Xavier Blanc
email: xavier.blanc@u-paris.fr

Philippe Hoch
email: philippe.hoch@cea.frclement.lasuen@cea.fr

Clément Lasuen

Proof of uniform convergence for a cell-centered AP discretization of the hyperbolic heat equation on conical meshes

, where the same result is proved for polygonal meshes. We also give some geometrical results on conical meshes.

1

Introduction

This work focuses on the properties of a numerical scheme that discretizes the following system, called the P 1 system:

     ∂ t Ẽε + 1 ε div Fε = 0, ∂ t Fε + 1 ε ∇ Ẽε = - σ ε 2
Fε .

(

) 1
where ε and σ > 0 are positive constants. The space variable is x ∈ R 2 and the time variable is t ∈ R + . The unknowns of system (1) are the radiative energy Ẽε (t, x) and the radiative flux Fε (t, x) ∈ R 2 . This system is L 2 -stable:

d dt Ω Ẽε 2 + Ω Fε 2 ≤ 0.
Moreover, as ε vanishes, the flux Fε goes to 0 and Ẽε converges toward the solution of the following diffusion equation:

∂ t Ẽ0 -div 1 σ ∇ Ẽ0 = 0. ( 2
)
This property can be derived using a standard Hilbert expansion. A rigorous proof is given in [START_REF] Buet | Proof of uniform convergence for a cell-centered AP discretization of the hyperbolic heat equation on general meshes[END_REF]. Lemma 1.1 below gives the details of this result. A numerical scheme that discretises system (1) which is consistent with this diffusion limit is called asymptotic preserving (AP). Figure 1 illustrates this property. The space discretisation parameter is h, ε > 0 reflects the convergence of the model P ε (system (1)) toward the limit model P 0 (equation ( 2)). A scheme P ε h consistent with P ε is said to be asymptotic preserving if the scheme P 0 h computed in the limit ε → 0 is consistent with the limit model P 0 .

In this work, we study the AP scheme described in [START_REF] Blanc | Asymptotic preserving schemes on conical unstructured 2d meshes[END_REF] for conical meshes and in [START_REF] Buet | Design of asymptotic preserving finite volume schemes for the hyperbolic heat equation on unstructured meshes[END_REF] polygonal meshes. Our purpose is to prove the uniform convergence of the scheme with respet to ε. More precisely, we want to prove an estimate of the form:

P ε h -P ε ≤ Ch η , ( 3
)
where P ε h -P ε represents the error between the numerical solution and the exact solution for a given ε, and η is a positive constant that does not depend on ε. See Theorem 4. [START_REF] Bernard-Champmartin | Stabilité locale et montée en ordre pour la reconstruction de quantités volumes finis sur maillages coniques non-structurés en dimension 2[END_REF] for the details of this result. In order to prove 3), we adapt the work [START_REF] Buet | Proof of uniform convergence for a cell-centered AP discretization of the hyperbolic heat equation on general meshes[END_REF] to conical meshes. In that work, the authors proved (3) for η = 1/4 on polygonal meshes. We propose here a proof adapted to conical meshes and which is simpler, yet leading to a rate of convergence of η = 1/6. Anyway, in practice, the rate of convergence is 1 (and 2 for the limit scheme P 0 h ).

The structure of the proof we propose is similar to the one of [START_REF] Buet | Proof of uniform convergence for a cell-centered AP discretization of the hyperbolic heat equation on general meshes[END_REF]. We first prove a naive estimate of the form:

P ε h -P ε ≤ C h ε . ( 4
)
The proof of ( 4) is given in Section 7. Then we derive another estimate using the triangle inequality:

P ε h -P ε ≤ P ε h -P 0 h + P 0 h -P 0 + P 0 -P ε , (5)
where:

• P ε h -P 0 h is the error between the numerical solution for a given ε and the solution computed with the limit scheme,

• P 0 h -P 0 is the error between the numerical solution computed with the limit scheme and the exact solution to (2),

• P 0 -P ε is the difference between the exact solutions of ( 2) and [START_REF] Buet | Proof of uniform convergence for a cell-centered AP discretization of the hyperbolic heat equation on general meshes[END_REF].

We compute each term in the right hand side of (5) (see respectively Sections 6 and 5 and Lemma 1.1) and end up with:

P ε h -P ε ≤ C √ εh + ε √ h + h + ε ,
thus leading to:

P ε h -P ε ≤ C min h ε , √ εh + ε √ h + h + ε . (6)
Eventually, Lemma 1.2 gives the desired result (3) for η = 1/6.

Even though the structure of our proof is similar to the one of [START_REF] Buet | Proof of uniform convergence for a cell-centered AP discretization of the hyperbolic heat equation on general meshes[END_REF], we emphasize that it is simpler even in the case of a polygonal scheme. Indeed, in [START_REF] Buet | Proof of uniform convergence for a cell-centered AP discretization of the hyperbolic heat equation on general meshes[END_REF], the authors introduced another scheme (namely the Diffusion Approximation scheme DA ε h ) which depends on h and ε and for which they prove the convergence toward P 0 . It comes down to writing:

P ε h -P ε ≤ P ε h -DA ε h + DA ε h -P 0 + P 0 -P ε .
Our proof does not need the introduction of an additional scheme and directly relies on the limit scheme and model.

In order to make a clear distinction between the numerical and exact solution, we denote by Ẽε , Fε = Ẽ, F the solution to [START_REF] Buet | Proof of uniform convergence for a cell-centered AP discretization of the hyperbolic heat equation on general meshes[END_REF], Ẽ0 the solution to [START_REF] Blanc | Asymptotic preserving schemes on conical unstructured 2d meshes[END_REF]. The numerical solution is (E, F) (which depends on ε but we do not write it so as to simplify the algebra) and E 0 is the solution to the limit scheme. Moreover, vectors are denoted in bold in the rest of the paper. In the following, the constants (C m ) m that appear in the inequalities we prove are said to be universal : they do not depend on h, ε, the numerical solution or the final time (denoted as T ), but they may depend on σ and Ẽ0 and its derivatives.

Lemma 1.1.

There exists a universal constant C 1.1 such that, for any T ≥ 0 and any p ∈ N, one has:

Ẽ0 -Ẽε L 2 ([0,T ];H p (Ω)) + Fε L 2 ([0,T ];H p (Ω)) ≤ C 1.1 (1 + T )ε, and: d dt Ẽε 2 H p (Ω) + Fε 2 H p (Ω) ≤ 0, d dt Ẽ0 2 H p (Ω) ≤ 0.
The proof of Lemma 1.1 is given in [START_REF] Buet | Proof of uniform convergence for a cell-centered AP discretization of the hyperbolic heat equation on general meshes[END_REF].

Lemma 1.2. Let (h, ε) ∈]0, 1] 2 , then: min h ε , √ εh + ε √ h + h + ε ≤ 4h 1 6 .
Proof. One has:

min h ε , √ εh + ε √ h + h + ε ≤ min h ε , √ εh +min h ε , ε √ h +min h ε , h +min h ε , ε .
Since min(a, b) ≤ √ ab and ε ≤ 1 and h ≤ 1, the first and last terms can be bounded by:

min h ε , √ εh ≤ h ε √ εh 1 2 = h 1 2 ≤ h 1 6 , min h ε , ε ≤ h ε ε 1 2 ≤ (hε) 1 4 ≤ h 1 6 .
Besides, the third term is bounded by:

min h ε , h ≤ h ≤ h 1 6 .
Eventually, using min(a, b) ≤ a , the second term satisfies:

min h ε , ε √ h ≤ h ε 2 3 ε √ h 1 3 = h 1 6 .

Conical meshes

In this section, we define conical meshes. We follow the presentation of [START_REF] Blanc | Asymptotic preserving schemes on conical unstructured 2d meshes[END_REF][START_REF] Bernard-Champmartin | Stabilité locale et montée en ordre pour la reconstruction de quantités volumes finis sur maillages coniques non-structurés en dimension 2[END_REF]. Important geometrical properties are presented and a quadrature formula is given in order to compute the integral of a smooth function on a conical cell.

