxes and present a goal-oriented error representation in terms of reconstructed equilibration fluxes. We first recall the space H(div, Ω) of vector-valued functions

H(div, Ω) = {t ∈ [L 2 (Ω)] 2 : ∇ • t ∈ L 2 (Ω)}, (15)
and the Raviart-Thomas finite element space of order m ∈ N 0 :

RT m (T h ) = {t h ∈ H(div, Ω) : t h | T ∈ [P m (T )] 2 + x P m (T ), ∀T ∈ T h }. ( 16
) Definition 2. Let u h ∈ V k h , k ∈ N 0 , and let l = max{0, k -1}. Let π l h : L 2 (Ω) → V l h denote the L 2 -orthogonal projection operator. A vector t h (u h ) ∈ RT l (T h ), reconstructed from the approximation u h , is said to be an equi- librated flux with respect to Problem (1) if ∇ • t h (u h ) = π l h (f ). ( 17
)
Note that we will use the notation t l h (u h ) = t h (u h ) when the order of the Raviart-Thomas space in which the flux is reconstructed needs to be mentioned explicitly.

The concept of equilibrated fluxes has been widely used to construct accurate a posteriori error estimates for conforming finite element approximations since such estimates are exempt of unknown constants, see for example [START_REF] Ainsworth | A posteriori error estimation in finite element analysis[END_REF][START_REF] Braess | Finite Elements. Theory, Fast Solvers, and Application in Solid Mechanics[END_REF]. In fact, various flux reconstruction techniques have been developed over the years in the case of elliptic problems and some of the techniques have been extended to non-conforming conservative methods [START_REF] Kim | A posteriori error estimators for locally conservative methods of nonlinear elliptic problems[END_REF][START_REF] Ern | An accurate H(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems[END_REF][START_REF] Braess | An equilibrated a posteriori error estimator fort he interior penalty discontinuous Galerkin method[END_REF], to first-order conforming finite element method [START_REF] Braess | Equilibrated residual error estimator for edge elements[END_REF], or to the finite volume and finite difference methods [START_REF] Vohralík | Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients[END_REF].

We propose here to investigate how reconstructed equilibrated fluxes could be used in goal-oriented error estimation. We start by deriving the following error representation with respect to the goal functional, using ( 9), the dual solution p to problem [START_REF] Babuška | The Finite Element Method and its Reliability[END_REF], and integration by parts,

E = Q(u) -Q(u h ) = F (p) -Q(u h ) = Ω f p + Ω ∇ • D∇p u h = T ∈T h T f p -D∇u h • ∇p + ∂T n • D∇p u h . (18)
Introducing the jump [START_REF] Becker | An optimal control approach to a posteriori error estimation in finite element methods[END_REF] in u h across edge E, the sum of the edge integrals can be recast as:

E = T ∈T h T f p -D∇u h • ∇p + E∈E h E n E • D∇p [[u h ]]. ( 19
)
Note that for an arbitrary t h ∈ RT l (T h ), above relation can be rewritten, by simple addition and subtraction of the term t h • ∇p, as:

E = T ∈T h T f p + t h • ∇p + T ∈T h T -t h -D∇u h • ∇p + E∈E h E n E • D∇p [[u h ]] = T ∈T h T f -∇ • t h p + E∈∂T E t h • n p + T ∈T h T -t h -D∇u h • ∇p + E∈E h E n E • D∇p [[u h ]].
Since, owing to the homogeneous Dirichlet boundary condition on the dual solution and the continuity of the normal component of t h ,

T ∈T h E∈∂T E (t h • n)p = E∈E h [[t h • n]] p = 0
we then obtain:

E = T ∈T h T f -∇ • t h p - T ∈T h T t h + D∇u h ) • ∇p + E∈E h E n E • D∇p [[u h ]].
We are now ready to establish the following error representation in terms of the goal functional.

Theorem 1. Let u h ∈ V k h , k ∈ N 0 , be an approximation of the solution u ∈ H 1 0 (Ω) to the primal problem (1) and let p h ∈ V m h , m > k, be an approx- imation of the solution p ∈ H 1 0 (Ω) ∩ H 2 (Ω)
to the dual problem [START_REF] Babuška | The Finite Element Method and its Reliability[END_REF]. Suppose that u h and p h admit reconstructed equilibrated fluxes

t h (u h ) ∈ RT l (T h ), l = max{0, k -1} and t h (p h ) ∈ RT m-1
, respectively. Then the error in the goal functional Q can be represented as

E = Q(u) -Q(u h ) = η(u h , p h ; t h (u h ), t h (p h )) + R(u, p; u h , p h ), (20)
where the error estimator η is defined as:

η = η(u h , p h ; t h (u h ), t h (p h )) = T ∈T h T (f -π l h (f )) p h + T ∈T h T t h (u h ) + D∇u h • D -1 t h (p h ) - E∈E h E (n E • t h (p h ))[[u h ]], (21)
and the remainder term is:

R(u, p, f ; u h , p h ) = T ∈T h T (f -π l h (f ))(p -p h ) - T ∈T h T t h (u h ) + D∇u h • D -1 D∇p + t h (p h ) + E∈E h E n E • (D∇p + t h (p h )) [[u h ]]. ( 22
)
Remark 1. The error estimator η can be easily decomposed into a sum of local element error contributions. Indeed, let us note that

E∈E h E t h (p h ) • n[[u h ]] = T ∈T h E∈∂T E χ E h (E)(n • t h (p h ))[[u h ]], (23)
where χ E h denotes the edge indicator function on

E h , that is χ E h (E) = 1/2, if E ∈ E i h , and χ E h = 1, if E ∈ E ∂ h .
Therefore, in that case, the decomposition would take the form:

η = T ∈T h η T (T ) = T ∈T h η O (T ) + η ∇ (T ) + η H (T ) (24)
with

η O (T ) = T (f -π l h (f )) p h η ∇ (T ) = T t h (u h ) + D∇u h • D -1 t h (p h ) η H (T ) = E∈∂T E χ E h (E)[[u h ]](n E • t h (p h ))
Here η O (T ) represents the data oscillation in element T with respect to the primal problem, weighted by the dual approximate solution; the flux estimator η ∇ (T ) measures the deviation of the discrete gradient -D∇u h from the reconstructed flux t h (u h ) ∈ H(div, T h ); and η H (T ) measures the deviation of u h from H 1 0 (Ω), both weighted by the reconstructed equilibrated flux of the dual approximate solution. Note that the error estimator η is fully computable once the flux reconstructions for t h (u h ) and t h (p h ) are established.

Remark 2. Depending on the properties of the data in the primal problem, the error estimator (21) may take a simpler form; e.g. the oscillation term disappears for f ∈ V l h . Remark 3. Depending on the discretization method used, the finite element approximation u h may weakly (or strongly) satisfy additional continuity properties along the interior edges and specific boundary conditions on the boundary edges of the mesh. Values of the different components of the error estimator will thus strongly depend on the choice of the numerical method used in the approximation of the primal problem. For example, if conforming finite element methods are used, η H = 0.

In the remainder of the paper, we show how the error representation [START_REF] Di Pietro | Analysis of a discontinuous Galerkin method for heterogeneous diffusion problems with low-regularity solutions[END_REF] can provide an asymptotically exact goal-oriented estimator of the error in approximations of the primal problem. One crucial ingredient of the estimator is the choice of the finite element method for the approximation of the dual solution. If one prefers to keep the same mesh to solve the dual problem as that used for the primal finite element solution, while retaining an efficient error estimator, higher-order methods should be employed, see e.g. [START_REF] Ainsworth | Guaranteed computable bounds on quan-tities of interest in finite element computations[END_REF]. Furthermore, it is desirable that the discretization method for the dual problem allows for computationally inexpensive equilibrated-flux reconstruction. Based on these facts, we propose here to use a high-order discontinuous Galerkin discretization method to approximate the dual solution. Advantages of the dG method are that high-order elements in any space dimension can be easily implemented and that it is locally conservative, which yields in a straightforward manner equilibrated fluxes. In fact, it has been shown to provide cheap and local reconstruction algorithms [START_REF] Kim | A posteriori error analysis for locally conservative mixed methods[END_REF][START_REF] Ern | An accurate H(div) flux reconstruction for discontinuous Galerkin approximations of elliptic problems[END_REF][START_REF] Carstensen | Effective postprocessing for equilibration a posteriori error estimators[END_REF][START_REF] Braess | An equilibrated a posteriori error estimator fort he interior penalty discontinuous Galerkin method[END_REF].

