B Achchab
email: achchab@yahoo.fr

A Agouzal
email: agouzal@univ-lyon1.fr

N Debit

K Bouihat
email: khalidbouihat@yahoo.fr

Star-based a posteriori error estimators Star-based a posteriori error estimates for elliptic problems

Keywords: A posteriori error estimator, nonconforming finite element method, diffusion reaction equations, Stokes equations AMS subject classification: 65D05, 65D15, 65N50

We give an a posteriori error estimator for nonconforming finite element approximations of diffusionreaction and Stokes problems, which relies on the solution of local problems on stars. It is proved to be equivalent to the energy error up to a data oscillation, without requiring Helmholtz decomposition of the error nor saturation assumption. Numerical experiments illustrate the good behavior and efficiency of this estimator for generic elliptic problems.

Introduction

During the last two decades a large amount of work has been devoted to a posteriori error estimation for solution approximated either by conforming [START_REF] Achchab | Robust hierarchical a posteriori error estimates for stabilized convection-diffusion problem[END_REF][START_REF] Verfürth | A review of a posteriori error estimation and adaptive mesh-refinement techniques[END_REF] or nonconforming [START_REF]A posteriori error estimator for nonconforming finite element methods[END_REF][START_REF] Dari | Error estimators for nonconforming finite element approximations of the Stokes problem[END_REF][START_REF] Dari | A posteriori error estimator for nonconforming finite element methods[END_REF] finite element methods. In the nonconforming context, two main approaches have been considered for constructing an a posteriori error estimator. In residual estimators some extra terms have to be added to well-known a posteriori error estimator used in conforming framework. In [START_REF]A posteriori error estimator for finite element discretizations of quasi-Newtonian Stokes flows[END_REF][START_REF]A posteriori error estimator for nonconforming approximation[END_REF][START_REF] Karakashian | A posteriori error estimates for discontinuous Galerkin approximation of second-order elliptic problems[END_REF], these extra terms are the jumps across the element edges of the tangential derivatives of the finite element approximation with respect to element edges.

One of the most successful estimators proposed by Bank and Weiser and extended by many authors ( [START_REF] Achchab | Robust hierarchical a posteriori error estimates for stabilized convection-diffusion problem[END_REF][START_REF] Bank | Some a posteriori error estimators for elliptic partial differential equations[END_REF][START_REF] Bank | A posteriori error estimates for the Stokes problem[END_REF][START_REF] Nobile | A posteriori error estimates for the finite element approximation of the Stokes problem[END_REF]), is based on the solution of local Neumann problems on elements, which seems to allow for cancellation and yields better effectivity indices than residual estimators in numerical tests performed in [START_REF] Morin | Local Problems on Stars: A Posteriori Error Estimators, Convergence, and Performance[END_REF]. The classical proof of equivalence with the energy error require the saturation assumption which states that this solution can be approximated asymptotically better with quadratic than with linear finite elements. The saturation assumption is shown to be superfluous by Nochetto in [START_REF] Nochetto | Removing the saturation assumption in a posteriori error analysis[END_REF]. However, removing this assumption requires comparison with residual estimators. More recently, an approach based on the solution of local problems on stars was proposed in [START_REF] Morin | Local Problems on Stars: A Posteriori Error Estimators, Convergence, and Performance[END_REF][START_REF] Prudhomme | Analysis of a subdomain-based error estimator for finite element approximations of elliptic problems[END_REF], and the proof of the equivalence with energy error applies directly without reference to residual estimators. This approach is applied in [START_REF] Agouzal | Local problem error estimator for nonconforming finite element approximations[END_REF][START_REF] Parés | Subdomain-based flux-free a posteriori error estimators[END_REF] to nonconforming approximations of two-dimensional second order elliptic problems, where the equivalence between the exact error and the estimator on star, is based there on Helmholtz decomposition of the error, which is no more valid in general three dimensional geometries due to convexity requirement.

In this paper, an alternative approach for constructing an a posteriori error estimator for nonconforming approximation of scalar second order elliptic problem, based on the solution of local problems on stars, is given. We prove in general dimensions the efficiency and the reliability of this estimator, without saturation assumption. Moreover, explicit constants for transfer operator ( [START_REF]A posteriori error estimator for nonconforming approximation[END_REF][START_REF] Schieweck | A general transfer operator for arbitrary finite element spaces[END_REF]) are given, which proves that this estimator is robust in suitable norms. The outline of the paper is as follows. In section 2 we introduce the functional framework and introduce the diffusionreaction problem with non nonconforming finite element approximation. In section three, we introduce the star-based a posteriori error estimator and perform the analysis for diffusion reaction problem. In section four, we extend the analysis to Stokes equation by adapting introduced arguments. Numerical results are given in section five to illustrate the good behavior and the efficiency of the given estimator on examples involving smooth and less-smooth solutions.

Setting of the problem

We consider the diffusion-reaction problem

(P ) -∆u + σu = f in Ω, u = 0 on Γ := ∂Ω,
where we assume that σ ∈ L ∞ (Ω) and f ∈ L 2 (Ω), and Ω ⊂ R d , d = 2, 3, is a simply connected polygonal domain.

Let T h be a family of conforming shape-regular triangulations of Ω by d-simplexes. We denote by E I the set of interior edges (faces) and by E f the set of all edges (faces) included in Γ. Let V h be the lowest order nonconforming Crouzeix-Raviart finite element space defined by

V h = {v h ∈ L 2 (Ω); ∀T ∈ T h , v h|T ∈ P 1 (T ), ∀E ∈ E I , E [v h ] E dγ = 0 and ∀E ∈ E f , E v h dγ = 0},
where [.] E denotes the jump of the argument across E.

We denote by {x i } i∈N the set of all nodes of the triangulation T h . In the paper, by i ∈ N we will refer to the node x i . For each i ∈ N , φ i denotes the canonical continuous piecewise linear basis function associated to x i . The star ω i is the interior relative to Ω of the support of φ i , and h i is the maximal size (diameters) of the elements constituting ω i . Finally, Γ i denotes the union of the edges (faces) touching x i that are contained in Ω, and Γ i the union of the edges (faces) touching x i that are contained in Ω. h E denotes the size (diameter) of an edge (face) E. For each star ω i , i ∈ N , we introduce the space V (ω i ) defined by

V (ω i ) = {v ∈ H 1 loc (ω i ) : ωi vφ i dx = 0}, if x i is an interior node, and
V (ω i ) = {v ∈ H 1 loc (ω i ) : v = 0 on ∂ω i ∩ Γ}, if x i is a boundary node.
There exists a constant C, only depending on the minimum angle of the triangulation but independent of the star being considered, such that (see Prop. 2.4 of [START_REF] Morin | Local Problems on Stars: A Posteriori Error Estimators, Convergence, and Performance[END_REF]) :

∀v ∈ V (ω i ), v 0,ωi ≤ Ch i ωi |∇v| 2 φ i dx 1/2 . (2.1)
We define the finite dimensional local spaces P 2 (ω i ) and P 2 0 (ω i ) as follows, Definition 1. For i ∈ N , let P 2 (ω i ) denote the space of continuous piecewise quadratic functions on the star ω i that vanish on ∂ω i . The space P 2 0 (ω i ) is defined by

P 2 0 (ω i ) = P 2 (ω i ) ∩ V (ω i ).
Let us introduce the usual H 1 -norm on ω i ,

||u|| 2 1,ωi = ∇u 2 0,ωi + u 2 0,ωi .
Let v h ∈ V h be fixed. We denote by ∇ h v h the vector field belonging to (L 2 (Ω)) d , defined by

∀T ∈ T h , ∇ h v h = ∇v h on T.
Let u N C h ∈ V h be a solution of the nonconforming approximation problem:

(P h ) N C ∀v h ∈ V h ∩ H 1 0 (Ω), a(u N C h , v h ) := T ∈T h T [∇u N C h .∇v h + σu N C h v h ]dx = Ω f v h dx.
We will now turn to the construction of a u C h belonging to

V h ∩ H 1 0 (Ω) such that ||u -u C h || 1,Ω i∈N (||u -u N C h || 2 1,ωi ) 1 2
.

