Robert Eymard
email: robert.eymard@univ-mlv.fr

Pierre Féron
email: pierre.feron@univ-mlv.fr

Thierry Gallouët
email: thierry.gallouet@univ-amu.fr

Raphaèle Herbin
email: raphaele.herbin@univ-amu.fr

Cindy Guichard
email: cindy.guichard@unice.fr

Gradient schemes for the Stefan problem

Keywords: Stefan problem, gradient schemes, uniqueness result, convergence study

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

We are interested here in the approximation of ū, solution to the so-called Stefan problem:

∂ t ū -∆ζ(ū) = f, in Ω × (0, T ) (1)
with the following initial condition: ū(x, 0) = u ini (x), for a.e. x ∈ Ω,

together with the homogeneous Dirichlet boundary condition:

ζ(ū(x, t)) = 0 on ∂Ω × (0, T ), (3)
under the following assumptions:

Ω is an open bounded connected polyhedral subset of R d , d ∈ N ⋆ and T > 0, (4a)

u ini ∈ L 2 (Ω) (4b) f ∈ L 2 (Ω × (0, T )), (4c)
The Stefan Problem (1)-( 2)- [START_REF] Amin | Convergence of American option values from discreteto continuous-time financial models[END_REF] arises in particular in the study of the heat equation in a nonmobile medium with two thermodynamical states, say solid and liquid. Denoting, for (x, t) ∈ Ω × (0, T ), by Θ(x, t) the temperature and by X(x, t) the normalised mass of liquid phase per unit volume (X(x, t) = 0 means that the medium is solid at point (x, t) and X(x, t) = 1 means that it is liquid), the internal energy u(x, t) can be modelled by ū(x, t) = H c Θ(x, t)+L f X(x, t), where H c denotes the heat capacity (assumed to be constant and identical for the liquid and the solid phases) and L f denotes the latent heat of fusion at the given fusion temperature Θ f . The heat equation can then be expressed by ∂ t ūdiv(λ∇Θ(x, t)) = f (x, t), in Ω × (0, T ), [START_REF] Bertsch | Degenerate diffusion and the Stefan problem[END_REF] where λ is the heat conductivity (assumed to be constant, isotropic and identical for the liquid and the solid phases). Dirichlet boundary conditions are assumed to be given on the temperature Θ, together with an initial condition given on ū. The thermodynamical equilibrium is then assumed to be expressed by (Θ(x, t) ≤ Θ f and X(x, t) = 0) or (Θ(x, t) = Θ f and 0 ≤ X(x, t) ≤ 1) or (Θ(x, t) ≥ Θ f and X(x, t) = 1)

a.e. (x, t) ∈ Ω × (0, T ).

(

) 6
International Journal on Finite Volumes

We may then remark that, under condition [START_REF] Carrillo | Entropy solutions for nonlinear degenerate problems[END_REF], X(x, t) and Θ(x, t) can be expressed by X(x, t) = ξ(ū(x, t)) and Θ(x, t) = 1 λ ζ(ū(x, t)), with

ξ(s) = min(max( s -H c Θ f L f , 0), 1) and ζ(s) = λ s -L f ξ(s) H c , ∀s ∈ R.
Plugging the preceding expression of Θ(x, t) as function of ū(x, t) in (5) leads to [START_REF] Alt | Quasilinear elliptic-parabolic differential equations[END_REF], in which the function ζ is Lipschitz continuous (it is in fact continuous and piecewise affine), nondecreasing, and constant on the interval [

H c Θ f , H c Θ f + L f ].
Many results are known in this situation, in particular the fact that, if f = 0 and if the measure of the set {x ∈ Ω, ū(x, t) ∈ [H c Θ f , H c Θ f + L f ]} (called the "mushy region") is equal to zero at t = 0, then it remains equal to zero for all t > 0, and a discontinuity of ū between the values H c Θ f and H c Θ f + L f may move inside the domain (see [START_REF] Bertsch | Degenerate diffusion and the Stefan problem[END_REF]). Therefore, Problem (1)-( 2)-(3) has to be considered in a weak sense, which includes the Rankine-Hugoniot condition for the conservation of ū in the case of discontinuities. A function ū is said to be a weak solution of Problem ( 1)-( 2)-( 3) if the following holds: We also recall that, in mathematical finance, some derivative of the price of an American option is the solution of some Stefan problem [START_REF] Amin | Convergence of American option values from discreteto continuous-time financial models[END_REF][START_REF] Berton | Finite volume methods for the valuation of american options[END_REF], whose computation is equivalent to the resolution of a variational inequality.

ū ∈ L 2 (Ω × (0, T )), ζ(ū) ∈ L 2 (0, T ; H 1 0 (Ω)), T 0 Ω (-ū(x, t)∂ t ϕ(x, t) + ∇ζ(ū)(x, t) • ∇ϕ(x, t)) dxdt - Ω u ini (x)ϕ(x, 0)dx = T 0 Ω f (x, t)ϕ(x, t)dxdt, ∀ϕ ∈ C ∞ c (Ω × [0, T [), (7)
Let us first mention that the first proof of existence of a solution to Problem [START_REF] Droniou | A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods[END_REF] has been provided in [START_REF] Alt | Quasilinear elliptic-parabolic differential equations[END_REF]. This proof relies on the convergence, as ε > 0 tends to zero, of the solution ūε of the following strictly parabolic regularization of (1):

∂ t ūε -∆(ζ(ū ε ) + εū ε ) = f (x, t), in Ω × (0, T ). ( 8
)
It is easy to prove that ∂ t ūε remains bounded in L 2 (0, T ; H -1 (Ω)), and that ζ(ū ε ) remains bounded in L 2 (0, T ; H 1 0 (Ω)) for all ε > 0. But no compactness can be deduced from these two bounds. One remarkable idea in [START_REF] Alt | Quasilinear elliptic-parabolic differential equations[END_REF]

is to show that ζ(ū ε ) - ζ(ū ε )(•, • + τ ) L 2 (Ω×(0,T ))
tends uniformly to 0 with τ . Then Kolmogorov's theorem allows to build a sequence (ε m ) m∈N converging to 0 such that the sequence (ζ(ū εm )) m∈N converges to some function χ in L 2 (Ω × (0, T )). A L ∞ (0, T ; L 2 (Ω)) bound on ūε allows to extract a subsequence from the sequence (ε m ) m∈N (identically denoted) such that there exists ū ∈ L ∞ (0, T ; L 2 (Ω)) such that (ū εm ) m∈N weakly converges to ū. Then Minty's trick (which is available thanks to the monotonicity of ζ) provides that χ = ζ(ū) (this is detailed in Lemma B.1).

These ideas have been used in the study of the convergence of a finite volume method, done in [START_REF] Eymard | Finite volumes and nonlinear diffusion equations[END_REF]. In this paper, ∆-admissible meshes are used, in the sense that in each control volume K, there exists a given point x K such that, for two neighbouring control volumes K and L sharing the interface σ KL , the line (x K , x L ) is parallel to the normal vector n K,L to σ KL , oriented from x K to x L . Then the approximation of ū in the control volume K (resp. L) is denoted by u K (resp. u L ), and the approximation of ∇ζ(ū) • n K,L at this interface is given by the so-called two-point flux approximation

ζ(u L )-ζ(u K ) d(x K ,x L )
. A great advantage can be drawn from this approximation: multiplying the discrete scheme by u K and summing on the control volumes leads to expressions such as

(ζ(u L ) -ζ(u K ))(u L -u K ).
Thanks to the Cauchy-Schwarz inequality, it is easy to prove that this expression is greater than (η(u L )η(u K )) 2 , where η is a primitive of (ζ ′ ) 1/2 . Then functional properties (including a discrete maximum principle), similar to that of the continuous problem, can be shown, and the weak convergence of u and ∇η(u) in L 2 , as well as the strong convergence of η(u) in L 2 thanks to monotony arguments and Kolmogorov's theorem, are proved. Stronger convergence properties (i.e. convergence of u and ∇η(u) in L 2 ) could then easily be shown, following the ideas developed in the present paper (see Lemma 3.2, Theorem 3.3 and Remark 1).

Unfortunately, if the diffusion term ∆ζ(ū) is approximated by another method than two-point flux approximation, say a general finite element method or mixed finite element method, all the results obtained from the multiplication by u no longer hold.

Concerning the uniqueness of the solution of ( 7), many results are given in the literature under various hypotheses. A uniqueness theorem was proved in [START_REF] Eymard | Finite volumes and nonlinear diffusion equations[END_REF] with more restrictive assumptions on Ω than (4), and a uniqueness theorem for nonlinear convection-diffusion problems was proved in [START_REF] Carrillo | Entropy solutions for nonlinear degenerate problems[END_REF]. This uniqueness result was extended to the notion of entropy process solution in [START_REF] Eymard | Convergence of a finite volume scheme for nonlinear degenerate parabolic equations[END_REF], useful to prove the convergence of a numerical scheme, which extends the two-point flux approximation [START_REF] Eymard | Finite volumes and nonlinear diffusion equations[END_REF]. Therefore we have to check the uniqueness of the solution under the precise assumptions (4) made here. This is done in Theorem 4.1, the proof of which provides the opportunity to study the convergence of some gradient schemes to regular linear parabolic problems.

The purpose of this paper is to study the convergence of the so-called gradient schemes for the approximation of the Stefan problem given by its weak form [START_REF] Droniou | A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods[END_REF]. This framework includes, for example, the general case of the conforming finite elements (see [START_REF] Amiez | On a numerical approach to Stefan-like problems[END_REF], [START_REF] Elliott | Error analysis of the enthalpy method for the stefan problem[END_REF], [START_REF] Nochetto | Approximation of degenerate parabolic problems using numerical integration[END_REF] for the use of this specific method). It also includes the case of mixed finite element methods Gradient schemes have been studied in [START_REF] Eymard | Small-stencil 3d schemes for diffusive flows in porous media[END_REF] for linear elliptic problems, and in [START_REF] Droniou | Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations[END_REF] in the case of nonlinear Leray-Lions-type elliptic and parabolic problems. For such general methods, the monotony properties obtained from the two-point flux approximation cannot be used, and the multiplication by the solution u of the discrete scheme is of no use: in order to get estimates, one can only multiply by ζ(u). We have therefore introduced the additional hypothesis (4e) (which is not restrictive in practice, since it concerns the values of ζ(u) for large u, whereas u remains generally bounded) in order that a L 2 estimate on ζ(u) implies one on u.

This paper is organised as follows. We first apply the gradient discretisation tools (described in an appendix) to the Stefan Problem in Section 2, and derive some estimates, which are used in Section 3 for the convergence analysis. Then, turning to the uniqueness of the solution of [START_REF] Droniou | A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods[END_REF], we first establish the existence of the continuous solution of a linear parabolic problem, showing the regularity which is further needed in the course of the uniqueness proof (see Section 4). Finally, numerical examples show the behaviour of the VAG scheme [START_REF] Eymard | Small-stencil 3d schemes for diffusive flows in porous media[END_REF] which presents some interesting characteristics for coupled flows in porous media (see Section 5).