Rational quadratic Bezier curve

A rational quadratic Bezier curve is a curve {M ω (q), q ∈ [0, 1]} such that:

M ω (q) = (1 -q) 2 M 0 + 2ωq(1 -q)M 1 + q 2 M 2 (1 -q) 2 + 2ωq(1 -q) + q 2 , ( 7
)
where M 0 and M 2 are the extremities, M 1 is the control point and ω ≥ 0 a scalar weight. The curve is said to be:

• degenerate if ω = 0, • elliptic if ω ∈ ]0, 1[, • parabolic if ω = 1, • hyperbolic if ω > 1.
In the first case (ω = 0), the curve is equal to the segment [M 0 , M 2 ]. Note that the control point M 1 almost never belongs to the curve (except in some very particular cases). This is the main drawback of this parametrisation. This is why we prefer to parametrise the curve in a different way, using a point that lies on the curve. This point is named the shoulder point and is defined by:

S = M ω (0.5) = 1 2 (Q 0 + Q 2 ), Q 0 = 1 1 + ω (ωM 1 + M 0 ), Q 2 = 1 1 + ω (ωM 1 + M 2 ). (8)
M 0 M 0

(M 1 , ω) M 2 ω = 0 ω = +∞ ω = 0.5 ω = 1 ω = 3 M 0 (M 1 , ω) M 2 M ω (q)
(M 1 , ω) M 2 S Q 0 Q 2

Computing the area of a conical cell

Let Ω j be a conical cell which center is denoted by x j . The area of Ω j can be computed using the following formula:

|Ω j | = Ωj dx = 1 2 ∂Ωj x(s) -x j , N(x(s)) ds, (9)
where N(x(s)) is the unit normal vector to the edge at the curvilinear coordinate s ∈ ∂Ω j and ds is the surface measure on ∂Ω j . In the following, we express this area as a function of the degrees of freedom (dof ) which are the vertices and the shoulder points. We define the following notation:

• (x r ) r the coordinates of the vertices of the cell j,

• (M 1,r+1/2 ) r+1/2 the coordinates of the control points of the cell j,

• (x r+1/2 ) r+1/2 the coordinates of the shoulder points of the cell j,

• r g r j : sum over all the vertices of the cell j of the quantity g (g r j being the evaluation of the function g on the vertex r in cell j),

• r+1/2 g r+1/2 j
: sum over all the shoulder points of the cell j of the quantity g,

• dof g dof j = r g r j + r+1/2 g r+1/2 j
: sum over all the degrees of freedom (dof ) (nodes and shoulder points) of the cell j of the quantity g,

• N dof = i 1: number of cells that contains the given degree of freedom dof ,

• i g dof i : sum, for a given degree of freedom, over all the cells that contains this degree of freedom,

• j * g j : sum over all the cells of the mesh, • r * g r : sum over all the nodes of the mesh, • r+1/2 * g r+1/2 : sum over all the shoulder points of the mesh,

• dof * g dof = r * g r + r+1/2 * g r+1/2 : sum over all the degrees of freedom (nodes and shoulder points) of the mesh,

•

dof g dof -g dof : sum over the neighboring degrees of freedom of dof (those which share a common cell with dof ).

The integral (9) can be computed exactly using the coordinates of the vertices (x r ) r and the control points (M 1,r+1/2 ) r+1/2 (cf [START_REF] Bernard-Champmartin | Stabilité locale et montée en ordre pour la reconstruction de quantités volumes finis sur maillages coniques non-structurés en dimension 2[END_REF]):

|Ω j | = 1 2 r C r j , x r -x j + 1 2 r+1/2, control points C r+1/2 j , M 1,r+1/2 -x j , ( 10
)
it can also be expressed in terms of (x r ) r and (x r+1/2 ) r+1/2 (cf [START_REF] Bernard-Champmartin | Stabilité locale et montée en ordre pour la reconstruction de quantités volumes finis sur maillages coniques non-structurés en dimension 2[END_REF]):

|Ω j | = 1 2 r Cr j , x r -x j + 1 2 r+1/2, shoulder points Cr+1/2 j , x r+1/2 -x j = 1 2 dof Cdof j , x dof -x j . ( 11
)
The coefficients C r j , C r+1/2 j , Cr j and Cr+1/2 j depend on the geometry of the cell and are given by:

C r j = 1 2 (1 -f (ω r-1/2 ))N r-1,r + (1 -f (ω r+1/2 ))N r,r+1 + f (ω r-1/2 )N r-1/2,r + f (ω r+1/2 )N r,r+1/2 , C r+1/2 j = f (ω r+1/2 ) 2 N r,r+1 , Cr j = 1 2 (1 -h(ω r-1/2 ))N r-1,r + (1 -h(ω r+1/2 ))N r,r+1 + h(ω r-1/2 ) Ñr-1/2,r + h(ω r+1/2 ) Ñr,r+1/2 , ( 12
)
and:

Cr+1/2 j = h(ω r+1/2 ) 2 N r,r+1 , (13)
and:

N r,r+1 = (x r+1 -x r ) ⊥ , N r,r+1/2 = (M 1,r+1/2 -x r ) ⊥ , N r+1/2,r+1 = (x r+1 -M 1,r+1/2 ) ⊥ , Ñr,r+1/2 = (x r+1/2 -x r ) ⊥ , Ñr+1/2,r+1 = (x r+1 -x r+1/2 ) ⊥ ,
with:

h(ω) = f (ω) 1 + ω ω , f (ω) =                  2ω 1 -ω 2 1 √ 1 -ω 2 arctan 1 -ω 1 + ω - ω 2 if ω ∈ [0, 1[, 2 3 if ω = 1, ω ω 2 -1 ω + 1 √ ω 2 -1 log ω -ω 2 -1 if ω > 1, (14)
and for any vector ξ ∈ R 2 :

ξ = ξ 1 ξ 2 , ξ ⊥ = -ξ 2 ξ 1 . M r+1 S ω r+1/2 M r S ω r-1/2 M r-1 Ñr + 1 , r
Ñr , r -1

Ñr + 1 / 2 , r + 1
Ñr, Equations ( 10) and (11) are a consequence of Theorem 2.1. Moreover, the following properties are satisfied:

r+ 1/ 2 Ñr-1/ 2, r Ñr -1 , r -1 / 2 Cr + 1 / 2 , ω j Cr, ω j C r -1 / 2 , ω j Ω j
• for any cell j:

dof Cdof j = 0,
• for any inner degree of freedom dof :

i Cdof i = 0. ( 15
)
For a given quantity defined on the degrees of freedom (u dof ) dof , we define its L 2 norm by:

u dof L 2 (Ω) = dof * h 2 |u dof | 2 1 2
, For a quantity that is defined on the cells (u j ) j<J and for a given function ũ, we define:

u L 2 (Ω) =   j * |Ω j ||u j | 2   1 2 , u -ũ L 2 (Ω) =   j * |Ω j | |u j -ũ(x j )| 2   1 2
.

Similarly, for a given quantity given on the shoulder points (u r+1/2 ) r+1/2 :

u r+1/2 L 2 (Ω) =   r+1/2 * h 2 |u r+1/2 | 2   1 2
.

We also define:

u dof -u j L 2 (Ω) = dof * h 2 i |u dof -u i | 2 1 2 =   j * h 2 dof |u dof -u j | 2   1 2 , ( 16
)
To conclude, the following theorem is used in the proof of the consistency of the limit scheme. It states that the quadrature formula (11) is of order 2:

Theorem 2.1. Let g ∈ C 2 (R 2 , R 2
) be a vector-valued function. Then:

1 |Ω j | ∂Ωj g, N = 1 |Ω j | dof g(x dof ), Cdof j + O(h).
More precisely:

∂Ωj g, N - dof g(x dof ), Cdof j ≤ Ch|Ω j | ∇ 2 g L ∞ (Ωj ) ,
where C is independant of the h, Ω j and g. The result is also true for any scalar function g ∈ C 2 (R 2 , R) :

1 |Ω j | ∂Ωj gN = 1 |Ω j | dof g(x dof ) Cdof j + O(h), ∂Ωj gN - dof g(x dof ) Cdof j ≤ Ch|Ω j | ∇ 2 g L ∞ (Ωj ) .
The following result is a consequence of the previous theorem.

Corollary 2.2. The following equality holds true for any cell j:

dof β dof j = |Ω j |I 2 ,
where

β dof j = Cdof j ⊗ (x dof -x j ).

Numerical method

We refer to [START_REF] Blanc | Asymptotic preserving schemes on conical unstructured 2d meshes[END_REF] for the details of the construction of this scheme (JLb scheme). It writes:

         |Ω j | d dt E j + 1 ε dof F dof , Cdof j = 0, |Ω j | d dt F j + 1 ε dof α dof j (F j -F dof ) = 0, (17)
where F dof satisfies:

α dof + σ ε β dof F dof = i E i Cdof i + α dof i F i . ( 18
)
The matrices α dof j and β dof j are defined as:

α dof j = Cdof j ⊗ ñdof j , β dof j = Cdof j ⊗ (x dof -x j ), ñdof j = Cdof j dof j , dof j = Cdof j , (19) and:
α dof = i α dof i , β dof = i β dof i . ( 20
)
For any node r, this reads as:

α r + σ ε β r F r = i E i Cr i + α r i F i . ( 21
)
We present in the next sections somme sufficient conditions on the mesh so that the matrix of ( 21) is positive definite, and thus invertible. However, for any shoulder point r + 1/2, the matrix of system (18) has rank 1 and thus it is not invertible. The vector F r+1/2 is computed as the solution to:

   F r+1/2 , D r+1/2 = s r+1/2 , F r+1/2 , D ⊥ r+1/2 = 1 2 F r + F r+1 , D ⊥ r+1/2 ,
with:

D r+1/2 = 2ñ r+1/2 j + σ ε (x i -x j ), s r+1/2 = E j -E i + ñr+1/2 j , F j + F i . ( 22
)
where i and j are the two cells that contain the shoulder point r + 1/2. The limit scheme writes:

|Ω j | d dt E 0 j + dof F 0 dof , Cdof j = 0, (23)
with, for any node r and any shoulder point r + 1/2:

σβ r F 0 r = i E 0 i Cr i ,        F 0 r+1/2 , x i -x j = E 0 j -E 0 i σ , F 0 r+1/2 , (x i -x j ) ⊥ = 1 2 F 0 r + F 0 r+1 , (x i -x j ) ⊥ . ( 24
)
Remark 1. Notice that, for any (ξ 1 , ξ 2 ) ∈ (R 2 ) 2 :

α dof j ξ 1 , ξ 2 = dof j ξ 1 , ñdof j ξ 2 , ñdof j , β dof j ξ 1 , ξ 2 = dof j ξ 1 , x dof -x j ξ 2 , ñdof j ,
4 Geometric properties

Assumptions on the mesh

In this section, we present the assumptions on the mesh that are used in the rest of the paper.