The star-based error estimate

For each i ∈ N , we consider the local problems :

(P i )    Find η i ∈ P 2 0 (ω i ) such that ∀µ i ∈ P 2 0 (ω i ), ωi (∇η i .∇µ i )φ i dx = ωi ∇ h u N C h .∇(µ i φ i )dx + ωi σu N C h µ i φ i dx - ωi f µ i φ i dx.
Using Lax-Milgram Theorem, we can prove that each discrete problem (P i ) admits a unique solution η i .

Now we introduce the local error indicators,

∀i ∈ N , ∀u N C h ∈ V h , E 2 1,i (u N C h ) = ωi |∇η i | 2 φ i dx. and ∀i ∈ N , ∀u N C h ∈ V h , E 2 2,i (u N C h ) = E∈ωi h -1 E ||[u N C h ] E || 2 0,E .

Upper bound

We consider first the upper bound of the error without oscillation, and we step the process to the main theorem by the following intermediate lemmas.

The first lemma is an adaptation of arguments given in [START_REF] Morin | Local Problems on Stars: A Posteriori Error Estimators, Convergence, and Performance[END_REF] and so the proof will be skipped.

Lemma 2. For all i ∈ N , there exists an operator Π i : V (ω i ) -→ P 2 0 (ω i ), such that for any v ∈ V (ω i ) the following conditions hold :

1. For all edge (face) E ⊂ Γ i , E (v -Π i v)φ i dγ = 0. 2. Moreover, ωi (v -Π i v)φ i dx = 0, if x i is an interior node. 3. ωi |∇Π i v| 2 φ i dx 1 2 ≤ C ωi |∇v| 2 φ i dx 1 2 .
where C is a positive constant only depending on the minimum angle of T h .

Lemma 3. For each i ∈ N , each v ∈ V (ω i ) and u h ∈ V h , we have ωi ∇ h u h .∇((Π i v)φ i )dx = ωi ∇ h u h .∇(vφ i )dx.
Proof. If we denote by [ ∂u h ∂n E ] ∈ P 0 (E) the jump of the normal derivative across E, we have by applying Green formula and subsequently using the property 1. of Lemma 2,

ωi ∇ h u h .∇((Π i v)φ i )dx = E⊂ωi E [ ∂u h ∂n E ](Π i v)φ i dγ = E⊂ωi E [ ∂u h ∂n E ]vφ i dγ. (3.1)
Applying again Green formula yields the result. Now we define the data oscillation by

osc(f ) = i∈N h 2 i (f -f i -σu N C h )φ 1 2 i 2 0,ωi 1 2 ,
where

f i = ωi f φ i dx ωi φ i dx
for i interior nodes and 0 otherwise.

We have the following result about the a posteriori error estimate for any conforming approximation;

Theorem 4. Let u N C h ∈ V h be a solution of (P N C h ) and u C h ∈ V h ∩ H 1 0 (Ω).
There exists a positive constant C only depending on the minimum angle of T h such that

u -u C h 1,Ω ≤ C   i∈N E 2 1,i (u N C h ) 1 2 + i∈N u N C h -u C h 2 1,ωi 1 2 + osc(f )   . (3.2)
Proof. Let v be an element of H 1 0 (Ω) and set ṽ :

= i∈N v i φ i , where v i = ωi vφ i dx ωi φ i dx
for interior nodes and 0 otherwise. We have by adapting standard arguments used in the analysis of finite element approximation of finite approximation of elliptic problems and introducing

u N C h , u -u C h 1,Ω ≤ C sup v∈H 1 0 (Ω) |a(u -u N C h , v) + a(u N C h -u C h , v)| v 1,Ω . Since ṽ ∈ V h ∩ H 1 0 (Ω), a(u N C h -u, ṽ) = 0. This gives, a(u N C h -u, v) = a(u N C h -u, v -ṽ), = i∈N ωi ∇ h u N C h .∇(v -ṽ)dx + ωi σu N C h (v -ṽ)dx - ωi f (v -ṽ)dx , Stating that v -ṽ = i∈N (v -v i )φ i , and using i∈N φ i (x) = 1 gives a(u N C h -u, v) = i∈N ωi ∇ h u N C h .∇[(v -v i )φ i ]dx + ωi σu N C h (v -v i )φ i dx - ωi f (v -v i )φ i dx . Since (v -v i ) ∈ V (ω i )
, adding and removing same quantities in the two last terms give

a(u N C h -u, v) = i∈N ωi ∇ h u N C h .∇[Π i (v -v i )φ i ]dx + ωi σu N C h Π i (v -v i )φ i dx - ωi f Π i (v -v i )φ i dx - i∈N ωi (f -σu N C h )(v -v i -Π i (v -v i ))φ i dx.
Using the definition of local problems

(P i ), a(u N C h -u, v) = i∈N ωi ∇η i .∇[Π i (v -v i )]φ i dx - i∈N ωi (f -σu N C h )(v -v i -Π i (v -v i ))φ i dx .
We now process successively with each term of the right-hand side. On one hand, using Cauchy-Schwarz and item 2.

of Lemma 2 we have i∈N ωi

∇η i .∇Π i (v -v i )φ i dx ≤ i∈N ωi |∇η i | 2 φ i dx 1 2 i∈N ωi |∇Π i (v -v i )| 2 φ i dx 1 2 , ≤ C i∈N E 2 1,i (u N C h ) 1 2 i∈N ωi |∇(v -v i )| 2 φ i dx 1 2 , ≤ C i∈N E 2 1,i (u N C h ) 1 2 v 1,Ω .
On the other hand, since both of (v -v i ) and Π i (v -v i ) belong to V (w i ), using definition of V (ω i ) and coefficients

f i give i∈N ωi (f -σu N C h )(v -v i -Π i (v -v i ))φ i dx = i∈N ωi (f -f i -σu N C h )(v -v i -Π i (v -v i ))φ i dx ,
Using Cauchy-Schwarz then inequality (2.1) and once more i∈N φ i (x) = 1, we get

i∈N ωi (f -σu N C h )(v -v i -Π i (v -v i ))φ i dx ≤ osc(f )( i∈N h -2 i (v -v i -Π i (v -v i ))(φ i ) 1 2 2 0,ωi ) 1 2 , ≤ C osc(f ) v 1,Ω .
C is a generic constant only depending on the minimum angle of triangulation. Finally, summing up the different contributions in the estimate of u -u C h 1,Ω and using the continuity of a(., .) yield the result.