Approximation of the Stefan problem by space-time gradient discretisations

Let D = (X D,0 , Π D , ∇ D , (t (n) ) n=0,...,N ) be a space-time discretisation in the sense of Definition A.9 such that Π D is a piecewise constant function reconstruction in the sense of Definition A.8. We define the following (implicit) scheme for the discretisation of Problem [START_REF] Droniou | A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods[END_REF]. We consider a sequence (u (n) ) n=0,...,N such that:

                     u (0) ∈ X D,0 , u (n+1) ∈ X D,0 , δ (n+ 1 2 ) D u = Π D u (n+1) -u (n) δt (n+ 1 2 ) , Ω δ (n+ 1 2 ) D u(x)Π D v(x) + ∇ D ζ(u (n+1) )(x) • ∇ D v(x) dx = 1 δt (n+ 1 2 ) t (n+1) t (n) Ω f (x, t)Π D v(x)dxdt, ∀v ∈ X D,0 , ∀n = 0, . . . , N -1. (9)
We again use the notations Π D and ∇ D for the definition of space-time dependent functions (note that we define these functions for all t ∈ [0, T ]):

Π D u(x, 0) = Π D u (0) (x)for a.e. x ∈ Ω, Π D u(x, t) = Π D u (n+1) (x) Π D ζ(u)(x, t) = Π D ζ(u (n+1) )(x) ∇ D ζ(u)(x, t) = ∇ D ζ(u (n+1) )(x), for a.e. x ∈ Ω, ∀t ∈ (t (n) , t (n+1) ], ∀n = 0, . . . , N -1. (10)
We also denote

δ D u(x, t) = δ (n+ 1 2 ) D u(x), for a.e. (x, t) ∈ Ω × (t (n) , t (n+1) ), ∀n = 0, . . . , N -1. (11)
We finally introduce the function

Z(s) = s 0 ζ(x)dx, ∀s ∈ R. ( 12
)
which is used several times in the convergence proofs. We then have

Z(s) = s 0 ζ(x)dx = s 0 (ζ(x) -ζ(0))dx ≤ L ζ s 0 xdx = L ζ s 2 2 , (13)
and, from Hypotheses (4d) and (4e),

Z(s) ≥ s 0 ζ(x) ζ ′ (x) L ζ dx ≥ ζ(s) 2 2L ζ ≥ (a|s| -b) 2 2L ζ . ( 14
)
Lemma 2.1 (L ∞ (0, T ; L 2 (Ω)) estimate, discrete L 2 (0, T ; H 1 0 (Ω)) estimate and existence of a discrete solution)

Under Hypotheses (4), let D = (X D,0 , Π D , ∇ D , (t (n) ) n=0,...,N ) be a space-time gradient discretisation in the sense of Definition A.9 such that Π D is a piecewise constant function reconstruction in the sense of Definition A.8. Then there exists at least one solution to Scheme [START_REF] Elliott | Error analysis of the enthalpy method for the stefan problem[END_REF], which satisfies

T 0 Ω |∇ D ζ(u)(x, t)| 2 dxdt + Ω (Z(Π D u N (x)) -Z(Π D u 0 (x)))dx ≤ T 0 Ω f (x, t)Π D ζ(u)(x, t)dxdt. (15)
Moreover, there exists

C 1 > 0, only depending on L ζ , a, b, C P > C D , C ini > u ini -Π D u (0) L 2 (Ω)
, f such that, for any solution u to this scheme,

Π D ζ(u) L ∞ (0,T ;L 2 (Ω)) ≤ C 1 , and Π D u L ∞ (0,T ;L 2 (Ω)) ≤ C 1 , (16)
and

∇ D ζ(u) L 2 (Ω×(0,T )) d ≤ C 1 . (17)

Proof

Before showing the existence of at least one discrete solution to Scheme [START_REF] Elliott | Error analysis of the enthalpy method for the stefan problem[END_REF], let us first prove [START_REF] Nochetto | Approximation of degenerate parabolic problems using numerical integration[END_REF] and (17). From the properties of function Z defined by [START_REF] Eymard | Finite volumes and nonlinear diffusion equations[END_REF], and using

δ (n+ 1 2 ) D u Π D ζ(u (n+1) ) ≥ Π D Z(u (n+1) ) -Π D Z(u (n) ).
We then let v = δt (n+ 1 2 ) ζ(u (n+1) ) in ( 9), and we sum the obtained equation on n = 0, . . . , m -1 for a given m = 1, . . . , N . Accounting for the above inequality yields [START_REF] Gallouët | Compactness of discrete approximate solutions to parabolic pdes-application to a turbulence model[END_REF]. Thanks to the Cauchy-Schwarz inequality, we get that

Π D Z(u (m) ) L 1 (Ω) + t (m) 0 ∇ D ζ(u)(•, t) 2 L 2 (Ω) d dt ≤ f L 2 (Ω×(0,t (m) )) Π D ζ(u) L 2 (Ω×(0,t (m) )) + Π D Z(u (0) ) L 1 (Ω) ,
which in turn yields, thanks to the Young inequality, ( 13) and ( 14),

1 2L ζ Π D ζ(u (m) ) 2 L 2 (Ω) + t (m) 0 ∇ D ζ(u)(•, t) 2 L 2 (Ω) d dt ≤ C 2
Lemma 2.2 (Uniqueness results on the discrete solution) Under Hypotheses (4), let D = (X D,0 , Π D , ∇ D , (t (n) ) n=0,...,N ) be a space-time gradient discretisation in the sense of Definition A.9 such that Π D is a piecewise constant function reconstruction in the sense of Definition A.8. Let u (0) ∈ X D,0 be given, and, for n = 0, . . . , N -1, let u (n+1) ∈ X D,0 be such that (9) holds. Then, for all n = 0, . .

. , N -1, Π D u (n+1) ∈ L 2 (Ω) and ζ(u (n+1) ) ∈ X D,0 are unique.

Proof

Let us consider two solutions, denoted u (n+1) , u (n+1) ∈ X D,0 , for some n = 0, . . . , N -1, such that (9) holds with Π D u (n) (x) = Π D u (n) (x), for a.e. x ∈ Ω. We then subtract the corresponding equation with u (n+1) to that with u (n+1) . We get 18). Using Hypothesis (4d), we may write that

Ω Π D (u (n+1) -u (n+1) )(x) δt (n+ 1 2 ) Π D v(x) +∇ D (ζ(u (n+1) ) -ζ( u (n+1) ))(x) • ∇ D v(x) dx = 0, ∀v ∈ X D,0 . (18) We let v = ζ(u (n+1) ) -ζ( u (n+1) ) in (
(Π D (u (n+1) -u (n+1) )(x))Π D (ζ(u (n+1) ) -ζ( u (n+1) ))(x) = (Π D u (n+1) (x) -Π D u (n+1) (x))(ζ(Π D u (n+1) (x)) -ζ(Π D u (n+1) (x))) ≥ 0, which implies that Ω |∇ D (ζ(u (n+1) ) -ζ( u (n+1) ))(x)| 2 dx = 0,
and therefore that ζ(u (n+1) ) = ζ( u (n+1) ). We then get, from (18), that

Ω Π D (u (n+1) -u (n+1) )(x) δt (n+ 1 2 ) Π D v(x)dx = 0, ∀v ∈ X D,0 . It now suffices to let v = u (n+1) -u (n+1) in the preceding equation, to get that Π D u (n+1) (x) = Π D u (n+1) (x) for a.e. x ∈ Ω.
In order to fulfil the hypotheses of discrete Alt-Luckhaus theorem B.3, let us study the time translates.

Lemma 2.3 (Estimate on the time translates)

Under Hypotheses (4), let D = (X D,0 , Π D , ∇ D , (t (n) ) n=0,...,N ) be a space-time gradient discretisation in the sense of Definition A.9 such that Π D is a piecewise constant function reconstruction in the sense of Definition A.8. Then there exists

C 2 > 0, only depending on L ζ , a, b, C P > C D , C ini > u ini -Π D u (0)
L 2 (Ω) , f such that, for any solution u to Scheme [START_REF] Elliott | Error analysis of the enthalpy method for the stefan problem[END_REF],

Π D ζ(u)(•, • + τ ) -Π D ζ(u)(•, •) 2 L 2 (Ω×(0,T -τ )) ≤ C 2 (τ + δt), ∀τ ∈ (0, T ). ( 19
)

Proof

In order to make the proof clear, let us give its principle, assuming that a solution ū of the continuous equation ( 1) is regular enough. We write the time translate of this solution in L 2 (Ω × (0, Tτ )), for a step τ ∈ (0, T ). We first note that

(ζ(ū(x, t + τ )) -ζ(ū(x, t))) 2 ≤ L ζ (ζ(ū(x, t + τ )) -ζ(ū(x, t)))(ū(x, t + τ ) -ū(x, t)), which gives, using (1), (ζ(ū(x, t + τ )) -ζ(ū(x, t))) 2 ≤ L ζ (ζ(ū(x, t + τ )) -ζ(ū(x, t))) τ 0 ∂ t ū(x, t + s)ds ≤ L ζ (ζ(ū(x, t + τ )) -ζ(ū(x, t))) τ 0 (∆ζ(ū(x, t + s)) + f (x, t + s))ds.

Therefore we have

T -τ 0 Ω (ζ(ū(x, t + τ )) -ζ(ū(x, t))) 2 dxdt ≤ L ζ τ 0 T -τ 0 Ω (ζ(ū(x, t + τ )) -ζ(ū(x, t))) (∆ζ(ū(x, t + s)) + f (x, t + s))dxdtds ≤ L ζ τ 0 T -τ 0 Ω (-∇ζ(ū(x, t + τ )) + ∇ζ(ū(x, t))) • ∇ζ(ū(x, t + s))dxdtds +L ζ τ 0 T -τ 0 Ω (ζ(ū(x, t + τ )) -ζ(ū(x, t)))f (x, t + s)dxdtds.
Each product ab of the above right hand side is then bounded by 1 2 (a 2 + b 2 ), which allows to conclude, thanks to the continuous estimates similar to [START_REF] Nochetto | Approximation of degenerate parabolic problems using numerical integration[END_REF], that

T -τ 0 Ω (ζ(ū(x, t + τ )) -ζ(ū(x, t))) 2 dxdt ≤ τ C.
Let us now use the same ideas for the proof of (19). Let τ ∈ (0, T ). Similarly using that L ζ is a Lipschitz constant of ζ and ζ is nondecreasing, and using (58), the following inequality holds:

Ω×(0,T -τ ) Π D ζ(u)(x, t + τ ) -Π D ζ(u)(x, t) 2 dxdt ≤ L ζ T -τ 0 A(t)dt, (20)
where, for almost every t ∈ (0, Tτ ),

A(t) = Ω Π D ζ(u)(x, t + τ ) -Π D ζ(u)(x, t) Π D u(x, t + τ ) -Π D u(x, t) dx. Let t ∈ (0, T -τ ). Denoting n 0 (t), n 1 (t) = 0, . . . , N -1 such that t (n 0 (t)) ≤ t < t (n 0 (t)+1) and t (n 1 (t)) ≤ t + τ < t (n 1 (t)+1
) , we may write

A(t) = Ω Π D ζ(u (n 1 (t)+1) )(x) -Π D ζ(u (n 0 (t)+1) )(x) × n 1 (t) n=n 0 (t)+1 δt (n+ 1 2 ) δ (n+ 1 2 ) D u(x) dx,
which also reads