The dG method

We briefly introduce in this section the symmetric interior penalty dG method and present some basic results on the stability and approximation properties of the method. We first introduce the bilinear form B h :

V k h ×V k h → R as B h (u h , v h ) = T ∈T h T D∇u h • ∇v h - E∈E h E {{n • D∇u h }}[[v h ]] + E∈E h E (-{{n • D∇v h }} + γ E [[v h ]]) [[u h ]], (25)
where γ E > 0 denotes the interior penalty parameter to be specified later.

For the primal boundary value problem (1), the symmetric interior penalty dG method is defined as follows (see e.g. [START_REF] Di Pietro | Mathematical Aspects of Discontinuous Galerkin Methods[END_REF]):

Find u h ∈ V k h such that B h (u h , v h ) = F (v h ), ∀v h ∈ V k h . (26)
Since symmetric dG method is adjoint consistent, the corresponding dual problem reads:

Find p h ∈ V m h such that B h (v h , p h ) = Q(v h ), ∀v h ∈ V m h . (27)
For completeness, we present below some basic analytical results referring to the symmetric interior penalty dG method; owing to the symmetry of the method and of the primal and dual problems, we can limit the analysis to the primal problem only. Firstly, let us note that, for sufficiently large penalty parameters, problem (26) has a unique solution, see e.g. [START_REF] Di Pietro | Mathematical Aspects of Discontinuous Galerkin Methods[END_REF]. We introduce here a computable lower bound γ(T h ) on γ E , as presented in [START_REF] Ainsworth | Technical note: A note on the selection of the penalty parameter for discontinuous Galerkin finite element schemes[END_REF] for a general setting, that is sufficient to show existence and uniqueness of the dG method. Therefore, we suppose from now on that the penalty parameters satisfy

γ E ≥ γ(T h ) ∀E ∈ E h . ( 28
)
Convergence properties of high-order dG methods are summarized in the following theorem, see e.g. [START_REF] Di Pietro | Mathematical Aspects of Discontinuous Galerkin Methods[END_REF].

Theorem 2. Let us suppose that Ω is a convex polygonal domain in R 2 and that the solution to Problem (1) belongs to H k+1 (Ω). Let T h , h > 0, be a family of matching shape regular triangular meshes on Ω and assume that parameters γ E satisfy the stability condition [START_REF] Ern | A unified framework for a posteriori error estimation in elliptic and parabolic problems with application to finite volumes[END_REF]. Then, the following estimates hold for the dG solution to (26) with a positive constant C that depends only on the shape-regularity of T h :

|u -u h | h ≤ Ch k u H k+1 (Ω) (29)
u -u h L 2 (Ω) ≤ Ch k+1 u H k+1 (Ω) , (30)
where the dG energy norm is defined on V k h as

|v h | 2 h = T ∈T h ∇v h 2 (L 2 (K)) 2 + E∈E h γ E [[v h ]] 2 L 2 (E) .
In the following sections, we will show that, depending on the approximation properties of the primal method and on the problem data, the order of the dG method, used to solve the dual problem, can be chosen in such a way that the respective error estimator is asymptotically exact. We will consider the following discretizations of the primal problem: high-order discontinuous Galerkin method, the first-order conforming finite element method, and the lowest order mixed Raviart-Thomas finite element method. Other non-conforming methods, finite volume and finite difference methods can be considered in a similar fashion using corresponding flux reconstruction algorithms, see e.g. [START_REF] Vohralík | Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients[END_REF][START_REF] Ern | Four closely related equilibrated flux reconstructions for nonconforming finite elements[END_REF].

Goal-oriented error estimates for dG methods

We consider here the symmetric dG method of order k ∈ N for the discretization of Problem [START_REF] Ainsworth | A posteriori error estimation for discontinuous Galerkin finite element approximation[END_REF]. We construct equilibrated fluxes t k-1 h (u h ) ∈ RT k-1 in terms of the dG solution u h ∈ V k h , depending on the value of k, as follows (see e.g. [START_REF] Kim | A posteriori error estimators for locally conservative methods of nonlinear elliptic problems[END_REF][START_REF] Di Pietro | Analysis of a discontinuous Galerkin method for heterogeneous diffusion problems with low-regularity solutions[END_REF]):

k ≥ 1 : E n E • t k-1 h (u h ) q h = E -{{n • D∇u h }} + γ E [[u h ]] q h , ∀q h ∈ P k-1 (E), ∀E ∈ E h ; (31) k ≥ 2 : T t k-1 h (u h ) • r h = - T D∇u h • r h + E∈∂T E χ E h (E)n • Dr h [[u h ]], ∀r h ∈ (P k-2 (T )) d , ∀T ∈ T h . ( 32
)
Note that the flux reconstruction using ( 31)-( 32) is computationally inexpensive and is relatively easy to implement as an element-wise procedure.

Definition 3. A function f ∈ L 2 (Ω) is said to have oscillations of order l ∈ N 0 on V k h if
there exists a constant C, that depends on f and on the shape regularity of T h , such that

f -π k h f L 2 (Ω) ≤ Ch l ∀T h , h > 0. ( 33
)
For example, it follows from the optimality of the orthogonal projection that a function f ∈ H s (Ω) has oscillations of order l = min{k + 1, s} on

V k h . Theorem 3. Let f ∈ L 2 (Ω) in Problem (1) have oscillations of order k -1 in V k-1 h . Let u ∈ H k+1 (Ω) and u h ∈ V k
h be the solutions to (1) and (26), respectively. Moreover, let t h (u h ) ∈ RT k-1 be the equilibrated flux with respect to u h given by (31)- [START_REF] Harriman | hp-version discontinuous Galerkin methods with interior penalty for partial differential equations with nonnegative characteristic form[END_REF]. Then there exists a constant C > 0, that depends on the problem data {D, f } and on the shape-regularity of T h , such that

t k-1 h (u h ) + D∇ h u h [L 2 (Ω)] 2 ≤ Ch k . ( 34
)
Proof: Equality (17) follows from ( 31)-( 32) and ( 26) by straightforward calculations. It is also known (see e.g. [START_REF] Kim | A posteriori error estimators for locally conservative methods of nonlinear elliptic problems[END_REF][START_REF] Di Pietro | Analysis of a discontinuous Galerkin method for heterogeneous diffusion problems with low-regularity solutions[END_REF]) that the following estimate holds for t k-1 h (u h ) given by ( 31)-( 32):

t k-1 h (u h ) + D∇ h u h [L 2 (Ω)] 2 ≤ |u -u h | h + h f -π k-1 h f L 2 (Ω) .
Therefore, inequality [START_REF] Karakashian | A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems[END_REF] follows from above inequality owing to the a priori estimate (29) and oscillation estimate [START_REF] Houston | Energy norm a posteriori error estimation of hp-adaptive discontinuous Galerkin methods for elliptic problems[END_REF].

The next theorem provides the key result to investigate exactness of the error estimator [START_REF] Alaoui | Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems[END_REF] for the primal dG methods.

Theorem 4. Let f ∈ L 2 (Ω) in Problem (1) have oscillations of order k -1 in V k-1 h
and let u ∈ H k+1 (Ω) be the solution to [START_REF] Ainsworth | A posteriori error estimation for discontinuous Galerkin finite element approximation[END_REF]. Let p ∈ H m+1 (Ω) be the solution to the dual problem [START_REF] Babuška | The Finite Element Method and its Reliability[END_REF], with m > k, and suppose that the Riesz representer q of the goal functional has oscillations of order m -

1 in V m-1 h . Let u h ∈ V k
h and p h ∈ V m h be the dG finite element solutions to (26) and ( 27), respectively. Then, there exists a constant C, depending only on the shape regularity of T h , the data {D, f, q} in the primal and dual problems, and the exact solutions {u, p}, such that

R(u, p, f ; u h , p h ) ≤ Ch k+m . ( 35
)
Proof: Using the Cauchy-Schwarz inequality, the oscillation estimate for f , and the a priori estimate for the dual solution, there holds

T ∈T h T (f -π k-1 h (f ))(p -p h ) ≤ f -π k-1 h (f ) L 2 (Ω) p -p h L 2 (Ω) ≤ Ch k-1 h m+1 = Ch k+m .
We infer from Theorem 3 that the reconstructed equilibrated flux t m-1 h (p h ) with respect to the dual dG solution satisfies

t m-1 h (p h ) + D∇p) (L 2 (Ω)) 2 ≤ Ch m . (36)
It follows that

T ∈T h T (t k-1 h (u h ) + D∇u h ) • (∇p + D -1 t m-1 h (p h )) ≤ t k-1 h (u h ) + D∇u h [L 2 (Ω)] 2 D -1 D∇p + t m-1 h (p h ) [L 2 (Ω)] 2 ≤ Ch k h m .
Now, owing to the smoothness of the exact solution and using the divergence theorem, one gets

E∈E h E n E • (D∇p + t m-1 h (p h )) [[u h ]] = E∈E h E n E • (-D∇p -t m-1 h (p h )) [[u -u h ]] = T ∈T h T (-D∇p -t m-1 h (p h )) • ∇(u -u h ) + T ∈T h T (q -π m-1 h (q))(u -u h ).
Finally, using Theorem 3, the a priori estimates for the dG solutions, and the data oscillation estimate for the dual problem, one concludes that

E∈E h E n • (D∇p + t m-1 h (p h )) [[u h ]] ≤ Ch k+m (37)
which completes the proof.