We denote by h the characteristic length of the mesh (h = ∆x = ∆y for a cartesian mesh). We assume that there exists a numerical constant C 1 > 0 independent of h such that, for any cell j and any dof :

1 C 1 h 2 ≤ |Ω j | ≤ C 1 h 2 , 1 C 1 h ≤ Cdof j ≤ C 1 h, N dof ≤ C 1 .
We define: α j = dof α dof j . The matrices α r and α j are symetric positive, we also assume that they are invertible, meaning that:

∀ξ ∈ R 2 , α r ξ, ξ ≥ 1 C 1 h ξ 2 , α j ξ, ξ ≥ 1 C 1 h ξ 2 , (25)
In addition, we assume that, for any cell j and any dof of j:

1 C 1 h ≤ x dof -x j ≤ C 1 h, ( 26
)
and for any neighboring cells i and j:

1 C 1 h ≤ x i -x j ≤ C 1 h. ( 27
)
Eventually, we define:

γ 1 = 1 h 2 min j, dof Cdof j , x dof -x j , γ 2 = 1 h 2 max j, dof Cdof j , (x dof -x j ) ⊥ , ( 28
)
and we assume that:

γ 1 > C1 γ 2 , 1 C1 -C1 (γ 2 + γ 2 2 ) > 0, C1 = 8C 3 1 . ( 29
)

Important results

Under the assumptions of the previous section, we can prove some very important results that are essential for the rest of the paper. The proofs are given in the Appendix. The first one is the fact that the matrices (β r ) r are positive definite:

Theorem 4.1. There exists a constant C such that, for any node r:

∀ξ ∈ R 2 , β r ξ, ξ ≥ C 4.1 h 2 ξ 2 . ( 30
)
The following theorem if very useful since it allows to bound from below the sum dof β dof u dof , u dof by

u dof 2 L 2 (Ω)
. This inequality is easy to prove for the polygonal scheme since the matrices (β r ) r are positive definite. However, in the conical case, the matrices (β r+1/2 ) r+1/2 are never positive definite. Theorem 4.2. Let (u dof ) dof , (u r+1/2 ) r+1/2 be such that, for any shoulder point r + 1/2:

u r+1/2 , D⊥ r+1/2 = 1 2 u r + u r+1 , D⊥ r+1/2 + u r+1/2 ,
with:

Dr+1/2 = D r+1/2 D r+1/2 = 2εñ r+1/2 j + σ(x i -x j ) 2εñ r+1/2 j + σ(x i -x j )
.

Then there exists a universal constant C 4.2 > 0 such that:

dof * β dof u dof , u dof ≥ 1 C 4.2 u dof 2 L 2 (Ω) -C 4.2 u r+1/2 2 L 2 (Ω) .
The following lemma is used in the proof of Theorem 4.2.

Lemma 4.3. For any (σ, ε) ∈ (R * + ) 2 , one has:

C D 1 + σh ε ≥ D r+1/2 ≥ 1 C D 1 + σh ε , with C D = max(2, C 1 ).
Eventually, the aim of the rest of the paper is to prove the following:

Theorem 4.4. We assume that the assumptions of Section 4.1 are fulfilled. We also assume that the initial energies Ẽ(t = 0) and E(t = 0) do not depend on ε and that they satisfy:

E(t = 0) -Ẽ(t = 0) L 2 (Ω) ≤ C init h, ( 31
)
and that the initial fluxes satisfy:

F(t = 0) -F(t = 0) L 2 (Ω) ≤ C init h, F(t = 0) L 2 (Ω) + F(t = 0) L 2 (Ω) ≤ C init ε. ( 32
)
Then there exists a constant C such that, for any final time T ≥ 0:

E -Ẽ L 2 ([0,T ]×Ω) + F -F L 2 ([0,T ]×Ω) ≤ C 1 + T 3 2 h 1 6
. 13

Convergence of the limit diffusion scheme (23)

In this section, we prove the convergence of the limit diffusion scheme (23) toward the solution of (2) for sufficiently smooth data. To this aim, we adapt the proof of [START_REF] Franck | Construction et analyse numérique de schema asymptotic preserving sur maillages non structurés[END_REF]. We prove an estimate of the form:

P 0 h -P 0 ≤ Ch,
where the meaning of this notation is explained in the Introduction. More precisely, we define the following errors:

e 0 j (t) = E 0 j (t) -Ẽ0 (t, x j ), f 0 dof (t) = F 0 dof (t) + 1 σ ∇ Ẽ0 (t, x dof ).
Therefore, using the notation of Section 2.2:

e 0 (t) 2 L 2 (Ω) = j * |Ω j | E 0 j (t) -Ẽ0 (t, x j ) 2 , f 0 dof 2 L 2 (Ω) = dof * h 2 f 0 dof 2 ,
and we prove the following theorem:

Theorem 5.1. Under the assumptions of Theorem 4.4, the diffusion limit scheme ( 23) is first order convergent:

e 0 L 2 ([0,T ]×Ω) + f 0 dof L 2 ([0,T ]×Ω) ≤ C 5.1 h 1 + T 3 2
.

To this aim, we first prove that the limit scheme ( 23) is stable and consistent.

Lemma 5.2. The limit scheme ( 23) is L 2 stable:

d dt E 0 2 L 2 (Ω) + C 5.2 F 0 dof 2 L 2 (Ω) ≤ 0, (33)
and:

E 0 (t) 2 L 2 (Ω) + C 5.2 σ t 0 F 0 dof 2 L 2 (Ω) ≤ E 0 (t = 0) 2 L 2 (Ω) . ( 34
)
Moreover, defining:

a j (t) = ∂ t Ẽ0 (t, x j ) - 1 |Ω j | 1 σ dof ∇ Ẽ0 (t, x dof ), Cdof j , b dof (t) = 1 σ β dof ∇ Ẽ0 (t, x dof ) + i Ẽ0 (t, x i ) Cdof i ,
and:

b r+1/2 (t) = ∇ Ẽ0 (t, x r+1/2 ) - 1 2 ∇ Ẽ0 (t, x r ) + ∇ Ẽ0 (t, x r+1 ) , (x i -x j ) ⊥ x i -x j ,
we have the following result:

Lemma 5.3. The limit scheme is consistent of order 1, that is to say, there exists a universal constant C 5.3 > 0 such that:

a L 2 (Ω) ≤ C 5.3 h, b dof L 2 (Ω) ≤ C 5.3 h 3 , b r+1/2 L 2 (Ω) ≤ C 5.3 h 2 .
The idea is to prove that β -1

r i Ẽ0 (t, x i ) Cdof i is an approximation to -(∇ Ẽ0 ) r .

Proof of Lemma 5.2

Using (23), one has:

1 2 d dt E 0 2 L 2 (Ω) = j * |Ω j |E 0 j d dt E 0 j = - j * dof E 0 j F 0 dof , Cdof j = - dof * i E 0 i F 0 dof , Cdof i .
According to (24), one can write, for any dof :

i E 0 i F 0 dof , Cdof i = σ β dof F 0 dof , F 0 dof .
Therefore, exchanging the sum over the cells and the sum over the degrees of freedom, one can write:

1 2 d dt E 0 2 L 2 (Ω) = -σ dof * β dof F 0 dof , F 0 dof ,
which is negative by virtue of Theorem 4.2. This gives equation (33). Equation ( 34) is obtained by integrating (33) in time.