A(t) = Ω Π D ζ(u (n 1 (t)+1) )(x) -Π D ζ(u (n 0 (t)+1) )(x) × N -1 n=1 χ n (t, t + τ )δt (n+ 1 2 ) δ (n+ 1 2 ) D u(x) dx, (21) with χ n (t, t + τ ) = 1 if t (n) ∈ (t, t + τ ] and χ n (t, t + τ ) = 0 if t (n) / ∈ (t, t + τ ]. Letting v = ζ(u (n 1 (t)+1) ) -ζ(u (n 0 (t)+1
) ) in Scheme ( 9), we get from ( 21)

A(t) = N -1 n=1 χ n (t, t + τ ) × Ω t (n+1) t (n) f (x, t)dt Π D ζ(u (n 1 (t)+1) )(x) -Π D ζ(u (n 0 (t)+1) )(x) dx - N -1 n=1 χ n (t, t + τ )δt (n+ 1 2 ) × Ω ∇ D ζ(u (n+1) )(x) • ∇ D ζ(u (n 1 (t)+1) )(x) -∇ D ζ(u (n 0 (t)+1) )(x) dx.
Using the inequality ab ≤ 1 2 (a 2 + b 2 ), this yields:

A(t) ≤ 1 2 A 0 (t) + 1 2 A 1 (t) + A 2 (t) + A 3 (t), (22)
with

A 0 (t) = N -1 n=1 χ n (t, t + τ )δt (n+ 1 2 ) Ω |∇ D ζ(u (n 0 (t)+1) )(x)| 2 dx, A 1 (t) = N -1 n=1 χ n (t, t + τ )δt (n+ 1 2 ) Ω |∇ D ζ(u (n 1 (t)+1) )(x)| 2 dx, A 2 (t) = N -1 n=1 χ n (t, t + τ )δt (n+ 1 2 ) Ω |∇ D ζ(u (n+1) )(x)| 2 dx,
and

A 3 (t) = N -1 n=1 χ n (t, t + τ ) Ω t (n+1) t (n) f (x, t)dt Π D ζ(u (n 1 (t)+1) )(x) -Π D ζ(u (n 0 (t)+1) )(x) dx.
International Journal on Finite Volumes Applying [14, Proposition 9.3] yields

T -τ 0 A 0 (t)dt ≤ (τ + δt) ∇ D ζ(u) 2 L 2 (Ω×(0,T ))
and

T -τ 0 A 1 (t)dt ≤ (τ + δt) ∇ D ζ(u) 2 L 2 (Ω×(0,T )) , (23)
as well as

T -τ 0 A 2 (t)dt ≤ τ ∇ D ζ(u) 2 L 2 (Ω×(0,T )) , (24)
and, with again the application of [14, Proposition 9.3], and using the Young inequality as well as ( 16), we obtain

T -τ 0 A 3 (t)dt ≤ (τ + δt)T C 2 1 + τ f 2 L 2 (Ω×(0,T )) . (25)
Using inequalities (20), ( 22), ( 23), ( 24) and (25), we conclude the proof of (19).

Convergence results

Theorem 3.1 Let Hypotheses (4) be fulfilled. Let (D m ) m∈N be a consistent sequence of spacetime gradient discretisations in the sense of Definition A.10, such that the associated sequence of approximate gradient approximations is limit-conforming (Definition A.4) and compact (Definition A.5, it is then coercive in the sense of Definition A.2), and such that, for all m ∈ N, Π Dm is a piecewise constant function reconstruction in the sense of Definition A.8. For any m ∈ N, let u m be a solution to Scheme [START_REF] Elliott | Error analysis of the enthalpy method for the stefan problem[END_REF], such that

u ini -Π Dm u (0) m L 2 (Ω) → 0 as m → ∞. Then there exists u ∈ L 2 (Ω × (0, T )) such that 1. Π Dm u m weakly converges in L 2 (Ω × (0, T )) to u as m → ∞, 2. Π Dm ζ(u m ) converges in L 2 (Ω × (0, T )) to ζ(u) as m → ∞, 3. ζ(u) ∈ L 2 (0, T ; H 1 0 (Ω)) and ∇ Dm ζ(u m ) weakly converges in L 2 (Ω × (0, T )) d to ∇ζ(u) as m → ∞,
and u is the unique weak solution of Problem [START_REF] Droniou | A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods[END_REF].

Proof

We consider, for all m ∈ N, the spaces B m = Π Dm X Dm,0 ⊂ L 2 (Ω), embedded with the norm

w Bm = inf{ u Dm , Π Dm u = w}, ∀w ∈ B m , ∀m ∈ N.
The compactness hypothesis of (D m ) m∈N allows to enter into the framework of discrete Alt-Luckhaus' theorem B.3.

Thanks to Lemma 2.1, we get that Hypothesis (h1) of Theorem B.3 is satisfied. We classically identify L 2 (0, T ; L 2 (Ω)) and L 2 (Ω × (0, T )), and we define, for τ ∈ (0, T ),

g m (τ ) = Π D ζ(u)(•, • + τ ) -Π D ζ(u)(•, •) L 2 (Ω×(0,T -τ )) and g(τ, t) = (C 2 (τ + δt)) 1/2
. Thanks to Lemma 2.3 and to the continuity in means theorem (which implies that g m is continuous in 0), we may apply Lemma B.2 and deduce that hypothesis (h2) of Theorem B.3 also holds. Therefore, there exists χ ∈ L 2 (Ω × (0, T )) such that Π Dm ζ(u m ) converges, up to the extraction of a subsequence, to χ in L 2 (Ω × (0, T )). Again applying Lemma 2.1, we get that there exists u ∈ L 2 (Ω × (0, T )) such that Π Dm u m weakly converges, up again to the extraction of a subsequence, to u in L 2 (Ω × (0, T )). Thanks to Lemma B.1, we conclude that χ(x, t) = ζ(u(x, t)) for a.e. (x, t) ∈ Ω × (0, T ). It now remains to prove that u is the weak solution of Problem [START_REF] Droniou | A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods[END_REF].

Let m ∈ N, and let us denote D = D m (belonging to the above subsequence) and drop some indices m for the simplicity of the notation.

Let

ϕ ∈ C ∞ c ([0, T )) and w ∈ C ∞ c (Ω) , and let v ∈ X D,0 be such that v = argmin z∈X D,0 S D (w).
We take as test function v in (9) the function δt (n+ 1 2 ) ϕ(t (n) )v, and we sum the resulting equation on n = 0, . . . , N -1. we get

T (m) 1 + T (m) 2 = T (m) 3 , (26)
with

T (m) 1 = N -1 n=0 δt (n+ 1 2 ) ϕ(t (n) ) Ω δ (n+ 1 2 ) D u(x)Π D v(x)dx, T (m) 2 = N -1 n=0 δt (n+ 1 2 ) ϕ(t (n) ) Ω ∇ D ζ(u (n+1) )(x) • ∇ D v(x)dx, and
T (m) 3 = N -1 n=0 ϕ(t (n) ) t (n+1) t (n) Ω f (x, t)Π D v(x)dxdt.
Writing

T (m) 1 = - T 0 ϕ ′ (t) Ω Π D u(x, t)Π D v(x)dxdt -ϕ(0) Ω Π D u (0) (x)Π D v(x)dx, we get that lim m→∞ T (m) 1 = - T 0 ϕ ′ (t) Ω u(x, t)w(x)dxdt -ϕ(0) Ω u ini (x)w(x)dx.
We also immediately get that Since the set

lim m→∞ T (m) 2 = T 0 ϕ(t) Ω ∇ζ(u)(x, t) • ∇w(x)
T = { q i=1 ϕ i (t)w i (x) : q ∈ N, ϕ i ∈ C ∞ c [0, T ), w i ∈ C ∞ c (Ω)} is dense in C ∞ c (Ω × [0, T ))
, we conclude the proof of Theorem 3.1 thanks to the uniqueness of the limit solution proved in Theorem 4.1 below.

The next lemma states a continuous property, which is used below for proving that the convergence of ∇ Dm ζ(u m ) to ∇ζ(u) is in fact strong.

Lemma 3.2

Under Hypotheses (4), let u be a solution of [START_REF] Droniou | A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods[END_REF]. Then the following property holds:

T 0 Ω |∇ζ(u)(x, t)| 2 dxdt + Ω (Z(u(x, T )) -Z(u ini (x)))dx = T 0 Ω f (x, t)ζ(u(x, t))dxdt. (27)

Proof

We first notice that [START_REF] Droniou | A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods[END_REF] implies that ∂ t u ∈ L 2 (0, T ; H -1 (Ω)) (and therefore u ∈ C([0, T ], H -1 (Ω)) with u(0) = u ini ) and that we can write

T 0 ∂ t u(t), w(t) + Ω ∇ζ(u) • ∇wdx dt = T 0 Ω f wdxdt, ∀w ∈ L 2 (0, T ; H 1 0 (Ω)), (28)
denoting by •, • the duality product (H -1 (Ω), H 1 0 (Ω)). We prolong u by u(t) = u ini for all t ≤ 0, and by u(t) = u(T ) for all t ≥ T .

Let h ∈ (0, T ). We consider α h ∈ L 2 (R; H -1 (Ω)) defined by

α h (t), w = 1 h t t-h ∂ t u(s), w ds = Ω 1 h (u(x, t) -u(x, t -h))w(x)dx, for t ∈ R, ∀w ∈ H 1 0 (Ω). Then α h tends to ∂ t u in L 2 (R; H -1 (Ω)) as h → 0, which implies that lim h→0 R Ω 1 h (u(x, t) -u(x, t -h))w(x, t)dxdt + T 0 Ω ∇ζ(u) • ∇wdxdt = T 0 Ω f wdxdt, ∀w ∈ L 2 (R; H 1 0 (Ω)).
Let us take w = ζ(u) in the above equation. We get

lim h→0 R Ω 1 h (u(x, t) -u(x, t -h))ζ(u(x, t))dxdt + T 0 Ω |∇ζ(u)| 2 dxdt = T 0 Ω f ζ(u)dxdt. International Journal on Finite Volumes Again observing that b a ζ(x)dx = Z(b) -Z(a) = ζ(b)(b -a) - b a ζ ′ (x)(x -a)dx, which implies Z(b) -Z(a) ≤ ζ(b)(b -a), we get that R Ω 1 h (u(x, t) -u(x, t -h))ζ(u(x, t))dxdt ≥ R Ω 1 h (Z(u(x, t)) -Z(u(x, t -h)))dxdt. Since R Ω 1 h (Z(u(x, t)) -Z(u(x, t -h)))dxdt = 1 h T +h T Ω Z(u(x, T ))dxdt - 1 h h 0 Ω Z(u ini (x))dxdt = Ω (Z(u(x, T )) -Z(u ini (x)))dx.
We may then pass to the limit h → 0. We then obtain

Ω (Z(u(x, T )) -Z(u ini (x)))dx + T 0 Ω |∇ζ(u)| 2 dxdt ≤ T 0 Ω f ζ(u)dxdt.
(29) We then follow the same reasoning, defining w = ζ(u) and

β h ∈ L 2 (R; H -1 (Ω)) by β h (t), w = 1 h t+h t ∂ t u(s), w ds, for t ∈ R, ∀w ∈ H 1 0 (Ω).
Remarking that

b a ζ(x)dx = Z(b) -Z(a) = ζ(a)(b -a) + b a ζ ′ (x)(b -x)dx, which implies Z(b) -Z(a) ≥ ζ(a)(b -a), we get that R Ω 1 h (u(x, t + h) -u(x, t))ζ(u(x, t))dxdt ≤ R Ω 1 h (Z(u(x, t + h)) -Z(u(x, t)))dxdt. Since R Ω 1 h (Z(u(x, t + h)) -Z(u(x, t)))dxdt = 1 h T T -h Ω Z(u(x, T ))dxdt - 1 h 0 -h Ω Z(u ini (x))dxdt = Ω (Z(u(x, T )) -Z(u ini (x)))dx.
We may then pass to the limit h → 0. We thus get

Ω (Z(u(x, T )) -Z(u ini (x)))dx + T 0 Ω |∇ζ(u)| 2 dxdt ≥ T 0 Ω f ζ(u)dxdt,
which, in addition to (29), concludes the proof of (27).