Summarizing the previous results gives the following result about the a posteriori error estimate for the nonconforming approximation:

Theorem 5. Let u N C h ∈ V h be a solution of (P N C h ) and u C h be an arbitrary function of ∈ V h ∩ H 1 0 (Ω). We have i∈N u -u N C h 2 1,ωi 1 2 ≤ C i∈N u N C h -u C h 2 1,ωi 1 2 + i∈N E 2 1,i (u N C h ) 1 2 + osc(f ) , (3.3)
where C only depends on the minimum angle of T h .

In order to prove now the reliability of the estimator, we need the following lemma [START_REF] Karakashian | A posteriori error estimates for discontinuous Galerkin approximation of second-order elliptic problems[END_REF],

Lemma 6. There exists a linear operator I: V h -→ V h ∩ H 1 0 (Ω), satisfying the following estimate

∀u N C h ∈ V h , ∀ω i ∈ T h , k = 0, 1, u N C h -Iu N C h k,ωi ≤ C E∈E I ,E∩ωi =∅ h 1 2 -k E [u N C h ] E 0,E , (3.4)
where h E is the diameter of face (edge) E.

Lower bound

In this section we prove a lower bound of the error without oscillation.

Theorem 7. Let u N C h ∈ V h , there exist generic positive constant C depending on the minimum angle of the triangulation such that, for any i ∈ N ,

E 1,i (u N C h ) ≤ C||u -u N C h || 1,ωi , and E 2,i (u N C h ) ≤ C( i∈N u -u N C h 2 1,ωi ) 1 2
.

Proof. We refer for a proof of second estimate to [START_REF] Karakashian | A posteriori error estimates for discontinuous Galerkin approximation of second-order elliptic problems[END_REF], and proceed with the first one. For each i ∈ N , by definition of E 1,i (u N C h ) and taking test function

µ i = η i in local problem (P i ) give E 2 1,i (u N C h ) = ωi (|∇η i | 2 φ i dx), = ωi ∇u N C h .∇(η i φ i )dx + ωi (σu N C h η i )φ i dx - ωi f η i φ i dx, (3.5)
Since

(η i φ i ) ∈ H 1 0 (ω i ), we have ωi ∇u.∇(η i φ i )dx + ωi σuη i φ i dx = ωi f η i φ i dx. This gives E 2 1,i (u N C h ) = ωi (∇ h u N C h -∇u)∇(η i φ i )dx + ωi (σ(u N C h -u)η i φ i dx, = ωi (∇ h u N C h -∇u)∇(η i )φ i dx + ωi (∇ h u N C h -∇u)(η i )∇(φ i )dx + ωi (σ(u N C h -u)η i φ i dx.
Applying Cauchy-Schwarz inequality gives

E 2 1,i (u N C h ) ≤ u -u N C h 1,ωi E 1,i (u h ) + u -u N C h 1,ωi η i 0,ωi φ i W 1,∞ (wi) + σ L ∞ (ωi) u -u N C h 0,ωi η i 0,ωi , φ i being bounded in ω i . Now since η i ∈ V (ω i ), using (2.

1), we have

η i 0,ωi ≤ C h i E 1,i (u N C h ).

Finally using the property |φ

i | W 1,∞ (ωi) ≤ C h i , we get E 1,i (u N C h ) ≤ C 1 + σ L ∞ (Ω) h i u -u N C h 1,ωi .
which concludes the proof.

Extension to Stokes problem

Let us now extend the ideas given above to the Stokes equations. We will define an error estimator for this problem and prove that it is equivalent with the energy error. Given a simply connected domain Ω ⊂ R d d = 2, 3, we consider then the Stokes problem,

(SP )            -∆u + ∇p = f in Ω, div u = 0 in Ω, u = 0 on Γ = ∂Ω, where f ∈ (L 2 (Ω)) d .
V h being defined in section 2, we set

Q h = {q h ∈ L 2 0 (Ω), q h|T ∈ P 0 (T ), ∀T ∈ ω i and ω i ∈ T h }.
and consider the approximate solution

(u N C h , p h ) ∈ (V h ) d × Q h defined by              ∀v h ∈ (V h ) d , T ∈T h T ∇u N C h : ∇v h dx - T p h div v h dx = Ω f.v h dx, ∀q h ∈ Q h , T ∈T h T q div u N C h dx = 0
Note that the second equation means that for every

T ∈ T h , div (u N C h | T ) = 0.
Let v h ∈ (V h ) d be fixed. We define ∇ h v h and div h v h by :

∀T ∈ T h , ∇ h v h = ∇v h on T,
and

∀T ∈ T h , div h v h = div v h on T.
We introduce the following local problems :

(SP i )                    Find i ∈ (P 2 0 (ω i )) d such that ∀µ i ∈ (P 2 0 (ω i )) d , ωi (∇ i : ∇µ i )φ i dx = ωi ∇ h u N C h : ∇(µ i φ i )dx - ωi p h div (µ i φ i )dx - ωi (f.µ i )φ i dx.
It is obvious that these local problems admit unique solutions.

We introduce for all i ∈ N the three indicators,

η 1,i (u N C h , p h ) = ( T ∈ωi div h u N C h φ 1 2 i 2 0,T ) 1 2 , B. Achchab et al.
Star-based a posteriori error estimators

η 2,i (u N C h , p h ) = ( ωi |∇ i | 2 φ i dx) 1 2 , η 2 3 (u N C h , p h ) = E∈E I h -1 E ||[u N C h ] E || 2 0,E ,
and set the problem data oscillation,

osc(f ) = ( i∈N h 2 i (f -f i )φ 1 2 i 2 0,ωi ) 1 2
,

where

f i = ωi f φ i dx ωi φ i dx
for interior nodes, and f i = 0 otherwise.

As previously, we give the first lemma which proof is similar to Lemma 2 one. Lemma 8. For each i ∈ N , there exists an operator Π i : (V (ω i )) d -→ (P 2 0 (ω i )) d , such that for any v ∈ (V (ω i )) d the following assumptions hold :

1. For all edge E ⊂ Γ i , E (v -Π i v)φ i dγ = 0, 2. For all v ∈ (V (ω i )) d and v h ∈ (V h ) d , ωi ∇ h v h : ∇((Π i v -v)φ i )dx = 0, 3. For all q h ∈ Q h , ωi q h div (v -Π i v)φ i dx = 0.
The following theorem gives the a posteriori error estimate for the nonconforming finite element approximation of Stokes problem solution.

Theorem 9. There exists a positive constant C depending on the minimum angle of the triangulation such that :

i∈N u -u N C h 2 1,ωi 1 2 + p -p h 0,Ω ≤ C i∈N η 2 1,i + η 2 2,i + η 2 3 1 2 + osc(f ) , (4.1)
where, for more readability, we have skipped the arguments of η 1,i , η 2,i and η 3 , and so will be done in the sequel.

Proof. Since (Iu N C h , p h ) ∈ (H 1 0 (Ω)) d × L 2 0 (Ω)
, by standard finite element analysis arguments we state

i∈N u -Iu N C h 2 1,ωi 1 2 + p -p h 0,Ω ≤ C sup (v,q)∈(H 1 0 (Ω))) d ×L 2 0 (Ω)) |a((u, p); (v, q)) -a((Iu N C h , p h ); (v, q))| |v| 1,Ω + q 0,Ω
, where a(.; .) is defined by

∀(u, p), (v, q) ∈ (H 1 0 (Ω)) d × L 2 0 (Ω), a((u, p); (v, q)) = Ω ∇u : ∇v dx - Ω p div v dx + Ω q div u dx. then a((u, p); (v, q))-a((Iu N C h , p h ); (v, q)) = Ω (∇u-∇Iu N C h ) : ∇v dx- Ω (p-p h ) div vdx+ Ω q div h (u-Iu N C h ) dx.
On one hand, since div u = 0 on Ω, i φ i (x) = 1 and φ i being bounded, we have

| Ω q div (u -Iu N C h ) dx| ≤ C ωi∈T h q 0,ωi div Iu N C h φ 1 2 i 0,ωi .
By virtue of Lemma 6 with k = 1, we have

∀ω i ∈ T h , div Iu N C h φ 1 2 i 0,ωi ≤ div h u N C h φ 1 2 i 0,ωi + (div h u N C h -div Iu N C h )φ 1 2 i 0,ωi , ≤ div h u N C h 0,ωi + C E∈E I h -1 2 [u N C h ] E 0,E .
Summing up the contributions, using given indicators definitions and the inequality i

α i β i ≤ ( i α 2 i ) 1/2 ( i β 2 i ) 1/2 , we get Ω q div (u -Iu N C h ) dx ≤ C ωi∈T h q 2 0,ωi i∈N η 2 1,i + η 2 3 1 2 .
On the other hand,

A : = Ω (∇u -∇Iu N C h ) : ∇v dx - Ω (p -p h )div v dx = ωi∈T h ωi (∇u -∇ h u N C h ) : ∇v dx + ωi (∇ h u N C h -∇Iu N C h ) : ∇v dx - ωi (p -p h )div v dx , = - ωi∈T h ωi (∇ h u N C h ) : ∇v dx + ωi p h div v dx + ωi f.v dx + ωi (∇ h u N C h -∇Iu N C h ) : ∇v dx, Introducing the field ṽ ∈ (V h ) d ∩ (H 1 0 (Ω)) d
, in the same manner as in proof of theorem 4 in order to involve (v -v i ) ∈ (V (ω i )) d and use item 2. of Lemma 8 we get

A = - ωi∈T h ωi (∇ h u N C h ) : ∇[Π i (v -v i )φ i ] dx + ωi p h [div Π i (v -v i )φ i ] dx + ωi f.Π i (v -v i )φ i dx + ωi p h [div h (v -v i -Π i (v -v i ))φ i ] dx + ωi (∇ h u N C h -∇Iu N C h ) : ∇v dx + ωi f.(v -v i -Π i (v -v i ))φ i dx,
Adapting arguments used in Theorem 4 and using successively item 3. of Lemma 8, definition of local problems (SP i ), Lemma 6 with k = 1, we get

A ≤ i∈N η 2 2,i + η 2 3 1 2 + osc(f ) |v| 1,Ω .
Summing up the contributions gives,

|a((u, p); (v, q)) -a((Iu N C h , p h ); (v, q))| ≤ i∈N η 2 1,i + η 2 3 1 2 q 0,Ω + i∈N η 2 2,i + η 2 3 1 2 + osc(f ) |v| 1,Ω . (4.2)
Finally stating

ωi∈T h u -u N C h 2 1,ωi ≤ ωi∈T h u -Iu N C h 2 1,ωi + ωi∈T h u N C h -Iu N C h 2 1,ωi ≤ Cη 2 3 + ωi∈T h u -Iu N C h 2 1,ωi ,
yield the result.

Efficiency of the estimator:

Theorem 10. ∀ω i ∈ T h , we have the following inequalities,

div h u N C h 0,ωi ≤ C u -u N C h 1,ωi , (4.3)
η 2,i ≤ C u -u N C h 1,ωi + p -p h 0,ωi (4.4)
and

η 3 ≤ C u -u N C h 1,ωi . (4.5)
Proof. The first inequality is obvious and the third one has already been proved in Theorem 7. So we proceed with the second estimation. Using the definition indicator and local problems (SP i ),

η 2 2,i (u N C h , p h ) = ωi (|∇ i | 2 φ i dx), = ωi ∇ h u N C h : ∇( i φ i )dx - ωi p h div ( i φ i )dx - ωi f. i φ i dx,
As in section 3, since

( i φ i ) ∈ (H 1 0 (ω i )) d , we can state η 2 2,i (u N C h , p h ) = ωi (∇ h u N C h -∇u) : ∇( i φ i )dx - ωi (p h -p)div ( i φ i )dx, = ωi (∇ h u N C h -∇u) : ∇( i )φ i dx + ωi h u N C h -∇u)( i ) : ∇(φ i )dx - ωi (p h -p)div ( i φ i )dx,
and following same steps as in proof of Theorem 7, we retrieve the second estimation.

Numerical experiments Diffusion reaction example:

For the numerical illustration of the efficiency of the error estimator and the based adaption process, we consider a model problem with homogeneous data on the computational domain [0, 1] 2 , with the source term f given by the exact solution, u = xy(x -1)(y -1)e -100(x-0.5) 2 -100(y-0.117) 2 , which presents sharp curvature in the vicinity of point (0.5, 0.117), and we perform a nonconforming finite element discretization on it. Successive iterations of adaptive mesh are represented in Figure 1. Computed and Exact solution are given in Figure 2, where the scaling of the height is the same for both pictures. Table 1 and Figure 3 give the evolution of the error indicator value and the error solution versus the number of degrees of freedom (ndof). We notice that the estimator and the error have analogous behavior, and the estimator under-estimates the energy norm error. Figure 3 illustrates quasi-optimality of the estimator, the dashed line of slope (-1/2) showing a numerical (ndof) (-1/2) asymptotic decay of the error estimator.

Algorithm 1 Based adaption procedure 1: Generate an initial mesh and compute the solution. Refine the mesh in the areas where the indicators are bigger than their mean value and compute solution.

Stokes problem example with analytic smooth solution :

We consider the test case proposed par Bercovier and Engelman [START_REF] Bercovier | A finite Element for the Numerical Solution of Viscous Incompressible Flows[END_REF], defined on the unit square [0, 1] 2 as follows, v(x, y) = -256x 2 (x -1) 2 y(y -1)(2y -1)

u(x, y) = v(x, y) -v(x, y) p(x, y) = (x - 1 2 ).(y - 1 2 ) f (x, y) =    -νv(x, y) + (y - 1 2 ) νv(x, y) + (x - 1 2 )

  

We perform nonconforming finite element discretization on it, and we report on Figure 4, Figure 5 and Figure 6 a sequence of adapted meshes using the proposed refinement indicators and corresponding computed velocity and pressure respectively. Lid-driven cavity problem example:

The two-dimensional Stokes driven cavity problem has been thoroughly studied in numerous references (eg. [START_REF] Shankar | The eddy structure in Stokes flow in a cavity[END_REF]). The main difficulty of this problem comes from the discontinuity of the velocity boundary data at corners. The problem configuration corresponds to a flow in a square cavity [0, 1] 2 .The top of the cavity moves from left to right, imparting motion to the fluid via the no-slip boundary condition, u = (1, 0) on the top. The velocity on all other boundaries is zero, u = (0, 0). We perform nonconforming finite element discretization on it, and we give below a sequence of adaptive meshes. Furthermore, we present the corresponding approximate velocity and pressure contour lines.

Conclusion

We presented and analyzed an a posteriori error estimator for nonconforming approximations of reaction diffusion and Stokes equations. The construction of this so-called star-based error estimator is based on the solution of local subproblems. We proved that it is equivalent to the energy error up to a data oscillation, without requiring Helmholtz decomposition of the error nor saturation assumption. The proof is valid in general space dimensions. Two-dimensional numerical experiments illustrated the good behavior and confirmed the quasi-optimal predicted asymptotic rate of decay of this error estimator.

Figure 1 :

 1 Figure 1: Adaptive mesh refinement using the error indicator.

2 : loop 3 :

 23 Calculate local error indicators and their sum.

5 :

 5 If stopping criterium is satisfied, then exit the loop . 6: end loop

Figure 2 :

 2 Figure 2: Computed solution (left) and exact solution (right) for diffusion reaction example (same scaling).

Figure 3 :

 3 Figure 3: Decay of error indicator and energy error. The dashed line has slope of -1/2.

Figure 4 :

 4 Figure 4: Adaptive mesh refinement using the error indicator.

Figure 5 :

 5 Figure 5: Adaptive computed velocity using the error indicator.

Figure 6 :

 6 Figure 6: Adaptive computed pressure using the error indicator.

Figure 7 :

 7 Figure 7: Error-indicator based refined meshes for Lid-driven Cavity problem.

Figure 8 :

 8 Figure 8: Adaptive computed velocity of Lid-driven Cavity problem.

Figure 9 :

 9 Figure 9: Adaptive computed pressure of Lid-driven Cavity problem.

Table 1 :

 1 Error and indicator values for diffusion reaction problem.

	ndof	Error indicator	u -u N C h
	449	3.4773e-001	1.6271e-001
	762	2.5113e-001	1.1980e-001
	1408	1.7055e-001	9.4704e-002
	5065	8.8906e-002	5.3381e-002
	9843	6.5920e-002	3.6450e-002
	19576	4.7111e-002	2.8596e-002
	38653	3.5282e-002	1.9497e-002
	77469	2.5526e-002	1.5080e-002
	153644	1.8019e-002	1.0170e-002

Acknowledgement

This work is supported in part by HydroMed Project, and CNRST (Projet d'établissement, Université Hassan 1 er Settat, Ministère de l'enseignement supérieur, Maroc). These numerical simulations have been carried out using the software FreeFem++ developed by Frédéric Hecht and O. Pironneau : http://www.freefem.org/.


A K Van Der Bij

R Van Mansfeld

G Peirano

W H F Goessens

J A Severin

J D D Pitout

R Willems

M Van Westreenen

First outbreak of VIM-2 metallo--lactamase-producing Pseudomonas aeruginosa in The Netherlands: microbiology, epidemiology and clinical outcomes

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A c c e p t e d M a n u s c r i p t A c c e p t e d M a n u s c r i p t

Introduction

In The Netherlands, identification of metallo--lactamases (MBLs) among

Pseudomonas aeruginosa has been extremely rare. A recent study at a large centre in The Netherlands that screened 650 isolates for MBL genes found only 2 isolates that produced MBLs [START_REF] Al Naiemi | Emergence of metallo--lactamases in The Netherlands[END_REF]. Worldwide, however, there has been a substantial increase in the reporting of MBLs among P. aeruginosa isolates since the 1990s [START_REF] Walsh | Clinically significant carbapenemases: an update[END_REF].

MBLs are capable of hydrolysing most of the -lactam antibiotics, including carbapenems, and are inhibited by metal chelators such as ethylene diamine tetraacetic acid (EDTA). MBL-encoding genes are frequently located on integrons, which are associated with mobile elements and often carry genes encoding for additional resistance determinants. As a consequence, MBL-producing P. aeruginosa isolates are often resistant to different groups of antimicrobial agents and this resistance can be transmitted to other P. aeruginosa strains or to unrelated types of bacteria. In addition, MBL-producing P. aeruginosa cause significant infections that are associated with higher case fatality rates compared with infections caused by carbapenem-resistant P. aeruginosa that do not produce MBLs [START_REF] Laupland | Population-based epidemiological study of infections caused by carbapenemresistant Pseudomonas aeruginosa in the Calgary Health Region: importance of metallo--lactamase (MBL)-producing strains[END_REF][START_REF] Zavascki | The influence of metallo--lactamase production on mortality in nosocomial Pseudomonas aeruginosa infections[END_REF].

Until now, six types of MBL genes have been described among P. aeruginosa (IMP, VIM, SPM, SIM, GIM and AIM). Although the last four MBL genes have rarely been detected beyond their region of initial detection, i.e. the spread of SPM types is limited to Brazil, whilst GIM, SIM and AIM types are rare and restricted to Germany, Korea and Australia, respectively [START_REF] Walsh | Clinically significant carbapenemases: an update[END_REF][START_REF] Walsh | Metallo--lactamases: the quiet before the storm?[END_REF][START_REF] Gupta | Metallo--lactamases in Pseudomonas aeruginosa and Acinetobacter species[END_REF], VIM and IMP genes show a global distribution and continue to be detected at increasing rates worldwide [START_REF] Walsh | Metallo--lactamases: the quiet before the storm?[END_REF]. In and VIM-2 MBL-producing P. aeruginosa has been responsible for clonal spread and nosocomial outbreaks [START_REF] Walsh | Clinically significant carbapenemases: an update[END_REF][START_REF] Corvec | Long-term evolution of a nosocomial outbreak of Pseudomonas aeruginosa producing VIM-2 metallo-enzyme[END_REF][START_REF] Pitout | Molecular epidemiology of the metallo--lactamase-producing Pseudomonas aeruginosa in the Calgary Health Region: emergence of VIM-2-producing isolates[END_REF][START_REF] Peňa | Nosocomial spread of Pseudomonas aeruginosa producing the metallo-lactamase VIM-2 in a Spanish hospital: clinical and epidemiological implications[END_REF][START_REF] Pitout | Metallo--lactamase-producing Pseudomonas aeruginosa isolated from a large tertiary care centre in Kenya[END_REF][START_REF] Elias | Nosocomial outbreak of VIM-2 metallo--lactamase-producing Pseudomonas aeruginosa associated with retrograde urography[END_REF].

In 2008-2009, we noticed a high prevalence of imipenem-resistant P. aeruginosa isolates at the Erasmus University Medical Centre in Rotterdam (The Netherlands), a 1300-bed teaching hospital. A retrospective study was therefore undertaken to assess the molecular and clinical epidemiology of imipenem-resistant isolates at this hospital.

Material and methods

Bacterial isolates

Consecutive imipenem-resistant P. aeruginosa [imipenem minimum inhibitory concentration (MIC) >8 g/mL or disk zone diameter ≤13 mm] isolated from clinical specimens during the period January 2008 to November 2009 were selected for this study. Only one isolate per patient was included. Resistance to imipenem was confirmed by Etest (bioMérieux, Marcy-l'Étoile, France) for all isolates. Isolates with an imipenem MIC ≤ 8 g/mL were excluded.

Bacterial identification and antimicrobial susceptibility

Identification to species level and antibiotic susceptibility were determined using VITEK ® 2 (Vitek AMS; bioMérieux Vitek Systems Inc., Hazelwood, MO). Identification of isolates from lower respiratory tract specimens of patients with cystic fibrosis (CF) Results were interpreted according to Clinical and Laboratory Standards Institute (CLSI) criteria for broth dilution and disk diffusion [START_REF]Performance standards for antimicrobial susceptibility testing; 18th informational supplement[END_REF].

Identification of metallo-β-lactamases

The presence of bla VIM and bla IMP genes was determined by polymerase chain reaction (PCR) for the simultaneous detection of bla VIM and bla IMP -lactamase genes using primers and conditions as described previously [START_REF] Pitout | Detection of Pseudomonas aeruginosa producing metallo--lactamases in a large centralized laboratory[END_REF]. Sequencing of the MBL genes was performed using the class 1 integron primers 5CS (5'-GGC ATC CAA GCA GCA AG-3') and 3CS (5'-AAG CAG ACT TGA CCT GA-3') in combination with VIM or IMP primers on a Thermal Cycler 9700 instrument (Applied Bio-systems, Norwalk, CT). Two different amplicons per isolate were sequenced in an overlapping fashion to ensure that the entire MBL allele was obtained [START_REF] Pitout | Molecular epidemiology of the metallo--lactamase-producing Pseudomonas aeruginosa in the Calgary Health Region: emergence of VIM-2-producing isolates[END_REF]. Automated sequencing was performed on the PCR products using an ABI Prism 3100 Genetic Analyzer (Applied Biosystems, Norwalk, CT) and sequence analysis software.

Multiple-locus variable-number tandem-repeat (VNTR) analysis (MLVA)

Isolates were cultured on trypticase soy agar II + 5% sheep blood plates (Becton Dickinson, Breda, The Netherlands) and were incubated overnight at 37 C. Cell lysates were made from a few colonies as described previously [START_REF] Van Mansfeld | The population genetics of Pseudomonas aeruginosa isolates from different patient populations exhibits high-level host specificity[END_REF]. Subsequently, using the MLVA9-Utrecht protocol, a touchdown PCR with 2.5 L of each lysate was A c c e p t e d M a n u s c r i p t 7 performed with primers for variable-number tandem-repeats (VNTRs) as described previously [START_REF] Van Mansfeld | The population genetics of Pseudomonas aeruginosa isolates from different patient populations exhibits high-level host specificity[END_REF][START_REF] Vu-Thien | Multiple-locus variable-number tandem-repeat analysis for longitudinal survey of sources of Pseudomonas aeruginosa infection in cystic fibrosis patients[END_REF]. PCR products were separated on a 2% agarose gel by electrophoresis next to a 100 bp DNA ladder (Invitrogen, Breda, The Netherlands) and the number of repeats was deduced by applying the MLVA alleles assignment table for P. aeruginosa genotyping (http://minisatellites.u-psud.fr/MLVAnet/). Loci that repetitively did not yield a PCR product were assigned allele '99' in order to include these isolates in the genotype analysis.

Epidemiological and clinical data

Microbiological data (i.e. site of isolation and susceptibility to antimicrobial agents)

were obtained for all imipenem-resistant isolates.

Results

Bacterial isolates

During the study period, 5167 P. aeruginosa isolates were cultured from 1491 patients. A total of 161 patients (11%) were infected or colonised with an imipenemresistant isolate; an isolate was retrospectively available for MBL testing from 111 (69%) of the 161 patients. Imipenem resistance was confirmed by Etest in 106 isolates: 74 (70%) were isolated from respiratory tract specimens, 6 (6%) from urine, 5 (5%) from blood, 8 (8%) from soft tissue or bone, 7 (7%) from intra-abdominal specimens and 6 (6%) from various other specimens. The majority of isolates (69%)

were hospital-acquired, 23% were acquired in the community but were healthcareassociated, and 9% of the isolates were community-acquired.

Of the 106 isolates confirmed by Etest, 35 (33%) were positive for bla VIM . None of the isolates were positive for bla IMP . VIM MBL-producing isolates were more often isolated than non-MBL-producing isolates from urine (14% vs. 1%), blood (11% vs.

1%) and intra-abdominal specimens (20% vs. 0%), whilst non-MBL producing isolates were more often isolated from the respiratory tract (78% vs. 22%) and soft tissue or bone specimens (11% vs. 0%). Almost all of the MBL-producing isolates were hospital-acquired (32/35; 91%).

A c c e p t e d

M a n u s c r i p t 10

Antimicrobial susceptibility

Results of antimicrobial susceptibility testing are shown in Table 1. Except for susceptibility to aztreonam, MBL-producing isolates were more resistant to the antibiotics tested than non-MBL-producing isolates. Colistin resistance was detected in one MBL-producing isolate and one non-MBL-producing isolate. Resistance to colistin was confirmed by Etest in both isolates.

Multiple-locus variable-number tandem repeat (VNTR) analysis (MLVA)

Among the MBL-producing isolates, MLVA identified one large cluster of 24 patients (MLVA 1), a cluster of 7 patients (MLVA 2), a cluster of 3 patients (MLVA 3) and one patient with a non-related isolate (Fig. 1A). Twelve (50%) of the isolates belonging to MLVA 1 and six (86%) of the seven isolates belonging to MLVA 2 were isolated from patients admitted to one of the two ICUs. The patient with the unrelated isolate and one patient belonging to MLVA 3 were also admitted to the ICU at the time of isolation (Fig. 1B). Most of the other patients with VIM MBL-producing isolates were admitted to various internal medical or surgical wards.

Identification of metallo--lactamases and the integron

PCR with integron primers 5C and 3C and the different combinations of VIM primers of random selected isolates among all three distinct MLVA genotypes (n = 12) amplified various amplicons ranging from 800 bp to 1.2 kb. Sequence analysis of the products revealed 100% identity with the sequence of the bla VIM-2 allele [START_REF] Poirel | Characterization of VIM-2, a carbapenem-hydrolyzing metallo--lactamase and its plasmid-and integron-borne gene from a Pseudomonas aeruginosa clinical isolate in France[END_REF]. The following gene cassettes were detected in all 12 VIM-2 MBL-producing P.

A c c e p t e d M a n u s c r i p t 11 aeruginosa: aacA29a-bla VIM-2 -aacA29b in the same order as previously described [START_REF] Poirel | Characterization of class 1 integrons from Pseudomonas aeruginosa that contain the bla VIM-2 carbapenem-hydrolyzing -lactamase gene and of two novel aminoglycoside resistance gene cassettes[END_REF].

Patient characteristics

The majority of patients were male (62%) and their median age was 43 years [interquartile range (IQR) 21-64 years]; 77% of patients had already been admitted at the time of specimen collection and the median length of hospitalisation at that time was 12 days (IQR 0-37 days). A large proportion of patients (37%) was diagnosed with CF. Of the 106 patients with an imipenem-resistant P. aeruginosa isolate, only 31 (29%) had clinical evidence of an infection at the site of isolation. In three patients, the first imipenem-resistant P. aeruginosa was isolated from blood culture. Eleven other patients developed bacteraemia with the imipenem-resistant isolate during their admission. The overall mortality rate 30 days after specimen collection was 15% (Table 2).

Table 3 shows the risk factors for isolation of a VIM-2 MBL-producing P. aeruginosa isolate in the univariate analysis. In multivariate analysis, admission to the ICU [odds ratio (OR) = 2.6, 95% confidence interval (CI) 1.0-7.3), clinical evidence of infection at the isolation site (OR = 2.6, 95% CI 1.0-6.9) and antimicrobial therapy with fluoroquinolones in the 14 days before isolation (OR = 3.0, 95% CI 1.0-9.2) were borderline significantly associated with isolation of a VIM-2 MBL-producing isolate.

Patients who were diagnosed with CF as an underlying disease were less likely to have a VIM-2 MBL-producing P. aeruginosa isolate than patients with a malignancy or cardiac, renal, neurological or gastrointestinal disease as an underlying disease A c c e p t e d M a n u s c r i p t 12 (OR = 0.1, 95% CI 0.1-0.6). However, two CF patients with a VIM-2 MBL-producing isolate were identified. One of the patients belonged to the large cluster MLVA 1.

However, this patient had no history of previous ICU admission and had only been admitted for 5 days to the Pulmonary Department at the time of specimen collection.

The other patient was a child transferred from another hospital with a VIM-2 MBLproducing isolate that belonged to the small cluster MLVA 3.

Clinical outcome

In the univariate logistic regression analysis, isolation of a VIM-2 MBL-producing P.

aeruginosa was significantly associated with infection (OR = 4.7, 95% CI 1.9-11.6).

However, statistical significance was not maintained in the multivariate analysis (OR = 2.5, 95% CI 0.9-6.8). There was a strong association between isolation of a VIM-2 MBL-producing isolate and development of bacteraemia in the univariate analysis (OR = 41, 95% CI 5-334). Multivariate analysis to assess the association between isolation of a VIM-2 MBL-producing isolate and development of bacteraemia was, however, not possible owing to the small numbers of patients with bacteraemia by a non-MBL-producing isolate. There was no significant association between isolation of a VIM-2 MBL-producing isolate and mortality after 30 days in univariate and multivariate analysis (OR = 2.3, 95% CI 0.8-6.9). Results were similar when analysis was restricted to non-CF patients only.

A c c e p t e d M a n u s c r i p t 13

Discussion

This is the first outbreak of VIM-2 MBL-producing P. aeruginosa in The Netherlands, a country that is renowned for its low level of antimicrobial resistance [START_REF]European Antimicrobial Resistance Surveillance System. EARSS annual report[END_REF]. In this study, a high prevalence of VIM-2 MBL-producing P. aeruginosa among imipenemresistant isolates was found. The majority of the P. aeruginosa isolates belonged to a single cluster, suggesting patient-to-patient transmission. VIM-2 MBL-producing isolates were mostly identified in patients with clinical evidence of infection and with underlying diseases other than CF. However, two CF patients with an isolate that tested positive for VIM-2 by PCR and sequencing were identified, a finding that is extremely rare since carbapenem resistance among P. aeruginosa isolates in CF patients is usually caused by mutational changes in the regulation of AmpC lactamase, OprD porin and efflux pumps and not by acquisition of MBL genes [START_REF] Tomás | Efflux pumps, OprD porin, AmpC -lactamase, and multiresistance in Pseudomonas A c c e p t e d M a n u s c r i p t 20 aeruginosa isolates from cystic fibrosis patients[END_REF]. A recent case report from Portugal in combination with our remarkable finding, however, suggest that the VIM-2 gene has now spread into the CF P. aeruginosa population [START_REF] Cardoso | Metallo--lactamase VIM-2 in Pseudomonas aeruginosa isolates from a cystic fibrosis patient[END_REF].

Worldwide, VIM-2 is the dominant MBL gene associated with nosocomial outbreaks due to MBL-producing P. aeruginosa [START_REF] Walsh | Clinically significant carbapenemases: an update[END_REF][START_REF] Corvec | Long-term evolution of a nosocomial outbreak of Pseudomonas aeruginosa producing VIM-2 metallo-enzyme[END_REF][START_REF] Pitout | Molecular epidemiology of the metallo--lactamase-producing Pseudomonas aeruginosa in the Calgary Health Region: emergence of VIM-2-producing isolates[END_REF][START_REF] Peňa | Nosocomial spread of Pseudomonas aeruginosa producing the metallo-lactamase VIM-2 in a Spanish hospital: clinical and epidemiological implications[END_REF][START_REF] Pitout | Metallo--lactamase-producing Pseudomonas aeruginosa isolated from a large tertiary care centre in Kenya[END_REF][START_REF] Elias | Nosocomial outbreak of VIM-2 metallo--lactamase-producing Pseudomonas aeruginosa associated with retrograde urography[END_REF]. These outbreaks are usually restricted to specific patient groups and locations, such as the ICU and haematology wards where patients are often exposed to broad-spectrum antimicrobial agents and with an increased risk of cross-transmission of resistant bacteria between patients [START_REF] Fridkin | Increasing prevalence of antimicrobial resistance in intensive care units[END_REF]. At our hospital, the outbreak was mainly among patients admitted to the ICU, but ten different hospital departments were also affected, and cluster MLVA 1 caused a small outbreak involving three patients admitted to a surgical ward in 2008. In addition, a All of the VIM-2 MBL-producing P. aeruginosa isolates harboured an integron structure that showed remarkable similarity to the integron-associated structure for VIM-2 MBL-producing P. aeruginosa, i.e. In59, previously described in France and Sweden [START_REF] Poirel | Characterization of class 1 integrons from Pseudomonas aeruginosa that contain the bla VIM-2 carbapenem-hydrolyzing -lactamase gene and of two novel aminoglycoside resistance gene cassettes[END_REF][START_REF] Samuelsen | Molecular epidemiology of metallo--lactamase-producing Pseudomonas aeruginosa isolates from Norway and Sweden shows import of international clones and local clonal expansion[END_REF]. Interestingly, the VIM-2 MBL-producing P. aeruginosa that was responsible for a nosocomial outbreak during 2003 and 2004 in Canada [START_REF] Pitout | Molecular epidemiology of the metallo--lactamase-producing Pseudomonas aeruginosa in the Calgary Health Region: emergence of VIM-2-producing isolates[END_REF] showed an MLVA genotype similar to our major outbreak strain (data not shown). However, the bla VIM-2 from the Canadian strain was associated with a different type of integron, i.e. In100, that was first described in Portugal [START_REF] Quinteira | Characterization of In100, a new integron carrying a metallo--lactamase and a carbenicillinase, from Pseudomonas aeruginosa[END_REF]. Previous reports from different areas in the world have shown that bla VIM among P. aeruginosa strains are associated with various integron-associated structures in specific major clonal complexes [START_REF] Giske | Establishing clonal relationships between VIM-1-like metallo--lactamaseproducing Pseudomonas aeruginosa strains from four European countries by multilocus sequence typing[END_REF][START_REF] Libisch | Molecular typing indicates an important role for two international clonal complexes in dissemination of VIM-producing Pseudomonas aeruginosa clinical isolates in Hungary[END_REF][START_REF] Tsakris | Large dissemination of VIM-2-metallo--lactamase-producing Pseudomonas aeruginosa strains causing health care-associated community-onset infections[END_REF][START_REF] Edalucci | Acquisition of different carbapenem resistance mechanisms by an epidemic clonal lineage of Pseudomonas aeruginosa[END_REF]. This suggests that the emergence of VIM MBL-producing P.

aeruginosa might occur independently in various parts of the world via the acquisition of different bla VIM -containing integrons in isolates of similar clonal lineage in combination with clonal dissemination.

Limited studies have addressed the effect of a MBL gene on morbidity and mortality [START_REF] Laupland | Population-based epidemiological study of infections caused by carbapenemresistant Pseudomonas aeruginosa in the Calgary Health Region: importance of metallo--lactamase (MBL)-producing strains[END_REF][START_REF] Zavascki | The influence of metallo--lactamase production on mortality in nosocomial Pseudomonas aeruginosa infections[END_REF][START_REF] Hirakata | Clinical and bacteriological characteristics of IMP-type metallo--lactamaseproducing Pseudomonas aeruginosa[END_REF][START_REF] Zavascki | Reappraisal of Pseudomonas aeruginosa hospital-acquired pneumonia mortality in the era of metallo--lactamase-mediated multidrug resistance: a prospective observational study[END_REF][START_REF] Zavascki | Nosocomial bloodstream infections due to metallo--lactamase-producing Pseudomonas aeruginosa[END_REF]. These studies showed higher infection and mortality rates among patients with a MBL-producing P. aeruginosa compared with patients with a non-

Competing interests

None declared.

Ethical approval

Not required.

A c c e p t e d M a n u s c r i p t

First

  outbreak of VIM-2 metallo-β-lactamase-producing in The Netherlands: microbiology, epidemiology and clinical outcomes A.K. van Der Bij, R. van Mansfeld, G. Peirano, W.H.F. Goessens, J.A. Severin, J.D.D. Pitout, R. Willems, M. van Westreenen To cite this version: A.K. van Der Bij, R. van Mansfeld, G. Peirano, W.H.F. Goessens, J.A. Severin, et al.. First outbreak of VIM-2 metallo-β-lactamase-producing in The Netherlands: microbiology, epidemiology and clinical outcomes. International Journal of Antimicrobial Agents, 2011, 10.1016/j.ijantimicag.2011.02.010. hal-00690061 Please cite this article as: Van der Bij AK, Van Mansfeld R, Peirano G, Goessens WHF, Severin JA, Pitout JDD, Willems R, van Westreenen M, First outbreak of VIM-2 metallo-␤-lactamase-producing Pseudomonas aeruginosa in The Netherlands: microbiology, epidemiology and clinical outcomes, International Journal of Antimicrobial Agents (2010), doi:10.1016/j.ijantimicag.2011.02.010

A c c e p t e d M a n u s c r i p t 2 Metallo*

 2 Corresponding author. Tel.: + 31 10 703 3867; fax: +31 10 703 3875.

A c c e p t e d M a n u s c r i p t 5 particular

 5 , VIM-2 emerged as the predominant MBL variant among P. aeruginosa,

A c c e p t e d M a n u s c r i p t 6 was

 6 performed by oxidase activity (DrySlide TM Oxidase; Becton Dickinson, Franklin Lakes, NJ) and resistance to C390 (DIATABS™; Rosco Diagnostica A/S, Taastrup, Denmark). Antibiotic susceptibility of these isolates was determined by disk diffusion.

10 9 /

 9 L or <4.5  10 9 /L). A confounding variable remained in the final model if its inclusion resulted in a >10% change in the effect size of the primary association of A c c e p t e d M a n u s c r i p t 9 interest. Additionally, the same analyses were performed restricted to patients without a diagnosis of CF.

A c c e p t e d M a n u s c r i p t 14 retrospective

 14 analysis of P. aeruginosa isolates cultured from blood specimens from 2000 to 2007 was performed and demonstrated that VIM-2 MBL-producing P. aeruginosa isolates have been present at our hospital since 2003. The eight retrospective isolates that tested positive for bla VIM-2 belonged to the major cluster MLVA 1 (data not shown) and were isolated from patients admitted to various departments.

A c c e p t e d M a n u s c r i p t 15 MBL

 15 -producing P. aeruginosa. The current results also suggest that isolation of VIM-2 MBL-producing isolates is associated with invasive infection, including bacteraemia. However, this study lacked power in multivariate analysis to reach statistical significance. To further evaluate the threat of MBL-producing P. aeruginosa and its potential to cause severe infections, well designed prospective epidemiological studies are urgently required. Given the limited therapeutic options, these studies should also include clinical predictors that are associated with improved survival.This study shows that the global dissemination of VIM-2 MBL-producing P. aeruginosa has reached The Netherlands and that these isolates are an important cause of carbapenem resistance. Results from molecular typing illustrate that once VIM-2 MBL-producing clusters are introduced into a hospital setting they are easily spread within the hospital. In the absence of new agents for the treatment of infections caused by multiresistant Gram-negative bacteria, the spread of these multiresistant bacteria will lead to treatment failures with increased morbidity and mortality. Clinical microbiological laboratories must therefore be capable of distinguishing MBL-producing strains from those with other mechanisms responsible for carbapenem resistance in order to implement timely and adequate infection control practices.

A c c e p t e d M a n u s c r i p t 16 Funding

 16 RvM is supported by The Netherlands Organization for Scientific Research (VICI NWO Grant 918.76.611).

Fig. 1 .

 1 Fig. 1. (A) Multiple-locus variable-number tandem repeat (VNTR) analysis (MLVA)

  Differences between MBL-and non-MBL-producing isolates as well as between patients with a MBL-producing isolate and patients without a MBL-producing isolate were analysed univariately using Student's t-test or Wilcoxon signed-rank test in the case of a non-normal distribution for continuous variables, and  2 tests with Fisher's exact test in the case of small numbers for non-continuous variables. Logistic regression analysis was performed to assess patient predictors for isolation of a

	2.6. Statistical analysis
	M a n u s c r i p t
	In multivariate analysis, the following potentially confounding variables were corrected for: age; gender; hospital admission, including the number of days admitted; Intensive Care Unit (ICU) admission; primary diagnosis; type of A c c e p t e d antimicrobial treatment in the previous 14 days; and elevated C-reactive protein
	(CRP) and leukocytosis or leukopenia (i.e. CRP > 10 mg/L and leucocytes >10 
	Patients were deemed to have a community-acquired isolate if the culture was
	performed on an outpatient visit or if the culture was performed within the first 2 days
	of hospital admission; these patients were further classified as having either

The electronic medical records of all patients with an imipenem-resistant isolate were reviewed systematically and included the collection of demographic information (i.e. age, sex, outpatient visit or hospital admission and hospital department) and clinical information (i.e. infection or colonisation, development of bacteraemia with the imipenem-resistant isolate, underlying disease, antimicrobial treatment in the 14 days before specimen collection and 30-day mortality after specimen collection). Infection was defined as a positive sample together with clinical evidence of infection according to standard US Centers for Disease Control and Prevention (CDC) criteria

[START_REF] Horan | CDC/NHSN surveillance definition of health care-associated infection and criteria for specific types of infections in the acute care setting[END_REF]

. Colonisation was defined as a positive sample without clinical evidence of infection. Patients were deemed to have an hospital-acquired isolate if the culture was performed >48 h after admission. 8 community-acquired isolates or healthcare-associated community-acquired isolates

[START_REF] Friedman | Health care-associated bloodstream infections in adults: a reason to change the accepted definition of community-acquired infections[END_REF]

.

MBL-producing P. aeruginosa isolate. A multivariate logistic regression model was constructed using backward selection to identify significant variables. Variable selection was based on the likelihood ratio test. A P-value of <0.05 was considered statistically significant. In addition, it was determined whether isolation of a MBLproducing P. aeruginosa isolate was a predictor of infection, development of bacteraemia or mortality by univariate and multivariate logistic regression analyses.

Acknowledgments

The authors thank Damien Bierschenk, Marco Mol and Nicole Lemmens for their technical support of this study as well as Roel Streefkerk for his electronic data support.

Resistance rates according to Clinical and Laboratory Standards Institute (CLSI) guidelines [START_REF]Performance standards for antimicrobial susceptibility testing; 18th informational supplement[END_REF]