Proof of Lemma 5.3

Since the solution to the diffusion equation ( 2) is smooth (see [START_REF] Evans | Partial differential equations[END_REF] for instance), one can write:

∂ t Ẽ0 (t, x j ) = 1 |Ω j | Ωj ∂ t Ẽ0 (t, x)dx + O(h 2 ),
since x j is the barycenter of cell Ω j . Therefore:

∂ t Ẽ0 (t, x j ) = 1 |Ω j | Ωj div 1 σ ∇ Ẽ0 (t, x)dx + O(h 2 ) = 1 |Ω j | ∂Ωj 1 σ ∇ Ẽ0 (t, x), N dx + O(h 2 ),
and according to Theorem 2.1:

∂ t Ẽ0 (t, x j ) = 1 |Ω j | 1 σ dof ∇ Ẽ0 (t, x dof ), Cdof j + O(h 2 ).
Therefore:

|a j (t)| ≤ C 35 h 2 , a L 2 (Ω) ≤ C 35 h 2 , ( 35
)
here the constant C 30 depends on the gradient of Ẽ0 , which is bounded uniformly in time according to Lemma 1.1. In addition, we have:

b dof = i Ẽ0 (t, x i ) -Ẽ0 (t, x dof ) -∇ Ẽ0 (t, x dof ), x i -x dof =O(h 2 ) Cdof j = O(h 3 ),
which leads to:

b dof L 2 (Ω) ≤ C 36 h 3 . ( 36
)
Here again the constant C 31 depends on the gradient of Ẽ0 . Eventually, one has:

∇ Ẽ(t, x r+1/2 ) - 1 2 (∇ Ẽ(t, x r ) + ∇ Ẽ(t, x r+1 )) = O(h 2 ), b r+1/2 L 2 (Ω) ≤ C 37 h 2 . ( 37
)

Proof of Theorem 5.1

First and foremost, one can write, using the linearity of the scheme:

                   d dt e 0 j + 1 |Ω j | dof Cdof j , f 0 dof = -a j , β dof f 0 dof = 1 σ i e 0 i Cdof i -b dof , f 0 r+1/2 , (x i -x j ) ⊥ x i -x j = f 0 r + f 0 r+1 2 , (x i -x j ) ⊥ x i -x j -b r+1/2 . ( 38
)
Using the first equation of (38), one has:

1 2 d dt e 0 2 L 2 (Ω) = j * |Ω j |e 0 j d dt e 0 j = - j * |Ω j |e 0 j a j - j * e 0 j dof Cdof j , f 0 dof .
Exchanging the sum over the cells and th sum over the degrees of freedom leads to:

1 2 d dt e 0 2 L 2 (Ω) = - j * |Ω j |e 0 j a j -σ dof * β dof f 0 dof , f 0 dof - dof * b dof , f 0 dof .
Moreover, applying Theorem 4.2 to u dof = f 0 dof and u r+1/2 = b r+1/2 , one can write:

dof * β dof f 0 dof , f 0 dof ≥ 1 C 4.2 f 0 dof 2 L 2 (Ω) -C 4.2 b r+1/2 2 L 2 (Ω) .
Using Lemma 5.3, one has:

dof * β dof f 0 dof , f 0 dof ≥ 1 C 39 f 0 dof 2 L 2 (Ω) -C 39 h 4 , (39)
and:

j * |Ω j |e 0 j a j ≤ C 40 h e 0 L 2 (Ω) , dof * b dof , f 0 dof ≤ C 40 h f 0 dof L 2 (Ω) . ( 40
)
This leads to:

d dt e 0 2 L 2 (Ω) + 1 C 41 f 0 dof 2 L 2 (Ω) ≤ C 41 h e 0 L 2 (Ω) + h 4 + h f 0 dof L 2 (Ω) , (41)
for some universal constant C 41 > 0. Moreover, one can write, for any γ > 0:

h f 0 dof L 2 (Ω) ≤ γ f 0 dof 2 L 2 (Ω) + 1 γ h 2 .
Thus choosing γ small enough (and independently of h), equation (41) implies:

d dt e 0 2 L 2 (Ω) + 1 C 42 f 0 dof 2 L 2 (Ω) ≤ C 42 h e 0 L 2 (Ω) + h 2 + h 4 , ( 42
)
for some universal constant C 42 > 0. Using h ≤ 1, we thus have:

d dt e 0 2 L 2 (Ω) ≤ C 42 h e 0 L 2 (Ω) + h .
Therefore, applying Lemma 9.1 to f = e 0 L 2 (Ω) , k 1 = Ch and k 2 = h and using the assumption on the initial data (31), there exists a constant C 43 > 0 such that:

e 0 (t) 2 L 2 (Ω) ≤ C 43 h 2 (1 + t 2 ), e 0 2 L 2 ([0,T ]×Ω) ≤ C 43 h 2 (1 + T 3 ). ( 43
)
Integrating equation (42) in time gives:

f 0 dof 2 L 2 ([0,T ]×Ω) ≤ C 44 h 2 (T 2 + 1), (44)
for some universal constant C 44 > 0.

Study of P 0 h -P ε h

In this section, we prove an estimate of the form:

P 0 h -P ε h ≤ C √ hε + ε √ h .
More precisely, we denote by (E, F) the solution to (17) (which depends on ε but we do not make it explicit). We denote by E 0 the solution to the limit diffusion scheme (23). We define: e = E -E 0 and:

F 0 j = α -1 j dof α dof j F 0 dof , that is to say: dof α dof j (F 0 j -F 0 dof ) = 0, (45)
as well as:

f dof = F dof -εF 0 dof , f j = F j -εF 0 j .
We prove the following: Theorem 6.1. Under the assumptions of Theorem 4.4, there exists a constant C 6.1 > 0 such that:

e 2 L 2 ([0,T ]×Ω) + f 2 L 2 ([0,T ]×Ω) ≤ C 6.1 (1 + T 3 ) εh + ε 2 h .
In order to prove Theorem 6.1, we prove the following lemmas: Lemma 6.2. The scheme ( 17) is L 2 stable, that is, the solution (E, F) to ( 17) satisfies:

d dt E 2 L 2 (Ω) + F 2 L 2 (Ω) + C 6.2 1 ε 2 F dof 2 L 2 (Ω) ≤ 0, (46)
and:

E(t) 2 L 2 (Ω) + F(t) 2 L 2 (Ω) + C 6.2 1 ε 2 t 0 F dof 2 L 2 (Ω) ≤ E(t = 0) 2 L 2 (Ω) + F(t = 0) 2 L 2 (Ω) . ( 47
)
Lemma 6.3. Under the assumptions of Theorem 4.4, there exists a universal constant C 6.3 > 0 such that, for any T ≥ 0, the solution (E, F) to (17) satisfies:

F L 2 ([0,T ]×Ω) ≤ C 6.3 ε.
Lemma 6.4. Under the assumptions of Theorem 5.1, there exists a constant C 6.4 > 0 such that:

F 0 i -F 0 dof L 2 ([0,T ]×Ω) ≤ C 6.4 (1 + T )h, ( 48
)
where 16), and:

F 0 i -F 0 dof L 2 ([0,T ]×Ω) is defined in (
F 0 j L 2 ([0,T ]×Ω) ≤ C 6.4 , ( 49
)
d dt F 0 j L 2 ([0,T ]×Ω) ≤ C 6.4 h . ( 50
)

Proof of Lemma 6.2

This proof is done in the polygonal case in [START_REF] Franck | Construction et analyse numérique de schema asymptotic preserving sur maillages non structurés[END_REF]. Using (17), one has:

1 2 d dt E 2 L 2 (Ω) + F 2 L 2 (Ω) = j |Ω j | E j d dt E j + F j , d dt F j = - 1 ε   j dof E j F dof , Cdof j + dof F j , α dof j (F j -F dof )   .
According to (18), one can write, for any dof :

i E i Cdof i , F dof = σ ε β dof F dof , F dof - i α dof i (F i -F dof ), F dof ,
and therefore:

1 2 d dt E 2 L 2 (Ω) + F 2 L 2 (Ω) = - σ ε 2 dof β dof F dof , F dof - 1 ε dof i α dof i (F i -F dof ), F i -F dof ≥0 ,
which is nonpositive in virtue of Theorem 4.2. Using Theorem 4.2 leads to (46). Integrating (46) gives (47).