We may now state the strong convergence of ∇ D ζ(u).

International Journal on Finite Volumes Theorem 3.3 (Strong convergence of the numerical scheme) Under the same hypotheses as those of Theorem 3.1, Then there exists u ∈

L 2 (Ω × (0, T )) ∩ C([0, T ], H -1 (Ω)) such that 1. Π Dm u m (t) weakly converges in L 2 (Ω) to u(t), for all t ∈ [0, T ], as m → ∞, 2. Π Dm ζ(u m )(t) converges in L 2 (Ω) to ζ(u)(t), for all t ∈ [0, T ], as m → ∞, 3. ζ(u) ∈ L 2 (0, T ; H 1 0 (Ω)) and ∇ Dm ζ(u m ) converges in L 2 (Ω × (0, T )) d to ∇ζ(u) as m → ∞,
and u is the unique weak solution of Problem [START_REF] Droniou | A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods[END_REF]. We get from Scheme (9) and from the estimates given in Lemma 2.1 the following property: there exists C 3 , only depending on the data introduced in 4, such that, for all 0 ≤ s ≤ t,

Proof

Ω (Π Dm u m (x, t) -Π Dm u m (x, s))Π Dm w m (x)dx ≤ (t -s + 2δt m ) 1/2 C 3 w m Dm ,
which gives, thanks to [START_REF] Nochetto | Approximation of degenerate parabolic problems using numerical integration[END_REF],

Ω (Π Dm u m (x, t) -Π Dm u m (x, s))ϕ(x)dx ≤ (t -s + 2δt m ) 1/2 C 3 w m Dm + 2C 1 ϕ -Π Dm w m L 2 (Ω) . Using √ a + b ≤ √ a + √ b, we get Ω (Π Dm u m (x, t) -Π Dm u m (x, s))ϕ(x)dx ≤ g(t -s, h ϕ m ), with g(a, b) = √ aC 4 + b, with C 4 = C 3 max m w m Dm and h ϕ m = (2δt m ) 1/2 C 4 + 2C 1 ϕ -Π Dm w m L 2 (Ω)
. We then may apply Theorem B.4 (given in the Appendix), proving that, for all t ∈ [0, T ], Π Dm u m (t) tends to u(t) for the weak topology of L 2 (Ω).

Thanks to the convexity of Z, we then get

Ω Z(u(x, T ))dx ≤ lim inf m→∞ Ω Z(Π Dm u m (x, T ))dx,
and we classically have, from the weak convergence property of

∇ Dm ζ(u m ), T 0 Ω |∇ζ(u)(x, t)| 2 dxdt ≤ lim inf m→∞ T 0 Ω |∇ Dm ζ(u m )(x, t)| 2 dxdt.
International Journal on Finite Volumes Therefore, we may pass to the limit sup as m → ∞ in [START_REF] Gallouët | Compactness of discrete approximate solutions to parabolic pdes-application to a turbulence model[END_REF], and subtract (27). We thus obtain lim sup

m→∞ T 0 Ω |∇ Dm ζ(u m )(x, t)| 2 dxdt + Ω Z(Π Dm u m (x, T ))dx ≤ T 0 Ω |∇ζ(u)(x, t)| 2 dxdt + Ω Z(u(x, T ))dx.
This shows that

lim m→∞ T 0 Ω |∇ Dm ζ(u m )(x, t)| 2 dxdt = T 0 Ω |∇ζ(u)(x, t)| 2 dxdt,
which concludes the proof of the convergence of

∇ Dm ζ(u m ) to ∇ζ(u) in L 2 (Ω × (0, T )) d , and lim m→∞ Ω Z(Π Dm u m (x, T ))dx = Ω Z(u(x, T ))dx. ( 30
)
Note that the preceding limit result holds in fact for all t 0 ∈ [0, T ] instead of T . We then remark that, thanks to the monotony of ζ, there holds [10, Lemma 2.3]

1 2L ζ (ζ(a) -ζ(b)) 2 ≤ b a (ζ(s) -ζ(a))ds = Z(b) -Z(a) -ζ(a)(b -a), ∀a, b ∈ R.
We then deduce

1 2L ζ Ω (ζ(Π Dm u m (x, t 0 )) -ζ(u(x, t 0 ))) 2 dx ≤ Ω (Z(Π Dm u m (x, t 0 )) -Z(u(x, t 0 )))dx -Ω ζ(u(x, t 0 ))(Π Dm u m (x, t 0 ) -u(x, t 0 ))dx.
Since the right hand side tends to zero using (30) and the weak convergence of

Π Dm u m (•, t 0 ) to u(•, t 0 ), we conclude the convergence in L 2 (Ω) of ζ(Π Dm u m (•, t 0 )) to ζ(u(•, t 0 )), hence concluding the proof.
Remark 1 In the case of where a two-point flux approximation is used instead of a gradient scheme, one can get with the same arguments that the approximation of u is strongly convergence at all times to the weak solution.

Proof of uniqueness by a regularised adjoint problem

Let us state and prove the uniqueness theorem, admitting some existence theorem proven below. The method is similar to that of [START_REF] Eymard | Finite volumes and nonlinear diffusion equations[END_REF], where the existence result for the adjoint problem is given under some regularity hypotheses on Ω which are not done in this paper.

Theorem 4.1 Under Hypotheses (4), there exists at most one solution to [START_REF] Droniou | A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods[END_REF].

International Journal on Finite Volumes

Proof

Let u 1 and u 2 be two solutions of Problem [START_REF] Droniou | A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods[END_REF]. We set u d = u 1u 2 . Let us also define, for all (x, t)

∈ Ω × R ⋆ + , q(x, t) = ζ(u 1 (x, t)) -ζ(u 2 (x, t)) u 1 (x, t) -u 2 (x, t) if u 1 (x, t) =
u 2 (x, t), else q(x, t) = 0. For all T ∈ R ⋆ + and for all ψ ∈ L 2 (0, T ; H 1 0 (Ω)) with ∂ t ψ ∈ L 2 (Ω × (0, T )) and ∆ψ ∈ L 2 (Ω × (0, T )), we deduce from [START_REF] Droniou | A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods[END_REF], approximating

ψ by regular functions ϕ ∈ C ∞ c (Ω × [0, T )), that T 0 Ω u d (x, t) ∂ t ψ(x, t) + q(x, t)∆ψ(x, t) dxdt = 0. ( 31
) Let w ∈ C ∞ c (Ω × (0, T )). Let us denote, for ε > 0, q ε = q + ε. We have ε ≤ q ε (x, t) ≤ L ζ + ε, for all (x, t) ∈ Ω × (0, T ), and (q ε (x, t) -q(x, t)) 2 q ε (x, t) ≤ ε. (32)
Let ψ ε be given by lemma 4.2 below, with g = q ε . Substituting ψ by ψ ε in (31) and using (36) give

| T 0 Ω u d (x, t)w(x, t)dxdt| ≤ | T 0 Ω
u d (x, t)(q ε (x, t)q(x, t))∆ψ ε (x, t)dxdt|.

(33) The Cauchy-Schwarz inequality, (37) and (32) imply

T 0 Ω u d (x, t)(q ε (x, t) -q(x, t))∆ψ ε (x, t)|dxdt 2 ≤ T 0 Ω u d (x, t) 2 (q(x, t) -q ε (x, t)) 2 q ε (x, t) dxdt T 0 Ω q ε (x, t) ∆ψ ε (x, t) 2 dxdt ≤ ε T 0 Ω u d (x, t) 2 dxdt 4T T 0 Ω |∇w(x, t)| 2 dxdt. ( 34
)
We deduce that the right hand side of (34) tends to zero as ε → 0. Hence the left hand side of (33) also tends to zero as ε → 0, which gives

| T 0 Ω u d (x, t)w(x, t)dxdt| = 0. ( 35
)
Since (35) holds for any function w ∈ C ∞ c (Ω × (0, T )), we get that u d (x, t) = 0 for a.e. (x, t) ∈ Ω × (0, T ), which concludes the proof of Theorem 4.1.

Let us now prove the properties of the function ψ, used in the course of the proof of Theorem 4.1.

Lemma 4.2

Under Hypothesis (4a), let w ∈ L 2 (0, T ; H 1 0 (Ω)) and g ∈ L ∞ (Ω × (0, T )) with g(x, t) ∈ [g min , g max ] with given g max ≥ g min > 0 for a.e. (x, t) ∈ Ω × (0, T ). Then there exists at least one function ψ such that,

1. ψ ∈ L ∞ (0, T ; H 1 0 (Ω)), ∂ t ψ ∈ L 2 (Ω × (0, T )), ∆ψ ∈ L 2 (Ω × (0, T )) (hence ψ ∈ C 0 (0, T ; L 2 (Ω))), 2. ψ(•, T ) = 0, 3. the following holds ∂ t ψ(x, t) + g(x, t)∆ψ(x, t) = w(x, t),
for a.e. (x, t) ∈ Ω × (0, T ), (36)

4. and

T 0 Ω g(x, t) ∆ψ(x, t) 2 dxdt ≤ 4T T 0 Ω |∇w(x, t)| 2 dxdt. ( 37
)

Proof

We first apply Lemma 4.4, which states the convergence of a gradient scheme to ψ ∈ L 2 (0, T ; H 1 0 (Ω)) with ∂ t ψ ∈ L 2 (Ω × (0, T )) and ∆ψ ∈ L 2 (Ω × (0, T )) such that (36) holds, setting ν = 1/g, f = w/g, µ(s) = s, ψ ini = 0 and changing t in -t (this ensures that Hypotheses (45) are fulfilled). Therefore the existence of ψ satisfying (36) follows. Let us prove that it satisfies (37). Approximating ψ by a sequence of regular functions and passing to the limit, we get that ∇ψ(

•) L 2 (Ω) d ∈ C 0 ([0, T ]) and that τ s Ω ∂ t ψ(x, t)∆ψ(x, t)dxdt = - 1 2 Ω |∇ψ(x, τ )| 2 dx + 1 2 Ω |∇ψ(x, s)| 2 dx,
for all s < τ ∈ [0, T ] and

τ s Ω w(x, t)∆ψ(x, t)dxdt = - τ s Ω ∇w(x, t) • ∇ψ(x, t)dxdt.
We thus obtain, multiplying (37) by ∆ψ(x, t) and integrating on Ω × (0, τ ) for any τ ∈ [0, T ],

