Mohamed Farhloul
email: mohamed.farhloul@umoncton.ca

Abdelmalek Zine
email: abdel-malek.zine@ec-lyon.fr

A posteriori error estimation for a dual mixed finite element method for quasi-Newtonian flows whose

Keywords: A posteriori Error Analysis, Dual-Mixed Formulations, Quasi-Newtonian, Power law, Carreau law AMS (MOS) subject classification: 65N30; 65N15;

come

A posteriori error estimation for a dual mixed finite element method for quasi-Newtonian flows whose viscosity obeys a power law or Carreau law

Introduction

Governed by the classical Stokes problem, the Newtonian fluid flows are a reasonable approximation of the more realistic non-Newtonian fluids (quasi-Newtonian or Viscoelastic). In the case of quasi-Newtonian fluids, the viscosity is a function of strain rate tensor, temperature, time, etc. For a steady and creeping flow of an incompressible quasi-Newtonian fluid, the most used formulation, see Bird et al. [START_REF] Bird | Dynamics of polymeric liquids[END_REF], is based on the strain rate tensor. In that case, for Ω a bounded domain of R 2 with a Lipschitz boundary Γ and a given mass forces f defined on Ω, the combination of the constitutive an conservation equations leads to the following Nonlinear Stokes problem:

-div 2µ(|d(u)|) d(u) + ∇p = f in Ω, div u = 0 in Ω, (1)
where u and p, the unknowns of the problem, are the velocity and pressure, respectively.

d(u) = 1 2 ∇ u +(∇ u) t
is the strain rate tensor, and |d(u

)| 2 = 2 i,j=1 d(u) 2 ij .
For µ 0 > 0 a reference viscosity and r a fluid characteristic real parameter verifying 1 < r < ∞, the viscosity function µ(•), depending on |d(u)|, is usually given by one of the two following famous models: µ(x) = µ 0 x r-2 , ∀x ∈ R + , for the Power law model, or µ(x) = µ 0 1 + x 2 (r-2)/2

, ∀x ∈ R + , for the Carreau model.

Finally, system (1) is supplemented by a set of boundary conditions. The generalized Stokes problem [START_REF] Baranger | Analyse numérique des écoulements quasi-newtoniens dont la viscosité obéit à la loi puissance ou la loi de Carreau[END_REF] and its approximation by standard finite elements was first studied in Baranger and Najib [START_REF] Baranger | Analyse numérique des écoulements quasi-newtoniens dont la viscosité obéit à la loi puissance ou la loi de Carreau[END_REF]. Extensions and improvements of the error bounds have been obtained in Sandri [START_REF] Sandri | Sur l'approximation numérique des écoulements quasi-Newtoniens dont la viscosité suit la loi puissance ou la loi de Carreau[END_REF] and Barrett and Liu [2,[START_REF] Barrett | Quasi-norm error bounds for the finite element approximation of a non-Newtonian flow[END_REF].

In these works, only the primal variables velocity and pressure are taken into account. But, for various reasons, one may need information on other (dual) variables such as velocity gradients ∇ u, strain rate tensor d(u), and extra-stress tensor σ = 2µ(|d(u)|) d(u). For these reasons, it is necessary to build appropriate mixed formulations.

On the other hand, in connection with the use of the gradient tensor ∇ u which corresponds to the Ladyzhenskaya model [START_REF] Ladyzhenskaya | New equations for the description of the viscous incompressible fluids and solvability in the large of the boundary value problems for them[END_REF]:

µ(|∇ u|) = (µ 0 + µ 1 |∇ u|) r-2 , µ 0 ≥ 0, µ 1 > 0, r > 1,
a large amount of work is available in the literature. Among these works, there may be mentioned Manouzi and Farhloul [START_REF] Manouzi | Mixed finite element analysis of a non-linear three-fields Stokes model[END_REF], Farhloul and Zine [START_REF] Farhloul | A mixed finite element method for a Ladyzhenskaya model[END_REF], Gatica et al. [START_REF] Gatica | A low-order mixed finite element method for a class of quasi-Newtonian Stokes flows. Part I: a priori error analysis[END_REF][START_REF] Gatica | A low-order mixed finite element method for a class of quasi-Newtonian Stokes flows. Part II: a posteriori error analysis[END_REF] and Ervin et al. [START_REF] Ervin | A dual-mixed approximation method for a three-field model of a nonlinear generalized Stokes problem[END_REF]. The major drawback of formulations using the gradient lies in the fact that we can not deal with natural boundary conditions. To overcome this drawback related to the boundary conditions, we have introduced and analyzed a dual-mixed finite element method for quasi-Newtonian fluid flow obeying to the Power law, in Farhloul and Zine [START_REF] Farhloul | A mixed finite element method for a quasi-Newtonian fluid flow[END_REF][START_REF] Farhloul | A posteriori error estimation for a dual mixed finite element approximation of non-Newtonian fluid flow problems[END_REF]. A priori error estimates for the finite element approximation were proved in the first paper, while a posteriori error estimation was provided in the second work. In both papers, our analysis is based on the fact that the equation describing the extra-stress tensor in terms of the rate of strain tensor is invertible and give the rate of strain tensor as a function of the stress tensor. In a recent work Farhloul-Zine [START_REF] Farhloul | A dual-mixed finite element method for quasi-Newtonian flows whose viscosity obeys a power law or the Carreau law[END_REF], we developed a mixed formulation to overcome this constraint of inversibility of the viscosity law. The main advantage of this formulation is that it makes it possible to consider differently viscosity functions obeying the Power law or Carreau Law.

The aim of this work is to give an a posteriori error estimates for the mixed formulation developed in [START_REF] Farhloul | A dual-mixed finite element method for quasi-Newtonian flows whose viscosity obeys a power law or the Carreau law[END_REF]. In the next section we recall the mixed formulation developed in [START_REF] Farhloul | A dual-mixed finite element method for quasi-Newtonian flows whose viscosity obeys a power law or the Carreau law[END_REF] and then we give the a posteriori error estimates in section 3. This will be done by extending our investigations by avoiding the assumption of expressing the rate of strain tensor as function of the stress tensor. We may be then able to deal with both problems associated with Power law and Carreau model.