Proof of Lemma 6.3

Multiplying the second equation of system (17) by F j leads to:

1 2 d dt |Ω j | F j 2 + 1 ε F j , α j F j - 1 ε dof F j , α dof j F dof = 0.
According to assumption (25), one can write:

1 2 d dt (|Ω j | F j 2 ) + 1 C 1 h ε F j 2 ≤ C 1 h ε F j dof F dof . ( 51
)
Moreover, using:

F j F dof ≤ γ F j 2 + 1 γ F dof 2 ,
and chosing γ small enough (γ = 1/(2C 2 1 N j )), equation (51) gives:

d dt |Ω j | F j 2 + 1 C 52 h ε F j 2 ≤ C 52 h ε dof F dof 2 , ( 52
)
for some universal constant C 31 > 0. Summing (52) over the cells gives:

d dt F 2 L 2 (Ω) + 1 C 53 1 εh F 2 L 2 (Ω) ≤ C 53 1 εh F dof 2 L 2 (Ω) . ( 53
)
Equation ( 53) can be written as:

d dt F(t) 2 L 2 (Ω) exp 1 C 53 t εh ≤ C 53 1 εh F dof (t) 2 L 2 (Ω) exp 1 C 53 t εh . ( 54
)
Integrating (54) twice leads to:

F 2 L 2 ([0,T ]×Ω) -εhC 53 F(0) 2 L 2 (Ω) 1 -exp - 1 C 53 t εh ≤ C 53 1 εh T 0 t 0 F dof (s) 2 L 2 (Ω) exp - 1 C 53 t -s εh ds.
(55) According to the Young inequality for convolution products (see [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF]):

T 0 t 0 F dof (s) 2 L 2 (Ω) exp - 1 C 53 t -s εh dsdt ≤ T 0 F dof (s) 2 L 2 (Ω) ds T 0 exp 1 C 53 -s εh ds ≤ C 53 εh T 0 F dof (s) 2 L 2 (Ω) ds.
Therefore, owing to Lemma 6.2, equation (55) comes down to:

F 2 L 2 ([0,T ]×Ω) -εhC 53 F(t = 0) 2 L 2 (Ω) ≤ C 2 53 T 0 F dof (s) 2 L 2 (Ω) ds ≤ C 2 53 ε 2 E(t = 0) 2 L 2 (Ω) + F(t = 0) 2 L 2 (Ω) .
As a conclusion, since εh ≤ 1 and according to assumption (31) and (32), there exists a universal constant C 6.3 > 0 suh that:

F 2 L 2 ([0,T ]×Ω) ≤ C 6.3 ε 2 .

Proof of lemma 6.4

We first show (48). In fact F 0 j converges toward σ -1 (∇ Ẽ0 )(t, x j ). According to (45), we have:

F 0 j -F 0 dof = α -1 j dof α dof j (F 0 dof -F 0 dof ),
thus:

F 0 j -F 0 dof = α -1 j dof α dof j [F 0 dof + σ -1 ∇ Ẽ0 (t, x dof )] -(σ -1 ∇ Ẽ0 (t, x dof ) + F 0 dof ) +α -1 j dof α dof j [σ -1 ∇ Ẽ0 (t, x dof ) -σ -1 ∇ Ẽ0 (t, x dof )]
The last term is O(h) since the exact solution Ẽ0 is smooth. The two firsts terms are also bounded by O(h) according to Theorem 5.1. More precisely, there exists a positive constant C 37 such that:

F 0 j -F 0 dof L 2 ([0,T ]×Ω) ≤ C 56 F 0 dof + σ -1 ∇ Ẽ0 (t, x dof ) L 2 ([0,T ]×Ω) (56) +C 56 T 0 h 2 dof * dof ∇ Ẽ0 (t, x dof ) -∇ Ẽ0 (t, x dof ) 2 dt 1 2
.

Hence:

F 0 j -F 0 dof L 2 ([0,T ]×Ω) ≤ C 57 1 + T 3 2 h + C 57 h √ T ≤ 2C 57 1 + T 3 2 h, ( 57
)
for some positive constant C 57 . For the inequality (49), we use the L 2 stability of the limit scheme (Lemma 5.2):

F 0 dof L 2 ([0,T ]×Ω) ≤ C 58 . ( 58
)
20 Using (25) and ( 45), there exists a constant C 59 such that, for any cell j:

F 0 j ≤ C 59 dof F 0 dof , (59)
thus one can write:

F 0 j L 2 ([0,T ]×Ω) ≤ C 60 . ( 60
)
Eventually, for the inequality (50), we compute the time derivative of (45) and we use (25):

d dt F 0 j L 2 ([0,T ]×Ω) ≤ C 61 d dt F 0 dof L 2 ([0,T ]×Ω) . ( 61
)
Since the scheme is linear and its coefficients are constant, the time derivative of the unknowns, namely d dt E 0 j j and d dt F 0 dof dof satisfy the same equations as (E 0 j ) j and (F 0 dof ) dof (only the initial conditions change). Therefore we can apply Lemma 5.2 to the quantity d dt F 0 dof dof :

d dt F 0 dof L 2 ([0,T ]×Ω) ≤ C 62 d dt E 0 (t = 0) L 2 (Ω) . ( 62
)
The right hand side of (62) can be computed as follows:

d dt E 0 j (t = 0) = - 1 |Ω j | dof F 0 dof (t = 0), Cdof j ,
thus leading to:

d dt E 0 (t = 0) L 2 (Ω) ≤ C 63 h F 0 dof (t = 0) L 2 (Ω) . ( 63
)
Moreover, by assumption (31), at initial time on can write: E 0 j (t = 0) = Ẽ0 (t = 0, x j ) + O(h). This leads to:

dof * i E 0 i (t = 0) Cdof i 2 ≤ dof * i Ẽ0 (t = 0, x i ) Cdof i 2 + C 64 h 2 . ( 64
)
Using i Cdof i = 0 and the fact that Ẽ is smooth:

dof * i Ẽ0 (t = 0, x i ) Cdof i 2 = dof * i Ẽ0 (t = 0, x i ) -Ẽ0 (t = 0, x dof ) =O(h) Cdof i 2 ≤ C 65 h 2 . ( 65
)
Therefore, according to the definition (24) of F 0 dof (t = 0) , one can write:

F 0 dof (t = 0) 2 L 2 (Ω) ≤ C 66 dof * 1 h 2 i Ẽ0 i (t = 0) Cdof i 2 ≤ C 66 . ( 66
)
Hence, in view of (63):

d dt E 0 (t = 0) L 2 (Ω) ≤ C 67 h . ( 67
)
Collecting ( 61), ( 62) and (67), we find (50).

Proof of theorem 6.1

Using equations ( 18) and ( 24), one can write:

σ ε β dof f dof = i e i Cdof i + i α dof i (F i -F dof ). ( 68
)
In addition, ( 17) and ( 23) imply:

1 2 d dt |Ω j |e 2 j + 1 ε dof f dof , e j Cdof j = 0. ( 69
)
Therefore, summing (69) over the cells gives:

1 2 d dt e 2 L 2 (Ω) = - 1 ε dof * f dof , i e i Cdof i ,
which, using (68), can be written as:

1 2 d dt e 2 L 2 (Ω) = - σ ε 2 dof * β dof f dof , f dof + 1 ε dof * i f dof , α dof i (F i -F dof ) .
Hence:

1 2 d dt e 2 L 2 (Ω) = - σ ε 2 dof * β dof f dof , f dof + 1 ε dof * i f dof , α dof i (f i -f dof ) + dof * i f dof , α dof i (F 0 i -F 0 dof ) .
Using the definition (45) of F 0 i , this reads as:

1 2 d dt e 2 L 2 (Ω) = - σ ε 2 dof * β dof f dof , f dof + 1 ε dof * i f dof , α dof i (f i -f dof ) + dof * i f dof -f i , α dof i (F 0 i -F 0 dof ) .
Moreover, the second line of (17) gives:

1 2 d dt f 2 L 2 (Ω) = - 1 ε dof * i f i , α dof i (F i -F dof ) -ε j * |Ω j | f j , d dt F 0 j . ( 70
)
Using ( 45), one has:

dof * i f i , α dof i (F 0 i -F 0 dof ) = j * dof f j , α dof j (F 0 j -F 0 dof ) = 0.
Therefore (70) reads as:

1 2 d dt f 2 L 2 (Ω) = - 1 ε dof * i f i , α dof i (f i -f dof ) -ε j * |Ω j | f j , d dt F 0 j ,
thus leading to:

1 2 d dt e 2 L 2 (Ω) + f 2 L 2 (Ω) = - σ ε 2 dof * β dof f dof , f dof - 1 ε dof * i f i -f dof , α dof i (f i -f dof ) + dof * i f dof -f i , α dof i (F 0 i -F 0 dof ) -ε j * |Ω j | f j , d dt F 0 j .
Now, for any γ > 0, we have:

f dof -f i , α dof i (F 0 i -F 0 dof ) ≤ γ ε f i -f dof , α dof i (f i -f dof + ε γ F 0 i -F 0 dof , α dof i (F 0 i -F 0 dof ) .
Hence, choosing γ small enough (independently of h and ε) and using Theorem 4.2, one can finally write:

d dt e 2 L 2 (Ω) + f 2 L 2 (Ω) + 1 C 71 1 ε 2 f dof 2 L 2 (Ω) + 1 C 71 1 ε dof * i f i -f dof , α dof i (f i -f dof ) ≥0 (71) ≤ C 71 ε dof * i F 0 i -F 0 dof , α dof i (F 0 i -F 0 dof ) + ε f j L 2 (Ω) d dt F 0 j L 2 (Ω)
.