1 2 Ω |∇ψ(x, 0)| 2 dx - 1 2 Ω |∇ψ(x, τ )| 2 dx + τ 0 Ω g(x, t) ∆ψ(x, t) 2 dxdt = - τ 0 Ω ∇w(x, t) • ∇ψ(x, t)dxdt. (38) Since ∇ψ(•, T ) = 0, letting τ = T in (38) leads to 1 2 Ω |∇ψ(x, 0)| 2 dx + T 0 Ω g(x, t) ∆ψ(x, t) 2 dxdt = - T 0 Ω ∇w(x, t) • ∇ψ(x, t)dxdt. (39)
Integrating (38) with respect to τ ∈ (0, T ) leads to

1 2 T 0 Ω |∇ψ(x, τ )| 2 dxdτ ≤ T 2 Ω |∇ψ(x, 0)| 2 dx + T T 0 Ω g(x, t) ∆ψ(x, t) 2 dxdt + T T 0 Ω |∇w(x, t) • ∇ψ(x, t)|dxdt. ( 40
)
International Journal on Finite Volumes Using (39) and (40), we get

1 2 T 0 Ω |∇ψ(x, τ )| 2 dxdτ ≤ 2T T 0 Ω |∇w(x, t) • ∇ψ(x, t)|dxdt. ( 41
)
Thanks to the Cauchy-Schwarz inequality, the right hand side of (41) may be estimated as follows:

T 0 Ω |∇w(x, t) • ∇ψ(x, t)|dxdt 2 ≤ T 0 Ω |∇ψ(x, t)| 2 dxdt T 0 Ω |∇w(x, t)| 2 dxdt.
With (41), this implies

T 0 Ω |∇w(x, t) • ∇ψ(x, t)|dxdt 2 ≤ 4T T 0 Ω |∇w(x, t) • ∇ψ(x, t)|dxdt T 0 Ω |∇w(x, t)| 2 dxdt.
Therefore,

T 0 Ω |∇w(x, t) • ∇ψ(x, t)|dxdt ≤ 4T T 0 Ω |∇w(x, t)| 2 dxdt,
which, together with (39), yields (37).

In Lemma 4.2, we have used a result of existence of ū ∈ L 2 (0, T ; H 1 0 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)), such that ∆ū ∈ L 2 (Ω × (0, T )), solution to the following problem:

ν(x, t)∂ t ū(x, t) -∆ū(x, t) = f (x, t), for a.e. (x, t) ∈ Ω × (0, T ) (42)
with the following initial condition:

ū(x, 0) = u ini (x), for a.e. x ∈ Ω, (43)
together with the homogeneous Dirichlet boundary condition:

ū(x, t) = 0 for a.e. (x, t) ∈ ∂Ω × (0, T ), ( 44
)
under the following assumptions (which are not exactly the standard ones done in the literature):

Ω is an open bounded connected polyhedral subset of R d , d ∈ N ⋆ and T > 0, (45a)

u ini ∈ H 1 0 (Ω) (45b) f ∈ L 2 (Ω × (0, T )), (45c)
and

ν ∈ L ∞ (Ω × (0, T )) and ν(x, t) ∈ [ν min , ν max ] with given ν max ≥ ν min > 0 for a.e. (x, t) ∈ Ω × (0, T ). ( 45d
)
This problem, issued from (31), is called the regularised adjoint problem to Problem [START_REF] Alt | Quasilinear elliptic-parabolic differential equations[END_REF]. In order to prove the existence of a solution to Problem (42)-( 43)-(44) under hypotheses (45), we consider an approximation of this solution, using a gradient scheme. Let D = (X D , Π D , ∇ D , (t (n) ) n=0,...,N ) be a space-time discretisation in the sense of Definition A.9. We define the fully implicit scheme for the discretisation of Problem (51) by the sequence (u (n) ) n=0,...,N ⊂ X D,0 such that:

                         u (0) ∈ X D,0 , u (n+1) ∈ X D,0 , δ (n+ 1 2 ) D u = Π D u (n+1) -u (n) δt (n+ 1 2 ) , t (n+1)
t (n) Ω ν(x, t)δ (n+ 1 2 ) D u(x)Π D v(x)dxdt +δt (n+ 1 2 ) Ω ∇ D u (n+1) (x) • ∇ D v(x)dx = t (n+1) t (n) Ω f (x, t)Π D v(x)dxdt, ∀v ∈ X D,0 , ∀n = 0, . . . , N -1. (46)
We then denote by u D (t) ∈ X D,0 the function defined by

u D (t) = u (n+1) , for a.e. (x, t) ∈ Ω × (t (n) , t (n+1) ), ∀n = 0, . . . , N -1. (47)
We then use the notations Π D and ∇ D for the definition of space-time dependent functions, defining

Π D u(x, t) = Π D u (n+1) (x) and ∇ D u(x, t) = ∇ D u (n+1) (x), for a.e. (x, t) ∈ Ω × (t (n) , t (n+1) ), ∀n = 0, . . . , N -1. (48)
and

δ D u(x, t) = δ (n+ 1 2 ) D u(x), for a.e. (x, t) ∈ Ω × (t (n) , t (n+1) ), ∀n = 0, . . . , N -1. (49)
Let us state some estimates and the existence and uniqueness of the solution to the scheme.

Lemma 4.3 L 2 (Ω × (0, T )) estimate on δ D u and L ∞ (0, T ; X D,0 ) estimate on u.
Under Hypotheses (45), let D be a space-time gradient discretisation in the sense of Definition A.9. Then, for any solution u to Scheme (46), it holds:

ν min t (m) 0 Ω (δ D u(x, t)) 2 dxdt + ∇ D u m 2 L 2 (Ω) d ≤ ∇ D u (0) 2 L 2 (Ω) d + 1 ν min f 2 L 2 (Ω×(0,T )) , ∀m = 1, . . . , N. (50)
As a result, there exists one and only one solution u to Scheme (46).

Proof

We set v = u (n+1)u (n) in (46) and we sum on n = 0, . . . , N -1. We can then write

1 2 |∇ D u (n+1) (x)| 2 - 1 2 |∇ D u (n) (x)| 2 ≤ ∇ D u (n+1) (x) • (∇ D u (n+1) (x) -∇ D u (n) (x)).
Thanks to the Young inequality applied to the right hand side, we conclude (50), which ensures the existence and uniqueness of the solution to the linear Scheme (46), which leads to square linear systems.

We then have the following convergence lemma.

Lemma 4.4 (A convergence property of the fully implicit scheme)

Let Hypotheses (45) be fulfilled. Let (D m ) m∈N be a consistent sequence of spacetime gradient discretisations in the sense of Definition A.10, such that the associated sequence of approximate gradient approximations is limit-conforming (Definition A.4) and compact (Definition A.5, it is then coercive in the sense of Definition A.2). For any m ∈ N, let u m be the solution to Scheme (46) for a given u

(0) m ∈ X Dm,0 , such that ∇u ini -∇ Dm u (0) m ) L 2 (Ω) d → 0 as m → ∞.
Then there exist a sub-sequence of (D m ) m∈N , again denoted (D m ) m∈N , and a function ū ∈ L 2 (0, T ;

H 1 0 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)) such that 1. Π Dm u m converges in L 2 (Ω × (0, T )) to ū as m → ∞, 2. δ Dm u m weakly converges in L 2 (Ω × (0, T )) to ∂ t ū as m → ∞, 3. ∇ Dm u m weakly converges in L 2 (Ω × (0, T )) d to ∇ū as m → ∞.

Proof

Thanks to (50), ∇ Dm u m remains bounded in L ∞ (0, T ; L 2 (Ω) d ) and Π Dm u m remains bounded in L 2 (Ω × (0, T )). Since (50) also provides an L 2 (Ω × (0, T )) estimate of δ Dm u m , which immediately provides an L ∞ (0, T ; L 2 (Ω)) estimate on (Π Dm u m ) m∈N , thanks to

Π D u (n) -Π D u (p) 2 L 2 (Ω) = n-1 k=p δt (k+ 1 2 ) δ (k+ 1 2 ) D u 2 L 2 (Ω) ≤ (t (n) -t (p-1) ) δ Dm u m 2 L 2 (Ω×(0,T )) .
Classically applying the Kolmogorov theorem (prolonging the functions by 0 outside of Ω×(0, T )), we deduce that there exists a function ū ∈ L 2 (Ω×(0, T )) such that, up to the extraction of a subsequence, (Π Dm u m ) m∈N converges to ū in L 2 (Ω × (0, T )).

We then get that ū is such that

ū ∈ L 2 (0, T ; H 1 0 (Ω)) ∩ H 1 (0, T ; L 2 (Ω)), ū(x, 0) = u ini (x) for a.e. x ∈ Ω, T 0 Ω (ν ∂ t ū v + ∇ū • ∇v) dxdt = T 0 Ω f vdxdt, ∀v ∈ L 2 (0, T ; H 1 0 (Ω)). (51)
Then (51) shows that ∆ū ∈ L 2 (Ω × (0, T )) and that ( 42)-( 43)-( 44) hold.

International Journal on Finite Volumes

5 Numerical examples 5.1 The Vertex Approximate Gradient scheme

In the numerical tests proposed in this section, We use the Vertex Approximate Gradient scheme [START_REF] Eymard | Small-stencil 3d schemes for diffusive flows in porous media[END_REF]. In this scheme, a primary mesh M in polyhedra is given. We assume that each element K ∈ M is strictly star-shaped with respect to some point x K . We denote by E K the set of all interfaces K ∩ L, for all neighbours of K denoted by L ∈ M and, for a boundary control volume, E K also contains the element K ∩∂Ω. Each σ ∈ E K is assumed to be the reunion of d -1 simplices (segments if d = 2, triangles if d = 3) denoted τ ∈ S σ . We denote by V σ the set of all the vertices of σ, located at the boundary of σ, and by V 0 σ the set of all the internal vertices of σ. We assume that, for all v ∈ V 0 σ , there exists coefficients (α

x v ) x∈Vσ , such that v = x∈Vσ α x v x, with x∈Vσ α x v = 1.
Therefore, the d vertices of any τ ∈ S σ are elements of V 0 σ ∪ V σ . We denote by

V = σ∈E V σ ,
and by V K the set of all elements of V which are vertices of K. For any K ∈ M, σ ∈ E K , τ ∈ S σ , we denote by S K,τ the d-simplex (triangle if d = 2, tetrahedron if d = 3) with vertex x K and basis τ .

• We then define X D as the set of all families u = ((u K ) K∈M , (u v ) v∈V ) and X D,0 the set of all families u ∈ X D such that u v = 0 for all v ∈ V ∩ ∂Ω.

• Disjoint arbitrary domains V K,v ⊂ v∈τ S K,τ are defined for all v ∈ V K . Then the mapping Π D is defined, for any u ∈ X D , by Π D u(x) = u K , for a.e.

x ∈ K \ v∈V K V K,v , and Π D u(x) = u v for a.e. x ∈ V K,v . It is important to notice that it is not in general necessary to provide a more precise geometric description of V K,v than its measure.

• The mapping ∇ D is defined, for any u ∈ X D , by ∇ D u = ∇ Π D u, where Π D u is the continuous reconstruction which is affine in all S K,τ , for all K ∈ M, σ ∈ E K and τ ∈ S σ , with the values u K at x K , u v at any vertex v of τ which belongs to V σ , and x∈Vσ α x v u x at any vertex v of τ which belongs to V 0 σ .