Dual-mixed formulation

In order to obtain a dual-mixed formulation of (1), first the problem (1) is formulated as follows:

   -div σ -p I = f in Ω, div u = 0 in Ω, u = 0 on Γ, (2)
and then, we introduce two new variables t = d(u), the strain rate tensor, (3)

A(t) = 2µ (|t|) t = σ, the extra stress tensor. ( 4
) Suppose f ∈ [L r (Ω)] 2 , 1 < r < ∞. Let ω = ω(u) = 1 2 (∇ u -(∇ u) t
) be the vorticity tensor. Then, for all (τ , q) ∈ [L r (Ω)] 2×2 × L r 0 (Ω), such that div τ -q I ∈ [L r (Ω)] 2 , and for all u ∈ [W 1,r (Ω)] 2 such that div u = 0, it is easy to see that

(t, τ ) = (d(u), τ ) = -(div(τ -q I), u) -(ω, τ ),
where, from now on, ( • , • ) denotes the duality pairing between L r (Ω) and L r (Ω), and

L r 0 (Ω) = q ∈ L r (Ω); Ω q dx = 0 .
In order to derive the mixed formulation of (2), we define the following spaces:

T = [L r (Ω)] 2×2 , Σ = (τ , q) ∈ [L r (Ω)] 2×2 × L r 0 (Ω); div(τ -q I) ∈ [L r (Ω)] 2 , M = (v, η) ∈ [L r (Ω)] 2 × [L r (Ω)] 2×2 ; η + η t = 0 ,
equipped with the following norms:

s T = s 0,r,Ω = Ω |s| r 1 r , (τ , q) Σ = τ r 0,r ,Ω + q r 0,r ,Ω + div(τ -q I) r 0,r ,Ω 1 r , (v, η) M = v r 0,r,Ω + η r 0,r,Ω 1 r .
The dual-mixed formulation of problem (2) reads as follows: Find t ∈ T , (σ, p) ∈ Σ and (u,

ω) ∈ M such that      (A(t), s) -(σ, s) = 0 ∀ s ∈ T , (t, τ ) + (div(τ -q I), u) + (τ , ω) = 0 ∀τ ∼ = (τ , q) ∈ Σ, (div(σ -p I), v) + (σ, η) + (f , v) = 0 ∀v ∼ = (v, η) ∈ M . (5)
Remark 2.1. From the last equation of (5),

(div(σ -p I), v) + (σ, η) + (f , v) = 0, ∀(v, η) ∈ M , one gets (σ, η) = 0, ∀η ∈ [L r (Ω)] 2×2 such that η + η t = 0.
This corresponds to the symmetry relaxation of the extra-stress tensor σ by a Lagrange multiplier.

Remark 2.2. As stated above, the use of the rate of strain tensor enables to handle different types of boundary conditions, such as mixed boundary conditions. More precisely, assuming that we consider the following boundary conditions:

   u = 0 on Γ D and (2µ(|d(u)|) d(u) -p I) n = 0 on Γ N ,
where Γ = Γ D ∪ Γ N , Γ D = ∅ and n is the unit outward normal vector field along the boundary of Ω.

Then, the only change to be made is to replace the space Σ by the following one:

Σ = (τ , q) ∈ [L r (Ω)] 2×2 × L r 0 (Ω); div(τ -q I) ∈ [L r (Ω)] 2 ; (τ -q I) n = 0 on Γ N .
However, for the sake of clearness, we developed in [START_REF] Farhloul | A dual-mixed finite element method for quasi-Newtonian flows whose viscosity obeys a power law or the Carreau law[END_REF] the analysis in the case of Dirichlet boundary conditions only.

To formally rewrite (5) as a twofold saddle-point problem, we define the following operators:

A : T -→ T , B : T -→ Σ and C : Σ -→ M ,
where for a Banach space X, X denotes the dual space with associated norm • X .

[

A(t), s] = (A(t), s) , ∀ s, t ∈ T , (6)
[B(s), τ ∼ ] = -(s, τ ) , ∀ s ∈ T , ∀τ ∼ = (τ , q) ∈ Σ, (7)
[C(τ ∼ ), v ∼ ] = -(div(τ -q I), v) -(τ , η), ∀τ ∼ ∈ Σ, ∀v ∼ ∈ M . (8)
Remark 2.3. Recall that the operator A is defined by

∀ t ∈ T , A(t) = 2µ(|t|) t,
µ, the viscosity function, being given by either Power or Carreau law.

Problem ( 5) is then written in the following twofold saddle-point form:

Find t ∈ T , σ ∼ ∈ Σ and u ∼ ∈ M such that        [A(t), s] + [s, B (σ ∼ )] = 0 ∀ s ∈ T , [B(t), τ ∼ ] + [τ ∼ , C (u ∼ )] = 0 ∀τ ∼ ∈ Σ, [C(σ ∼ ), v ∼ ] = [F , v ∼ ] ∀v ∼ ∈ M , (9)
where, [F , v ∼ ] = (f , v) , ∀v ∼ ∈ M , and B and C denote the dual operators of B and C, respectively. The existence, uniqueness and stability of t, σ ∼ , u 9) are obtained under some assumptions on the operators A, B and C. These hypotheses are verified by some technical lemmas that we recall here. These assumptions concern the properties of the operators A, B and C. Mainly:

∼ = t, (σ, p), (u, ω) ∈ T × Σ × M , solution to the problem (
• A is bounded, continuous and strictly monotone,

• B verifies the inf-sup condition on the Kernel of C,

• C verifies the inf-sup condition.

Several technical lemmas that establish the appropriate conditions on the operators A are given in [START_REF] Farhloul | A dual-mixed finite element method for quasi-Newtonian flows whose viscosity obeys a power law or the Carreau law[END_REF].

We also recall here the inf -sup condition in the continuous framework. Its discrete form is recalled below, see Lemma 2.3. These two conditions are used for a posteriori estimates.

Lemma 2.1. There exists a positive constant

β 2 such that inf v ∼ ∈M sup τ ∼ ∈Σ [C(τ ∼ ), v ∼ ] τ ∼ Σ v ∼ M ≥ β 2 . ( 10
) Lemma 2.2. Let Z * 1 = τ ∼ = (τ , q) ∈ Σ; (div(τ -q I), v) = 0, ∀ v ∈ [L r (Ω)] 2 .
Then, there exists a positive constant

β * 1 such that inf τ ∼ ∈Z * 1 sup s∈T (s, τ ) s T τ ∼ Σ ≥ β * 1 .
Proof. The proof of this result is similar to the one of Lemma 3.6 in [START_REF] Farhloul | A dual-mixed finite element method for quasi-Newtonian flows whose viscosity obeys a power law or the Carreau law[END_REF].

The following result giving the existence, uniqueness and stability is also established in [START_REF] Farhloul | A dual-mixed finite element method for quasi-Newtonian flows whose viscosity obeys a power law or the Carreau law[END_REF].

Theorem 2.1. Problem (9) admits a unique solution (t, σ ∼ , u ∼ ) ∈ T × Σ × M satisfying the following stability condition, t T + σ ∼ Σ + u ∼ M ≤ C(f ),
where C(f ) is a positive constant depending on f .

Let us now recall the discrete problem. We assume that the boundary Γ of the domain Ω is polygonal. We first give some finite element notations. Let h > 0 and T h a triangulation of Ω into triangles. We assume that the triangulation T h is regular in the sense of Ciarlet [START_REF] Ciarlet | The Finite Element Methods for Elliptic Problems[END_REF]. Let K ∈ T h be an element of the triangulation, we denote by b K the bubble function defined by

b K (x) = λ 1 (x)λ 2 (x)λ 3 (x), ∀x ∈ K, λ i , i = 1, • • • , 3
being the barycentric co-ordinates with respect to the element K. For k ∈ N, let P k (K) denote the space of polynomials of degree less than or equal to k on K, and

R(K) = P 1 (K) 2 ⊕ R curl b K , where curl b K = ( ∂b K ∂x 2 , - ∂b K ∂x 1
).