Theorem 5. Let the conditions of Theorem 4 hold and let us suppose that the goal functional error is properly of order h 2k . Then error estimator (21) is asymptotically exact, that is,

lim h→0 + η(u h , p h ; t k-1 h (u h ), t m-1 h (p h )) Q(u) -Q(u h ) = 1.
Proof: From Theorem 4, we infer that R(u, p, f ;

u h , p h ) Q(u) -Q(u h ) ≤ ch m-k with m > k, so that lim h→0 + η(u h , p h ; t k-1 h (u h ), t m-1 h (p h )) Q(u) -Q(u h ) = lim h→0 + 1 - R(u, p, f ; u h , p h ) Q(u) -Q(u h ) = 1.
Remark 4. The order doubling in the rate of convergence for the approximation of the goal functional is typical in the case of the symmetric version of the discontinuous Galerkin method, see e.g. [START_REF] Harriman | hp-version discontinuous Galerkin methods with interior penalty for partial differential equations with nonnegative characteristic form[END_REF].

Goal-oriented error estimates for finite element methods

We now consider the case in which Problem (1) is discretized by the lowest-order conforming finite element method:

Find u h ∈ V h such that Ω D∇u h • ∇v h = Ω f v h , ∀v h ∈ V h , (38)
where

V h = V 1 h ∩ H 1 0 (Ω).
Note that the gradient ∇u h is a piecewise constant vector field that may be discontinuous along the interelement edges. Following [START_REF] Braess | Finite Elements. Theory, Fast Solvers, and Application in Solid Mechanics[END_REF][START_REF] Braess | Equilibrated residual error estimator for edge elements[END_REF], we consider reconstructed equilibrated fluxes in the form t h (u h ) = -D∇u hσ(u h ), where σ(u h ) represents some correction to the broken flux D∇u h such that t h (u h ) belongs to RT 0 (T h ). Such a correction σ(u h ) is actually sought in the space

RT -1 (T h ) = s h ∈ [L 2 (Ω)] 2 : s h | T ∈ [P 0 (T )] 2 + x P 0 (T ), ∀T ∈ T h and should satisfy the conditions div σ(u h ) = -Π 0 (f ), [[σ(u h ) • n E ]] = -[[D∇u h • n E ]] ∀E ∈ E i
h , in order to guarantee that t h (u h ) be a zero-order reconstructed flux. The existence of such σ(u h ) is proven in [START_REF] Braess | Equilibrated residual error estimator for edge elements[END_REF] in a more general situation and a constructive proof in the two-dimensional case is presented in [START_REF] Braess | Finite Elements. Theory, Fast Solvers, and Application in Solid Mechanics[END_REF]. The proof uses the hypercircle method for flux equilibration in the star patches ω i associated with all vertices i in T h . An estimate for this correction is provided in the following theorem (see [START_REF] Braess | Finite Elements. Theory, Fast Solvers, and Application in Solid Mechanics[END_REF], p. 184): Theorem 6. There exists a constant C > 0 that depends only on the shape regularity of T h such that

σ(u h ) (L 2 (Ω)) 2 ≤ C ∇u -∇u h (L 2 (Ω)) 2 + h f -π 0 h (f ) L 2 (Ω) . ( 39
)
Theorem 7. Let u ∈ H 2 (Ω) be the solution to (1) and let p ∈ H 3 (Ω) be the solution to the dual problem [START_REF] Babuška | The Finite Element Method and its Reliability[END_REF]. Suppose that the Riesz representer q of the goal functional has first-order oscillations in V 1 h . Let u h ∈ V h be the finite element solution to [START_REF] Korotov | A posteriori error estimation of goal-oriented quantities for elliptic type BVPs[END_REF] and p h ∈ V 2 h be the second-order dual dG finite element solution to [START_REF] Ern | Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusionreaction problems[END_REF]. Then, there exists a constant C, depending only on the shape regularity of T h , the data {D, f, q} in the primal and dual problems, and the exact solutions {u, p}, such that

|R(u, p, f ; u h , p h )| ≤ Ch 3 . ( 40
)
Proof: The first term in ( 22) is estimated as

T ∈T h T (f -π 0 h (f ))(p -p h ) ≤ Ch 3 ,
since, owing to the elliptic regularity, one has

p -p h L 2 (Ω) ≤ Ch 3 .
From ( 36), [START_REF] Korotov | A posteriori error estimation of goal-oriented quantities by the superconvergence patch recovery[END_REF], and an energy norm a priori error estimate for the finite element solution, we obtain by applying Cauchy-Schwarz:

T ∈T h T (t h (u h ) + D∇u h ) • (∇p + D -1 t 1 h (p h )) = T ∈T h T σ(u h ) • (∇p + D -1 t 1 h (p h )) ≤ Ch 3 .
This completes the proof since the third term in the remainder vanishes owing to the consistency of the method.

Theorem 8. Let u h ∈ V h be the finite element solution to [START_REF] Korotov | A posteriori error estimation of goal-oriented quantities for elliptic type BVPs[END_REF] and suppose that the goal functional error is properly of order h 2 . Then, under the conditions of Theorem 7, the error estimator

η(u h , p h ; t h (u h ), t 1 h (p h )) = T ∈T h T (f -π 0 h (f )) p h - T ∈T h T σ(u h ) • D -1 t 1 h (p h ) (41)
is asymptotically exact.

Goal-oriented error estimates for the mixed RT 0 finite element

In this section, we consider the mixed form of the primal problem (1). The weak formulation then reads:

Find (u, r) ∈ L 2 (Ω) × H(div, Ω) such that Ω D -1 r • q - Ω u ∇ • q = 0 ∀q ∈ H(div, Ω) Ω v ∇ • r = Ω f v ∀v ∈ L 2 (Ω). ( 42
)
It is well known, see e.g. [START_REF] Brezzi | Mixed and Hybrid Finite Element Methods[END_REF], that there exists a unique solution (u, r) to the mixed problem and that the mixed formulation ( 42) is equivalent to [START_REF] Ainsworth | A posteriori error estimation for lowest order Raviart-Thomas mixed finite elements[END_REF] with u ∈ H 1 0 (Ω) and r = -D∇u. In addition, we have the following a priori estimates:

u L 2 (Ω) ≤ C f L 2 (Ω) r H(div,Ω) ≤ C f L 2 (Ω)
where constant C depends only on parameter D and domain Ω. We introduce the finite element space:

Q 0 (T h ) = {q h ∈ RT 0 (T h ) : q h • n = 0 on E ∂ h } (43)
The lowest-order Raviart-Thomas finite element method for the mixed formulation (42) reads:

Find (u h , r h ) ∈ P 0 × RT 0 (T h ) such that Ω D -1 r h • q h - Ω u h ∇ • q h = 0 ∀q h ∈ Q 0 (T h ) Ω v h ∇ • r h = Ω f v h ∀v h ∈ P 0 . (44)
Theorem 9. Let the Riesz representer q of the goal functional have oscillations of first-order in V 1 h . Let (u h , r h ) ∈ P 0 × RT 0 (T h ) be the solution to (44) and let p h ∈ V 2 h be the second-order dual dG finite element solution to [START_REF] Ern | Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusionreaction problems[END_REF]. Then, there exists a constant C, depending only on the shape regularity of T h , the data {D, f, q} in the primal and dual problems, and the exact solutions {(u, r), p}, such that

|R((u, r), D, f, q; (u h , r h ), p h )| ≤ Ch 2 . ( 45
)
Proof: The remainder ( 22) is estimated as follows. First, we determined the zeroth-order reconstructed flux r h with respect to u h . We then have

T ∈T h T (f -π 0 h (f ))(p -p h ) ≤ Ch 2 , (46)
owing to the (suboptimal) estimate

p -p h L 2 (Ω) ≤ Ch 2 . (47)
Next, from the elliptic regularity, we have u ∈ H 2 (Ω) and r ∈ (H 1 (Ω)) 2 so that the following a priori error estimates hold, see e.g. [START_REF] Brezzi | Mixed and Hybrid Finite Element Methods[END_REF],

r -r h (L 2 (Ω)) 2 ≤ Ch ∇r (L 2 (Ω)) 2 , u -u h L 2 (Ω) ≤ Ch ∇r (L 2 (Ω)) 2 + ∇u (L 2 (Ω)) 2 .
Therefore

r h + D∇u h (L 2 (Ω)) 2 = r h (L 2 (Ω)) 2 ≤ r -r h (L 2 (Ω)) 2 + r (L 2 (Ω)) 2 ≤ C(1 + h) r (L 2 (Ω)) 2 ,
which yields

T ∈T h T (r h + D∇u h ) • (∇p + D -1 t 1 h (p h )) ≤ Ch 2 .
Noting that

E∈E h E n E • (D∇p + t 1 h (p h ))[[u h ]] ≤ T ∈T h T (D∇p + t 1 h (p h )) • ∇u + T ∈T h T (q -π 1 h (q))(u -u h ) ≤ Ch 2 ∇u (L 2 (Ω)) 2 + Ch u -u h L 2 (Ω) ≤ Ch 2
allows us to conclude.