The advantage of this scheme is that it allows to eliminate all values (u K ) K∈M with respect to the values (u v ) v∈V , leading to linear systems which are well suited to domain decomposition and parallel computing.

We then have the following result.

Lemma 5.1 (Gradient scheme properties of the VAG scheme)

We assume that, for all m ∈ N, a gradient discretisation D m = (X Dm , Π Dm , ∇ Dm ) is defined as specified in this section, respecting a uniform bound on the maximum value of the ratio between the diameter of all K ∈ M and that of the greatest ball The Dirichlet boundary condition is given by ū = -1 on ∂Ω and the initial condition (2) is given by ū(x, 0) = 2. Four grids are used for the computations, a Cartesian grid with 32 2 = 1024 cells, the same grid randomly perturbed, a triangular grids with 896 cells and a "Kershaw mesh" with 1089 cells as illustrated for example on the Figure 3 (such meshes are standard in the framework of underground engineering). The time simulation is 0.1 for a constant given time step of 0.001.

Figures 3,4, 5 and 6 represent the discrete solution u(•, t) on all grids for t = .025, 0.05, 0.075 and 0.1. For a better comparison we have also plotted the interpolation of u along two lines of the mesh, the first line is horizontal and joins the two points (0, 0.5) and (1, 0.5), the second one is diagonal and joins points (0, 0) and (1, 1). The results thus obtained are shown in Figures 1 and2.

We can see that the obtained results are weakly dependent on the grid, and that the interface between the regions u < 0 and u > 1 are located at the same place for all grids. It is worth to notice that this remains true for the very irregular Kershaw mesh, although it presents high ratios between the radii of inscribed balls and the diameter of some internal grid blocks.

A Appendix: gradient discretisations for diffusion problems

A gradient scheme can be viewed as a general framework for nonconforming approximation of elliptic or parabolic problems. These methods have been studied in [START_REF] Eymard | Small-stencil 3d schemes for diffusive flows in porous media[END_REF] for linear elliptic problems, and in [START_REF] Droniou | Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations[END_REF] in the case of nonlinear Leray-Lions-type elliptic and parabolic problems. The interest of the notion of gradient schemes is that it includes conforming finite elements with mass lumping (see Remark 6 below), mixed finite elements, hybrid mixed mimetic methods [START_REF] Droniou | A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods[END_REF][START_REF] Droniou | Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations[END_REF], some discrete duality finite volume schemes, some particular Multi-point Flux Approximation and many other schemes. We begin with the discrete elements used for space partial differential equations.

Definition A.1 (Gradient discretisation) A gradient discretisation D for a spacedependent second order elliptic problem, with homogeneous Dirichlet boundary conditions, is defined by D = (X D,0 , Π D , ∇ D ), where:

1. the set of discrete unknowns X D,0 is a finite dimensional vector space on R, 2. the linear mapping Π D : X D,0 → L 2 (Ω) is the reconstruction of the approximate function, Remark 2 (Boundary conditions.) The definition of • D depends on the considered boundary conditions. Here for simplicity we only consider homogeneous Dirichlet boundary conditions, but other conditions can easily be addressed. For example, in the case of homogeneous Neumann boundary conditions, we will use the notation X D instead of X D,0 for the discrete space, and define

• D := ( Π D • 2 L 2 (Ω) + ∇ D • 2 L 2 (Ω) d ) 1/2 .
Definition A.2 (Coercivity) Let D be a gradient discretisation in the sense of Definition A.1, and let C D be the norm of the linear mapping Π D , defined by

C D = max v∈X D,0 \{0} Π D v L 2 (Ω) v D . (54)
A sequence (D m ) m∈N of gradient discretisations is said to be coercive if there exists

C P ∈ R + such that C Dm ≤ C P for all m ∈ N. Remark 3 (Discrete Poincaré inequality.) Equation (54) yields Π D v L 2 (Ω) ≤ C D ∇ D v L 2 (Ω) d .
The consistency is ensured by a proper choice of the interpolation operator and discrete gradient.

Π D v -ϕ L 2 (Ω) + ∇ D v -∇ϕ L 2 (Ω) d . ( 55
)
International Journal on Finite Volumes Thanks to [13, Lemma 2.4], we may check the consistency and limit-conformity properties of given gradient schemes, only using dense subsets of the test functions spaces. The following lemma, useful in Section 5, is an immediate consequence of [START_REF] Eymard | Small-stencil 3d schemes for diffusive flows in porous media[END_REF]Lemma 2.4] and of Kolmogorov's theorem.

Lemma A.7 (Sufficient conditions)

Let F be a family of gradient discretisations in the sense of Definition A.1. Assume that there exist C, ν ∈ (0, ∞) and, for all D ∈ F, a real value h D ∈ (0, +∞) such that:

S D (ϕ) ≤ Ch D ϕ W 2,∞ (Ω) , for all ϕ ∈ C ∞ c (Ω), (57a)
W D (ϕ) ≤ Ch D ϕ (W 1,∞ (R d )) d , for all ϕ ∈ C ∞ c (R d ) d , (57b)
max v∈X D,0 \{0} Π D v(• + ξ) -Π D v L p (R d ) v D ≤ C|ξ| ν , for all ξ ∈ R d , (57c)
We now take ξ ∈ R N such that |ξ| ≤ min(η, (η m ) m=0,...,m 0 ). We then get that, for all m = 0, . . . , m 0 , the inequality g m (ξ) ≤ ε holds, and for all m ∈ N such that m > m 0 , then g(ξ, µ m ) ≤ ε. Gathering the previous results gives (60).

We finally state a discrete version of Alt-Luckhaus theorem [START_REF] Alt | Quasilinear elliptic-parabolic differential equations[END_REF], whose proof is immediate following [START_REF] Gallouët | Compactness of discrete approximate solutions to parabolic pdes-application to a turbulence model[END_REF]. Theorem B.3 (Discrete Alt-Luckhaus theorem) Let T > 0, let B be a Banach space, and let p ∈ [1, +∞). Let (B m ) m∈N be a sequence of normed subspaces of B such that, for any sequence (w m ) m∈N such that w m ∈ B m and ( w m Bm ) m∈N is bounded, then the set {w m , m ∈ N} is relatively compact in B. Let (v m ) m∈N such that v m ∈ L p (0, T ; B m ) for all m ∈ N. We assume that (h1) the sequence ( v m L p (0,T ;Bm) ) m∈N is bounded, (h2) v m (• + h)v m L p (0,T -h;B) tends to 0 as h ∈ (0, T ) tends to 0, uniformly with respect to m ∈ N.

Then the set {v m , m ∈ N} is relatively compact in L p (0, T ; B).

Proof

Our aim is to apply Theorem 2.1 of [START_REF] Gallouët | Compactness of discrete approximate solutions to parabolic pdes-application to a turbulence model[END_REF]. We then prolong v m by 0 on (-∞, 0) ∪ (T, +∞), for all m ∈ N. Let us prove that v m (• + h)v m L p (R;B) tends to 0 as h ∈ (0, T ) tends to 0, uniformly with respect to m ∈ N. Let us first remark that there exists C N > 0 such that,

∀m ∈ N, ∀v ∈ B m , v B ≤ C N v Bm .
Indeed, otherwise one could, up to a subsequence of (B m ) m∈N , construct a sequence such that v m Bm = 1 and v m B tends to infinity, which is in contradiction with the relative compactness in B of {v m , m ∈ N}. Hence we can define Thanks to (62), the above inequality gives

C B = sup
We also assume that there exists a dense subset R of L 2 (Ω) such that, for all ϕ ∈ R, there exists a function g ϕ : R + × R + with g(0, 0) = 0, continuous in (0, 0) and a sequence (h ϕ m ) m∈N with h ϕ m ≥ 0 and lim m→∞ h ϕ m = 0 and such that u m (t 2 )u m (t 1 ), ϕ L 2 (Ω),L 2 (Ω) ≤ g ϕ (t 2t 1 , h ϕ m ), ∀m ∈ N, ∀a ≤ t 1 ≤ t

ζ

  ∈ C 0 (R) is non-decreasing, Lipschitz continuous with constant L ζ and s.t. ζ(0) = 0, (4d) and |ζ(s)| ≥ a|s|b for all s ∈ R for some given values a, b ∈ (0, +∞).

  where we denote by C ∞ c (Ω×[0, T [) the set of the restrictions of functions of C ∞ c (Ω×]-∞, T [) to Ω × [0, T [.

  b a ζ(x)dx = Z(b) -Z(a) = ζ(b)(ba) -b a ζ ′ (x)(xa)dx, we get, from Hypothesis (4d), that

f

  (x, t)w(x)dxdt.

  We first apply Theorem 3.1, which shows the weak convergence properties for Π Dm u m and ∇ Dm ζ(u m ), and strong convergence for Π Dm ζ(u m ). Let ϕ ∈ C ∞ c (Ω), and let w m such that w m = argmin z∈X D,0 S D (ϕ).

4 .

 4 ∆ū ∈ L 2 (Ω × (0, T )), 5. (42)-(43)-(44) hold.

  (a) t = 0.025 (b) t = 0.050 (c) t = 0.075 (d) t = 0.1

Figure 1 :

 1 Figure 1: Interpolation of u along the line [x, 0.5] of the mesh for each grids : the Cartesian in blue, the perturbed in red, the triangular in green and the Kershaw in black dashed.

  (a) t = 0.025 (b) t = 0.050 (c) t = 0.075 (d) t = 0.1

Figure 2 :

 2 Figure 2: Interpolation of u along a diagonal axe of the mesh for each grids : the Cartesian in blue, the perturbed in red, the triangular in green and the Kershaw in black dashed.

Figure 3 :

 3 Figure 3: Discrete solution u on all grids at t = 0.025.

Definition A. 3 (

 3 Consistency) Let D be a gradient discretisation in the sense of Definition A.1, and let S D : H 1 0 (Ω) → [0, +∞) be defined by ∀ϕ ∈ H 1 0 (Ω) , S D (ϕ) = min v∈X D,0

Figure 5 :Π

 5 Figure 5: Discrete solution u on all grids at t = 0.075.

  0,T ;B) .We have, for all h ∈ (0, T ),v m (• + h)v m p L p (R;B) = v m (• + h)v m p L p (0,T -h;B) + v m p L p (0,h;B) ε > 0. We first choose h 0 ∈ (0, T ) such that, for all h ∈ (0, h 0 ), v m (• + h)v m p L p (0,T -h;B) ≤ ε, ∀m ∈ N. (62)Let τ ∈ (0, Th 0 ), h ∈ (0, h 0 ) and m ∈ N be given. We haveτ 0 v m (t) p B dt ≤ 2 p-1 τ 0 v m (t + h)v m (t) p B dt + τ 0 v m (t + h) p B dt .

  h) p B dhdt ≤ C B τ.This proves thath 0 inf h∈(0,h 0 ) τ 0 v m (t + h) p B dt ≤ C B τ.Taking the infimum on h in (63), we get, for all τ ∈ (0, Th 0 ) and m ∈ N,τ 0 v m (t) p B dt ≤ 2 p-1 ε + C B τ h 0 . It now suffices to take τ ∈ (0, min(Th 0 , h 0 ε C B )) for getting τ 0 v m (t) p B dt ≤ 2 p ε, ∀m ∈ N.This concludes the proof of (61). A similar proof can be done for proving thatlim • + h)v m p L p (R;B) = 0,which enables to apply Theorem 2.1 of[START_REF] Gallouët | Compactness of discrete approximate solutions to parabolic pdes-application to a turbulence model[END_REF], hence providing the conclusion of the proof. Theorem B.4 Let Ω be an open bounded subset of R 2 , a < b ∈ R and (u m ) m∈N be a sequence of functions from [a, b] to L 2 (Ω), such that there exists C 1 > 0 with u m (t) L 2 (Ω) ≤ C 1 , ∀m ∈ N, ∀t ∈ [a, b].

  2 ≤ b. (65) Then there exists u ∈ L ∞ (a, b; L 2 (Ω)) with u ∈ C w ([a, b], L 2 (Ω)) (where we denote by C w ([a, b], L 2 (Ω)) the set of functions from [a, b] to L 2 (Ω), continuous for the weak topology of L 2 (Ω)) and a subsequence of (u m ) m∈N , again denoted (u m ) m∈N , such that, for all t ∈ [a, b], u m (t) converges to u(t) for the weak topology of L 2 (Ω).

International Journal on Finite Volumes

with centre x K inscribed in K, and the ratio between the diameter of all S K,τ and that of the greatest ball inscribed in S K,τ , for K ∈ M, σ ∈ E K and τ ∈ S σ . We also assume that h Dm , the maximum diameter of all K ∈ M, tends to 0 as m → ∞. Then the sequence (D m ) m∈N is consistent, limit-conforming and compact (and therefore coercive).

Proof

For all u ∈ X D , the following property

is resulting from Π D u(x) -Π D u(x) = (xy(x)) • ∇ D u(x), for all x ∈ S K,τ , where y(x) ∈ S K,τ is the point of the mesh M defined by y(x) = x K if x ∈ S K,τ \ v∈V K V K,v , and by y(x)

Let us check that the hypotheses of Lemma A.7 are satisfied, for some C only depending on regularity factors specified in the statement of the lemma, for D m = (X Dm , Π Dm , ∇ Dm ). Then (57a) results from the interpolation results on the P 1 finite element under the regularity factor of the mesh, (57b) results from

and (57c) results from

Therefore we obtain that the sequence ( D m ) m∈N is consistent, limit-conforming and compact. From this result and thanks to (52), it is immediate to check that the sequence (D m ) m∈N is consistent and limit-conforming. We then remark that

which leads, using (52) and (53), to

The application of (60) proved in Lemma B.2 leads to the relative compactness in B of any sequence (Π Dm u m ) m∈N , if u m ∈ X Dm,0 is such that u m Dm remains bounded. This completes the proof that the sequence (D m ) m∈N is compact.

A 2D test case on a variety of meshes

In this 2D test case, we approximate Stefan's problem (1) by using the VAG scheme previously described in the domain Ω = (0, 1) 2 with the following definition of ζ(ū),

otherwise. A sequence (D m ) m∈N of gradient discretisations is said to be consistent if, for all ϕ ∈ H 1 0 (Ω), S Dm (ϕ) tends to 0 as m → ∞. Since we are dealing with nonconforming methods, we need that the dual of the discrete gradient be "close to" a discrete divergence.

Definition A.4 (Limit-conformity) Let D be a gradient discretisation in the sense of Definition A.1. We let

A sequence (D m ) m∈N of gradient discretisations is said to be limit-conforming if, for all ϕ ∈ H div (Ω), W Dm (ϕ) tends to 0 as m → ∞.

Dealing with generic non-linearity often requires compactness properties on the scheme.

Let us state an important relation between compactness and coercivity.

Lemma A.6 (Compactness implies coercivity)

Let (D m ) m∈N be a compact sequence of gradient discretisations in the sense of Definition A.5. Then it is coercive in the sense of Definition A.2. where C D , S D , W D are defined in this appendix. Then, any sequence (D m ) m∈N ⊂ F such that h Dm → 0 as m → ∞ is consistent, limit-conforming and compact (and therefore coercive).

Remark 4 In several cases, h D stands for the mesh size: this is the case for the numerical schemes used in Section 5.

Definition A.8 (Piecewise constant function reconstruction)

Let D = (X D,0 , Π D , ∇ D ) be a gradient discretisation in the sense of Definition A.1, and I be the finite set of the degrees of freedom, such that X D,0 = R I . We say that Π D is a piecewise constant function reconstruction if there exists a family of open subsets of Ω, denoted by

is not requested to be a norm on X D,0 . Indeed, in many examples that can be considered, some degrees of freedom are involved in the reconstruction of the gradient of the function, but not in that of the function itself. Hence it can occur that some of the Ω i are empty. Remark 6 An important example of gradient discretisation D = (X D,0 , Π D , ∇ D ) in the sense of Definition A.1, such that Π D is a piecewise constant function reconstruction in the sense of Definition A.8, is the case of the mass-lumping of conforming finite elements. Indeed, assuming that (ξ i ) i∈I is the basis of some finite-dimensional space V h ⊂ H 1 0 (Ω), we consider a family (Ω i ) i∈I , chosen such that

International Journal on Finite Volumes

We then define Π D as in Definition A.8, and ∇ D u = i∈I u i ∇ξ i . This is easily performed, considering P 1 conforming finite element, splitting each simplex in subsets defined by the highest barycentric coordinate, and defining Ω i by the union of the subsets of the simplices connected to the vertex indexed by i.

Remark 7 Note that we have the two important following properties, in the case of a piecewise constant function reconstruction in the sense of Definition A.8:

where for any continuous function g ∈ C(R) and u = (u i ) i∈I ∈ X D,0 , we classically denote by g(u) = (g(u i )) i∈I ∈ X D,0 and

where, for u = (u i ) i∈I and v = (v i ) i∈I ∈ X D,0 , we denote by uv = (u i v i ) i∈I ∈ X D,0 .

Definition A.9 (Space-time gradient discretisation) Under Hypothesis (4a), we say that

) is a gradient discretisation of Ω, in the sense of Definition A.1,

• t (0) = 0 < t (1) . . . < t (N ) = T .

We then set δt (n+ 1 2 ) = t (n+1)t (n) , for n = 0, . . . , N -1, and δt D = max n=0,...,N -1 δt (n+ 1 2 ) .

Definition A.10 (Space-time consistency) A sequence (D m ) m∈N of space-time gradient discretisations of Ω × (0, T ), in the sense of Definition A.9, is said to be consistent if it is consistent in the sense of Definition A.3 and if δt Dm tends to 0 as m → ∞.

B Appendix: technical results

The next result, which is known in the literature as the Minty trick, is used in the proof of the convergence theorem.

Then χ(x) = ζ(u(x)), for a.e. x ∈ ω.

Proof

We consider, for a given v ∈ L 2 (Ω),

Since ζ is a nondecreasing, we have A n ≥ 0. By weak/strong convergence, we get that lim

The above inequality holds in particular for v = utϕ, with t > 0 and ϕ ∈ C ∞ c (ω). Then we get, dividing by t > 0,

Letting t → 0 in the above equation, we get, by dominated convergence, that

Since the same inequality holds for -ϕ instead of ϕ, we get ω (χ(x)ζ(u(x)))ϕ(x)dx = 0.

Since the above inequality holds for all ϕ ∈ C ∞ c (ω), the conclusion of the lemma follows.

The following result is used in the convergence proof, for proving the compactness of a particular scheme.

Lemma B.2 (Uniform limit.)

Let N ∈ N ⋆ and (g m ) m∈N be a sequence of functions from R N to R + such that g m (0) = 0 and g m is continuous in 0. We assume that there exists a function g : R N × R + → R + , with g(0, 0) = 0, continuous in (0, 0), and for all m ∈ N, there exists µ m ∈ R + verifying lim m→∞ µ m = 0, such that

Proof

Let ε > 0. Let η > 0 be such that, for all (ξ, t) ∈ B(0, η) × [0, η], g(ξ, t) ≤ ε. Let m 0 ∈ N such that, for all m > m 0 , µ m ≤ η. For all m = 0, . . . , m 0 , thanks to the continuity of g m , there exists η m > 0 such that, for all ξ verifying |ξ| ≤ η m , we have g m (ξ) ≤ ε.

Proof

The proof follows that of Ascoli's theorem. Let (t p ) p∈N be a sequence of real numbers, dense in [a, b]. Due to (64), for each p ∈ N, we may extract from (u m (t p )) m∈N a subsequence which is convergent to some element of L 2 (Ω) for the weak topology of L 2 (Ω). Using a diagonal method, we can choose a sub-sequence, again denoted (u m ) m∈N , such that (u m (t p )) m∈N s is weakly convergent for all p ∈ N. For any t ∈ [a, b] and v ∈ L 2 (Ω), we then prove that the sequence ( u m (t), v L 2 (Ω),L 2 (Ω) ) m∈N is a Cauchy sequence. Indeed, let ε > 0 be given. We first choose ϕ ∈ R such that ϕv L 2 (Ω) ≤ ε. Let η > 0 such that, for all (s, t) ∈ [0, η] 2 , we have g ϕ (s, t) ≤ ε. Then, we choose p ∈ N such that |tt p | ≤ η. Since ( u m (t p ), ϕ L 2 (Ω),L 2 (Ω) ) m∈N is a Cauchy sequence, we choose n 0 ∈ N such that, for k, l ≥ n 0 , u k (t p )u l (t p ), ϕ L 2 (Ω),L 2 (Ω) ≤ ε, and such that h ϕ k , h ϕ l ≤ η. We then get, using (65),


Noam Faust

Introduction and problem

The Modern Hebrew (henceforth MH) data in (1) contrasts two types of basic Hebrew nouns. The nouns in (1a) have penultimate stress; whereas all other nouns in (1b) have final stress (not marked). In addition, the first vowel of the singular nouns in (1b) is not present in the plural, and instead we find a vowel a where the singular has an unstressed vowel e, alongside the regular plural suffix -im. The same plural suffix is concatenated to the forms in (1b). An a in the initial open syllable of bisyllabic nouns (1b) is deleted; this is the only stem-internal modification in such nouns.

( Nouns like those in (1a) are traditionally called "segholates" (after the name of the symbol for the vowel [e] in Biblical Hebrew). In MH, they constitute a moderately productive class. This paper will initially ask the following question:

(2) 1st Study Question Why is it that only penultimately stressed masculine nouns (i.e. masculine segholates) regularly have templatic plurals?

Furthermore, we observe that the vocalizations of the three noun types in (1) constitute the bulk of attested CVCVC vocalizations in Hebrew. We may make the following descriptive generalization: when there is a stable V2 (e.g) vowel, the Vowel in V1 (if there is one) will be [a]. One may also generalize that when there is only a V1 vowel, there may only be a vowel [e] in V2. As we will see below, this [e] is not a "real" vowel: it is there to save the illicit final consonant cluster. The correct generalization about these nouns is thus the following: when there is only a V1 vowel, there is no vowel in V2. This asymmetric state-of affairs is presented in (3a), and the second study question, which stems from it, is stated in (3b) Segholates

(3) Vocalization: overview and question 1 a. vocalization Qv1T(e)L Qa1Tv2L

QTv2L *Qv1Ta2L

b. 2nd study question: Why don't we find cases of QVTaL (*QiTaL, QuTaL etc.)?

We will see that both study questions can be answered together.

More data: determining the lexical site

Consider the masculine-feminine forms in (4).

( QaTVL nouns become QTVL before the feminine suffix -a, just as before the plural suffix in (1) above. 2 The vowel in V2 is again stable (kaxol, kxulim). Segholates (i.e. Qv1T(e)L nouns), on the other hand, do not behave as they did before the plural prefix; instead, no vowel is attested in V2 before the feminine suffix and it is the V1 position which is maintained. In addition, we observe a neutralization in the quality of V1: both a and i in the feminine segholate may correspond to e in the free segholate. To summarize, segholates have three stems, where QaTVL nouns have two:

1 I assume that the widely attested [QiTuL] is actually /qittul/. I assume the same for the poorly attested QiToL, QuTaL, and QiTaL: they are all case of /QvTTvL/. These assumptions are confirmed by both spirantization data and diachrony, and -as we will see -confirm to the analysis in the body of the paper. One small class, QoTaL, necessitates further investigation: historically, this class is underlyingly /QvvTaL/. 2 This has been analyzed in OT in Bat-EL (200?) in the following manner: QaTVL nouns have two stems, QaTVL and QTVL. The speaker strives to unify the syllable-size of all members of a paradigm, and so she selects the shorter stem for the suffixed noun. Bat-El has a problem with QaTaL nouns that do not alternate with QTaL (i.e. those for which I have assumed an underlying geminate): she solves this problems by stating that these simply do not have two allomorphs. In other words, the present paper treats the phenomenon as phonological, while Bat-El's views it as morphological. Since Morphology is not as well defined as phonology (for some, anything can be "morphological"), I consider the present analysis, if it is correct, preferable by far. Segholates

(5) Stems Free Bound ms. fm. pl.

QaTVL pakid pkid-

Qv 1 T(e)L pérec pirc- prac-
The unstressed vowel e alternates with zero when followed by another vowel. Unsurprisingly, the vowel e is breaks other impermissible consonant clusters in Modern Hebrew (li-srok 'scan' => srika 'a scan' ; limxok 'erase' => mexika 'erasure' *mxika). In Government Phonology (GP; Kaye et al 1990), such a behaviour is typical of empty nuclei. The following diagram demonstrates how this is represented in the CV-option of that theory, which assumes that all "syllables" are of the CV type (instead of Onset-Nucleus pairs):

(6) Vowel-Zero alternation in CV-GP a. /kalb/ => (kálev) => [kélev] 'dog' b. /kalb+a/ => [kalba] 'bitch' k a l b | | | | C V C V2 C V3 k a l b a | | | | | C V C V2 C V3
In (6a) the empty V-slot is not properly-governed by a vocalic element in the following V-slot; an empty nucleus is thus not licensed in this position, and the position has to be realized. It is realized as the less marked vowel of the (specific) language; in MH, this vowel is e. In (6b), the position is well-governed, and the empty nucleus is licensed and not pronounced. The lexical vowel of segholates is thus clearly V1. This is the vowel speakers have to memorize when deriving a segholate from the root. For the remainder of this paper, I leave aside the alternations in the quality of this vowel (for my view on these, see Faust, in preparation). The implications for the data in (6) are clear: segholates do not have three stems. Like QaTVL nouns, they have two stems (at best): QVT(e)L for singulars and QTaL for the plural. We will now see that, in fact, both segholates and QaTVL have only one stem.

Stress in Modern Hebrew

Stress in the native vocabulary of MH seems to be final. But if stress were final, we'd expect MH to behave like e.g. French, where even vowels that alternate with zero consistently get stressed (appel 'call (imper.)' [appel], but appeler [aple] 'to call'). In a view of MH as stress final, segholates (along with all loanwords) would constitute a class apart. Indeed, all accounts blah blah blah. The true nature of MH lies in the distinction between final and iambic, or right-headed. I propose the following rule for Hebrew: (7) Modern Hebrew stress Build an iambic foot from the lexical vowel. Reduce remaining well-governed vowels. Segholates

Of course, there are some complications to this rule3 ; nevertheless, for the data discussed in this paper it is all we need. As we have seen, QaTVL nouns present a bisyllabic free stem and a monosyllabic bound stem. The bisyllabic stem always has a vowel [a] in V1, and it is this [a] that disappears when the stem is bound. This change is not explicable in terms of licensing or government alone: the same relation holds between V2 and V1 in both free (QV1TV2Lim) and bound (QV1TV2L) stems. This is where the stress rule is important:

(8) vowel deletion in pkidim a. plural suffixation b. footing c. erasure

C V C V C V | | | | | p a k i d C V C V C V C V | | | | | p a k i d i m C V C V C V C V | | | | | p a k i d i m w s F C V C V C V C V | | | | p k i d i m w s F
In the singular form QaTVL, stress fall Stress is placed on the lexical vowel of the head, namely the plural suffix. An iambic (=right-headed) foot is built and the vowel a, which is in a governed position, is reduced. However, sometimes reducing this vowel creates an illicit cluster of descending sonority. As phonology cannot look ahead,4 the vowel is still reduced in these cases, and the V-slot is empty. That this nucleus surfaces as e is unsurprising, since (as we have just seen) this is the realization of illicit empty nuclei in MH.

(9) Vowel deletion resulting in illicit cluster: lexicim 'buttons' a. plural suffixation b. footing c. erasure

C V C V C V | | | | | l a x i c C V C V C V C V | | | | | l a x i c i m C V C V C V C V | | | | | l a x i c i m w s F C V C V C V C V | | | | l x i c i m w s F
QaTVL nouns thus have one single unique "stem" for both singular and plural: this stem is the one apparent in the singular form. Now consider the case of the free segholate. In these, I repeat, the lexical vocalic position is V1. The foot built for such an item has a degenerate left branch. Moreover, because the foot built on V1 is iambic, whatever vowel there might be in V2 is always reduced. V2 is an empty nucleus. Because (the native vocabulary of) MH does not tolerate any final consonant clusters, this empty nucleus is realized as e. Segholates (10) footing of singular segholate footing erasure

C V C V C V | | | | | p e r ? c C V C V C | | | | | p e r ? c w s F C V C V C | | | | p i r c s F [pérec]
Because it is outside the foot, the identity of the vowel in V2 is not detectable in the free segholate. Now let us go back to our second study question, namely "why don't we find cases of QVTaL" in MH. It is now tempting to say that there are cases of QiTaL, but their V2 vowel is always reduced.

Stress and segholate plurals/feminines

The hypothesis is stated in ( 11):

(11) Segholate V2 hypothesis

All segholates are underlyingly QVTaL, i.e.:

Q T L | | | C V C V C V | | v a
This hypothesis is proved in the plural segholate stem. Just as in the case of p(a)kidim, stress is placed on the lexical vowel of the head, i.e. the plural suffix. An iambic foot is built, and its left branch falls on the vowel a. The first CV is now outside the foot, and its is now deleted.

(12) Segholate plurals a. plural suffixation b. footing c. erasure

C V C V C V | | | | | p e r a c C V C V C V C V | | | | | p e r a c i m C V C V C V C V | | | | | p e r a c i m w s F C V C V C V C V | | | | p r a c i m w s F
Segholates are thus the exact mirror image of QaTVL nouns.

We have thus answered the two questions above in one move: segholates, i.e. QVTeL nouns, are the missing cases of QVTaL. However, the quality of their V2 vowel is only revealed in the plural, because only then is this vowel footed. This is why only segholates have "templatic" plurals.

We still have one more question to solve. We have seen that the feminine of a QaTVL noun is Q{TVLa}. If segholates are underlyingly QiTaL nouns, why isn't their feminine form *Q{TaLa}, Segholates but rather QvTLa? This, again, is a case of misguiding appearances. Here, -a seems to be a gender marker. But if one looks at the other gender markers in MH, this -a stands apart: all the other feminine markers are -(V)t. Another aspect with respect to which -a stands apart is its form in the construct state.

(

s v f r t s v f r t | | | | | | | [sifra] C V C V C V -C V C V C V C V -C V a a
The situation in (14a), prior to the linking of the root consonants or the non-lexical a to the skeleton, has three consecutive empty nuclei. 5 The last nucleus is always empty in MH. The penultimate is unlicensed and has to be filled. The templatic a lands in this position. The antepenultimate V-slot is now a licensed empty nucleus. There is no need to express the final /t/ in these cases, because its presence is detectable by the stem final a. This works the same for the pair pakidpkida 'clerk':

(15) Masculine-Feminine QaTVL a. ms. b. fm.

p k i d p k i d t | | | | [pakid] | | | | | [pkida] C V C V C V C V C V C V -C V a a
5 That the a floats is not just some trick. There is no sense in which the feminine segholate is derived from the masculine correspondent: a masculine-feminine relation only when the masculine is animate. Segholates

If so, the case of pkida is slightly different from that of pkidim. In the latter, the singular's [a] is reduced (*pakidim => pkidim), whereas in the former the a never lands in V1 in the first place.

Conclusions:

The segholate class of nouns have a vowel a where there is no vowel in the singular. This is a reflex of an infixed plural marker a found in the Broken plurals of other Semitic languages languages. As the many studies (Angoujard, McCarthy, Kihm, Palmer, LeSège, etc.) conducted on those topic show, several aspects of the singular form are respected in the plural form, so that however broken the plural is, it cannot be claimed that no reference is made to the singular. However, In Hebrew (both Modern and Biblical), this historical infixed marker appears alongside the external plural markers -im (or -ot in case the free segholate is lexically feminine). Whether this is the case of reanalysis (already in Biblical Hebrew) or preservation of some earlier stages where all plurals were concatenative is insignificant. In MH, as I hope to have made clear, the phonology of the language is perfectly compatible with an analysis of segholates as being underlyingly QVTaL. This is clearly preferable to assuming some kind of stem/root allomorphy (although such cases are not ruled out in principle).

It is standard practice in Semitic morphology (if not in linguistics in general) to both base and bolster an abstraction with the non-existence of its straightforward realization. Thus, one of the most striking evidence for assuming a root 2rcy for the verb raca 'want' (cf. katav 'write') is that *racay is impossible. Similarly, that there are no independent QiTaL, QuTaL, QaTaL (with v~zero alternation in V2 rather than V1) is proof for the legitimacy of such an abstraction for the segholate stem.

This analysis relied on a distinction between two lexical sites, V1 and V2. Items with a lexical site in V2 can be either nouns or adjectives (though the latter is not true for QTVL item), and constitute the bulk of MH native nouns; items with V1 are exclusively segholate nouns. Moreover, the same vowel root can appear with two different vowels in QaTVL (e.g. amud 'column' amid 'durable'), but the few equivalent cases for segholates are only distantly related cases (róga 'calmness' réga 'moment'; there are no QiTL~QaTL pairs) . I would like to ask a question for future research: what grammatical information is provided by the position of the vowel? Or, alternatively: what structure does the position convey?

"Well-governed" is added for these cases, mos of which will be analyzed as in fn.1, i.e. as having a medial virtual geminate (S&S). For loanwords (and some native words), the rule would probably have to be atennuated to "reduce remaining (predictable) well-governed vowels".

Nor does it "evaluate candidates", as is current practice in Optimality Theory.