To write the discrete mixed formulation, we introduce the following finite dimensional spaces:

T h = s h ∈ T ; s h| K ∈ R(K), ∀K ∈ T h , Σ h = τ h ∼ = (τ h , q h ) ∈ Σ; τ h| K ∈ [R(K)] 2 , q h| K ∈ P 1 (K), ∀K ∈ T h , M h = v h ∼ = (v h , η h ) ∈ M ; v h| K ∈ P 0 (K) 2 , η h = θ h χ, θ h| K ∈ P 1 (K), ∀K ∈ T h ,
where

χ = 0 -1 1 0 .
The discrete mixed formulation of problem ( 9) is given by the following: Find

t h ∈ T h , σ h ∼ ∈ Σ h and u h ∼ ∈ M h such that          [A(t h ), s h ] + [s h , B (σ h ∼ )] = 0 ∀ s h ∈ T h , [B(t h ), τ h ∼ ] + [τ h ∼ , C (u h ∼ )] = 0 ∀τ h ∼ ∈ Σ h , [C(σ h ∼ ), v h ∼ ] = [F , v h ∼ ] ∀v h ∼ ∈ M h . ( 11
)
The analysis of the above discrete problem [START_REF] Farhloul | A mixed finite element method for a quasi-Newtonian fluid flow[END_REF], as well as the a priori estimates of discrete errors, are given in Farhloul-Zine [START_REF] Farhloul | A dual-mixed finite element method for quasi-Newtonian flows whose viscosity obeys a power law or the Carreau law[END_REF]. These results are, in part, based on the some lemmas that we recall here because we need them further to establish the posteriori estimates. For the details on proofs, see Farhloul-Zine [START_REF] Farhloul | A dual-mixed finite element method for quasi-Newtonian flows whose viscosity obeys a power law or the Carreau law[END_REF].

Lemma 2.3. There exists a positive constant β * 2 independent of h, such that

inf v h ∼ ∈M h sup τ h ∼ ∈Σ h [C(τ h ∼ ), v h ∼ ] τ h ∼ Σ v h ∼ M ≥ β * 2 . ( 12
) Lemma 2.4. Let Z h 1 = τ h ∼ = (τ h , q h ) ∈ Σ h ; [C(τ h ∼ ), v h ∼ ] = 0, ∀v h ∼ ∈ M h .
Then, there exists a positive constant C, independent of h, such that

∀τ h ∼ ∈ Z h 1 , q h 0,r ,Ω ≤ C τ h 0,r ,Ω . (13)
Due to the previous Lemma, the discrete inf-sup conditions ( 10) and ( 12), we obtain, as for the continuous problem, the existence, uniqueness and stability of the discrete solution. More precisely, Theorem 2.2. Problem (11) admits a unique solution

(t h , σ h ∼ , u h ∼ ) ∈ T h × Σ h × M h satisfying the fol- lowing stability condition, t h T + σ h ∼ Σ + u h ∼ M ≤ C(f ).
where C(f ) is a positive constant depending on f and independent of h.

A posteriori error estimates

Let (t, σ ∼ , u ∼ ) = t, (σ, p), (u, ω) ∈ T × Σ × M and (t h , σ h ∼ , u h ∼ ) = t h , (σ h , p h ), (u h , ω h ) ∈ T h × Σ h × M h
be the solutions of ( 9) and [START_REF] Farhloul | A mixed finite element method for a quasi-Newtonian fluid flow[END_REF], respectively. On T , Σ and M , one define the residuals R 1 , R 2 and R 3 by

< R 1 , s > = (A(t h ), s) -(σ h , s), ∀ s ∈ T , (14)
< R 2 , τ ∼ > = (t h , τ ) + (div(τ -q I), u h ) + (τ , ω h ), ∀ τ ∼ = (τ , q) ∈ Σ, (15)
< R 3 , v ∼ > = (div(σ h -p h I), v) + (σ h , η) + (f , v), ∀ v ∼ = (v, η) ∈ M . ( 16
)
We denote by R 1 * , R 2 * and R 3 * the dual norms of R 1 , R 2 and R 3 , respectively.

R 1 * = sup s∈T | < R 1 , s > | s T , R 2 * = sup τ ∼ ∈Σ | < R 2 , τ ∼ > | τ ∼ Σ and R 3 * = sup v ∼ ∈M | < R 3 , v ∼ > | v ∼ M .
In the sequel, our goal is to obtain upper bounds of the errors t -

t h T , σ ∼ -σ h ∼ Σ and u ∼ -u h ∼ M
as functions of the above dual norms R As we will see later, these results depend on the parameter r. We have then to distinguish two cases:

1 < r < 2 and r ≥ 2.
First, let us give the following estimate of A(t h ) -A(t), t h -t in terms of R 1 * , R 2 * and R 3 * .

Proposition 3.1. There exists a constant C independent of h such that

A(t h ) -A(t), t h -t ≤ C (R 1 * + R 3 * ) t h -t T + R 2 * f -P 0 h f 0,r ,Ω + (R 1 * + R 3 * ) R 2 * + R 2 * sup s∈T A(t) -A(t h ), s s T , (17)
where P 0 h f is the L 2 -projection of f onto

K∈T h P 0 (K) 2 .
Proof.

From the first equation of ( 9), one gets, ∀ s ∈ T ,

A(t h ) -A(t), s = (A(t h ), s) -(σ, s) = (A(t h ), s) -(σ h , s) + (σ h , s) -(σ, s) = R 1 , s + (σ h -σ, s) . Thus, A(t h ) -A(t), s = R 1 , s + (σ h -σ, s) , ∀ s ∈ T . (18)
From the second equation of ( 9), one gets

(t h -t, τ ) + (div(τ -q I), u h -u) + (τ , ω h -ω) = R 2 , τ ∼ , ∀ τ ∼ ∈ Σ . ( 19
)
Using the last equation of ( 9), we obtain

(div[(σ h -p h I) -(σ -p I)], v) + (σ h -σ, η)) = R 3 , v ∼ , ∀ v ∼ ∈ M . (20)
Now, taking τ

∼ = σ h ∼ -σ ∼ in (19) and v ∼ = u h ∼ -u ∼ in (20), we obtain (t h -t, σ h -σ) + (div[(σ h -p h I) -(σ -p I)], u h -u) + (σ h -σ, ω h -ω) = R 2 , σ h ∼ -σ ∼ and, (div[(σ h -p h I) -(σ -p I)], u h -u) + (σ h -σ, ω h -ω) = R 3 , u h ∼ -u ∼ .
These two last equations imply

(t h -t, σ h -σ) = R 2 , σ h ∼ -σ ∼ -R 3 , u h ∼ -u ∼ .
Finally, substituting s, by s = t h -t, in (18) and using the above last equation, we get

A(t h ) -A(t), t h -t = R 1 , t h -t + R 2 , σ h ∼ -σ ∼ -R 3 , u h ∼ -u ∼ . (21)
By the inf -sup condition [START_REF] Farhloul | A mixed finite element method for a Ladyzhenskaya model[END_REF] and [START_REF] Sandri | Sur l'approximation numérique des écoulements quasi-Newtoniens dont la viscosité suit la loi puissance ou la loi de Carreau[END_REF], it follows