Therefore, since α dof j ≤ Ch:

d dt e 2 L 2 (Ω) + f 2 L 2 (Ω) ≤ C 72 ε h F 0 i -F 0 dof 2 L 2 (Ω) + ε f L 2 (Ω) d dt F 0 L 2 (Ω) . ( 72
)
Integrating (72) over [0, T ] and using the assumptions on the initial data (31) and (32) gives:

e 2 L 2 (Ω) + f 2 L 2 (Ω) ≤ C 73 ε h F 0 i -F 0 dof 2 L 2 ([0,T ]×Ω) + ε f L 2 ([0,T ]×Ω) d dt F 0 L 2 ([0,T ]×Ω) + C 73 ε 2 . ( 73
)
As a conclusion, using Lemma 6.4:

e 2 L 2 ([0,T ]×Ω) + f 2 L 2 ([0,T ]×Ω) ≤ C 74 (1 + T 3 ) εh + ε 2 h . ( 74
)
7 Proof of the naive estimation (4)

We recall that we remove the exponent ε for clarity and we note ( Ẽ, F) = ( Ẽε , Fε ) the solution to (1). We define:

e 2 = 1 2 Ω (E -Ẽ) 2 + F -F 2 = j * Ωj (E j -Ẽ) 2 + F j -F 2 ,
and:

Ẽj = 1 |Ω j | Ωj Ẽ, Fj = 1 |Ω j | Ωj F, Ẽdof = Ẽ(x dof ), Fdof = F(x dof ), δ j,dof ( Ẽ) = Ẽj -Ẽdof , δ j,dof ( F) = Fj -Fdof , δj ( Ẽ) = ∂Ωj Ẽn - dof Ẽdof Cdof j , δj ( F) = ∂Ωj F, n - dof Fdof , Cdof j .
Lemma 7.1. The following results hold true:

Ω F 2 ≥ j * |Ω j | Fj 2 , F L 2 (Ω) ≤ C 7.1 ε. (75) δ( Ẽ) L 2 (Ω) + δ( F) L 2 (Ω) ≤ C 7.1 h, δ( F) L 2 (Ω) ≤ C 7.1 εh (76) δ( Ẽ) L 2 (Ω) + δ( F) L 2 (Ω) ≤ C 7.1 h 3 . ( 77
)
Theorem 7.2. Under the assumptions of Theorem 4.4, there exists a constant C 7.2 such that:

e L 2 ([0,T ]) ≤ C 7.2 (1 + T ) h ε .
Proof of Lemma 7.1

For (75), we use the Cauchy-Schwarz inequality:

Ω F 2 = j * Ωj F 2 ≥ j * Ωj F 2 1 |Ω j | = j * |Ω j | Fj 2 = F 2 L 2 (Ω) .
According to Lemma 1.1, one has:

Ω F 2 ≤ C 1.1 ε 2 ,
which gives (75). For (76), we use the fact that Ẽ and F are smooth:

δ j,dof ( Ẽ) = O(h), δ j,dof ( F) = O(h).
Moreover:

δ( F) L 2 (Ω) ≤ C 78 h Ω ∇ F 2 1 2 , ( 78
)
and according to Lemma 1.1, one has:

Ω ∇ F 2 1 2 ≤ C 1.1 ε.
The estimate (77) is proved using Theorem 2.1.

Proof of Theorem 7.2

One has:

d dt e 2 = D 1 + D 2 + D 3 + D 4 ,
with:

D 1 = Ω E d dt E + d dt F, F = j * |Ω j | E j d dt E j + F j , d dt F j , D 2 = Ω Ẽ∂ t Ẽ + F, ∂ t F ,
and:

D 3 = - Ω Ẽ d dt E + F, d dt F , D 4 = - Ω E∂ t Ẽ + F, ∂ t F .
According to Lemma 6.2:

D 1 = - σ ε 2 dof * β dof F dof , F dof - 1 ε dof * i α dof i (F i -F dof ), F i -F dof , (79)
and it can be easily shown that:

D 2 = - σ ε 2 Ω F 2 .

Study of D 3

Using (17), we have:

D 3 = 1 ε j * dof F dof , Cdof j Ẽj + α dof j (F j -F dof ), Fj .
Since, for any cell j, dof F j , Cdof j = 0, this implies:

D 3 = 1 ε j * dof F dof -F j , Cdof j Ẽj + α dof j (F j -F dof ), Fj .
Hence:

D 3 = 1 ε j * dof F dof -F j , Cdof j δ j,dof ( Ẽ) + α dof j (F j -F dof ), δ j,dof ( F) (80)
+ 1 ε j * dof F dof -F j , Cdof j Ẽdof + α dof j (F j -F dof ), Fdof . ( 81
)

Study of D 4

Using (1), one has:

D 4 = 1 ε j * E j ∂Ωj F, n + F j , ∂Ωj Ẽn + σ ε 2 j * |Ω j | F j , Fj .
Hence:

D 4 = 1 ε j * E j δj ( F) + F j , δj ( Ẽ) + σ ε 2 j * |Ω j | F j , Fj + 1 ε j * dof E j Fdof , Cdof j + Ẽdof F j , Cdof j . ( 82
)
Since i Ẽdof F dof , Cdof i = 0 for any dof , the last term of (82) is equal to:

j * dof Ẽdof F j , Cdof j = j * dof Ẽdof F j -F dof , Cdof j .
This leads to:

D 4 = 1 ε j * E j δj ( F) + F j , δj ( Ẽ) + σ ε 2 j * |Ω j | F j , Fj (83)
+ 1 ε j * dof E j Fdof , Cdof j + Ẽdof F j -F dof , Cdof j . ( 84
)
According to (18):

j * dof E j Fdof , Cdof j = dof * Fdof , i E i Cdof i = dof * i Fdof , α dof i (F dof -F i ) + σ ε dof * Fdof , β dof F dof .
(85) Eventually, summing (80)-( 81) and ( 83)-(84) and using (85):

D 3 +D 4 = 1 ε j * dof F dof -F j , Cdof j δ j,dof ( Ẽ) + α dof j (F j -F dof ), δ j,dof ( F) =D5 + 1 ε j * E j δj ( F) + F j , δj ( Ẽ) =D6 + σ ε 2   dof * Fdof , β dof F dof + j * |Ω j | F j , Fj   .
and:

d dt e 2 = D 5 + D 6 + σ ε 2 D 7 + D 8 , (86)
with:

D 7 = dof * Fdof , β dof F dof + j * |Ω j | F j , Fj - dof * F dof , β dof F dof - Ω F 2 . ( 87
)
D 8 = - 1 ε dof * i α dof i (F i -F dof ), F i -F dof . ( 88
)
Study of D 5 and D 6

Using Remark 1, one can write, for any γ > 0:

|D 5 | ≤ γ 1 ε j * dof dof j F dof -F j , ñdof j 2 + C 89 h γ 1 ε j * dof δ j,dof ( Ẽ) 2 + δ j,dof ( F) 2 . ( 89
)
According to Lemma 7.1, the right hand side of (89) can be bounded by:

h ε j dof δ j,dof ( Ẽ) 2 + δ j,dof ( F) 2 ≤ C 90 1 εh δ( Ẽ) 2 L 2 (Ω) + δ( F) 2 L 2 (Ω) ≤ C 90 h ε . ( 90
)
This leads to:

|D 5 | ≤ -γD 8 + C 90 h ε 1 γ . ( 91
)
Moreover, using the Cauchy-Schwarz inequality and Lemma 7.1, one can easily show:

|D 6 | ≤ C 92 h ε E L 2 (Ω) + F L 2 (Ω) ≤ C 92 h ε . ( 92
)

Study of D 7

Using Corollary 2.2 one can write:

D 7 = dof * Fdof -F dof , β dof F dof + i β dof i F i , Fi - Ω F 2 .
Using Lemma 7.1 and Corollary 2.2, we can notice that:

-

Ω F 2 ≤ - j * |Ω j | Fj 2 = - j * dof β dof j Fj , Fj .
Therefore we can write:

D 7 ≤ - dof * Fdof -F dof , β dof ( Fdof -F dof ) + D7 , (93)
with:

D7 = dof * i β dof i Fdof , Fdof -F dof -β dof i Fi , Fi -F i .
Applying Theorem 4.2 to u dof = Fdof -F dof and u r+1/2 = O(h 2 ), the sum in the right hand side of (93) can be bounded from below by:

dof * Fdof -F dof , β dof ( Fdof -F dof ) ≥ 1 C 94 Fdof -F dof 2 L 2 (Ω) -C 94 h 4 . ( 94
)
Moreover, one has:

D7 = dof * i β dof i δ i,dof ( F), Fdof -F dof + dof * i β dof i Fi , δ i,dof ( F) + dof * i β dof i Fi , F dof -F i . ( 95
)
The first sum of (95) can be bounded by:

dof * i β dof i δ i,dof ( F), Fdof -F dof ≤ dof * i β dof i δ i,dof ( F), δ i,dof ( F) + dof * i β dof i δ i,dof ( F), Fi -F dof (96) ≤ C 97 δ( F) 2 L 2 (Ω) ≤h 2 ε 2 +C 97 δ( F) L 2 (Ω) ≤hε    F L 2 (Ω) ≤ε + F dof L 2 (Ω) ≤ε    ≤ C 97 hε 2 . ( 97
)
The second sum of (95) can be bounded by:

dof * i β dof i Fi , δ i,dof ( F) ≤ C 98 F L 2 (Ω) ≤ε δ( F) L 2 (Ω) ≤hε ≤ C 98 hε 2 . ( 98
)
Using Remark 1, the third sum of (95) can be bounded by:

dof * i β dof i Fi , F dof -F i ≤ γε dof * i dof i F dof -F i , ñdof i 2 + C 99 h εγ F 2 L 2 (Ω) ≤Cε 2 . ( 99
)
Summing equations ( 96), (97), ( 98) and (99) ans using (88) leads to:

D 7 ≤ C 100 1 + 1 γ hε -γε 2 D 8 . ( 100
)

Conclusion

Using (91), ( 92), ( 100) and (86), one can write, for any γ > 0:

d dt e 2 ≤ C 101 h ε 1 + 1 γ + D 8 (1 -C 101 γ). ( 101
)
Choosing γ small enough and using D 8 ≤ 0 leads to:

d dt e 2 (t) ≤ C 102 h ε . ( 102
)
Integrating this inequality on [0, T ], we conclude the proof of Theorem 7.2.

Conclusion and proof of Theorem 4.4

Using Theorem 7.2, the exists a constant C 103 such that:

E -Ẽ L 2 ([0,T ]×Ω) + F -F L 2 ([0,T ]×Ω) ≤ C 103 (1 + T ) h ε . ( 103
)
Moreover, using the triangle inequality, we have:

E -Ẽ L 2 ([0,T ]×Ω) + F -F L 2 ([0,T ]×Ω) ≤ E -E 0 L 2 ([0,T ]×Ω) + E 0 -Ẽ0 L 2 ([0,T ]×Ω) + Ẽ0 -Ẽ L 2 ([0,T ]×Ω) + F -εF 0 L 2 ([0,T ]×Ω) + ε F 0 L 2 ([0,T ]×Ω) + F L 2 ([0,T ]×Ω) .
Therefore, according to Theorem 5.1, Theorem 6.1 and Lemma 1.1 and using 1 + T ≤ 2(1 + T 3

2 ), we can write:

E -Ẽ L 2 ([0,T ]×Ω) + F -F L 2 ([0,T ]×Ω) ≤ C 104 1 + T 3 2 √ εh + ε √ h + h + ε (104)
Using Lemma 1.2 concludes the proof.

Appendix

9.1 Proof of lemma 4.3

One has, according to (22):

D r+1/2 ≤ 2 ñr+1/2 j + σ ε x i -x j ≤ 2 + C 1 σ ε h ≤ C D 1 + σ ε h ,
and:

D r+1/2 2 = 4 + σ ε 2 x i -x j 2 + 4 σ ε ñr+1/2 j , x i -x j .
Moreover:

ñr+1/2 j , x i -x j ≥ 2 hC 1 γ 1 ≥ 0. Indeed: ñr+1/2 j , x i -x j = ñr+1/2 j , x r+1/2 -x j - ñr+1/2 j , x r+1/2 -x i , since -ñ r+1/2 j = ñr+1/2 i
and using the definition of γ 1 (28):

ñr+1/2 j , x i -x j = ñr+1/2 j , x r+1/2 -x j + ñr+1/2 + , x r+1/2 -x i ≥ 1 hC 1 2γ 1 .
Therefore:

D r+1/2 2 ≥ 4 + σh C 1 ε 2 , and: D r+1/2 ≥ 1 C D 1 + σ ε h .

Proof of Theorem 4.2

We recall that, for any shoulder point r + 1/2:

Dr+1/2 = 1 D r+1/2 D r+1/2 .
One has:

u r+1/2 , β r+1/2 u r+1/2 = u r+1/2 , Cr+1/2 j u r+1/2 , x i -x j .
We decompose Cr+1/2 j and x i -x j on the basis ( Dr+1/2 , D⊥ r+1/2 ), finding:

u r+1/2 , β r+1/2 u r+1/2 = u r+1/2 , Dr+1/2 2 λ 0 + u r+1/2 , Dr+1/2 u r+1/2 , D⊥ r+1/2 λ 1 + u r+1/2 , D⊥ r+1/2 2 λ 2 ,
with:

λ 0 = Cr+1/2 j , Dr+1/2 x i -x j , Dr+1/2 , λ 2 = Cr+1/2 j , D⊥ r+1/2
x i -x j , D⊥ r+1/2 , and:

λ 1 = Cr+1/2 j , D⊥ r+1/2 x i -x j , Dr+1/2 + Cr+1/2 j , Dr+1/2 x i -x j , D⊥ r+1/2 .
Using ( 27), (28) and Lemma 4.3:

Cr+1/2 j , Dr+1/2 = 1 D r+1/2 2 Cr+1/2 j + σ ε Cr+1/2 j , x i -x j ≥ 1 C D C -1 1 h + σ ε h 2 γ 1 1 + σh ε ,
and:

x i -x j , Dr+1/2 = 1 D r+1/2 2 ñr+1/2 j , x i -x j + σ ε x i -x j 2 ≥ 1 C D C -1 1 hγ 1 + σ ε C -2 1 h 2 1 + σh ε .
This leads to:

λ 0 ≥ 1 C 2 1 C D h 2 1 + σ ε hγ 1 γ 1 + σ ε h 1 + σh ε 2 ≥ 1 C 2 1 C D h 2 γ 1 ≥ 0.
Hence:

u r+1/2 , β r+1/2 u r+1/2 ≥ u r+1/2 , Dr+1/2 2 (λ 0 -|λ 1 |) -u r+1/2 , D⊥ r+1/2 2 (|λ 2 | + |λ 1 |).
In addition, using Lemma 4.3 again:

|λ 1 | ≤ C D 1 1 + σh ε σ ε γ 2 h 2 C 1 h ≤ C D C 1 γ 2 h 2 , |λ 2 | ≤ C 2 D 1 1 + σh ε 2 σ ε γ 2 h 2 C 1 γ 2 h ≤ C 2 D C 1 γ 2 2 h 2 .
Using:

u r+1/2 , D⊥ r+1/2 2 ≤ 3 u r 2 + u r+1 2 + u 2 r+1/2 , ( 105
)
and noticing max

C 2 D C 1 , 3 2 C D C 2 1 ≤ 4C 3 1 , one can write: u r+1/2 , β r+1/2 u r+1/2 ≥ h 2 u r+1/2 , Dr+1/2 2 1 C γ 1 -Cγ 2 -C(γ 2 + γ 2 2 ) u r 2 + u r+1 2 -C(γ 2 + γ 2 2 )u 2 r+1/2 ,
with C = 8C 3 1 . Summing over the degrees of freedom and using (25), one finally has:

dof * u dof , β dof u dof ≥ h 2 1 C -C(γ 2 + γ 2 2 ) r * u r 2 (106) + h 2 1 C γ 1 -Cγ 2 r+1/2 * u r+1/2 , Dr+1/2 2 -h 2 C(γ 2 + γ 2 2 ) r+1/2 * u 2 r+1/2 . ( 107
)
Using the assumptions:

1 C -C(γ 2 + γ 2 2 ) > 0, 1 C γ 1 -Cγ 2 > 0,
and denoting

C = max 1 C -C(γ 2 + γ 2 2 ) -1 , 1 C γ 1 -Cγ 2 -1 , C(γ 2 + γ 2 2 ) ,
one can write (106)-(107) as:

dof * u dof , β dof u dof ≥ h 2 1 C   r * u r 2 + r+1/2 * u r+1/2 , Dr+1/2 2   -C h 2 r+1/2 * u 2 r+1/2 . ( 108
)
Using:

u r+1/2 2 = u r+1/2 , Dr+1/2 2 + u r+1/2 , D⊥ r+1/2 2 ≤ 3 u r+1/2 , Dr+1/2 2 + u r 2 + u r+1 2 + u 2 r+1/2 , ( 109
) one can eventually write:

dof * u dof , β dof u dof ≥ h 2 1 C   r * u r 2 + r+1/2 * u r+1/2 2   -h 2 C r+1/2 * u 2 r+1/2 ,
for some C ≥ 1, which is the desired result.