Theorem 10. Let (u h , r h ) ∈ P 0 ×RT 0 (T h ) be the solution to [START_REF] Neittaanmäki | New a posteriori error indicator in terms of linear functionals for linear elliptic problems[END_REF] and let the error in the goal functional be assumed properly of first order. Then, under the conditions of Theorem 9, the error estimator

η(u h , p h ; r h , t 1 h (p h )) = T ∈T h T (f -π 0 h (f )) p h + T ∈T h T r h • D -1 t 1 h (p h ) - E∈E h E n E • t 1 h (p h )[[u h ]] (48)
is asymptotically exact.

Numerical examples

We shall consider two classes of benchmark problems: one with large data oscillations and one with no data oscillation. The first problem, following [START_REF] Carstensen | Effective postprocessing for equilibration a posteriori error estimators[END_REF], is the homogeneous boundary-value problem (1) with D = 1 in Ω = (0, 1) 2 , for which the loading term f is given such that the exact solution reads

u(x, y) = 10 4 x(1 -x)y(1 -y) exp -100 (x -0.75) 2 + (y -0.75) 2 . (49)
The goal functional is chosen for the first problem as:

Q(u) = 1 |ω| ω u (50)
with

ω = {(x, y) ∈ Ω : 1.5 ≤ x + y ≤ 1.75} . (51)
In other words, the goal is to calculate the average of the solution over the strip ω, namely the region near the maximum of the solution, where the righthand side f , the solution u, and the gradient of u exhibit large oscillations. The exact value of the quantity of interest was evaluated analytically to be

Q(u) = 43.28448988.
The second benchmark problem is given by the homogeneous boundaryvalue problem [START_REF] Ainsworth | A posteriori error estimation for discontinuous Galerkin finite element approximation[END_REF] in Ω = (0, 1) 2 with uniform load f = 10 3 , for which the oscillations vanish. The goal functional is given by (50) evaluated over the subdomain ω = [0.75, 1] × [0.75, 1],

which corresponds to the averaged solution in the upper-right corner where the solution exhibits large gradient. A representation of the exact solution in terms of series was used to calculate the exact value of the functional. The objective of the numerical experiments is to study the behavior of the contributions to the error estimator η (21) in the case of three finite element methods described above. For each problem, we describe the numerical results that confirm the asymptotical exactness of the suggested error estimator and demonstrate the potential of such an estimator in goal-oriented mesh adaptation. In all examples presented below, meshes are adapted using a refinement strategy based on the method proposed by Dörfler [START_REF] Dörfler | A convergent adaptive algorithm for Poisson's equation[END_REF], whereby the elements in a minimal set M ⊂ T h , such that

θ T ∈M η(T ) 2 ≤ T ∈T h η(T ) 2 ,
are refined. The refinement parameter θ = 0.75 was used in the first benchmark problem while θ = 0.5 was considered in the second problem. Elements are refined using the longest edge bisection technique and additional refinements of the mesh are considered in order to eliminate hanging nodes.

Finally, the quality of the error estimator η = η(u h , p h ; t h (u h ), t h (p h )) will be assessed in terms of the effectivity index:

I η = η(u h , p h ; t h (u h ), t h (p h )) Q(u) -Q(u h ) (53)
evaluated on sequences of uniformly and adaptively refined meshes.

dG method

We consider here the first-order and second-order symmetric dG methods for the discretization of the primal and dual problems and reconstruct the equilibrated fluxes in RT 0 and RT 1 , respectively.

We show in Figure 1(a) the effectivity index of estimator η for a sequence of uniform and adaptive mesh refinements for the first test problem. For uniform refinement, the effectivity index converges to unity as the mesh size decreases, which confirms asymptotic exactness of the error estimator. In Figure 1(b) the true error E in the goal functional approximation as well as the error estimator and its different components, are shown for a sequence of adaptive mesh refinements. To illustrate the effect of error cancelation when summing up the three contributions, the positive and negative parts of each component are shown in black and red, respectively. Moreover, given a quantity f (i.e. e or η), we denote by f + = max{f, 0} (resp. f -= max{-f, 0}) the positive (resp. negative) part of f . We can see that the error estimate remains very close to the true error offering accurate information for mesh adaptation. For such a problem with large data oscillations, the error estimate is essentially quantified by the difference between the oscillation contribution η O and the flux contribution η ∇ , which both dominate in absolute value but with opposite sign, and is corrected by the relatively smaller η H component.

In the case of the second test problem, the effectivity index is shown in

Conforming finite element method

In the case of the first test problem, we consider the first-order conforming finite element method for the primal problem and dG methods of various orders for the dual problem. The equilibrated fluxes for the primal problem was constructed using the hypercircle method presented in [START_REF] Braess | Finite Elements. Theory, Fast Solvers, and Application in Solid Mechanics[END_REF]. In Figure 3(a), we present the effectivity index of estimator η for a sequence of uniform and adaptive mesh refinements. In the case of uniform refinements, we use dG methods of order first, second, and third for the calculation of the error estimates. As one can observe from the figure, the first-order dG method does not provide an asymptotically exact error estimator (the effectivity index tends to ≈ 1.9 on fine meshes), but asymptotic exactness is recovered in accordance with Theorem 8 in the case of the second-and third-order dG methods. We show in Figure 3(b) the exact error e in the goal functional approximation as well as the error estimate η and the different contributions (calculated using the second-order dG method) for a sequence of adaptive mesh refinements. The component η H vanishes here, so the difference between the oscillation estimate η O and the flux estimate η ∇ , which are of the same order but of opposite signs, accurately captures the goal functional (a) error.

In the case of the second benchmark problem, we use the same finite element method for the primal problem as before and restrict ourselves to the second-order symmetric dG method for the dual problem. Flux reconstruction is performed as in the previous test problem. The numerical results, shown in Figure [START_REF] Ainsworth | A posteriori error estimation in finite element analysis[END_REF], are as expected since in this case both η O ≈ 0 and η H ≈ 0.

Raviart-Thomas mixed finite element

We solve the first benchmark problem using RT 0 mixed finite elements, calculate the error estimate (48) on a sequence of uniform meshes using the first-and second-order symmetric dG methods, and reconstruct equilibrated fluxes in RT 0 and RT 1 , respectively. The results are shown in Figure 5(a). Theorem 10 is not valid in the first case and we can see that the effectivity index fails to converge to unity as expected. When the approximation order of both the dG method used for the solution of the dual problem and flux reconstruction method increases, the error estimate becomes again asymptotically exact. The exact error and the error indicator with its components, calcu-(a) The second problem is solved using RT 0 mixed finite elements as before. Here, the error estimate ( 48) is also evaluated using the second-order symmetric dG method and reconstructed fluxes in RT 1 . The results are presented in Figure 6. In absence of data oscillations, η O O ≈ 0 (η O O is actually not shown in Figure 6(b)), and the error estimate η amounts to the the difference between the flux and the jump estimates. Even in this case, we can observe that the error estimate η is asymptotically exact as demonstrated in Figure 6(a).

Conclusions

We have presented a flexible approach for the computation of a posteriori error estimates with respect to quantities of interest in the case of elliptic problems. The main ingredients of the method are the calculation of higherorder approximation of the dual problem by the discontinuous Galerkin method and the construction of equilibrated fluxes in Raviart-Thomas finite element spaces. The advantage of using a dG method for the dual problem lies in its simplicity to consider high-order approximations and to reconstruct equilibrated fluxes from its solution. The error estimator in the quantity of interest is decomposed into three contributions: 1) an error estimate due to the data oscillations in the primal problem, 2) an error estimate that measures the difference between the reconstructed fluxes and the finite element fluxes, and 3) a contribution from the jump of the finite element solution at the interface of the elements, all terms being weighted by the discontinuous Galerkin solution to the dual problem. Within this framework, the primal problem can be approximated by any finite element method as long as it satisfies standard a priori error estimates and its solution can be subjected to (a) the reconstruction of equilibrated fluxes. We have shown that, depending on the convergence properties of the underlying method, the order of the dual dG method can be chosen in such a way that the resulting error estimator be asymptotically exact. Numerical experiments, using either the dG method, the conforming finite element method, or the mixed Raviart-Thomas method for the solution of the primal problem, clearly confirm that the proposed error estimator has effectivity indices close to unity and is asymptotically exact.

Figure ( 2 )Figure 1 :

 21 Figure 1: First benchmark problem using the dG method for the solution of the primal problem: (a) effectivity index I η ; (b) exact error e in quantity of interest, error estimator η, and contributions η O , η ∇ , and η H to error estimate η on a sequence of adapted meshes (negative values are shown in red, positive values in black).

Figure 2 :

 2 Figure 2: Second benchmark problem using the dG method for the solution of the primal problem: (a) effectivity index I η ; (b) exact error e in quantity of interest, error estimator η, and contributions η O , η ∇ , and η H to error estimate η on a sequence of adapted meshes (negative values are shown in red, positive values in black).

Figure 3 :

 3 Figure 3: First benchmark problem using the conforming finite element method for the solution of the primal problem: (a) effectivity index I η ; (b) exact error e in quantity of interest, error estimator η, and contributions η O , η ∇ , and η H to error estimate η on a sequence of adapted meshes (negative values are shown in red, positive values in black).

Figure 4 :

 4 Figure 4: Second benchmark problem using the conforming finite element method for the solution of the primal problem: (a) effectivity index I η ; (b) exact error e in quantity of interest, error estimator η, and contributions η O , η ∇ , and η H to error estimate η on a sequence of adapted meshes (negative values are shown in red, positive values in black).

Figure 5 :

 5 Figure 5: First benchmark problem using the mixed RT 0 method for the solution of the primal problem and dG method for the dual problem: (a) effectivity index I η ; (b) exact error e in quantity of interest, error estimator η, and contributions η O , η ∇ , and η H to error estimate η on a sequence of adapted meshes (negative values are shown in red, positive values in black).

Figure 6 :

 6 Figure 6: Second benchmark problem using the mixed RT 0 method for the solution of the primal problem and dG method for the dual problem: (a) effectivity index I η ; (b) exact error e in quantity of interest, error estimator η, and contributions η O , η ∇ , and η H to error estimate η on a sequence of adapted meshes (negative values are shown in red, positive values in black).

  By definition, an edge E ∈ E h is said to be an interior edge if there exist two triangles T -, T + ∈ T h such that E = T

	-∩ T	+ and is said
	to be a boundary edge if there exists one and only one triangle T ∈ T h such
	that E = T ∩ ∂Ω. For a polygonal domain Ω, we have ∂Ω = ∪ E∈E ∂ h E. For
	any function v ∈ L 2 (Ω), assumed sufficiently smooth to admit a trace on all
	E ∈ E h (possibly different on either side of an edge E ∈ E i h ), let us denote
	the jump of

Acknowledgements

The first author gratefully acknowledges the partial support by CNPq, Brazil. The second author is grateful for the support by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. He is also a participant of the KAUST SRI center for Uncertainty Quantification in Computational Science and Engineering.


Building the urban Future and Transit Oriented Development (BUFTOD) Rail and other modes, connecting with urban and regional

Keywords: Firm location patterns, Transport policy, long term impacts assessment, tramway, land use and economic benefits

It is widely accepted that there are some factors that influence the location patterns of firms over space and over time. The above is especially relevant in the case of deprived regions because an increase of firms' location leads to higher employment opportunities which impacts on the regional economic growth. Moreover, the public transportation system may be efficiently exploited if the correct policy strategies of firms' location are enhanced.

The former reasons explain the usefulness to analyze the location patterns of firms over a certain region. In practice, the main problem of the empirical analysis of spatial and temporal patterns of firm location is related to the complexity to obtain detailed firmographic data due to both the lack of availability of micro data and confidentiality issues. Therefore, very few studies are available on these issues. Herein, we discuss different methodologies to obtain information on the location patterns of firms. As a case study, it is analyzed the region of the "bassin minier" of Nord-Pas-de-Calais, at the northern part of France. The Transport Authority is the SMT Artois-Gohelle and it has planned to build in the near future two new tramway lines that will improve the accessibility in the region. In this case, an analysis that understands the mobility patterns of firms in the area would be certainly useful to foster their location when the new tramways start operating.

INTRODUCTION

It is well known that transportation is a key factor for improving the socioeconomic conditions of a region. As commented by [START_REF] Lucas | Providing transport for social inclusion within a framework for environmental justice in the UK[END_REF], it influences the conditions of society by means of reducing health inequalities, increasing educational and employment opportunities, along with a renewal of neighborhood conditions and services. However, researchers have concluded that transport infrastructure is a necessary but not sufficient condition for economic development [START_REF] Button | Transport and industrial and commercial location[END_REF].

Therefore, when a new project of public infrastructure is under discussion for its further construction and implementation, it is important to propose in advance different methodologies in order to work along with the Authorities to record micro-level data over the long term for its further analysis and comparison. Indeed, it is necessary to measure the future impact of a new transport infrastructure using multiple appraisal methods in order to better comprehend the changes that are taking place, even before the construction of it. Therefore in this way, it will be possible to apply the correct policy strategies if changes are either not taking place or are negative for the community. For the sake of the analyses, the wider range of methodologies are used, the better information can be elicited, which means making use of quantitative and qualitative methodologies along with continuous microlevel datasets over a long period.

In general, the analysis carried out over a certain area where a new transport infrasturcture is going to be located are related to cross-sectional data, and focused on travel pattern impacts. These analyses do not take into account time factors at the short and long term. Literature affirms that if no temporal effects, such as habits and inertia, are taken into account, there may be overestimatation problems. Moreover, it is necessary to evaluate not only the effects related to changes in travel patterns but also the ones related to the built environment [START_REF] Banister | Transport investment and the promotion of economic growth[END_REF][START_REF] Banister | Quantification of the non-transport benefits resulting from rail investment[END_REF]Mejia-Dorantes et al., 2011;Mejia-Dorantes et al., forthcoming).

As it is well known, one of the most important problems related to the assessment of long term impacts of a new transport infrastructure, is related to the lack of availability of detailed datasets to analyze different factors that might be influenced by the transport infrastructure. There are only few studies that have succeeded to carry out a thorough ex-ante/ex-post assessment. Probably the most interesting one is the Jubilee line extension (London) in 1999. It comprised 2 years before operation and 4 years afterwards, where different aspects were evaluated, such as: Transport impacts and accessibility changes; Residential and commercial development; Economic impacts (employment); Residents and travel patterns (Arup Economics and Planning, 1999;[START_REF] Lucas | Methodology and study programme for an impact assessment of the effects of the Jubilee line extension[END_REF]Wofinden, 1998a;[START_REF] Wofinden | Report on Household Pilot Survey[END_REF].

In fact, the ideal analysis would be to follow the infrastructure before construction until the settle effects take place (See figure 1). The aim of this paper is to focus on an important aspect of long term assessment: firms' location analysis. Hence, we discuss different methodologies to assess the location of economic activities that may be used before the construction of a certain transport infrastructure and after it starts operations. We use as a case study the ex-coal mining area (ex-bassin minier), which is a deprived area of Nord-Pas-de-Calais, at the northern part of France. The Transport Authority is the SMT Artois-Gohelle which has planned to build in the near future two new tramway lines that will improve the accessibility in the region -although, by the time this paper was written, the status of project has changed and now it is indefinitely suspended-.

In the current state of the analysis (before construction takes place), perception of a future change is the main issue and an interesting problem that needs to be further investigated. An analysis that understands the mobility patterns of firms in the area, their needs, advantages, and problems, would be very useful to foster their location whenever the new tramway starts operations.

This paper is organized in the following form. After this introduction, the case study is presented. In section three, the methodologies proposed are discussed. Finally, section four presents the conclusions and discussion.

CASE STUDY

This study focuses on the region the ex-coal mining area (ex-bassin minier), which is a deprived area of Nord-Pas-de-Calais, at the northern part of France. Figure 2 and 3 shows the location of the territory under study.

It is known as an area which has different socio-economic problems: unemployment, the highest rate of mortality in France and poorly academic performances and failures, among others after the conclusion of the coal mining activity era at the end of the 80's.

To ameliorate the socioeconomic situation of this area, different policies are being implemented aimed at regenerating the socioeconomic activities of these towns. For example,e the case of the new Museum of Louvre in Lens, and the improvement of public transportation, which intends to promote the economic, social and environmental situation of the zone. The Transport Authority (SMT Artois-Gohelle) planned to build two new tramway lines (see figure 4 and5).

The region served by the SMT Artois Gohelle is composed of 115 towns and has around 600,000 inhabitants in an area of around 76,000 hectares. The two main towns are Lens with around 36,000 inhabitants and Béthune with around 26,000 inhabitants. The two tramways are not connected and they would replace two bus routes that count with the greatest number of ridership known as "Bulle (1 and 2)" See figures 4 and 5.

Moreover, it is interesting to note that this area is more dependent to car than in the rest of France, even though this is a more deprived area: Lens, Liévin and Hénin-Beaumont for example, have a level of car ownership close to the 63%; in the case of Béthune it is equal to 71%, whereas the national average is of 60%.

.maps.google.fr For the territories under study (known as "Communes") we present different facts. For example, how population has evolved during the last years for each Commune, and the type of population (men/women) per age (figure 6). It is interesting to note that the difference among the percentage of women and men is higher for the people older than 65 years. This is related to the fact that men used to work in the mines, which was a very tough job. Figure 7 shows the type of economic activities in the region. Most of the people are non-skilled workers, and jobs are focused on commerce and services (see figure 8). Moreover, figure 9 shows that the rate of unemployment in the area has not importantly diminished in a decade, although more firms have localized in the area (see figure 10). According to Caratini (2012), in Béthune, the owners of retail stores believe that due to the new tramway their customers will go away more easily to other areas for shopping, that in other towns, such as Bruay, the downtown will be even less crowded than nowadays.

METHODOLOGIES

Three different methodologies are proposed. They depend to some extent on the possibilities we have to obtain micro-level datasets. They are based on spatial analysis, which represents a collection of techniques that use spatial referencing associated with each data value under study. The importance of spatial analysis lies on the fact that results of any spatial analysis are not the same under a re-arrangement of the spatial distribution of the events under investigation [START_REF] Chorley | Spatial Analysis in Geomorphology[END_REF][START_REF] Haining | Spatial Data Analysis: Theory and Practice[END_REF].

The current state is "without infrastructure and works" also known as the "state zero" of firms' location. For that sake, it is proposed to use the methodology used by Mejia-Dorantes ( 2011) by means of spatial multinomial logit models and Kernel analysis, which are based on spatial analysis. Afterwards, we propose to use questionnaire surveys focusing on retail firms to elicit the perception of managers about the change in their living environment.

Spatial multinomial logit models

The methodology presented in the following paragraphs is based on the model developed by Mejia-Dorantes et al. (2011;forthcoming).

A model for unordered categorical variables was used, also known as polytomous variables. In the model, one of the outcome classes is arbitrarily set as the base outcome. For ease of interpretation, in this study "vacant" is chosen as the reference category. It is based on the possibility that each address has to be in used by a certain economic activity, which is named "portal".

The spatial version of the model incorporates a spatial weights matrix. The probability of a portal being in state j depends on a set of covariates x and also the presence/absence of other businesses in the neighborhood (an occupancy ratio).

The occupancy ratio at location i for activity class k gives a measure of agglomeration in the preceding time period. This ratio is calculated based on the spatial weights matrix and occupancy as follows:

' ' 1 1 1 k ji i k k j ii k xr j x r m j e P e           
(1)

Further information on the above formula may be found at Mejia Dorantes (2011) and Mejia-Dorantes et al. (forthcoming). Table 1 shows the sectors under study:

Kernel estimation method

There are different statistical methods that have proved their utility for urban analysis [START_REF] Maoh | Geographic clustering of firms and urban form: a multivariate analysis[END_REF]Mejia-Dorantes et al., forthcoming). One of them is the point pattern analysis, which evaluates the distribution of events over an urban area. It is useful to analyze the firm location evolution over time and its clustering. Kernel surfaces are a powerful tool for the exploration of the spatial and temporal characteristics of clustering especially useful when variables are a subset of a total [START_REF] Bailey | Interactive Spatial Data Analysis[END_REF][START_REF] Cressie | Statistics for Spatial Data[END_REF]. Figure 11 shows an example of the resulting map. The Survey Questionnaire

Literature is relative scarce on the use of questionnaire surveys to assess how a future infrastructure may impact on the urban environment. In this respect little is known on the use of surveys related to the location of economic activities and transportation. Survey questionnaires are generally applied in travel behavior modeling [START_REF] Gatersleben | Contemplating cycling to work: Attitudes and perceptions in different stages of change[END_REF][START_REF] Heinen | The role of attitudes toward characteristics of bicycle commuting on the choice to cycle to work over various distances[END_REF][START_REF] Ory | Exploring the cognitive and affective mechanisms behind subjective assessments of travel amounts[END_REF][START_REF] Ory | Exploring the cognitive and affective mechanisms behind subjective assessments of travel amounts[END_REF][START_REF] Ory | When is getting there half the fun? Modeling the liking for travel[END_REF][START_REF] Páez | Enjoyment of commute: A comparison of different transportation modes[END_REF][START_REF] Redmond | The positive utility of the commute: modeling ideal commute time and relative desired commute amount[END_REF]. To our knowledge, only few exist, for example, the study carried out for the Jubilee line extension (Wofinden, 1998a). Unfortunately, the former is a study which was not entirely published. On the other hand, the one from the Sevilla metro (Castillo-Manzano and López-Valpuesta, 2009), focuses on the retail sector during the construction phase. There is another study on the Hong Kong metro expansion [START_REF] Loo | How would people respond to a new railway extension? The value of questionnaire surveys[END_REF]. [START_REF] Button | Transport and industrial and commercial location[END_REF] present a remarkable research on the location of firms in a region of Scotland. Later, [START_REF] Leitham | The influence of transport on industrial location choice: a stated preference experiment[END_REF] present an interesting study on stated preference surveys for industrial location. In this case we are dealing with an infrastructure not currently available and our aim is to study the retail sector.

Different authors mention that a metro line may be an important instrument for revitalizing retail commerce [START_REF] Banister | Transport Investment and Economic Development[END_REF][START_REF] Banister | Transport investment and the promotion of economic growth[END_REF][START_REF] Banister | Quantification of the non-transport benefits resulting from rail investment[END_REF][START_REF] Castillo-Manzano | Urban retail fabric and the metro: A complex relationship. Lessons from middle-sized Spanish cities[END_REF][START_REF] Mejia-Dorantes | Transportation infrastructure impacts on house prices and firms' location: The effect of a new metro line in the suburbs of Madrid[END_REF]. However, as commented by Becky Loo (2002), little is known about the perception and response of people about a major transport infrastructure change. This comment should be extrapolated for economic activities towards a new transport infrastructure.

As concluded by [START_REF] Loo | How would people respond to a new railway extension? The value of questionnaire surveys[END_REF], questionnaires are a very useful tool to evaluate the people's perception about their living environment. [START_REF] Sell | Perception of land use change in a desert city[END_REF] mention that the availability, quality and timing of information are important for the perception of change. They also conclude that temporal factor affects awareness, therefore people is more concerned about a project when it is under execution rather than when it is under planning. Moreover, the process of imagining change is directly influenced by the level of information and certain socio-economic characteristics. For example, [START_REF] Loo | How would people respond to a new railway extension? The value of questionnaire surveys[END_REF] states that people's subjective feelings and evaluation on the impact of a certain infrastructure were significantly associated with their knowledge about the spatial location of it.

Existing literature also shows that personal characteristics are significant determinants of people's perceptions about their living environment [START_REF]The potential impacts of strategic highways on new town development: a case study of Route 3 in Hong Kong[END_REF].

Herein, we will be able to observe the differences between actual location and ideal. For that reason, we will interview managers/owners of retail firms in the region of study. We must be careful with chains, since they are many times located following certain corporate strategies.

The construction of questionnaires is a complicated task, and the definition, selection and ordering of questions should be deeply studied on beforehand. For example, as mentioned by E. Martin (2005) and [START_REF] Presser | Methods for testing and evaluating survey questions[END_REF], it is important to carry out pilot surveys to assure that further problems are controlled on beforehand.

Although different types of stratifications are possible (Transportation Catchment areas: in/out of them; and multiple stratifications: for example by re-grouping economic activities), by now it is planned to use a simple random sampling. The location of firms who have responded could be georeferenced to observe the spatial distribution of responses. Hence, it will be possible to use a Wmatrix for spatial relationships.

In this analysis we make use of both revealed and stated preference survey methods. The questionnaire (see table 2) intends to capture the following explanatory variables:

-Objective location characteristics (current)

Figure 2 :

 2 Figure 2: Nord-Pas-de-Calais Region

Figure 6 .

 6 Figure 6. Population in the study area (INSEE)

Figure 7 .

 7 Figure 7. Types of economic activities (INSEE)

Figure 11 .

 11 Figure 11. Kernel map from for retail activities. Source (Mejia-Dorantes et al., forthcoming)

-

  Subjective location characteristics (current) -Desired location characteristics -Attitudes regarding the future tramway projects -Socio-demographic characteristicsTable 2. Questionnaire information Desires Type if you could improve the characteristics of your current location (for the benefit of your business), what would you changepublic transport (bus stops) Categorical Closer location to public transport (train stations) Categorical Closer location to public transport (tramway stations) Categorical More cycling infrastructure (either parking or cycling lanes) Categorical Private transport restrictions (pedestrian only streets) Categorical More work force availability Categorical More firms around you (agglomeration economies) Categorical More safety Categorical PERCEPTION (of future posibilities) and Attitudes "some people think that the two tramway lines will improve the area, for others, this is not true. For you, to what extent you agree with the following statements (1-Totally agree / 2-Partially agree / 3-Partially disagree / 4-Totally disagree) Variable

Table 1 .

 1 Sectors description

	Sector Description
	BC1 Manufacturing firms
	BC2 Construction firms
	BC3 Retail and related business activities
	BC4 Food service and hotel business activities
	BC5 Transport, storage and communication firms
	BC6 Finance, insurance, real estate, rental and other enterprise services
	BC7 Health and vet; social services; cultural and recrational; sport activities
	BC8 Others
	BC9 Vacant

Building the urban Future and Transit Oriented Development (BUFTOD) Rail and other modes, connecting with urban and regional development, Paris, France, Avril 2012

Although at the beginning this study is a cross-sectional one, the final goal could be a panel study. As explained by [START_REF] Richardson | Survey Methods for Transport Planning[END_REF], when carrying out a panel study it is selected a sample at the start of the time-series and it is retained the same individuals within the sample for the entire series of surveys. Therefore, it is possible to follow the changes of perceptions of the sample during a period of time. Disadvantages on panel data are well known, therefore, it will be only possible to observe if this is possible, once the first wave is carried out. Different public organizations in the region could continue with the different waves.

CONCLUSION AND DISCUSSION

The lack of availability of datasets should not be an obstacle to carry out this kind of research. There is in fact, an opportunity to develop new ideas and propose new approaches.

Literature shows that urban changes are not automatically perceived to be good by the people affected by them. However, in the analysis carried out by Loo (2002), people perceived that the retail sector was one of those industries best benefited by the construction on a new infrastructure.

Herein, apart from the methodology proposed by Mejia Dorantes ( 2011) to analyze firm patterns, we have proposed a questionnaire to focus on the retail sector using stated and revealed preference surveys. Different hypothesis will be tested, such as factors that influence location choice among different types of retail activity. Moreover, we will analyze the choices they have made along with hypothetical situations. We look forward to understand their location patterns and to elicit if the future tramways may trigger certain economic impacts. We believe that this approach is quite interesting and that the information that may be elicited from it will contribute to the literature.


Lucia Mejia-Dorantes

Odile Heddebaut Land

Dr Lucía Mejía-Dorantes

Dr Odile Heddebaut

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

INTRODUCTION

A transport station/stop generates a benefit around it, which people living or working close to it take special advantage from. This benefit is noticed in the real estate market, especially in the areas connected by an infrastructure. However, many times it is possible that negative externalities appear where most of the times they are due to noise, barrier and landscape effects or even a rise in criminality [START_REF] Cohen | Spatial hedonic models of airport noise, proximity, and housing prices[END_REF].

There are basically two different methods in literature to assess those positive and negative externalities: Repeated sales model and hedonic models. Hybrid models, which are the combination of those methodologies, have also been used. Repeated sales deals with a . regression of the difference in sale prices for the same set of homes and a set of time dummies. It is known also as BMN-model, first presented by [START_REF] Bailey | A Regression Method for Real Estate Price Index Construction[END_REF]. Its main problem is caused by the small sample data available and the difficulties to collect the data due to the required time frame. Compared to hedonic models, only few studies have been carried out using this method [START_REF] Sommervoll | Temporal Aggregation in Repeated Sales Models[END_REF].

The hedonic models refer to the implicit characteristics that assemble goods [START_REF] Rosen | Hedonic Prices and Implicit Markets: Product Differentiation in Pure Competition[END_REF]. In the case of land values, the idea is that property prices can be decomposed into its attributes, mainly into its structural, neighborhood and accessibility characteristics. In order to estimate the implicit price for each attribute, they are mathematically modeled as a linear regression model and its parameters are estimated by the Ordinary Least Squares (OLS) method.

It is important to notice that a house unit is fixed in its location and at the same time, it is influenced by the characteristics of the neighborhood, which are basically a set of geographically distributed data. Data members may as well be related to each other, which is known as spatial autocorrelation [START_REF] Dubin | Estimation of Regression Coefficients in the Presence of Spatially Autocorrelated Error Terms[END_REF][START_REF] Anselin | Spatial Econometrics: Methods and Models[END_REF]. Therefore, regression analysis should account for the fact that it may be spatial dependence either in the substantive part of the model or in the residual pattern [START_REF] Farber | Topology, Dependency Tests and Estimation Bias in Network Autoregressive Models[END_REF]. In order to accurately evaluate all these types of relations, Geographical Information Systems (GIS) become an important tool to increase the model's accuracy.

As an empirical analysis, this paper presents the case study of the "Bassin Minier", the coal mining area located at the region of Nord Pas-de-Calais, at the northern part of France. This area is well connected to the metropolis of Lille by the motor way A1 and it is also well connected by many roads and the motorway A21 from east to west. In total, the transport authority deserves 115 communes where seven out of them are the most important towns.

The aim of this paper is to analyze residential land values within these seven most important towns such as: Béthune, Bruay-la-Buissière, Carvin, Hénin-Beaumont, Lens, Liévin, and Noeux-les-Mines. Using a hedonic approach we assess if transport supply has impacted during the last years or not the land market in this area and only for the case of apartments. House prices are divided into three main characteristics: location to public transport, house amenities and neighborhood characteristics. In order to obtain a more accurate model, we use the real network distance to different neighborhood amenities, which has proved to diminish problems of spatial autocorrelation [START_REF] Mejia-Dorantes | Analysis of House Prices to Assess Economic Impacts of New Public Transport Infrastructure[END_REF]. This paper is divided into 5 main sections. After the introduction, section 2 presents the case study. Section 3 presents the methodology and data used. Section 4 presents the results. Finally, section 5 presents the concluding remarks.

.

THE BASSIN-MINIER

Characteristics of the region

This study focuses on the region the ex-coal mining area (ex-bassin minier), which is a deprived area of Nord-Pas-de-Calais, at the northern part of France, next to Belgium. Figures 1 and2 show the location of the territory under study. It is known as an area which has different socio-economic problems, such as: unemployment, the highest rate of mortality in France and poorly academic performances and failures, which have taken place after the conclusion of the coal mining activity era at the end of the 80's. , 2011). For the territories under study (known as "Communes") we present different facts. For example, how population has evolved during the last years from 1968 to 2008 for each main Commune, and the type of population (men/women) per age (figure 3). It is interesting to note that the difference among the percentage of women and men is higher for the people older than 65 years. This is related to the fact that men used to work in the mines, which was a very tough job and had a direct repercussion on men mortality. Figure 4 shows the jobs' qualifications in the region. Most of the people are non-skilled workers, and jobs are focused on commerce and services (see figure 5). Moreover, figure 6 shows that the rate of unemployment in the area has not importantly diminished in a decade, although according to the INSEE (2011) more firms have localized in the area over the last years.

w.maps.google.fr

Transportation

To ameliorate the socioeconomic situation of this area, different policies are being implemented aimed at regenerating the socioeconomic activities of these towns. For example the case of the new Museum of Louvre in Lens, and the improvement of public transportation, which intend to improve the economic, social and environmental situation of the zone. The Transport Authority (SMT Artois-Gohelle) planned to build two new tramway lines (see figure 7 and8). The project was announced at the end of 2008. There are still some areas where there may be changes in the final route of the tramways.

10th Transport Engineering Conference "Innovative and sustainable Transportation for the 21st Century", Congresso de Ingenieria del Transporte (CIT), Grenade, Espagne, 20-22 juin 2012

. However, different publications on the media show that people living or working in the area are afraid of the negative effects of these new lines. For example, recently Caratini (2012) reported that the owners of retail stores believe that due to the new tramway their customers will go away more easily to other areas for shopping, that in other towns, such as Bruay, the downtown will be even less crowded than nowadays.

.

METHODOLOGY AND DATA

Hedonic regression analysis are very popular for land market studies. Literature using this technique is extremely extensive. Different examples include the assessment property values due to commuter rail proximity in California [START_REF] Cervero | Transit's value-added effects: Light and commuter rail services and commercial land values[END_REF], the land market value-added due to accessibility to a Bus Rapid Transit [START_REF] Rodriguez | Value of accessibility to Bogotas bus rapid transit system[END_REF], the benefits of air quality in house prices (Won [START_REF] Kim | Measuring the benefits of air quality improvement: a spatial hedonic approach[END_REF], the impact in house prices due to high-speed rail accessibility [START_REF] References Andersson | Does high-speed rail accessibility influence residential property prices? Hedonic estimates from southern Taiwan[END_REF] or the benefits of proximity to the beach [START_REF] Parsons | A value capture property tax for financing beach nourishment projects: an application to Delaware's ocean beaches[END_REF], among others. The model presented herein is based in the following equation:

ε + = i i i β X P
(1) P i (N x 1) vector of registered prices at Notary X i (N x K) matrix with observations on structural or neighborhood characteristics β i :

(K x 1) vector of unknown regression coefficients εi is assumed to be a vector of independent and identically distribute (i.i.d.) error terms

As noticed by different authors, many times hedonic models exclude locational variables (for example, distance to relevant points such as CBD) [START_REF] Paez | Spatial association and heterogeneity issues in land price models[END_REF] which have proven to increase the probability of spatial error autocorrelation or heterogeneity [START_REF] Mejia-Dorantes | Analysis of House Prices to Assess Economic Impacts of New Public Transport Infrastructure[END_REF]. Therefore, in order to minimize those problems, we analyzed several variables, as shown in table 1.

The dataset used was obtained through the Conseil Régional Nord-Pas de Calais which provided us with all the facilities to work at their place to treat the information they have. We are using a time-series dataset, from 2005 to 2010. This dataset has all the selling prices registered at notary from the entire region and for all types of buildings and terrains. In this case, we are analyzing only apartments for the seven different towns involved in the future project of the tramway lines.

We realized that the datasets needs to be deeply analyzed and has many problems such as lack of information or wrong data. In this paper we present a preliminary analysis taking into account that each apartment unit had the following characteristics:

• 20 < Real surface < 120 • 50000 € < Registered value < 250,000 € A Geographic Information System (GIS) was used to calculate the real distance through the street network to the closest transport station or other amenity (itinerary). Many hedonic models are based on Euclidean distance, however, this approach overestimates the real distance (and the population served) to get to the closest station through the street network . [START_REF] Gutiérrez | Distance-measure impacts on the calculation of transport service areas using GIS[END_REF]. Moreover, an analysis by [START_REF] Mejia-Dorantes | Analysis of House Prices to Assess Economic Impacts of New Public Transport Infrastructure[END_REF] proves that this approach may be an alternative to a spatial econometric approach.

As we said before, it is a time-series dataset, in order to analyze it we are using dummy variables accounting for the year of registered sale. We also use dummy variables to account for the location of the apartment by different Communes.

PRELIMINARY RESULTS

The results indicate that the entire dataset needs to be further analyzed. Probably it is necessary to analyze detached or semi-detached housing instead of apartments. Years ago, in the mining . area, detached and semi-detached dwellings were built for engineers and mine workers (its location and type of house depended on their working status). Therefore, more centrically lodgments areas have an important amount of this type of construction. These houses are not in the private market (where our database comes from) because they belong to a public society "SOGINORPA" that manages this patrimony. .

CONCLUSIONS

Although the analysis should include detached and semi-detached housing it does give some hints. For example, it shows the relevance of being close to a rail station (gare in French), which is a negative coefficient and significant at 5%. Although the distance to the closest bus stop is not significant, it has a negative sign. We found other interesting results, for example, being close to sport activities, green zones, cultural buildings, poles of economic activity (industry). Conversely, we find positive signs for the distance to parking lots, and to clinics and hospitals (santé). The former may be related to the fact that, were apartments are located (new areas), they may have less parking restrictions on the streets or the buildings may have their own garage. In the case of distance to the hospitals, it seems to be a negative externality, which is in the same line of [START_REF] Mejia-Dorantes | Analysis of House Prices to Assess Economic Impacts of New Public Transport Infrastructure[END_REF]. Finally, it is not conclusive about the announcement effect of the new tramway lines although it shows that the trend notably increases in 2009. It may be due to the general trend rather than due to the tramway (and Louvre) projects.

As we said before, this study leaves many doors opened for further research. Moreover, a more sophisticated study to analyze the time-series taking into account the market trend should be tested. It might help to be more conclusive about the announcement effect.

Figure 1 -

 1 Figure 1 -Location of the city of Lille and the region of Pas-de-Calais

Figure 3 -

 3 Figure 3 -Population in the study area (INSEE)

Figure 4 -.Figure 5 -Figure 6 -

 456 Figure 4 -Job qualification in the region (INSEE)

Figure 7 -Figure 8 -

 78 Figure 7 -Tramway project from Béthune to Bruay-la-Buissière (Source: SMT)

Table 1 -Descriptive statistics of the variables tested for the different analyses

 1

	Variable	Obs	Mean	Std.Dev. Min	Max
	year registered
	2006		1507 0.171201 0.37681	0	1
	2007		1507 0.173855 0.379111	0	1
	2008		1507 0.155275 0.362287	0	1
	2009		1507 0.163238 0.369706	0	1
	2010		1507 0.078301 0.268734	0	1
	RealEstateValue		1507 130607.8 49628.35	50500 245000
	Commune (base outcome= Bethune)
	Bruay la Buissiere		1507 0.084273 0.27789	0	1
	Carvin		1507 0.048441 0.214767	0	1
	Henin Beaumont		1507 0.095554 0.294076	0	1
	Lens		1507 0.352356 0.477863	0	1
	Lievin		1507 0.050431 0.218906	0	1
	Nouex les Mines		1507 0.047777 0.213365	0	1
	Realsurface		1507 54.61181 20.88656	21	119
	Rooms		1507 2.339748 1.001343	0	7
	it_sante		1507 1110.031 622.9978 27.12043 4797.618
	it_science		1507 340.4016 199.8037 18.35634 1475.432
	it_culture		1507 467.2102 378.3164 3.251444 2111.612
	it_spacena		1507 2017.884 592.6364 389.4961 4004.452
	it_industr		1507 1377.863 503.2491 91.11511 2903.628
	id_sports		1507 805.5978 380.3213 40.38059 2986.229
	it_bus		1507 168.3819 106.5183 1.714499 741.5867
	it_aerop		1507 6849.708 3778.152 409.6963 18912.86
	it_parking		1507 559.9523 515.9897 1.35194 5259.967
	it_gares		1507 2684.221 1267.941 0.993151 6685.238
	dummy_tramway		1507 0.24154 0.428159	0	1

Table 2 -

 2 Regression analysis10th Transport Engineering Conference "Innovative and sustainable Transportation for the 21st Century", Congresso de Ingenieria del Transporte (CIT),Grenade, Espagne, 20-22 juin 2012

	Number of obs = 1507
	R-squared = 0.1073
	RealEstateValue	Coef.
	RealSurface	352.3154 *
	year registered (base year=2005)
	2006	-1421.36
	2007	8335.966 *
	2008	2803.115
	2009	6707.186
	2010	362.3277
	Commune (base Commune= Bethune)
	Bruay la Buissiere	11629.66
	Carvin	20260.75 *
	Henin Beaumont	-5102.02
	Lens	17460.45 *
	Lievin	-35095 *
	Noeux les Mines	-1395.47
	it_bus	-17.7181
	it_gares	-6.98556 *
	it_parking	6.915502 *
	it_industr	-2.41506
	it_sante	7.991265 *
	it_culture	-8.54634
	it_spacena	-18.7729 *
	id_sports	-3.08583
	Constant	159492.4 *
	* Significant at 5%

10th Transport Engineering Conference "Innovative and sustainable Transportation for the 21st Century", Congresso de Ingenieria del Transporte (CIT),Grenade, Espagne, 20-22 juin 2012

AKNOWLEDGEMENTS

The authors would like to thank the Regional Council of Nord-Pas-Calais for their support to carry out this research which is part of the SUIM project.