β 2 u h ∼ -u ∼ M ≤ sup τ ∼ ∈Σ (div(τ -q I), u h -u) + (τ , ω h -ω) τ ∼ Σ ≤ sup τ ∼ ∈Σ < R 2 , τ ∼ > τ ∼ Σ + sup τ ∼ ∈Σ |(t h -t, τ )| τ ∼ Σ .
Thus,

u h ∼ -u ∼ M ≤ C R 2 * + t h -t T . (22)
Now, using the third equation of ( 9) and the third equation of [START_REF] Farhloul | A mixed finite element method for a quasi-Newtonian fluid flow[END_REF], one gets, ∀v ∈

[L r (Ω)] 2 , (div[(σ h -p h I) -(σ -p I)], v) + P 0 h f -f , v = div(σ h -p h I) + P 0 h f , v = 0. ( 23
)
On the other hand, since P 0 h ff ∈ [L r 0 (Ω)] 2 , there exists (see Galdi [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations[END_REF])

ξ ∈ {τ ∈ [L r (Ω)] 2×2 ; div τ ∈ [L r (Ω)] 2 }
such that div ξ = P 0 h ff in Ω and ξ 0,r ,Ω + div ξ 0,r ,Ω ≤ C P 0 h ff | 0,r ,Ω . Thus, from these last relations and (23), we get

div (σ h -p h I) + ξ -(σ -p I) , v = 0, ∀v ∈ [L r (Ω)] 2 ,
Hence, using the above mentioned inf -sup condition in Lemma 2.2, we obtain

β * 1 (σ h -σ + ξ, p h -p) Σ ≤ sup s∈T (s, σ h -σ + ξ) s T ≤ sup s∈T (s, σ h -σ) s T + sup s∈T (s, ξ) s T .
This last inequality together with [START_REF] Manouzi | Mixed finite element analysis of a non-linear three-fields Stokes model[END_REF], lead to

(σ h -σ + ξ, p h -p) Σ ≤ C R 1 * + sup s∈T (A(t h ) -A(t), s) s T + f -P 0 h f 0,r ,Ω . . ( 24
)
On the other hand,

σ h ∼ -σ ∼ Σ = (σ h -σ, p h -p) Σ ≤ (σ h -σ + ξ, p h -p) Σ + (ξ, 0) Σ . and (ξ, 0) Σ = ξ r 0,r ,Ω + div ξ r 0,r ,Ω 1/r ≤ C P 0 h f -f 0,r ,Ω .
Then, from (24), we get

σ h ∼ -σ ∼ Σ ≤ C R 1 * + P 0 h f -f 0,r ,Ω + sup s∈T (A(t h ) -A(t), s) s T . (25)
Finally, using (21), ( 22) and (25), we get

A(t h ) -A(t), t h -t ≤ R 1 * t h -t T + R 2 * σ h ∼ -σ ∼ Σ + R 3 * u h ∼ -u ∼ M ≤ C R 1 * t h -t T + R 2 * R 1 * + R 2 * P 0 h f -f 0,r ,Ω + R 2 * sup s∈T A(t h ) -A(t), s s T + R 3 * R 2 * + R 3 * t h -t T .
This ends the proof of the previous proposition.

As it was mentioned above, the upper bounds of the errors depend on the parameter r > 1. On the other hand, to distinguish the two models, we set: δ = 0 for Power law and δ = 1 for Carreau law. We first consider the case where the parameter r verify 1 < r < 2

Theorem 3.1. Let (t, σ ∼ , u ∼ ) and (t h , σ h ∼ , u h ∼
) be the solution of problems ( 9) and ( 11), respectively. Suppose that 1 < r < 2, then there exists a constant C independent of h such that

t -t h T ≤ C 3 i=1 R i * + R r/2 2 * + f -P 0 h f 0,r ,Ω , (26)
σ ∼ -σ h ∼ Σ ≤ C R 1 * + 3 i=1 R 2/r i * + R r/r 2 * + f -P 0 h f 0,r ,Ω + f -P 0 h f 2/r 0,r ,Ω , (27)
u ∼ -u h ∼ M ≤ C 3 i=1 R i * + R r/2 2 * + f -P 0 h f 0,r ,Ω . (28)
Proof. Owing to Lemma 3.1 and Lemma 3.3 in Farhloul-Zine [START_REF] Farhloul | A dual-mixed finite element method for quasi-Newtonian flows whose viscosity obeys a power law or the Carreau law[END_REF], we have

A(t h ) -A(t), t h -t ≥ C t h -t 2 0,r,Ω δ + t h 2-r 0,r,Ω + t 2-r 0,r,Ω + Ω |A(t h ) -A(t)| |t h -t| dx and, A(t h ) -A(t) 0,r ,Ω ≤ C Ω |A(t h ) -A(t)| |t h -t| dx 1/r . ( 29
)
Then, we get from [START_REF] Ladyzhenskaya | New equations for the description of the viscous incompressible fluids and solvability in the large of the boundary value problems for them[END_REF],

t h -t 2 T δ + t h 2-r T + t 2-r T + Ω |A(t h ) -A(t)| |t h -t| dx ≤ C (R 1 * + R 3 * ) t h -t T + R 2 * f -P 0 h f 0,r ,Ω + (R 1 * + R 3 * ) R 2 * + R 2 * sup s∈T (A(t) -A(t h ), s) s T
On the other hand, using stability conditions (see Farhloul-Zine [START_REF] Farhloul | A dual-mixed finite element method for quasi-Newtonian flows whose viscosity obeys a power law or the Carreau law[END_REF]),

t T ≤ C(f ) and t h T ≤ C(f ),
and (29), we get

t h -t 2 T + Ω |A(t h ) -A(t)| |t h -t| dx ≤ C (R 1 * + R 3 * ) t -t h T + R 2 * f -P 0 h f 0,r ,Ω + R 2 * (R 1 * + R 3 * ) + R 2 * Ω |A(t h ) -A(t)| |t h -t| dx 1/r .
Now, using the Young inequality, we obtain, ∀ ε > 0 and ∀ ε > 0,

t h -t 2 T + Ω |A(t h ) -A(t)| |t h -t| dx ≤ C ε -1 (R 1 * + R 3 * ) 2 + ε t -t h 2 T + R 2 * f -P 0 h f 0,r ,Ω + R 2 * (R 1 * + R 3 * ) + (ε) -r R r 2 * +(ε) r Ω |A(t) -A(t h )| |t -t h | dx .
Thus, for an adequate choice of ε and ε, we get

t h -t 2 T + Ω |A(t h ) -A(t)| |t h -t| dx ≤ C (R 1 * + R 3 * ) 2 + R 2 * f -P 0 h f 0,r ,Ω + R 2 * (R 1 * + R 3 * ) + R r 2 * .
This implies, in particular, the expected result in equation ( 26), namely:

t -t h T ≤ C 3 1 R i * + R r/2 2 * + f -P 0 h f 0,r ,Ω ,
and

Ω |A(t h ) -A(t)| |t -t h | dx ≤ C 3 1 R 2 i * + R r 2 * + f -P 0 h f 2 0,r ,Ω . (30)
Finally, to obtain the expected estimate (27), it suffices to use the inequalities (25), ( 29) and (30). And the estimate (28) is a direct consequence of ( 22) and (26).

After the study of the case 1 < r < 2, we will now consider the case r ≥ 2.

Theorem 3.2. Let (t, σ ∼ , u ∼ ) and (t h , σ h ∼ , u h ∼
) be the solution of problems ( 9) and [START_REF] Farhloul | A mixed finite element method for a quasi-Newtonian fluid flow[END_REF], respectively. Suppose that r ≥ 2, then there exists a constant C independent of h such that

t -t h T ≤ C 3 i=1 R r /r i * + R 2 * + R 2/r 2 * + f -P 0 h f 0,r ,Ω , (31)
σ ∼ -σ h ∼ Σ ≤ C R r /2 1 * + R r /2 3 * + 3 i=1 R i * + f -P 0 h f 0,r ,Ω , (32)
u ∼ -u h ∼ M ≤ C 3 i=1 R r /r i * + R 2 * + R 2/r 2 * + f -P 0 h f 0,r ,Ω . (33)
Proof. Using Lemma 3.2 in Farhloul-Zine [START_REF] Farhloul | A dual-mixed finite element method for quasi-Newtonian flows whose viscosity obeys a power law or the Carreau law[END_REF], we get

A(t h ) -A(t), t h -t ≥ C t h -t r 0,r,Ω + Ω δ + |t h | r-2 + |t| r-2 |t h -t| 2 dx
and

A(t h ) -A(t) 0,r ,Ω ≤ C Ω δ + |t h | r-2 + |t| r-2 |t h -t| 2 dx 1/2 (34)
× δ + t h (r-2)/2 0,r,Ω + t (r-2)/2 0,r,Ω
.

To simplify notations, we set

Λ = t -t h r T + Ω δ + |t h | r-2 + |t| r-2 |t -t h | 2 dx. Then, from (17), we have
Λ ≤ C (R 1 * + R 3 * )( t h -t T + R 2 * ) + R 2 * f -P 0 h f 0,r ,Ω + R 2 * A(t h ) -A(t) 0,r ,Ω
And then, from (34), we get

Λ ≤ C (R 1 * + R 3 * )( t h -t T + R 2 * ) + R 2 * f -P 0 h f 0,r ,Ω + R 2 * Ω δ + |t h | r-2 + |t| r-2 |t h -t| 2 dx 1/2 δ + t h (r-2)/2 T + t (r-2)/2 T
And again, using Young's inequality with two parameters ε and ε, we get

Λ ≤ C ε r t h -t r T + ε -r (R 1 * + R 3 * ) r + R 2 * f -P 0 h f 0,r ,Ω + (R 1 * + R 3 * ) R 2 * +ε Ω δ + |t h | r-2 + |t| r-2 |t h -t| 2 dx + (ε) -1 R 2 2 * δ + t h (r-2)/2 T + t (r-2)/2 T 2 .
Thus, using the stability conditions t T ≤ C(f ) and t h T ≤ C(f ), we obtain

Λ ≤ C (R 1 * + R 3 * ) r + R 2 * f -P 0 h f 0,r ,Ω + (R 1 * + R 3 * ) R 2 * + R 2 2 * . ( 35
)
This last inequality leads in particular to

t -t h r T ≤ C (R 1 * + R 3 * ) r + R r 2 * + f -P 0 h f r 0,r ,Ω + R r 2 * + R 2 2 * .
And then,

t -t h T ≤ C 3 i=1 R r /r i * + R 2 * + R 2/r 2 * + f -P 0 h f 0,r ,Ω ,
which is precisely the expected estimate (31).

On the other hand, from (35), we deduce the following estimate

Ω δ + |t h | r-2 + |t| r-2 |t -t h | 2 dx ≤ C (R 1 * + R 3 * ) r + (R 1 * + R 3 * ) 2 + R 2 2 * + f -P 0 h f 2 0,r ,Ω ,
and then, using (25), (34) and the fact that t T and t h T are bounded, we get

σ ∼ -σ h ∼ Σ ≤ C 3 i=1 R i * + R r /2 1 * + R r /2 3 * + f -P 0 h f 0,r ,Ω ,
which is the expected estimate (32). Finally, the estimation (33) is a consequence of the estimates ( 22) and (31).

Finally, the previous results show that to have the a posteriori error estimates of our problem, it suffices to estimate R i * , i = 1, 2, 3. To this end, we first precise some notations: for a tensor field τ ∈ R 2×2 and for a vector field

v = (v 1 , v 2 ) ∈ R 2 , • tr(τ ) = τ 11 + τ 22 , as(τ ) = τ 21 -τ 12 , rot(τ ) = ∂τ 12 ∂x 1 - ∂τ 11 ∂x 2 , ∂τ 22 ∂x 1 - ∂τ 21 ∂x 2 , • Curl(v) =    ∂v 1 ∂x 2 - ∂v 1 ∂x 1 ∂v 2 ∂x 2 - ∂v 2 ∂x 1    ,
• and g E stands for the jump of function g across an edge E.

We also recall the following Helmholtz decomposition of a tensor field in Σ.

Proposition 3.2. Let τ ∼ ∈ Σ. Then there exist z ∈ W 2,r (Ω) 2 and ψ ∈ W 1,r (Ω) 2 such that

τ -q I = ∇z + Curlψ, (36)
with the estimate

z 2,r ,Ω + ψ 1,r ,Ω ≤ C τ ∼ Σ . (37)
Proof. To prove this result it is sufficient to apply Theorem 1.1 of Creus et al. [START_REF] Creusé | A posteriori error estimation for the dual mixed finite element method for the p-Laplacian in a polygonal domain[END_REF] to each row of the tensor τ -q I, i.e. the two vector fields (τ 11 -q, τ 12 ) and (τ 21 , τ 22 -q). Lemma 3.1. For every τ ∼ ∈ Σ, we have

< R 2 , τ ∼ > = K∈T h (t h +ω h , ∇z -Π h (∇z)) + K∈T h (tr(t h )), q) + K∈T h (rot(t h +ω h ), ψ -I cl (ψ)) - E∈E h < (t h +ω h )t E , ψ -I cl (ψ) > E (38) where -(z, ψ) ∈ [W 2,r (Ω)] 2 × [W 1,r (Ω)] 2 denotes the Helmholtz decomposition of τ ∼ ∈ Σ, -I cl (ψ)
is the Clment interpolant of ψ (see Clment [START_REF] Clément | Approximation by finite element functions using local regularization[END_REF]), -E h denotes the set of all edges of the triangulation T h , -(t h +ω h )t E denotes the tangential jump of t h +ω h across the edge E., -Π h (∇z) is the Brezzi-Douglas-Marini interpolant of the lowest degree of ∇z (see Brezzi et al. [START_REF] Brezzi | Two families of mixed finite elements for second order elliptic problems[END_REF]).

where η 1 (K), η 2 (K) and η 3 (K) are the local estimators given by

η 1 (K) r = A(t h ) -σ h r 0,r ,K , η 2 (K) r = h r K t h +ω h r 0,r,K + rot(t h +ω h ) r 0,r,K + tr(t h ) r 0,r,K + E∈∂K h E t h +ω h )t E r 0,r,E , η 3 (K) r = f -P 0 h f r 0,r ,K + as(σ h ) r 0,r ,K .
Proof. It follows from ( 14) that for every s ∈ T ,

|< R 1 , s >| ≤ K∈T h A(t h ) -σ h r 0,r ,K 1/r s 0,r,Ω
and then,

sup s∈T |< R 1 , s >| s T ≤ K∈T h A(t h ) -σ h r 0,r ,K 1/r .
Which is precisely the estimate (41). To show the estimate (42), we will use (38) obtained in Lemma 3.1. This inequality leads, for every τ ∼ ∈ Σ, to

| < R 2 , τ ∼ > | ≤ K∈T h t h +ω h 0,r,K ∇z -Π h (∇z) 0,r ,K + K∈T h tr(t h ) 0,r,K q 0,r ,K + K∈T h rot(t h +ω h ) 0,r,K ψ -I cl (ψ) 0,r ,K + E∈E h (t h +ω h )t E 0,r,E ψ -I cl (ψ) 0,r ,E . (44)
Now, by Lemma 3.1 in Verfürth [START_REF] Verfürth | A review of a posteriori error estimation and adaptive mesh-refinement techniques[END_REF], we have

ψ -I cl (ψ) 0,r ,K ≤ Ch K |ψ| 1,r ,ω K and ψ -I cl (ψ) 0,r ,E ≤ Ch 1/r E |ψ| 1,r
,ω E , where ω K denotes the union of K with all the triangles from the triangulation T h adjacent to the triangle K, ω E denotes the union of at most two triangles of T h admitting E as a common edge and | • | 1,r ,ω , the semi-norm of W 1,r (ω).

Thus, using these two last estimates and the fact that ∇z -Π h (∇z) 0,r ,K ≤ Ch K |∇z| 1,r ,K , the above inequality (44) yield ( v r 0,r,Ω + η r 0,r,Ω ) 1/r .

| < R 2 , τ ∼ > | ≤ C K∈T h h K t h +ω h 0,
| < R 3 , v ∼ > | ≤ K∈T h div(σ h -p h I) + f r 0,r ,K 1
On the other hand, following the second equation of the discrete problem [START_REF] Farhloul | A mixed finite element method for a quasi-Newtonian fluid flow[END_REF], we have div(σ h -p h I) = -P 0 h f .

Therefore, for every v ∼ ∈ M ,

| < R 3 , v ∼ > | ≤ C( K∈T h
f -P 0 h f r 0,r ,K + as(σ h ) r 0,r ,K ) 1/r v ∼ M which implies R 3 * ≤ Cη 3 , and the proof is completed.

Conclusion

In this work, we have developed and analyzed a new a posteriori error estimator for a dual mixed finite element approximation of non-Newtonian fluid flow problems. Our mixed method allows to treat, in a unified approach, both the power law and the Carreau law. The estimator justifies an adaptive finite element scheme which refines a given grid only in regions where the error is relatively large. Furthermore, this estimator generalises the one that we have obtained in the particular case of power law (see, Farhloul and Zine [START_REF] Farhloul | A posteriori error estimation for a dual mixed finite element approximation of non-Newtonian fluid flow problems[END_REF]).

  h -p h I) + f r 0,r ,K + as(σ h ) r 0,r ,K 1/r

  1 * , R 2 * and R 3 * whose expressions involve only the data of the problem and the available computed quantities t h , σ h

∼ and u h ∼ . And then, we give upper bounds to the estimators R 1 * , R 2 * and R 3 * .

  r,K |∇z| 1,r ,K + C K rot(t h +ω h ) 0,r,K |ψ| 1,r ,ω K (t h +ω h )t E 0,r,E |ψ| 1,r ,ω E K∈T h h r K rot(t h +ω h ) r 0,r,K ) 1/r |ψ| 1,r ,Ω + C( E∈E h h E (t h +ω h )t E r 0,r,E ) 1/r |ψ| 1,r ,Ω ,

	and so
	| < R 2 , τ ∼ > | ≤ C	(h r K t h +ω h	r 0,r,K + tr(t h ) r 0,r,K
		K∈T h
	+ h r K rot(t h +ω h ) r 0,r,K +	h E (t h +ω h )t E	0,r,E ) r	1/r
				E⊂∂K
	×	|∇z| r 1,r ,Ω + q r 0,r ,Ω + |ψ| r 1,r ,Ω	1/r	.
	Therefore, using (37), we obtain
	| < R 2 , τ ∼ > | ≤ C	η 2 (K) r 1/r	τ
			K∈T h
					tr(t h ) 0,r,K q 0,r ,K
					K∈T h
	+ C h + C K∈T h h 1/r E
	E∈E h
	≤ C(

K∈T h h r K t h +ω h r 0,r,K ) 1/r |∇z| 1,r ,Ω + C( K∈T h tr(t h ) r 0,r,K ) 1/r q 0,r ,Ω + C( ∼ Σ

and (42) follows immediately. It remains to prove (43). By (

16

), we have for every v ∼ ∈ M ,

Proof. By [START_REF] Gatica | A low-order mixed finite element method for a class of quasi-Newtonian Stokes flows. Part I: a priori error analysis[END_REF], we get for every τ ∼ ∈ Σ, < R 2 , τ ∼ >= (t h +ω h , τ ) + (div(τ -q I), u h ).

Then, using the Helmholtz decomposition (36), we get

On the other hand, using the properties of Π h (∇z), the Brezzi-Douglas-Marini interpolant, we get

Thus, using this last relation and the fact that tr(ω h ) = 0, the equation (39) may be rewritten as follows:

Taking successively [START_REF] Farhloul | A mixed finite element method for a quasi-Newtonian fluid flow[END_REF], we obtain

Injecting these two last relations in the right-hand side of (40), we get < R 2 , τ ∼ > = (t h +ω h , ∇z -Π h (∇z)) + (tr(t h ), q) + (t h +ω h , Curl(ψ -I cl (ψ))).

Thus, using Green's formula, we obtain

We are now able to give upper bounds of R 1 * , R 2 * and R 3 * . These upper bounds will be functions of the error indicators η 1 , η 2 and η 3 . More precisely, we have the following results. Theorem 3.3. There exists a constant C independent of h such that


-C Richard

M Verin

Snejana Jurici

Isabelle Le Ber

Anne Boland

Jean-François Deleuze

Stéphanie David

Joana Ferreira

Olivier Quenez

Anne Rovelet-Lecrux

Anne-Claire Richard

Marc Vérin

Thierry Frebourg

João Ricardo Mendes De Oliveira

Didier Hannequin

Dominique Campion

Gaël Nicolas

Identification of partial SLC20A2 deletions

Keywords: Primary Familial Brain Calcification, SLC20A2, deletion, CNV, exome, CANOES, Fahr disease

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INTRODUCTION

Primary Brain Calcification (PBC), also known as Primary Familial Brain Calcification (PFBC, OMIM 213600) or formerly Fahr's disease, is defined by the presence of microvascular calcification affecting at least the basal ganglia with no cause following an extensive clinical, biological, and imaging etiological assessment. Calcification carriers may exhibit diverse neurologic or psychiatric symptoms. Among them, the most frequent three categories are: psychiatric disturbances (mainly mood, psychotic or personality disorders), cognitive impairment (involving mainly memory and executive functions) and movement disorders (mainly extrapyramidal signs) [START_REF] Nicolas | Brain calcification process and phenotypes according to age and sex: Lessons from SLC20A2, PDGFB, and PDGFRB mutation carriers[END_REF] . Patients may present other symptoms, such as gait disorder, cerebellar syndrome, dysarthria, and rarely seizures.

Cephalalgia and especially migraine are frequent circumstances allowing the identification of brain calcification. [START_REF] Nicolas | Brain calcification process and phenotypes according to age and sex: Lessons from SLC20A2, PDGFB, and PDGFRB mutation carriers[END_REF] PBC is inherited as an autosomal dominant trait. In this context, causative variants have been identified in 4 genes: SLC20A2 (OMIM 158378) [START_REF] Wang | Mutations in SLC20A2 link familial idiopathic basal ganglia calcification with phosphate homeostasis[END_REF] , PDGFRB (OMIM 173410) [START_REF] Nicolas | Mutation of the PDGFRB gene as a cause of idiopathic basal ganglia calcification[END_REF] , PDGFB (OMIM 190040) [START_REF] Keller | PDGFB mutations cause brain calcifications in humans and mice[END_REF] and XPR1 (OMIM 605237) [START_REF] Legati | Mutations in XPR1 cause primary familial brain calcification associated with altered phosphate export[END_REF] . Two Copy Number Variants (CNVs) were previously detected as PBC-causing: a partial deletion of PDGFB in one patient [START_REF] Nicolas | PDGFB partial deletion: a new, rare mechanism causing brain calcification with leukoencephalopathy[END_REF] and a deletion encompassing SLC20A2 entirely and 6 other genes in one family [START_REF] Baker | SLC20A2 and THAP1 deletion in familial basal ganglia calcification with dystonia[END_REF] . SLC20A2 and XPR1 encode inorganic phosphate transporters, the importer SLC20A2 (also known as PiT2) and the exporter XPR1, respectively. Loss of function of either gene leads to microvascular mural and perivascular calcification involving vascular smooth muscle cells and pericytes. Loss of function variants of PDGFB or PDGFRB are responsible for an alteration of the blood brain barrier (BBB). Whether the BBB alteration is the direct cause of microvascular calcification or there is other links with inorganic phosphate metabolism remains to be determined [START_REF] Vanlandewijck | Functional Characterization of Germline Mutations in PDGFB and PDGFRB in Primary Familial Brain Calcification[END_REF] . In our French series of PBC patients, more than 50% of probands or sporadic cases do not exhibit any causative variant in one of these genes after Sanger sequencing(unpublished data).

We performed whole exome sequencing (WES) in 24 patients with PBC negatively screened for the four genes. Although WES was developed to detect single nucleotide variants and short indels, we can now take advantage from WES data to detect CNVs. We report here the results of CNV analysis of these four genes in these 24 WES.

Patients

Patients were included following the previously described criteria [START_REF] Nicolas | Phenotypic spectrum of probable and genetically-confirmed idiopathic basal ganglia calcification[END_REF] . Briefly, probands exhibited (1) at least bilateral lenticular calcification, (2) the total calcification score (TCS) using our own visual rating scale was above the age-specific threshold, and (3) an extensive etiological assessment was normal.

Genetic analyses were performed after written, informed consent. This study was approved by our ethics committee. The entire coding sequence of SLC20A2, PDGFB, PDGFRB, and XPR1 was assessed by Sanger sequencing.

We included for WES 24 patients with PBC (14 probands of unrelated families, 10 apparently sporadic cases) negatively screened for all four genes.

Whole exome sequencing

Exomes were captured using the Agilent Sureselect All Exons Human V5 Kit (Agilent technologies, Santa Clara, CA, USA). Final libraries were sequenced on a HiSeq2000 with paired ends, 100-bp reads. Reads were mapped to the 1000 Genomes GRCh37 build using BWA 0.7.5a. [START_REF] Li | Fast and accurate long-read alignment with Burrows-Wheeler transform[END_REF] Picard Tools 1.101 was used to flag duplicate reads. We applied GATK for indel realignment, base quality score recalibration and SNPs and indels discovery using the Haplotype Caller across all samples simultaneously according to GATK 3.3 Best Practices recommendations. [START_REF] Mckenna | The Genome Analysis Toolkit: a MapReduce framework for analyzing next-generation DNA sequencing data[END_REF]

CNV calling

In order to detect CNVs from WES data, we used the CANOES suite [START_REF] Backenroth | CANOES: detecting rare copy number variants from whole exome sequencing data[END_REF] . Its algorithms are dedicated to the detection of CNVs using read depth comparisons between different samples. In a first step, we calculated the read depth for each target using bedtools. We merged close targets (less than 30 bp) to reduce the rate of false positives and removed all events localized in known regions of segmental duplication. As a second step, we used the GenotypeCNVs function of CANOES, which allows determining the probability of a specific event. The objective was to analyze each target intersecting any exon of SLC20A2, PDGFB, PDGFRB, and XPR1 by considering each target as a putative independent event.

For exon numbering, we considered as exon 1 the first one from 5' to 3' in the c.DNA following the given transcript. The three SLC20A2 partial deletions were submitted to the LOVD database (http://databases.lovd.nl/shared/genes/SLC20A2/) with following variant IDs: 0000089490, 0000089491, and 0000089492.

SLC20A2 QMPSF assay

We designed a Quantitative Multiplex PCR of Short Fluorescent Fragments (QMPSF) assay to confirm the presence of the deletion encompassing exon 2 of SLC20A2 in two unrelated individuals and to screen all other patients. This assay included one control amplicon in HMBS and one amplicon in each exon of SLC20A2 (NM_006749.4) plus one amplicon in alternative (non-coding) exon 1 (NM_001257180.1), which maps between non-coding exon 1 and (coding) exon 2 of transcript NM_006749.4 in the genome. Primers are available in Supplementary Table 1.

RESULTS

Among the 24 WES of patients with PBC, mean depth of coverage was 135x (a mean of 98.97% of targeted bases were covered more than 15x). We extracted all variants mapping to SLC20A2, PDGFB, PDGFB, and XPR1 genes and confirmed that no single nucleotide variant or short insertion or deletion probably affecting function had been missed by Sanger sequencing.

After exome-wide CNV calling using the CANOES tool, we detected a total of 273 events without any filtration on genes (11.4 The sensitivity of CNV detection has been estimated to be 74% in the first report of CANOES using WES data obtained with another capture kit [START_REF] Backenroth | CANOES: detecting rare copy number variants from whole exome sequencing data[END_REF] . With the aim to assess the presence or absence of CNV in each SLC20A2 exon, we developed a QMPSF assay including one amplicon in each exon of SLC20A2. After assessment of the 22 remaining patients, we detected two new heterozygous deletions: one encompassing exon 4 only (EXT 434, Fig. 1C) and the other one encompassing exons 4 and 5

(EXT 451, Fig. 1D). The deletion of exon 4 was also found in the the proband's affected sister. DNA from relatives of the patient carrying the exon 4-5 deletion was not available. RNA from these patients was not available. None of the three deletions is reported in the database of genomic variants.