Proof of theorem 4.1

Let ξ ∈ R 2 . ξ, β r j ξ = ξ, Cr j ξ, x r -x j .
We decompose x r -x j on the basis ñr j , (ñ r j ) ⊥ and get:

ξ, β r j ξ = ξ, Cr j ξ, ñr j x r -x j , ñr j + ξ, (ñ r j ) ⊥ x r -x j , (ñ r j ) ⊥ . ( 110
)
Owing to ( 27) and ( 28) , one has:

x r -x j , ñr j ≥ 1 C 1 h x r -x j , Cr j ≥ h C 1 γ 1 ,
Thus the first term of (110) can be bounded from below as follows:

ξ, Cr j ξ, ñr j x r -x j , ñr j = ξ, α r j ξ ≥0 x r -x j , ñr j ≥ ξ, α r j ξ γ 1 h C 1 .
Using (28) again, the second term of (110) is bounded by: ξ, Cr j ξ, (ñ r j ) ⊥ x r -x j , (ñ r j ) ⊥ = ξ, ñr j ξ, (ñ r j ) ⊥ x r -x j , ( Cr j ) ⊥ ≤ γ 2 h 2 ξ 2 . We thus have:

ξ, β r ξ ≥ ξ, α r ξ γ 1 h C 1 - i γ 2 h 2 ξ 2 ≥ h 2 γ 1 1 C 2 1 -C 1 γ 2 >0 ξ 2 .
9.4 Sufficient condition on the mesh (independent of ω) ensuring the assumptions of Theorem 4.1

The assumptions of Theorem 4.1 depend on the parameters (ω r+1/2 ) r+1/2 of the edges and on the coordinates of the vertices and the shoulder points (which also depend on the (ω r+1/2 ) r+1/2 . In this section, we present some sufficient conditions on the vertices and the control points that ensure the assumptions of Theorem 4.1. We define:

γ1 = 1 h 2 min j, r
min( (x r+1 -x r ) ⊥ , x r -x j , (x r -x r-1 ) ⊥ , x r -x j ), γ2 = 1 h 2 max j, r max(| x r+1 -x r , x r -x j |, | x r -x r-1 , x r -x j |),

δ = 1 h max r+1/2 M r+1/2 - x r+1 + x r 2 , ( 111
)
and we assume γ1 > 0.

According to the definition of the shoulder points [START_REF] Bernard-Champmartin | Extension of centered hydrodynamical schemes to unstructured deforming conical meshes : the case of circles[END_REF], it reads as a convex combination of the middle of the edge and of the control point: Eventually, the condition γ1 ≥ C(γ 2 + δ) for some C ≥ 1, implies the assumptions (29).

x

Proof of Corollary 2.2

Let ξ ∈ R 2 , one has:

|Ω j |ξ = Ωj ξ = ∂Ωj ξ, x -x j n
Moreover, according to Theorem 2.1:

∂Ωj ξ, x -x j n = dof ξ, x dof -x j Cdof j ,
and the relation ξ, x dof -x j Cdof j = β dof j ξ gives the result.

Technical lemma

Lemma 9.1. Let f : R + → R + be a C 1 function and k 1 , k 2 > 0 such that:

d dt (f 2 ) ≤ k 1 (f (t) + k 2 ) , then: ∀t ≥ 0, f 2 (t) ≤ 2 k 2 1 t 2 + f 2 (t = 0) + k 2 2
Proof. One has:

d dt f 2 + k 2 2 ≤ 2k 1 f 2 + k 2 2 , hence: d dt f 2 + k 2 2 ≤ k 1 ,
therefore:

f 2 (t) ≤ 2 k 2 1 t 2 + f 2 (t = 0) + k 2 2 .

Proof of Theorem 2.1

In order to prove Theorem 2.1, we prove that the quadrature formula is exact for affine vector-valued functions. More precisely:

∀A ∈ R 2×2 , ∂Ωj Ax, N = dof * Ax dof , Cdof j ( 116
)
To do so, we first write: Each edge e can be parametrised by the following: e = {M(q), q ∈ [0, 1]} with:

M(q) = 2 i=0
λ i (q)M i , λ i (q) ≥ 0,

2 i=0 λ i (q) = 1,
where M 0 = x r and M 2 = x r+1 are the vertices and M 1 = M 1,r+1/2 is the control point, and: λ 0 (q) = (1 -q) 2 (q) , λ 1 (q) = 2ω (1 -q)q (q) , λ 2 (q) = q 2 (q) , and:

(q) = (1 -q) 2 + 2ω(1 -q)q + q 2 .

Therefore we can write: We explain in the following how the computation of the integrals and for i = j:

1 0 λ i dλ i dq =            - 1 2 if i = 0 0 if i = 1 1 2 if i = 2
Moreover, using λ 0 (q) = λ 2 (1 -q) and λ 1 (q) = λ 1 (1 -q), one has : In addition, since λ 0 = 1 -λ 1 -λ 2 : Therefore we only need to compute the integral:

I 1 = 1 0 dλ 2 dq λ 1 .
The expression of dλ2 dq λ 1 is quite complicated. This is why we show in the following that computing I 1 is equivalent to computing a simpler integral.

Writing: 1 = q 2 + (1 -q) 2 + 2q(1 -q), we deduce: 1 (q) = λ 2 (q) + λ 0 (q) + 1 ω λ 1 (q). (117)

We define I 2 = 1 0 dλ2 dq 1 . Multiplying (117) by dλ2 dq and integrating, we find, using the above equalities:

Figure 1 :

 1 Figure 1: Definition of an AP scheme.

Figure 2 :

 2 Figure 2: Rational quadratic Bezier curve.

Figure 3 :Figure 4 :

 34 Figure 3: Shoulder point.

Figure 5 : 2 jFigure 6

 526 Figure 5: Normal vector from a degree of freedom defined on boundary cell Ω j . Two types: endpoints M r denoted by Cr j or shoulder point denoted by Cr+1/2 j

Figure 6 :

 6 Figure 6: functions f and h.

  to the computation of a single integral. Using:λ 0 (0) = 1 λ 1 (0) = λ 2 (0) = 0 λ 2 (1) = 1, λ 1 (1) = λ 0 (1) = 0,we deduce that, for i = j:

  ) ⊥ , x r -x j | ≤ C 113 γ2 h 2 + C 113 δh 2 . r+1/2 -x j ) ⊥ ≤ C 115 x r+1 -x r , x r+1 + x r 2 -x j + δh 2 ≤ C 115 h 2 (γ 2 + δ) (115)

	Cr+1/2
	r+1/2 =	x r+1 + x r 2	+ λ r+1/2 M r+1/2 -	x r+1 + x r 2	,	λ r+1/2 =	ω r+1/2 1 + ω r+1/2	.
	Using (12), one can write:
	2 Cr j = 1 -	h(ω r+1/2 ) 2	(x r+1 -x r ) ⊥ + 1 -(x 1 h(ω r-1/2 ) 2 C 112 γ1 h 2 -C 112 δh 2 .	(112)
	Besides, one has:
				| ( Cr					(113)
	According to (13), one has:
				Cr+1/2 j	=	h(ω r+1/2 ) 2	(x r+1 -x r ) ⊥ ,
	thus leading to:
	Cr+1/2

r -x r-1 ) ⊥ +h(ω r+1/2 )λ r+1/2 M r+1/2 -x r+1 + x r 2 ⊥ -h(ω r-1/2 )λ r-1/2 M r-1/2 -x r + x r-1 2 ⊥ .

Since γ1 > 0 and 1 -h(ω)/2 ≥ 1 -π/4 > 0, we have:

Cr j , x r -x j ≥ j j , x r+1/2 -x j ≥ 1 C 114 (x r+1 -x r ) ⊥ , x r+1 + x r 2 -x j -C 114 δh 2 ≥ 1 C 114 γ1 h 2 -C 114 δh 2 (114) j , (x

This leads to:

Hence we only need to compute I 2 . An integration by part leads to:

and:

.

Besides:

Eventually, we find:

We thus need to compute the integral I 5 = 1 0 1/ 2 . This can be done using a partial fraction decomposition, we do not give the details of this computation. In the end we find I 1 = f (ω)/2 where f is defined in (14). Now we explain why this gives the desired result. Using:

with:

we obtain:

with:

and f r+1/2 = f (ω r+1/2 ). Notice that B r+1/2 can also be written as a function of the vertices and the shoulder point:

and h r+1/2 = h(ω r+1/2 ). This leads to:

Moreover, one has:

To conclude, we notice that the coefficient in front of AM r+1/2 , x ⊥ r+1 writes exactly f r+1/2 /2. More generally, we can notice that the coefficient in front of AM i , M ⊥ j writes exactly as

. This leads to:

with:

and M 1 = M r+1/2 is the control point of the edge [x r , x r+1 ]. This concludes the proof for the vector-valued functions and proves (116).

For the scalar case, it is sufficent to show:

This can be done by choosing A = ξ ⊗ e for any e ∈ R 2 in (116).

Remark 2. (Quadrature formula using direct edge parametrization)

We recall the parametrization of conical curve using M 0 , M 1 (and a non negative scalar weight ω), M 2 in terms of CAD points is given by [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF]. Changing the view point and taking now the shoulder points S instead of control point M 1 as reference node, the parametrization of conical curve (using M 0 , S, M 2 ) (see Figure 3):

with:

Using this parametrization, we could have directly proved:

with: