A non-conforming finite volume element method of weighted upstream type for the two-dimensional...

Applied Numerical Mathematics 58 (2008) 615–634

www.elsevier.com/locate/apnum

A non-conforming finite volume element methodof weighted upstream type for the two-dimensional

stationary Navier–Stokes system

K. Djadel a, S. Nicaise b,∗

a Ecole Nationale des Ponts et Chaussée, 6 et 8 avenue Blaise Pascal, cité Descartes, Champs-sur-Marne, 77455 Marne La Vallée cedex 2, Franceb Université de Valenciennes et du Hainaut Cambrésis, MACS, ISTV, F-59313 Valenciennes cedex 9, France

Available online 15 February 2007

Abstract

We consider the discretization of the stationary Navier–Stokes system in a two-dimensional domain by a non-conforming finitevolume element method. We use the standard formulation of the Navier–Stokes system in the primitive variables and take asapproximation space the non-conforming P

1-elements for the velocity and piecewise constant elements for the pressure. The non-linear convective term is treated using an upstream approach with weight, based on the scheme from [F. Schieweck, L. Tobiska,A non-conforming finite element method of upstream type applied to the stationary Navier–Stokes equation, M2AN 23 (1989) 627–647]. For the proposed scheme, we prove existence and uniqueness results (under the standard assumption that the datum has to besufficiently small with respect to the viscosity parameter, cf. [R. Temam, Navier–Stokes Equations, North-Holland, Amsterdam,1984]). An error estimate in the energy norm is proved and is confirmed by different numerical tests.© 2007 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords: Finite volumes; Navier–Stokes system; Upstream scheme

1. Introduction

The finite volume element methods appear quite recently [3,5,6,17,22,25] and seem to be a good compromisebetween standard finite element methods and finite volume methods. Indeed finite volume methods are highly appre-ciated and particularly efficient in applications, like fluid dynamics, thermodynamics, . . . . The main interest of thesemethods stays on the principle of conservation of flux [18] that is not taken into account by finite element methods.On the other hand, one of their drawbacks is their lack of flexibility, due to some orthogonality conditions [18], whichleads to some difficulties for domains with a complex geometry or for refinement procedures. Some solutions to thisproblem can be cited:

• The diamond finite volume methods [9,10,4]: In this case no orthogonality conditions are imposed, and the normalflux of the exact solution is still approximated with the help of finite differences, using the values at the center ofthe boxes but also using reconstructed values at the nodes, allowing to define an approximated tangent flux. This

* Corresponding author.E-mail addresses: [email protected] (K. Djadel), [email protected] (S. Nicaise).

0168-9274/$30.00 © 2007 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.apnum.2007.01.012

616 K. Djadel, S. Nicaise / Applied Numerical Mathematics 58 (2008) 615–634

approach requires a supplementary computational cost and its convergence for arbitrary meshes is not yet proved.We also refer to [14] for a similar approach.

• The finite volume element methods [6,7,15,22–24,29]: The principle of this method is to use two meshes. Theprimal mesh, which is a triangulation of the domain, is used to approximate the exact solution (P1 conforming ornot, conforming-P1+ bubble for the velocity, P

0 for the pressure, . . . ), while the dual mesh, built using the primalmesh, allows to discretize the equation. In some aspects, the finite volume element methods are relatively close tofinite element methods.

The incompressible Navier–Stokes system is one of the main equation studied in fluid mechanics. There is anhuge literature concerning this system, let us quote [21] for finite difference methods, [16,20,26,27] for finite elementmethods and [15,19,23,24,12,13] for finite volume methods. The main goal of all these methods is to allow an accuratediscretization and a rapid implementation of the problem. One of the key points is to find an adequate discretizationof the non-linear (convective) term. In [23,24] an upstream scheme is used for finite volume element methods. Thisscheme allows to obtain a better convergence of the system (more precisely a better convergence of the discretesubproblems appearing during the Newton procedure, allowing to solve the non-linear discrete problem) and to takeinto account the flux of the exact solution. Nevertheless in practice such schemes are relatively diffusive. On theother hand, in [15], a centred scheme is introduced, which is less diffusive than the upstream scheme but for which noexistence and convergence results are available up to now. Therefore we here propose a scheme which is a combinationof both methods, that we called of weighted upstream type. We further decide to work with the pair (non-conformingP

1/P0) to discretize the velocity and the pressure. Indeed this pair presents the advantage of a weak algorithmic

cost with respect to other pairs like (conforming P1 + bubble/continuous P

1) or (conforming P2/continuous P

1) forinstance.

The schedule of the paper is as follows: In the second section, we introduce the stationary incompressible Navier–Stokes system and recall some notation and definitions used in the whole paper. Section 3 is devoted to the introductionof our numerical scheme. In Section 4 we give some preliminary results, notably some Poincaré or discrete Sobolevinequalities and some interpolation error estimates useful later on. Section 5 concerns the proof of the existence anduniqueness results of the discrete problem. In Section 6 we obtain the error estimate in the energy norm, extendingthe results from [23] (for the standard upstream scheme) and from [15], where no theoretical analysis was made.Finally numerical tests are presented in Section 7, which confirm our predicted order of convergence and underlinethe advantage of the use of a weighted scheme.

In the whole paper the spaces Hs(Ω), with any non-negative integer s, are the standard Sobolev spaces in Ω withnorm ‖ · ‖s,Ω and semi-norm | · |s,Ω . The space H 1

0 (Ω) is defined, as usual, by H 10 (Ω) := {v ∈ H 1(Ω) | v = 0 on Γ }.

Lp(Ω), p > 1, are the usual Lebesgue spaces with norm ‖ · ‖0,p,Ω (as usual we drop the index p for p = 2). In thesequel the symbol | · | will denote either the Euclidean norm in R

n (n = 1 or 2), or the length of a line segment orfinally the area of a plane region. The symbol v means that v is a vector valued function, i.e., v = (v1, v2)

�, vi beingits ith component. The notation a � b means here and below that there exists a positive constant C independent ofa and b (and of the meshsize of the triangulation) such that a � C b, while the notation a ∼ b means that a � b andb � a hold simultaneously.

2. Problem setting and notation

Let Ω be a polygonal domain of R2. Over the domain Ω , we consider the stationary incompressible Navier–Stokes

problem with Dirichlet boundary conditions: Given a vector function f = (f1, f2)�, find a solution (u,p) of{−ν�u + (u · ∇)u + ∇p = f in Ω,

div u = 0 in Ω,

u = 0 on ∂Ω,

(1)

where ν > 0 represents the viscosity of the fluid. The weak velocity-pressure formulation of (1) reads as follows: Findu ∈ V = (H 1

0 (Ω))2 and q ∈ Q = L20(Ω) = {r ∈ L2(Ω):

∫Ω

r = 0} solutions of{νa(u,v) + c(u,u,v) + b(v,p) = (f,v)Ω ∀v ∈ V,

b(u, q) = 0 ∀q ∈ Q,(2)

where (·, ·)Ω means the inner product in (L2(Ω))2 or in L2(Ω) according to the context and

K. Djadel, S. Nicaise / Applied Numerical Mathematics 58 (2008) 615–634 617

a(v,w) =∫Ω

∇v(x) : ∇w(x) dx,

b(v, q) = −∫Ω

q divv dx,

c(u,v,w) =∫Ω

u · ∇v · wdx, ∀u,v,w ∈ V,

when for a vector valued function v = (v1, v2)�, ∇v means the gradient row by row, i.e. the matrix(

∂1v1 ∂2v1∂1v2 ∂2v2

),

∇v : ∇w = ∑i,j=1,2 ∂ivj ∂iwj and u · ∇v · w = ∑2

i,j=1 ui Diuj wj .It is well known that (2) admits at least one solution satisfying

|u|1,Ω � ν−1‖f ‖0,Ω, (3)

and that this solution is unique provided that ν−2‖f ‖0,Ω is sufficiently small (see Chapter 2 in [26]).The idea of our finite volume element method is to approximate the velocity field u in the P

1 non-conforming finiteelement space and the pressure in the space of piecewise constant functions in the primal mesh [6,7,15] but with testfunctions which are piecewise constant on the dual and the primal mesh respectively. More precisely the primal meshconsists in a triangulation Th of Ω made of triangles, regular in Ciarlet’s sense [8], i.e., there exists ξ > 0 independentof h such that

hK

ρK

� ξ, ∀K ∈ Th, (4)

where we recall that hK (respectively ρK ) is the diameter of K (respectively the maximum of the diameter of the ballsincluded in K) and h := maxK∈Th

hK . For K ∈ Th denote by Eh(K) its set of edges. Let us define Eh as the set ofedges of the triangulation, Eint

h = {e ∈ Eh | e ⊂ Ω} the set of interior edges of Th, while Eexth = Eh \ Eint

h is the set ofboundary edges of Th. For any edge e we denote by me the midpoint of e and by Ve its set of neighbouring edges, i.e.,Ve = (Eh(K) ∪ Eh(L)) \ {e}, when e ∈ Eh(K) ∩ Eh(L).

The dual mesh is built as follows: for any K ∈ Th let zK be the barycenter of K . For any e ∈ Einth , there exist

two different elements K,L ∈ Th such that e ∈ Eh(K) ∩ Eh(L), then the box associated with e is defined by be :=Conv({zK, zL} ∪ e) (see Fig. 1). On the contrary for e ∈ Eext

h , there exists a unique K ∈ Th such that e ⊂ K ∩ ∂Ω , andits associated box is given by be := Conv(zK, e). The set of boxes is simply Bh := {be: e ∈ Eh}. Finally for e, l ∈ Eh,we define me,l as the midpoint of the segment ∂be ∩ ∂bl .

We further introduce some function spaces, notably the Crouzeix–Raviart finite element space S0h and some forms

that will be used in the sequel [8,11,26].

P0(Th) := {

qh ∈ L2(Ω): qh|K ∈ P0(K), ∀K ∈ Th

},

P0(Bh) := {

qh ∈ L2(Ω): qh|b ∈ P0(b), ∀b ∈ Bh

},

Fig. 1. Example of a box be .


Sh := {vh ∈ L2(Ω): vh|K ∈ P

1(K), ∀K ∈ Th, vh|K(me) = vh|L(me), ∀e ∈ Einth , K,L ∈ Th: e = K ∩ L

},

S0h := {

vh ∈ Sh: vh(me) = 0, ∀e ∈ Eexth

}.

Since S0h is not included into H 1

0 (Ω), the space S0h is equipped with the norm | · |1,h := (

∑K∈Th

| · |21,K)12 . To any

function vh ∈ S0h we associate the piecewise constant function vh ∈ P

0(Bh) defined by vh|be := vh(me),∀e ∈ Eh.

ah :(Sh + H 1(Ω)

)2 × (Sh + C0(Ω)

)2 → R,

(v,w) → −∑e∈Eh

∫∂be

∇vnbe · w(me) ds,

ah :(Sh + H 1(Ω)

)2 × (Sh + H 1(Ω)

)2 → R,

(v,w) →∑

K∈Th

∫K

∇v : ∇wdx,

bh :(Sh + C0(Ω)

)2 × L2(Ω) → R,

(v,p) →∑e∈Eh

∫∂be

v(me) · nbep ds,

bh :(Sh + H 1(Ω)

)2 × L2(Ω) → R,

(v,p) → −∑

K∈Th

∫K

div vp dx,

where nbe = (nbe,i)�i=1,2 denotes the unit outward normal vector along ∂be .

Let us now recall the relationship between ah and ah as well as bh and bh on the function space (S0h)2 × (S0

h)2 and(S0

h)2 × P0(Th). We refer to Lemmas 2.1 and 2.2 of [15] or to Lemma 3.1 of [12] for a detailed proof.

Lemma 2.1. The next identities hold:

ah(vh,wh) = ah(vh,wh), ∀(vh, wh) ∈ (S0

h

)2 × (S0

h

)2,

bh(vh, qh) = bh(vh, qh), ∀(vh, qh) ∈ (S0

h

)2 × P0(Th).

For e ∈ Einth , l ∈ Ve and vh ∈ (Sh)

2, we set

m+,vh

e,l :={

me if∫∂be∩∂bl

vh · nbe ds � 0,

ml if∫∂be∩∂bl

vh · nbe ds < 0.

For an arbitrary weight ω ∈ [0,1], we finally introduce the form

cωh : (Sh)

2 × (Sh)2 × (Sh)

2 → R,

(uh,vh,wh) →∑e∈Eh

∑l∈Ve

∫∂be∩∂bl

uh · nbe ds{ωvh

(m

+,uh

e,l

) + (1 − ω)vh(me,l)} · wh(me).

3. The numerical scheme

We are now ready to introduce the discretization of problem (1). Multiplying the first equation of (1) by wh, forwh ∈ (S0

h)2, integrating the result on Ω and applying Green’s formula on each box be we arrive at

−ν∑

e∈Ein

∫∂u

∂nbe

· wh ds +∑

e∈Ein

∫u · nbe u · wh ds +

∑e∈Ein

∫pwh · ne ds =

∫f · wh dx. (5)

h ∂be h ∂be h ∂be Ω


In the same way, multiplying the second equation of (1) by qh ∈ P0(Th), and integrating on Ω we obtain∫

Ω

qh div udx = 0. (6)

In view of (5) and (6) and of the Dirichlet boundary condition satisfied by the exact solution u, the finite volumeelement approximation of (1) is to find (uEV ,pEV ) ∈ (S0

h)2 × P0(Th) solution of{

νah(uEV ,vh) + cωh (uEV ,uEV ,vh) + bh(pEV ,vh) = (f, vh)Ω, ∀vh ∈ (S0

h)2,

bh(uEV , qh) = 0, ∀qh ∈ P0(Th).

(7)

Note that we should write (uωEV ,pω

EV ) instead of (uEV ,pEV ) but for shortness we omit this dependence wrt ω.The use of Lemma 2.1 allows to transform problem (7) in the following equivalent form:{

νah(uEV ,vh) + cωh (uEV ,uEV ,vh) + bh(pEV ,vh) = (f, vh)Ω, ∀vh ∈ (S0

h)2,

bh(uEV , qh) = 0, ∀qh ∈ P0(Th).

(8)

Remark 3.1. Note that in the case ω = 1, (8) reduces to the upstream scheme used in [15,13,23] while the case ω = 0is a variant, but very close, of the centred scheme introduced in [15], for which no existence result and a priori erroranalysis exist. Our analysis also covers the centred scheme of [15] (see Remark 4.10 below) and therefore fills thesegaps.

4. Some preliminaries

In this section, we collect some preliminary results that will be used later either for the existence and uniquenessresults or for the a priori error analysis.

Let us recall the so-called discrete Poincaré inequality (see Proposition II. 4.13 of [26]):

Lemma 4.1. If the triangulation Th is regular (i.e. satisfies (4)), then there exists a positive constant C(Ω,ξ) suchthat

‖vh‖0,Ω � C(Ω,ξ)|vh|1,h, ∀vh ∈ (S0

h

)2 + V.

In a similar manner, we have the discrete Sobolev inequalities (cf. Theorem II.2.3 and Remark II.2.6 of [26]):

Lemma 4.2. If the triangulation Th is regular, then for all q ∈ (1,∞), there exists C(Ω,ξ, q) > 0 such that

‖vh‖0,q,Ω � C(Ω,ξ, q)|vh|1,h, ∀vh ∈ (S0

h

)2 + V.

We further need a so-called inverse inequality under a realistic assumption on the meshes (see Lemma 9.1 of [12]or Lemma 6.4.4 of [13], where the optimality of this inequality is shown, see also the estimate (4.16) of [23]).

Lemma 4.3. Let Th be a regular triangulation of Ω satisfying

∃γ � 1: hK � hγ , ∀K ∈ Th. (9)

Then for any κ ∈ ]0,1], there exists a positive constant C(κ,Ω, ξ) such that

‖vh‖0,∞,Ω � C(κ,Ω, ξ)h−κ |vh|1,h, ∀vh ∈ (S0

h

)2. (10)

We go on with the inf-sup condition for the non-conforming pair P1/P

0 [11]:

Lemma 4.4. If the triangulation Th is regular, then there exists C(Ω,ξ) > 0 such that

infqh∈P 0(Th)

qh �=0

supvh∈(S0

h)2

vh �=0

|bh(vh, qh)||vh|1,h‖qh‖0,Ω

� C(Ω,ξ).


We now give some interpolation error estimates and properties of the Crouzeix–Raviart interpolant ICR [11]: Recallthat for u ∈ (H 1(Ω))2, ICRu is defined as the unique element in (Sh)

2 satisfying

ICRu(me) := 1

|e|∫e

uds, ∀e ∈ Eh.

Lemma 4.5. If u ∈ (H 1(Ω))2, then we have

(div ICRu)|K = 1

|K|∫K

div udx, (11)

|ICRu|1,h � |u|1,h. (12)

Assume furthermore that u ∈ (H 2(Ω))2 and that the triangulation Th is regular, then the next interpolation errorestimates hold:

|u − ICRu|1,h � h|u|2,Ω, (13)

‖u − ICRu‖0,Ω � h2|u|2,Ω . (14)

Finally for K ∈ Th, if z ∈ K and σ = ∂be ∩ ∂bl , for e, l ∈ Eh(K), then we have∥∥u − ICRu(z)∥∥

σ� hK |u|1,4,K , (15)

‖u − ICRu‖σ � hK |u|1,4,K . (16)

Proof. The first properties are based on the next identity: For any K ∈ Th and i, j = 1 or 2, by Green’s formula wehave (see [1])

(∂i(ICRu)j

)|K = 1

|K|∫K

∂i(ICRu)j dx

= 1

|K|∑

e∈Eh(K)

∫e

(ICRu)j nK,i ds

= 1

|K|∑

e∈Eh(K)

∫e

uj nK,i ds

= 1

|K|∫K

∂iuj dx,

which means that (∂i(ICRu)j )|K is the mean of ∂iuj on K . The identity (11) follows directly from this identity, while(12) follows from the well-known estimate∥∥∥∥ 1

|K|∫K

w

∥∥∥∥K

� ‖w‖K.

The global interpolation error estimates (13) and (14) are fully standard, we therefore omit the proof (see [11]). Wethen focus on the local ones (15) and (16). Fix an affine transformation FK(x) = BKx + bK , which maps the standardreference triangle K to K , where bK ∈ R

2 and BK is a 2 × 2 matrix which satisfies ‖BK‖K ∼ hK. Using the standardtransformation

u(x) = u(BKx + bK),

we see that∥∥u − ICRu(z)∥∥ � h

1/2∥∥u − ICRu(z)∥∥ , (17)

σ K σ


since |σ | � hK and |σ | � 1 (because σ is a segment joining the barycenter (1/3,1/3) to one vertex of K). Now weremark that for an arbitrary k ∈ R

2, we have

u − ICRu(z) = (I − ICR,z)(u − k), (18)

where I means the identity operator and ICR,z is defined by

ICR,zv := ICR v(z).

From the above identities we then need to show that I − ICR,z is a continuous mapping from W 1,4(K) into L2(σ ).The identity operator is clearly continuous from W 1,4(K) into L2(σ ), because a trace theorem yields W 1,4(K) ↪→W 3/4,4(σ ) and W 3/4,4(σ ) ↪→ L2(σ ). For the operator ICR,z, we remark that

ICR,zv =∑

e∈E(K)

1

|e|∫e

vds λe(z),

where E(K) is the set of edges of K and {λe}e∈E(K)is the basis of P

1(K) such that

1

|f |∫f

λe ds = δe,f , ∀f ∈ E(K).

As ‖λe‖∞,K� 1 we deduce that

|ICR,zv| �∑

e∈E(K)

∫e

|v|ds,

and again the embeddings W 1,4(K) ↪→ W 3/4,4(σ ) ↪→ L1(e), for all e ∈ E(K) lead to

‖ICR,z‖σ � |ICR,zv| � ‖v‖1,4,K.

This continuity properties, (17) and (18) yield∥∥u − ICRu(z)∥∥

σ� h

1/2K ‖u − k‖1,4,K

, ∀k ∈ R2.

By a Bramble–Hilbert argument, we get∥∥u − ICRu(z)∥∥

σ� h

1/2K |u|1,4,K

.

Using the inverse affine transformation F−1K , we arrive at (15).

The estimate (16) is proved similarly. �From the above lemma we notice that if u is divergence free, i.e., div u = 0 in Ω , then ICRu is also divergence free.We end up with some specific properties of the forms cω

h (·, ·, ·), ω ∈ [0,1]. We first recall the following propertyof cω=1

h proved in Section 4 of [23] (see also Lemma 6.3.2 of [13]). Introduce the space(S0

h

)2 := {vh ∈ (

S0h

)2: ∀K ∈ Th, div vh|K = 0}.

Lemma 4.6. For all (uh,vh) ∈ (S0h)2 × (S0

h)2, we have

cω=1h (uh,vh,vh) � 0.

Lemma 4.7. Let Th be a regular triangulation of Ω . Then for all uh,vh and wh ∈ (S0h)2, the next estimate holds:∣∣cω

h (uh,vh,wh)∣∣ � |uh|1,h|vh|1,h|wh|1,h, ∀ω ∈ [0,1].


Proof. We use some idea developed in [23,24], but adapted to our new form. For K ∈ Th, e, l ∈ Eh(K) and i ∈ {1,2},we may write (since ω ∈ [0,1])∣∣ωvh,i

(m

+,uh

e,l

) + (1 − ω)vh,i(me,l)∣∣ � ‖vh,i‖∞,K .

Scaling arguments and the fact that all norms are equivalent in a finite-dimensional space then lead to∣∣ωvh,i

(m

+,uh

e,l

) + (1 − ω)vh,i(me,l)∣∣ � h

−1/2K ‖vh,i‖0,4,K . (19)

Now from the definition of m+,uh

e,l , we may write

cωh (uh,vh,wh) =

2∑i=1

∑K∈Th

∑e,l∈Eh(K)

∫∂be∩∂bl

uh · nbe

{ωvh,i

(m

+,uh

e,l

) + (1 − ω)vh,i(me,l)}ds

× {wh,i(me) − wh,i(ml)

}=

2∑i=1

∑K∈Th

∑e,l∈Eh(K)

∫∂be∩∂bl

uh · nbe ds{ωvh,i

(m

+,uh

e,l

) + (1 − ω)vh,i(me,l)}−−−−→meml · (∇wh,i |K).

(20)

Now by scaling arguments we have∣∣∣∣∫

∂be∩∂bl

uh · nbe ds

∣∣∣∣ � hK

∫σ

|uh|.

Since all norms are equivalent in a finite-dimensional space we obtain∣∣∣∣∫

∂be∩∂bl

uh · nbe ds

∣∣∣∣ � hK‖uh‖0,4,K,

and going back to K , we arrive at∣∣∣∣∫

∂be∩∂bl

uh · nbe ds

∣∣∣∣ � h1/2K ‖uh‖0,4,K .

Using this estimate, the estimate (19) and the standard inverse inequality

|∇wh,i |K | � h−1K ‖∇wh,i‖K, (21)

in the estimate (20), we obtain

∣∣cωh (uh,vh,wh)

∣∣ �2∑

i=1

∑K∈Th

‖uh‖0,4,K‖vh,i‖0,4,K‖∇wh,i‖0,2,K .

Applying discrete Cauchy–Schwarz’s inequality, we arrive at∣∣cωh (uh,vh,wh)

∣∣ � ‖uh‖0,4,Ω‖vh‖0,4,Ω |wh|1,h.

We conclude with the help of Lemma 4.2 (with q = 4). �Lemma 4.8. Let Th be a regular triangulation of Ω . Then for all u1

h,u2h,vh and wh ∈ (S0

h)2, the next estimate holds∣∣cωh

(u1

h,vh,wh

) − cωh

(u2

h,vh,wh

)∣∣ �∣∣u1

h − u2h

∣∣1,h

|vh|1,h|wh|1,h, ∀ω ∈ [0,1].

Proof. By the definition of cωh , we write∣∣cω

h

(u1

h,vh,wh

) − cωh

(u2

h,vh,wh

)∣∣� ω

∣∣cω=1h

(u1

h,vh,wh

) − cω=1h

(u2

h,vh,wh

)∣∣ + (1 − ω)∣∣cω=0

h

(u1

h,vh,wh

) − cω=0h

(u2

h,vh,wh

)∣∣.


Now using the fact that cω=0h is trilinear, we get∣∣cω

h

(u1

h,vh,wh

) − cωh

(u2

h,vh,wh

)∣∣� ω

∣∣cω=1h

(u1

h,vh,wh

) − cω=1h

(u2

h,vh,wh

)∣∣ + (1 − ω)∣∣cω=0

h

(u1

h − u2h,vh,wh

)∣∣�

∣∣u1h − u2

h

∣∣1,h

|vh|1,h|wh|1,h,

using Lemma 2 of [23] to estimate the first term of this right-hand side and Lemma 4.7 to estimate the second term ofthis right-hand side. �Lemma 4.9. Let Th be a regular triangulation of Ω satisfying (9). Then for all uh,vh and wh ∈ (S0

h)2, the next estimateholds:∣∣cω

h (uh,vh,wh) − cω=1h (uh,vh,wh)

∣∣ � h1−κ |uh|1,h|vh|1,h|wh|1,h, ∀κ ∈ ]0,1].

Proof. From the definition of cωh , we may write∣∣cω


∣∣= (1 − ω)

∣∣∣∣ ∑K∈Th

∑e,l∈Eh(K)

∫∂be∩∂bl

uh · nbe ds{vh

(m

uh,+e,l

) − vh(me,l)} · wh(me)

∣∣∣∣= (1 − ω)

∣∣∣∣ ∑K∈Th

∑e,l∈Eh(K)

∫∂be∩∂bl

uh · nbe ds{vh

(m

uh,+e,l

) − vh(me,l)} · {wh(me) − wh(ml)

}∣∣∣∣�

∑K∈Th

h2K

∑e,l∈Eh(K)

∫∂be∩∂bl

|uh · nbe |ds∣∣(∇vh)

∣∣K

∣∣∣∣(∇wh)∣∣K

∣∣.By the inverse estimate (21), we obtain∣∣cω


∣∣ � h‖uh‖0,∞,Ω

∑K∈Th

‖∇vh‖0,K‖∇wh‖0,K .

Discrete Cauchy–Schwarz’s inequality and Lemma 4.3 lead to the conclusion. �Remark 4.10. The centred scheme introduced in [15] corresponds to the following choice for the trilinear form:

cC(uh,vh,wh) =∑e∈Eh

∑l∈Ve

∫∂be∩∂bl

uh(x) · nbe vh(x) · wh(me) ds(x).

To cover this case, we should take the form

cωh = ωcω=1

h + (1 − ω)cC.

But a careful inspection of our above arguments shows that the form cωh satisfies Lemmas 4.7–4.9. Since our analysis

below uses only the properties of cωh from these lemmas, our results also hold for the centred scheme from [15].

5. Existence and uniqueness of the approximated solution

We start with the existence result.

Theorem 5.1. Let Th be a regular triangulation of Ω satisfying (9). Then there exists h0 > 0 depending on ν and‖f‖0,Ω such that for h � h0, the system (8) has at least one solution (uEV ,pEV ) ∈ (S0

h)2 × P0(Th), which satisfies

|uEV |1,h � ν−1‖f‖0,Ω . (22)


Proof. Since the bilinear form bh satisfies the discrete inf-sup condition, (uEV ,pEV ) in (S0h)2 × P

0(Th) is solutionof (8) if and only if uEV belongs to (S0

h)2 and is solution of

νah(uEV ,vh) + cωh (uEV ,uEV ,vh) = (f, vh)Ω, ∀vh ∈ (

S0h

)2. (23)

Let us now prove the existence of a solution uEV ∈ (S0h)2 of (23): For that purpose introduce the mapping

Λ : (S0h)2 → (S0

h)2 defined by

ah(Λuh,vh) = νah(uh,vh) + cωh (uh,uh,vh) − (f, vh)Ω, ∀uh,vh ∈ (

S0h

)2.

Lemma 4.8 guarantees the continuity of Λ. Moreover applying Cauchy–Schwarz’s inequality and Lemma 4.1 weget for all uh ∈ (S0

h)2:

ah(Λuh,uh) � ν|uh|21,h + cωh (uh,uh,uh) − |uh|1,hC1(Ω, ξ)‖f‖0,Ω

� ν|uh|21,h + cω=1h (uh,uh,uh) − |uh|1,hC1(Ω, ξ)‖f‖0,Ω + cω

h (uh,uh,uh) − cω=1h (uh,uh,uh),

for some positive constant C1(Ω, ξ). Using Lemma 4.6, one obtains

ah(Λuh,uh) � ν|uh|21,h − |uh|1,hC1(Ω, ξ)‖f‖0,Ω + cωh (uh,uh,uh) − cω=1

h (uh,uh,uh).

By Lemma 4.9, for any κ ∈ ]0,1[, there exists a positive constant C0(Ω, ξ, κ) such that

ah(Λuh,uh) � |uh|1,h

(−C0(Ω, ξ, κ)h1−κ |uh|21,h + ν|uh|1,h − C1(Ω, ξ)‖f‖0,Ω

).

This left-hand side will be equal to zero if we choose |uh|1,h such that

−C0(Ω, ξ, κ)h1−κ |uh|21,h + ν|uh|1,h − C1(Ω, ξ)‖f‖0,Ω = 0.

We recognize a second order equation in |uh|1,h, whose solution are

|uh|1,h = ν ± √ν2 − 4C0(Ω, ξ, κ)h1−κC1(Ω, ξ)‖f‖0,Ω

2C0(Ω, ξ, κ)h1−κC1(Ω, ξ)‖f‖0,Ω

provided ν2 − 4C0(Ω, ξ, κ)h1−κC1(Ω, ξ)‖f‖0,Ω � 0. This last condition is equivalent to

4C0(Ω, ξ, κ)h1−κC1(Ω, ξ)‖f‖0,Ω � ν2,

or to h � h0, with h0 defined by

h1−κ0 = ν2

4C0(Ω, ξ, κ)C1(Ω, ξ)‖f‖0,Ω

.

Let us now set

C(f, ν) = ν − √ν2 − 4C0(Ω, ξ, κ)h1−κC1(Ω, ξ)‖f‖0,Ω

2C0(Ω, ξ, κ)h1−κC1(Ω, ξ)‖f‖0,Ω

,

the smallest (but positive) root of the above quadratic equation, we deduce that

ah(Λuh,uh) � 0,

for all |uh|1,h = C(f, ν). A simple application of Lemma II.1.4 of [26] yields at least one uEV ∈ (S0h)2 satisfying

|uEV |1,h � C(f, ν) (24)

and solution of ΛuEV = 0, or equivalently solution of (23).Let us finally notice that C(f, ν) may be equivalently written

C(f, ν) = 2C1(Ω, ξ)‖f‖0,Ω

ν + √ν2 − 4C0(Ω, ξ, κ)h1−κC1(Ω, ξ)‖f‖0,Ω

.

The estimate (22) directly follows. �


Theorem 5.2. Under the assumptions of Theorem 5.1, if ν−2‖f‖0,Ω is sufficiently small then problem (8) has a uniquesolution among those satisfying (22).

Proof. Let (uEV 1,pEV 1), (uEV 2,pEV 2) ∈ (S0h)2 × P

0(Th) be two solutions of (8) satisfying (22). Setting uEV :=uEV 1 − uEV 2, we remark that it satisfies

νah(uEV ,uEV ) = cωh (uEV 2,uEV 2,uEV ) − cω

h (uEV 1,uEV 1,uEV )

= cωh (uEV 2,uEV 2,uEV ) − cω

h (uEV 1,uEV 2,uEV ) − cωh (uEV 1,uEV ,uEV )

� cωh (uEV 2,uEV 2,uEV ) − cω

h (uEV 1,uEV 2,uEV ) + cω=1h (uEV 1,uEV ,uEV )

− cωh (uEV 1,uEV ,uEV ),

this last property following from Lemma 4.6. Therefore by Lemmas 4.8 and 4.9 we obtain

ν|uEV |21,h � |uEV |21,h

(C3(Ω, ξ)

(|uEV 2|1,h + |uEV 1|1,h

) + C0(Ω, ξ, κ)|uEV 1|1,h

).

According to the condition (22) we deduce the existence of a positive constant C2(Ω, ξ, κ) such that(ν − ν−1C2(Ω, ξ, κ)‖f‖0,Ω

)|uEV |21,h � 0.

Consequently if ν−2C2(Ω, ξ, κ)‖f‖0,Ω < 1, we can conclude that uEV = 0. �6. Error estimate

Theorem 6.1. Let the assumptions of Theorem 5.1 be satisfied and suppose that ν−2‖f‖0,Ω is sufficiently small. Thenproblem (1) and (8) has a unique solution (u,p) ∈ V × L2

0(Ω) and (uEV ,pEV ) ∈ (S0h)2 × P

0(Th) satisfying (22)respectively. If moreover (u,p) ∈ (H 2(Ω))2 × H 1(Ω), then the next error estimates hold:

|u − uEV |1,h + ‖p − pEV ‖0,Ω � C(Ω,ν, f, ξ)h, (25)

where C(Ω,ν, f, ξ) is positive constant which may depend on Ω , ξ , ν and ‖f‖0,Ω .

Proof. We start with the estimate of |u − uEV |1,h. Let us set vh = ICRu ∈ (S0h)2 (due to the identity (11) from

Lemma 4.5 and the fact that u is divergence free) and wh := uEV − vh ∈ (S0h)2. Note that from the estimate (12) of

Lemma 4.5 and the a priori bound (22), we get

|vh|1,h � ν−1‖f‖0,Ω . (26)

Now by (8) we may write

ν|wh|21,h = νah(uEV − vh,wh)

= (f, wh)Ω − cωh (uEV ,uEV ,wh) − νah(vh,wh).

Using the first identity of (1) and elementwise integration by parts, we get

ν|wh|21,h = (f, wh − wh)Ω + νah(u − vh,wh) + ch(u,u,wh) + bh(wh,p)

− cωh (uEV ,uEV ,wh) +

∑K∈Th

(−(∇unK,wh)∂K + (p,wh · nK)∂K

),

where we have set

ch(u,u,wh) =∑

K∈Th

∫K

u · ∇u · wh.

By Lemma 4.6 and the property

bh(wh,p) =∑

K∈Th

∫div whp dx =

∑K∈Th

div wh

∫p dx = 0,

K K


since wh belongs to (S0h)2, we deduce that

ν|wh|21,h � (f, wh − wh)Ω + νah(u − vh,wh) + ch(u,u,wh) − cωh (uEV ,uEV ,wh)

+∑

K∈Th

(−(∇unK,wh)∂K + (p,wh · nK)∂K

) + cω=1h (vh,wh,wh).

Introducing some artificial terms, we arrive at

|wh|21,h � 1

ν

6∑i=1

|Ti |, (27)

where we set

• T1 := (f, wh − wh)Ω ,• T2 := νah(u − vh,wh),• T3 := cω=1

h (vh,wh,wh) − cωh (vh,wh,wh),

• T4 := ch(u,u,wh) − cωh (vh,vh,wh),

• T5 := cωh (vh,uEV ,wh) − cω

h (uEV ,uEV ,wh),

• T6 := ∑K∈Th

(−ν(∇unK,wh)∂K + (p,wh · nK)∂K).

We now estimate each term |Ti |.Estimate of T1: Cauchy–Schwarz’s inequality and a Bramble–Hilbert argument directly yield

|T1| � h‖f‖0,Ω |wh|1,h. (28)

Estimate of T2: By Cauchy–Schwarz’s inequality one has

|T2| � ν|u − vh|1,h|wh|1,h,

and by the estimate (13) from Lemma 4.5, we obtain

|T2| � νh|u|2,Ω |wh|1,h. (29)

Estimate of T3: A simple application of Lemma 4.9 furnishes

|T3| � h1−κ |vh|1,h|wh|21,h,

and by the estimate (26) we obtain

|T3| � h1−κν−1‖f‖0,Ω |wh|21,h. (30)

Estimate of T4: We first transform ch(u,u,wh) in the following way:

ch(u,u,wh) =∑

K∈Th

∑e∈Eh(K)

∫be,K

u · ∇u · wh(me) dx

+∑

K∈Th

∑e∈Eh(K)

∫be,K

u · ∇u · {wh − wh(me)}dx,

where for shortness we have set be,K = be ∩ K . Applying elementwise Green’s formula in the first term of the aboveright-hand side, we get

ch(u,u,wh) =∑

K∈Th

∑e∈Eh(K)

∫∂be,K

u · nbe,Ku · wh(me) ds

−∑

K∈Th

∑e∈Eh(K)

∫b

(div u)u · wh(me) dx

e,K


+∑

K∈Th

∑e∈Eh(K)

∫be,K

u · ∇u · (wh − wh(me))dx

=∑e∈Eh

∑l∈Ve

∫∂be∩∂bl

u · nbe u · wh(me) ds

+∑

K∈Th

∑e∈Eh(K)

∫be,K

u · ∇u · (wh − wh(me))dx,

reminding that div u = 0. Consequently, we obtain

T4 = T41 + T42,

where

T41 =∑

K∈Th

∑e∈Eh(K)

∫be,K

u · ∇u · {wh − wh(me)}dx,

T42 =∑e∈Eh

∑l∈Ve

∫∂be∩∂bl

(u · nbeu − vh · nbe

{ωvh

(m

+,vh

e,l

) + (1 − ω)vh(me,l)}) · wh(me) ds.

A Taylor expansion of wh − wh(me) and the embedding H 2(Ω) ↪→ W 1,4(Ω), consequence of the Sobolev em-bedding theorem clearly lead to

|T41| � h‖u‖0,4,Ω |u|1,4,Ω |wh|1,h � h‖u‖22,Ω |wh|1,h. (31)

For the estimate of T42, using the definition of m+,vh

e,l , we rewrite T42 as follows

T42 =∑e∈Eh

∑l∈Ve

∫∂be∩∂bl

({u − {

ωvh

(m

+,vh

e,l

) + (1 − ω)vh(me,l)}}

vh · nbe

+ {u · nbe − vh · nbe }u) · (wh(me) − wh(ml)

)ds.

Using the estimate |wh(me) − wh(ml)| � ‖∇wh‖K , for e, l ∈ Eh(K) and Cauchy–Schwarz’s inequality, we obtain

|T42| �∑

K∈Th

‖∇wh‖K

∑e,l∈Eh(K):e �=l

(∥∥u − vh

(m

+,vh

e,l

)∥∥∂be∩∂bl

‖vh‖∂be∩∂bl

+ ∥∥u − vh(me,l)∥∥

∂be∩∂bl‖vh‖∂be∩∂bl

+ ‖u − vh‖∂be∩∂bl‖u‖∂be∩∂bl

). (32)

Now scaling arguments and the fact that all norms are equivalent in a finite dimensional space lead successively to

‖vh‖∂be∩∂bl� h

1/2K ‖vh‖σ � h

1/2K ‖vh‖0,4,K

� ‖vh‖0,4,K .

Similarly scaling arguments and a trace theorem yield

‖u‖∂be∩∂bl� h

1/2K ‖u‖σ � h

1/2K ‖u‖1,4,K

� h1/2K

(h

−1/2K ‖u‖0,4,K + h

1/2K |u|1,4,K

)� ‖u‖1,4,K .

These two estimates and (15), (16) allow to transform (32) as follows:

|T42| �∑

K∈Th

hK‖u‖1,4,K

(‖vh‖0,4,K + ‖u‖1,4,K

)‖∇wh‖K.

By discrete Cauchy–Schwarz’s inequality we arrive at

|T42| � h‖u‖1,4,Ω

(‖vh‖0,4,Ω + ‖u‖1,4,Ω

)|wh|1,h.


Making use of the discrete Sobolev inequality (Lemma 4.2), the estimate (13) and the embedding H 2(Ω) ↪→W 1,4(Ω), we conclude that

|T42| � h‖u‖1,4,Ω

(|vh|1,h + ‖u‖1,4,Ω

)|wh|1,h

� h‖u‖1,4,Ω

(‖u‖2,Ω + ‖u‖1,4,Ω

)|wh|1,h

� h‖u‖22,Ω |wh|1,h. (33)

The estimates (31) and (33) show that

|T4| � h‖u‖22,Ω |wh|1,h. (34)

Estimate of T5: Using Lemma 4.8 and the a priori bound (22), we get

|T5| � |uEV |1,h|wh|21,h � ν−1‖f‖0,Ω |wh|21,h. (35)

Estimate of T6: For e ∈ Eh, we denote by Me(v) = 1|e|

∫ev, the mean of v on e (corresponding to the L2(e)-

projection). Then the continuity of ∇un and p through the edges (in the sense of trace) allows to write

T6 = −∑

K∈Th

ν∑

e∈Eh(K)

(∇unK −Me(∇unK),wh −Me(wh))e

+∑

K∈Th

∑e∈Eh(K)

(p −Me(p),wh · nK −Me(wh · nK)

)e.

Cauchy–Schwarz’s inequality and Bramble–Hilbert arguments furnish

|T6| � h(ν|u|2,Ω + |p|1,Ω

)|wh|1,h. (36)

All together the estimates (28)–(36) of |Ti |, i = 1, . . . ,6, in the estimate (27) allow to conclude that

ν|wh|1,h � C3h(‖f‖0,Ω + ν‖u‖2,Ω + ‖p‖1,Ω + ‖u‖2

2,Ω

) + C4ν−1‖f‖0,Ω |wh|1,h,

where C3 and C4 are two positive constants which depends on Ω,ξ and κ . This equivalently means that(ν − C4ν

−1‖f‖0,Ω

)|wh|1,h � C3h(‖f‖0,Ω + ν‖u‖2,Ω + ‖p‖1,Ω + ‖u‖2

2,Ω

),

and we arrive at the estimate

|wh|1,h � 2C3

νh(‖f‖0,Ω + ν‖u‖2,Ω + ‖p‖1,Ω + ‖u‖2

2,Ω

), (37)

provided ν − C4ν−1‖f‖0,Ω � ν/2 or equivalently provided ν−2‖f‖0,Ω is small enough.

The requested estimate on the velocity, namely

|u − uEV |1,h � h (38)

follows from the estimate (13) and (37).We now pass to the estimate on ‖p − pEV ‖0,Ω . For that purpose denote by ph ∈ P

0(Th) the L2-projection of p onP

0(Th), which means that

ph|K = 1

|K|∫K

p,

is the mean of p on K ∈ Th. Recall that a standard Bramble–Hilbert argument yields

‖p − ph‖0,Ω � h|p|1,Ω . (39)

Thanks to Lemma 4.4, we have

‖pEV − ph‖0,Ω � supwh∈(S0

h)2

wh �=0

|bh(wh,pEV − ph)||wh|1,h

. (40)

We now transform the numerator of this right-hand side, using (8), (1) and elementwise integration by parts:


bh(wh,pEV − ph) = (f, wh − wh) + νah(u − uEV ,wh) + bh(p − ph,wh) + ch(u,u,wh)

− cωh (uEV ,uEV ,wh) +

∑K∈Th

(−ν(∇unK,wh)∂K + (p,wh · nK)∂K

). (41)

We estimate each term of this right-hand side as follows. Here for the sake of shortness, a � b mean that a � Cb,where C is positive constant which do not depend on a and b but which depends on Ω , ξ , κ , ν and ‖f‖0,Ω . The firstterm is estimated as T1 to get∣∣(f, wh − wh)

∣∣ � h‖f‖0,Ω |wh|1,h. (42)

For the next two terms, we use Cauchy–Schwarz’s inequality and the estimates (38) and (39) to obtain∣∣νah(u − uEV ,wh) + bh(p − ph,wh)∣∣ � h|wh|1,h. (43)

Now we write

ch(u,u,wh) − cωh (uEV ,uEV ,wh)

= ch(u,u,wh) − cωh (ICRu, ICRu,wh) + cω

h (ICRu, ICRu,wh) − cωh (uEV , ICRu,wh)

+ cωh (uEV , ICRu,wh) − cω

h (uEV ,uEV ,wh).

The first term of this right-hand side is estimated as T4 (cf. (34))∣∣ch(u,u,wh) − cωh (ICRu, ICRu,wh)

∣∣ � h‖u‖22,Ω |wh|1,h.

The second term is estimated with the help of Lemma 4.8:∣∣cωh (ICRu, ICRu,wh) − cω

h (uEV , ICRu,wh)∣∣ � |ICRu − uEV |1,h|ICRu|1,h|wh|1,h

and by the estimate (37) and (13), we obtain∣∣cωh (ICRu, ICRu,wh) − cω

h (uEV , ICRu,wh)∣∣ � h|wh|1,h.

For the third term we use Lemma 4.7 and again (37) to arrive at∣∣cωh (uEV , ICRu,wh) − cω

h (uEV ,uEV ,wh)∣∣ � |uEV |1,h|ICRu − uEV |1,h|wh|1,h

� h|wh|1,h.

These three estimates show that∣∣ch(u,u,wh) − cωh (uEV ,uEV ,wh)

∣∣ � h|wh|1,h. (44)

The last term of (41) is estimated as T6 (see (36)):∣∣∣∣ ∑K∈Th

(−ν(∇unK,wh)∂K + (p,wh · nK)∂K

)∣∣∣∣ � h|wh|1,h. (45)

Using the estimates (42)–(45) in (41), we conclude that∣∣bh(wh,pEV − ph)∣∣ � h|wh|1,h,

and by (40)

‖pEV − ph‖0,Ω � h.

Using again (39), the requested estimate on ‖p − pEV ‖0,Ω is proved. �Remark 6.2. By a simple application of Lemma 4.1, we deduce from the above Theorem 6.1 that

‖u − uEV ‖0,Ω � h,

this order of convergence being optimal in general (see Remark 3 of [28]).


7. Numerical tests

7.1. An analytical solution

Here we take Ω := ]0,1[2 the unit square and consider the problem (1) with ν := 10−1 and the exact solutionu := rot(xy(1 − x)(1 − y)) and p := x2 − y2, the datum f is fixed accordingly.

In order to illustrate the flexibility of our method, we consider a sequence of unstructured (but quasi-uniform)meshes as illustrated in Fig. 2. For this sequence we denote by N the number of elements and remark that h ∼ N−1/2.Then we compute the discrete solution (uEV ,pEV ) ∈ (Sh)

2 × P0(Th) of our discrete system (8) for ω := 1 (upstream

centred) and for ω := 0 (centred scheme). Table 1 furnishes the obtained errors that are graphically illustrated inFig. 3. We first observe that the results obtained by the centred or upstream approaches are approximately equivalent.We further observe an order of convergence for both methods in O(N−1/2) for the error |u−uEV |1,h +‖p−pEV ‖0,Ω .This is in accordance with the results from Theorem 6.1. Moreover we remark that the error ‖u − uEV ‖0,Ω is of orderO(N−1), which is not true in general (see Remark 6.2).

7.2. The lid-driven cavity

As a second test we consider a benchmark example in fluid mechanics, namely the lid-driven cavity [15,19,26,27].We again take Ω := ]0,1[2 and consider problem (1) with ν := 1/400 but with non-homogeneous boundary con-

ditions u = g, where g is defined by

g :={(1

0

)on ]0,1[ × {1},(0

0

)elsewhere.

We have implemented the centred (ω = 0) and upstream (ω = 1) schemes on a mesh made of 2626 elements refinednear the top corners of Ω in order to take into account the discontinuities of the solution and to capture adequately

Fig. 2. Representation of the used meshes with 36 elements.

Table 1Errors wrt N for the centred and upstream schemes

N ω = 0 (centred scheme) ω = 1 (upstream scheme)

‖u − uEV ‖0,Ω |u − uEV |1,h ‖p − pEV ‖0,Ω ‖u − uEV ‖0,Ω |u − uEV |1,h ‖p − pEV ‖0,Ω

36 8.80e−02 9.71e−01 1.09e−01 8.27e−02 9.12e−01 1.08e−01158 1.97e−02 4.57e−01 4.57e−02 1.91e−02 4.43e−01 4.56e−02604 5.31e−03 2.38e−01 2.28e−02 5.26e−03 2.34e−01 2.27e−02

2422 1.28e−03 1.16e−01 1.09e−02 1.28e−03 1.15e−01 1.09e−029746 3.26e−04 5.87e−02 5.51e−03 3.34e−04 5.85e−02 5.51e−03


Fig. 3. Rate of convergence wrt N of the centred scheme for ‖u − uEV ‖0,Ω (line (1)) and for |u − uEV |1,h + ‖p − pEV ‖0,Ω (line (2)).

Fig. 4. Velocity field (top) and isolines of pressure (bottom) for the centred scheme (left) and the upstream scheme (right).


Fig. 5. Length d(ω, ν)of the zone of recirculation wrt ω.

the secondary vortices of this solution. Fig. 4 illustrates the velocity field and the isolines of pressure obtained by bothschemes. The results appear to be approximatively identical and to be in accordance with the existing results fromRefs. [15,19,27]. Let us notice that, for the centred scheme, we encounter some instabilities in the convergence of thealgorithm (namely the non-linearity of the problem is solved via an over-relaxation method [27], each Oseen problembeing solved by a simple Uzawa algorithm [26]). Indeed, it is well known that centred schemes are unstable when thequantity ν/(h‖u‖0,Ω) becomes very small [15].

As already mentioned in the introduction, it is well known that upstream schemes have a better numerical conver-gence than centred schemes [15,19], but that they are more diffusive, they then require the use of finer meshes. Thenext test will illustrate this phenomenon.

7.3. The backward facing step

We here consider another benchmark example in fluid mechanics: the backward facing step. This test will allowus to illustrate the interest of our weighted upstream approach. We start with the description of the problem. We takethe domain Ω := ]0,2[×]0.5,1.5[ ∪ ]2,28[ × ]0,1.5[ and the viscosity parameter ν := 1/100. We still consider theproblem (1), but with the following boundary conditions:⎧⎨

⎩u = (−6(y−0.5)(y−1.5)

0

)on ]0.5,1.5[ × {0},

(∇u − pI) · n = 0 on ]0,1.5[ × {28},u = 0 elsewhere on ∂Ω,

where I denotes the identity matrix. In comparison with the formulation (1), we here consider “Neumann” boundarycondition on the exit of the domain, which physically means that the fluid is stable at the exit of the domain.

The experiences show that a principal zone of recirculation of the fluid develops near the backward facing step, thelength of this zone of recirculation increases when ν diminishes [2]. We therefore have an instrument of measure of theartificial diffusion of the used scheme, namely the length of the zone of recirculation. We then have implemented ourweighted upstream scheme for different values of the weight ω. The discrete solution (uEV ,pEV ) ∈ (Sh)

2 × P0(Th)

of (8) is computed on a structured mesh, refined near the backward facing step and made of 11 600 triangles. Fig. 5presents the length of the zone of recirculation as a function of ω. From this figure we can conclude that the schemeis less and less diffusive as the weight ω diminishes. Since the centred scheme is less diffusive, it is therefore moreprecise, but its main drawback is its instability. Its numerical convergence is sometimes difficult and for our problem,we even have no convergence for weights ω < 0.4. The use of a weighted upstream scheme appears to be particularlyinteresting in our case since it is a good compromise between the centred and upstream schemes. For ω = 0.4, Fig. 6


Fig. 6. Isolines of pressure (top) and velocity field (bottom) for the discrete solution in the domain Ω for ω = 0.4.

Fig. 7. The velocity field near the backward facing step for ω = 0.4.

presents the isolines of pressure and the velocity field on the whole domain, while Fig. 7 gives a zoom of the velocityfield near the backward facing step.

8. Conclusion

In this paper we have introduced a new finite volume element method of weighted upstream type for the approxi-mation of the incompressible stationary Navier–Stokes system in dimension two. This method is a generalization ofthe methods presented in [15,23,24,12,13]. We prove existence and uniqueness results as well as an error estimate inthe energy norm, under the standard assumption that the datum has to be sufficiently small with respect to the viscosityparameter. Some numerical tests illustrate the obtained theoretical error estimate and further underline the usefulnessof the introduced scheme.

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