As 2 of the 4 deletions were missed by the first CNV screening of CANOES, we went back to WES data. We used another tool from the CANOES suite, called GenotypeCNVs, which is designed to perform targeted detection of CNVs, to test whether it allowed identifying all four deletions. The GenotypeCNVs tool allowed detecting all four deletions only after decreasing quality thresholds.

However, this led to the detection of a false positive duplication of exon 4 of SLC20A2.

DISCUSSION

SLC20A2 is the major gene causing PBC [START_REF] Hsu | Mutations in SLC20A2 are a major cause of familial idiopathic basal ganglia calcification[END_REF] . It is clearly demonstrated that heterozygous loss of SLC20A2 function is sufficient to cause PBC 2 . However, only one genomic deletion has been published to date 7 . Among the 7 genes deleted in this family, the THAP1 gene also retained the attention as loss of function variants of this gene cause dystonia, which was a prominent feature associated with PBC in this family. We report here the first SLC20A2 partial deletions causing PBC.

Among 24 patients with genetically unexplained PBC, we found 4 (16.7%) SLC20A2 partial deletions, suggesting that such events should be primarily assessed, together with SLC20A2 coding sequence variants, during the genetic screening of PBC patients. In our series, SLC20A2 causative variants, including sequence variants and CNVs, are found in 28.5% of unrelated probands (40% when focusing on familial cases).

Interestingly, almost all patients with a partial SLC20A2 deletion exhibited the same radiological phenotype as patients carrying other loss of function variants, i.e. calcification of lenticular, caudate nuclei, thalami, cerebellar hemispheres and in particular the vermis and cortical sulci (Table 1, Supplementary Figure 1). All were symptomatic, which seems different from the 58% of symptomatic mutation carriers in our previous report [START_REF] Nicolas | Brain calcification process and phenotypes according to age and sex: Lessons from SLC20A2, PDGFB, and PDGFRB mutation carriers[END_REF] . Of note, in family ROU-1129, the grandmother developed cognitive decline only after the age of 86 (of unknown cause), and 2 affected relatives were asymptomatic by family interview. This suggests that the apparent high proportion of symptomatic patients in the present report could be an inclusion bias.

Exon 2 of SLC20A2 is the first coding exon and contains the translation initiation ATG codon as well as very conserved amino acid residues. This deletion probably results in a loss of function. The deletion of exon 4 causes a frameshift and therefore a probable loss of function. A splice variant at the acceptor site of intron 4 has already been described in a French family [START_REF] Nicolas | Mutation of the PDGFRB gene as a cause of idiopathic basal ganglia calcification[END_REF] and could result in the same consequences as the genomic deletion of exon 4, i.e. a total skipping of exon 4 in mRNA. However, RNA was not studied in the family with the splice variant and we cannot exclude that a cryptic acceptor site could be used. The deletion of exon 4-5 is in frame and results in the loss of two transmembrane domains (Fig. 1E). Exon 4 is highly conserved during evolution and it has been shown that an artificial deletion encompassing exon 5 and other exons resulted in decreased transport activity [START_REF] Bottger | Mapping of the minimal inorganic phosphate transporting unit of human PiT2 suggests a structure universal to PiT-related proteins from all kingdoms of life[END_REF] . Taken together, this suggests that this deletion might also result in a loss of SLC20A2 function.

Although WES was not developed with the primary aim to detect CNVs, several tools take advantage from multiple comparisons of depth of coverage allowing the detection of deletions as short as 300 bp (e.g., the deletions involving exon 2 of SLC20A2; size of the target in the capture kit: 316 bp). Exon 4 of SLC20A2 is only 85 bp long (size of the target in the capture kit: 175 bp) and was detected by CANOES from WES data only after targeted genotyping. This suggests that the first step of CANOES, performed exome-wide, might be associated with good specificity but false negatives could be misleading. In presence of a reduced number of candidate genes, the genotyping step used with lower quality thresholds should allow reducing false negatives with the disadvantage of detecting false positives. Sensitivity and specificity of bioinformatics tools aiming at calling CNVs from WES data remain to be determined in comparison with other techniques allowing such a high resolution. E. Schematic representation of the SLC20A2 protein as performed by the TOPO2 online software (http://www.sacs.ucsf.edu/TOPO2/) and according to ref. [START_REF] Bottger | Mapping of the minimal inorganic phosphate transporting unit of human PiT2 suggests a structure universal to PiT-related proteins from all kingdoms of life[END_REF] . Each color represents one different exon, the first coding exon being exon 2 (green). Monoexonic deletions of exon 2 or 4 are not represented as they are predicted to result in no protein or a frameshift; deletion of exons 4 and 5 is represented in the right part of the panel.

Figure 1 .

 1 Figure 1. Partial deletions of SLC20A2

  The joint variant calling file

	(VCF)	was	annotated	using	Annovar	(date	of	version	used:	2015/03/22

(http://www.openbioinformatics.org/annovar/).

  per individual on average [range: 6-26], with a mean estimated size of 50.5kb [range 0.099kb -796.27kb]; 96.7% overlapped known CNVs from the database of genomic variants (http://dgv.tcag.ca/dgv/app/home). In total we identified 120 duplications and 153 deletions. Of these, 110 duplications and 134 deletions were identified only once in a single patient i.e. unique CNVs. We then focused on PBC genes and identified a heterozygous deletion of exon 2 of SLC20A2 in two unrelated probands (ROU 1129 and EXT 383, Fig.1A and 1B, respectively). We confirmed the

presence of both deletions by QMPSF and showed that this deletion only encompassed exon 2 and not exon 1 (NM_006749.4)), alternative exon 1 (NM_001257180.1), or exon 3. In family 1129, the presence of the deletion was confirmed in an affected brother and his affected daughter. DNA was neither available for other affected relatives from this family nor for relatives from family EXT 383.

ROU 1129 and EXT 383 families both originate from the same region of France. However, pedigree information does not suggest that they are related and we confirmed the absence of close relatedness by estimating identity by descent through Plink software (see Supplementary Note 1). Clinical and imaging findings are summarized in Table

1

. A more detailed description of the phenotype is available in Supplementary Note 2.

ACKNOWLEDGEMENTS

We are grateful to Camille Charbonnier for her help in relatedness estimation and to patients and physicians involved in medical care and follow up of patients: Philippe Baize, Nolwen Delarue, Alexandra Durr, Patrick Louf, Yann Nadjar, Gilles Olivier, Stéphane Schaeffer and Frédéric Sedel. This study was funded by Conseil Régional de Haute Normandie -APERC 2014 n°2014-19 in the context of Appel d'Offres Jeunes Chercheurs (CHU de Rouen). SD thanks Fondation pour la Recherche Médicale (DEA20140630628) and JF thanks FACEPE (IBPG-0627-2.02/11; AMD-0047-2.00/15) for financial support.