Convergence of a finite volume scheme for the incompressible fluids
Untitled
Numerische Mathematik manuscript No.
(will be inserted by the editor)
Sébastien Zimmermann
Convergence of a finite volume scheme
for the incompressible fluids
Received: date / Revised: date
Abstract We consider a finite volume scheme for the two-dimensional in-
compressible Navier-Stokes equations. We use a triangular mesh. The un-
knowns for the velocity and pressure are respectively piecewise constant and
affine. We use a projection method to deal with the incompressibility con-
straint. The stability of the scheme has been proven in [15]. We infer from it
its convergence.
Mathematics Subject Classification (2000) Incompressible fluids ·
Navier-Stokes equations · projection methods · finite volume
1 Introduction
We consider the flow of an incompressible fluid in a open bounded polyhedral
set Ω ⊂ R2 during the time interval [0, T ]. The velocity field u : Ω× [0, T ] →
2 and the pressure field p : Ω × [0, T ] → R satisfy the Navier-Stokes equa-
tions
∆u+ (u ·∇)u+∇p = f , (1)
div u = 0 , (2)
with the boundary and initial condition
u|∂Ω = 0 , u|t=0 = u0.
The terms ∆u and (u ·∇)u are associated with the physical phenomena of
diffusion and convection, respectively. The Reynolds number Re measures the
S. Zimmermann
17 rue Barrème, 69006 Lyon - FRANCE
Tel.: (+33)0472820337
E-mail: Sebastien.Zimmermann@ec-lyon.fr
http://arxiv.org/abs/0704.0787v1
2 Sébastien Zimmermann
influence of convection in the flow. For equations (1)–(2), finite element and
finite difference methods are well known and mathematical studies are avail-
able (see [9] for example). For finite volume schemes, numerous computations
have been conducted ([12] and [1] for example). However, few mathematical
results are available in this case. Let us cite Eymard and Herbin [5] and
Eymard, Latché and Herbin [6]. In order to deal with the incompress-
ibility constraint, these works use a penalization method. Another way is
to use the projection methods which have been introduced by Chorin [4]
and Temam [13]. This is the case in Faure [8] where the mesh is made of
squares. In Zimmermann [14] the mesh is made of triangles, which allows
more complex geometries. In the present paper the mesh is also made of tri-
angles, but we consider a different discretisation for the pressure. It leads to
a linear system with a better-conditioned matrix. The layout of the article is
the following. We first introduce (section 2.1) some notations and hypotheses
on the mesh. We define (section 2.2) the spaces we use to approximate the
velocity and pressure. We define also (section 2.3) the operators we use to
approximate the differential operators in (1)–(2). By combining this with a
projection method, we build the scheme in section 3. In order to provide a
mathematical analysis, we state in section 4 that the differential operators in
(1)–(2) and their discrete counterparts share similar properties. In particular,
the discrete operators for the gradient and the divergence are adjoint. We
then prove in section 5 the convergence of the scheme.
We conclude with some notations. We denote by χI the characteristic func-
tion of an interval I ⊂ R. We denote by C∞0 = C∞0 (Ω) the set of the functions
with a compact support in Ω. The spaces (L2, |.|) and (L∞, ‖.‖∞) are the
usual Lebesgue spaces and we set L20 = {q ∈ L2 ;
q dx = 0}. Their vectorial
counterparts are (L2, |.|) and (L∞, ‖.‖∞) with L2 = (L2)2 and L∞ = (L∞).
For k ∈ N∗, (Hk, ‖ · ‖k) is the usual Sobolev space. Its vectorial counterpart
is (Hk, ‖.‖k) with Hk = (Hk)2. For k = 1, the functions of H1 with a null
trace on the boundary form the space H10. Also, we set ∇u = (∇u1,∇u2)T
if u = (u1, u2) ∈ H1. If X ⊂ L2 is a Banach space, we define C(0, T ;X)
(resp. L2(0, T ;X)) as the set of the applications g : [0, T ] → X such that
t → |g(t)| is continous (resp. square integrable). The norm ‖.‖C(0,T ;X) is
defined by ‖g‖C(0,T ;X) = sups∈[0,T ] |g(s)|. Finally in all calculations, C is a
generic positive constant, depending only on Ω, u0 and f .
2 Discrete setting
First, we introduce the spaces and operators needed to build the mesh.
2.1 The mesh
Let Th be a triangular mesh of Ω: Ω = ∪K∈ThK. For each triangle K ∈ Th,
we denote by |K| its area and EK the set of his edges. If σ ∈ EK , nK,σ is the
unit vector normal to σ pointing outwards of K.
The set of edges of the mesh is Eh = ∪K∈ThEK . The length of an edge σ ∈ Eh
is |σ|. The set of edges inside Ω (resp. on the boundary) is E inth (resp. Eexth ):
Convergence of a finite volume scheme for the incompressible fluids 3
Eh = E inth ∪ Eexth . If σ ∈ E inth , Kσ and Lσ are the triangles sharing σ as an
edge. If σ ∈ Eexth , only the triangle Kσ inside Ω is defined.
We denote by xK the circumcenter of a triangle K. We assume that the
measure of all interior angles of the triangles of the mesh are below π
that xK ∈ K. If σ ∈ E inth (resp. σ ∈ Eexth ) we set dσ = d(xKσ ,xLσ ) (resp.
dσ = d(xσ,xKσ )). We define for all edge σ ∈ Eh: τσ =
. The maximum
circumradius of the triangles of the mesh is h. We assume that there exists
C > 0 such that
∀σ ∈ Eh, d(xKσ , σ) ≥ C|σ| and |σ| ≥ Ch.
It implies that there exists a constant C > 0 such that for all edge σ ∈ Eh
τσ ≥ C (3)
and for all triangles K ∈ Th we have (with σ ∈ EK and hK,σ the matching
altitude)
|K| = 1
|σ|hK,σ ≥
|σ| d(xK ,xσ) ≥ C h2. (4)
2.2 The discrete spaces
We first define
P0 = {q ∈ L2 ; ∀K ∈ Th, q|K is a constant} , P0 = (P0)2.
For the sake of concision, we set for all qh ∈ P0 (resp. vh ∈ P0) and all
triangle K ∈ Th: qK = qh|K (resp. vK = vh|K). Although P0 6⊂ H1, we
define the discrete equivalent of a H1 norm as follow. For all vh ∈ P0 we set
‖vh‖h =
σ∈Eint
τσ |vLσ − vKσ |2 +
σ∈Eext
τσ |vKσ |2
. (5)
We have [7] a discrete Poincaré inequality for P0: there exists C > 0 such
that for all vh ∈ P0
|vh| ≤ C ‖vh‖h. (6)
From the norm ‖.‖h we deduce a dual norm. For all vh ∈ P0 we set
‖vh‖−1,h = sup
(vh,ψh)
‖ψh‖h
. (7)
For all uh ∈ P0 and vh ∈ P0 we have (uh,vh) ≤ ‖uh‖−1,h ‖vh‖h. We
define the projection operator ΠP0 : L
2 → P0 as follows. For all w ∈ L2,
ΠP0w ∈ P0 is given by
∀K ∈ Th , (ΠP0w)|K =
w(x) dx. (8)
4 Sébastien Zimmermann
We easily check that for all w ∈ L2 and vh ∈ P0 we have (ΠP0w,vh) =
(w,vh). We deduce from this that ΠP0 is stable for the L
2 norm. We define
also the operator Π̃P0 : H
2 → P0. For all v ∈ H2, Π̃P0v ∈ P0 is given by
∀K ∈ Th , Π̃P0v|K = v(xK).
According to the Sobolev embedding theorem, v ∈ H2 is a.e. equal to a
continuous function. Therefore the definition above makes sense. One checks
[14] that there exists C > 0 such that
|v − Π̃P0v| ≤ C h ‖v‖2 (9)
for all v ∈ H1. We introduce also the finite element spaces
1 = {v ∈ L2 ; ∀K ∈ Th, v|K is affine} ,
Pnc1 = {vh ∈ P d1 ; ∀σ ∈ E inth , vh|Kσ (xσ) = vh|Lσ(xσ) , Pnc1 = (Pnc1 )2.
If qh ∈ Pnc1 , we have usually ∇qh 6∈ L2. Therefore we define the operator
∇h : Pnc1 → P0 by setting for all qh ∈ P0 and all triangle K ∈ Th
∇hqh|K =
∇qh dx. (10)
We define the projection operator ΠPnc
. For all q ∈ H1, ΠPnc
q is given by
∀σ ∈ Eh ,
(ΠPnc
q) dσ =
q dσ. (11)
We also set ΠPnc
= (ΠPnc
)2. One checks that there exists C > 0 such that
∣∣∇q −∇h(ΠPnc
∣∣ ≤ C h ‖q‖2 , |v −ΠPnc
v| ≤ C h ‖v‖1 , (12)
for all q ∈ H1 and v ∈ H1.
We also use the Raviart-Thomas spaces
= {vh ∈ Pd1 ; ∀σ ∈ EK , vh|K · nK,σ is constant, and vh · n|∂Ω = 0} ,
RT0 = {vh ∈ RTd0 ; ∀K ∈ Th, ∀σ ∈ EK , vh|Kσ · nKσ,σ = vh|Lσ · nKσ,σ}.
For all vh ∈ RT0, K ∈ Th and σ ∈ EK we set (vh · nK,σ)σ = vh|K · nK,σ.
We define the operator ΠRT0 : H
1 → RT0. For all v ∈ H1, ΠRT0v ∈ RT0
is given by
∀K ∈ Th , ∀σ ∈ EK , (ΠRT0v · nK,σ)σ =
v dσ. (13)
One checks [3] that there exists a constant C > 0 such that for all v ∈ H1
|v −ΠRT0v| ≤ C h ‖v‖1. (14)
Convergence of a finite volume scheme for the incompressible fluids 5
2.3 The discrete operators
The equations (1)–(2) use the differential operators gradient, divergence and
laplacian. Using the spaces of section 2.2, we now define their discrete coun-
terparts. The discrete gradient ∇h : Pnc1 → P0 is defined by (10). The
discrete divergence operator divh : P0 → Pnc1 is built so that it is adjoint
to the operator ∇h (proposition 3 below). We set for all vh ∈ P0 and all
triangle K ∈ Th
∀σ ∈ E inth , (divh vh)(xσ) =
3 |σ|
|Kσ|+ |Lσ|
(vLσ − vKσ ) · nK,σ ;
∀σ ∈ Eexth , (divh vh)(xσ) = −
3 |σ|
|Kσ|+ |Lσ|
vKσ · nK,σ. (15)
The first discrete laplacian ∆h : P
1 → Pnc1 is given by
∀ qh ∈ Pnc1 , ∆hqh = divh(∇hqh).
The second discrete laplacian ∆̃h : P0 → P0 is the usual operator in finite
volume schemes [7]. We set for all vh ∈ P0 and all triangle K ∈ Th
∆̃hvh|K =
σ∈EK∩E
τσ (vLσ − vKσ)−
σ∈EK∩E
τσ vKσ . (16)
In order to approximate the convection term (u · ∇)u in (1), we define a
bilinear form b̃h : RT0 ×P0 → P0 using the well-known [7] upwind scheme.
For all uh ∈ P0, vh ∈ P0, and all triangle K ∈ Th we set
b̃h(uh,vh)
σ∈EK∩E
(u · nK,σ)+σ vK + (u · nK,σ)−σ vLσ
We have set a+ = max(a, 0), a− = min(a, 0) for all a ∈ R. Lastly, we define
the trilinear form bh : RT0 × P0 × P0 → R as follows. For all uh ∈ RT0,
vh ∈ P0, wh ∈ P0, we set
bh(uh,vh,wh) =
|K|wK · b̃h(uh,vh)
3 The scheme
We have defined in section 2 the discretization in space. We now have to
define the discretization in time, and treat the incompressibility constraint
(2). We use a projection method to this end. This kind of method has been
introduced by Chorin [4] and Temam [13]. The time interval [0, T ] is split
with a time step k: [0, T ] =
n=0[tn, tn+1] with N ∈ N∗ et tn = n k for all
n ∈ {0, . . . , N}. We start with the initial values
u0h ∈ P0 ∩RT0 , u1h ∈ P0 ∩RT0 , p1h ∈ Pnc1 ∩ L20.
For all n ∈ {1, . . . , N}, (ũn+1h , p
h ) is deduced from (ũ
h) as
follows.
6 Sébastien Zimmermann
– ũn+1h ∈ P0 is given by
3 ũn+1h − 4unh + u
∆̃hũ
+ b̃h(2u
h − un−1h , ũ
h ) +∇hp
h = f
h , (17)
– pn+1h ∈ Pnc1 ∩ L20 is the solution of
h − p
divh ũ
– un+1h ∈ P0 is given by
un+1h = ũ
∇h(pn+1h − p
h). (18)
We have proven in [14] that the scheme is well defined. In particular the term
b̃h(2u
h − u
h ) in (17) is defined thanks to the following result.
Proposition 1 For m ∈ {0, . . . , N} we have umh ∈ RT0.
Note also that for m ∈ {0, . . . , N} we have divumh = 0, since umh ∈ P0. Thus
the incompressibility condition (2) is fullfilled.
4 Properties of the discrete operators
The operators defined in section 2.3 have the following properties [14].
Proposition 2 There exists a constant C > 0 such that for all uh ∈ RT0
satisfying divuh = 0, vh ∈ P0, wh ∈ P0:
|bh(uh,vh,vh)| ≤ C |uh| ‖vh‖h ‖wh‖h.
Proposition 3 For all vh ∈ P0 and qh ∈ Pnc1 : (vh,∇hqh) = −(qh, divh vh).
Proposition 4 For all uh ∈ P0 and vh ∈ P0: −(∆̃huh,uh) = ‖uh‖2h and
−(∆̃huh,vh) ≤ ‖uh‖h ‖vh‖h.
If v ∈ H1 we have |divv| ≤ ‖v‖1. The operator divh has a similar property.
Proposition 5 There exists a constant C > 0 such that for all vh ∈ P0
|divh vh| ≤ C ‖vh‖h.
Proof. Using a quadrature formula we have
|divh vh|2 =
|(divh vh)(xσ)|2 .
Convergence of a finite volume scheme for the incompressible fluids 7
Let K ∈ Th. Using definition and (4) we have
∀σ ∈ EK ∩ E inth , |(divh vh)(xσ)|
|K| |vLσ − vK |
∀σ ∈ EK ∩ Eexth , |(divh vh)(xσ)|
2 ≤ C
|vK |2.
Thus: |divh vh|2 ≤ C
σ∈EK∩E
|vLσ − vK |2 +
σ∈EK∩E
|vK |2
Writing the sum over the triangles as a sum over the edges, we get
|divh vh|2 ≤ C
σ∈Eint
τσ |vLσ − vK |2 +
σ∈Eext
τσ |vK |2
 ≤ C ‖vh‖2h.
Proposition 6 If uh ∈ P0 and vh ∈ P0 we have (∆̃huh,vh) = (uh, ∆̃hvh).
Proof. Using definition (2.3) one checks that
(∆̃huh,vh) =
σ∈Eint
τσ (vLσ − vKσ ) · (uLσ − uKσ)−
σ∈Eext
τσ vKσ · uKσ
= (∆̃hvh,uh).
Proposition 7 There exists C > 0 such that for all v ∈ H2 satisfying
∇v · n|∂Ω = 0
‖ΠP0(∆v)−∆h(Π̃P0v)‖−1,h ≤ C h ‖v‖2.
Proof. Let ψh ∈ P0. We have
ΠP0(∆q)−∆h(Π̃P0v),ψh
ΠP0(∆q)−∆h(Π̃P0v)
·ψK .
For all K ∈ Th, using (2.3) and the divergence formula, we get
ΠP0(∆v) −∆h(Π̃P0v)
) ∣∣∣
σ∈EK∩E
∇v · nK,σ dσ − τσ
v(xLσ )− v(xK)
Thus, by writing the sum over the triangles as a sum over the edges, we get
ΠP0(∆v)−∆h(Π̃P0v),ψh
σ∈Eint
(ψLσ −ψKσ)Rσ
with Rσ =
∇v · nKσ,σ − 1dσ
v(xLσ ) − v(xKσ )
dσ. We denote by Dσ
the quadrilatere defined by xKσ , xLσ and the endpoints of σ. Using a Taylor
expansion and a density argument, we get as in [7] that
|Rσ| ≤ C h
|H(vi)(y)|2 dy
8 Sébastien Zimmermann
Thus, using the Cauchy-Schwarz inequality, we get
ΠP0(∆q)−∆h(Π̃P0v),ψh
≤ C h
σ∈Eint
|ψLσ −ψKσ |
σ∈Eint
|H(vi)(y)|2 dy
According to (3)
σ∈Eint
|ψLσ −ψKσ |
2 ≤ C
σ∈Eint
τσ |ψLσ −ψKσ |
2 ≤ C ‖ψ‖2h.
ΠP0(∆v)−∆h(Π̃P0v),ψh
)∣∣∣ ≤ C h ‖ψh‖h ‖v‖2. Using (7) we get
the result.
5 Convergence of the scheme
We first recall the stability result that has been proven in [15]. We deduce
from it an estimate on the Fourier transform of the computed velocity (lemma
1). Using a result on space P0, we infer from it the convergence of the
scheme (theorem 2). One shows that if the data u0 et f fulfill a compati-
bility condition [11], there exists a solution (u, p) to equations (1)–(2) such
that u ∈ C(0, T ;H2) , ∇p ∈ C(0, T ;L2). We assume from now on that there
exists C > 0 such that
(HI) |u0h−u0|+
‖u1h−u(t1)‖∞+|p1h−p(t1)| ≤ C h , |u1h−u0h| ≤ C k.
Let us recall the following result [15].
Theorem 1 We assume that the initial values of the scheme fulfill (HI).
There exists a constant C > 0 such that for all m ∈ {2, . . . , N}
|umh |+ k
‖ũnh‖2h +
|umh − um−1h |+
‖ũnh − ũn−1h ‖
h ≤ C
|pnh|2 + k |∇hpmh |+ |∇h(pmh − pm−1h )| ≤ C.
From now on we set ũ1h = u
h for the sake of conveniance. One deduces from
hypothesis (HI) [14] that |p1h| ≤ C and ‖ũ1h‖h ≤ C. Now, let ε = max(h, k).
We study the behaviour of the scheme as ε → 0. We define the applications
uε : R → P0 , ũε : R → P0 , ũcε : R → P0, pε : R → Pnc1 and fε : R → P0 as
follows. For all n ∈ {0, . . . , N − 1} and all t ∈ [tn, tn+1] we set
uε(t) = u
h , ũε(t) = ũ
h , ũ
ε(t) = ũ
(t− tn) (ũn+1h − ũ
pε(t) = p
h, fε(t) = ΠP0 f(tn+1) ,
Convergence of a finite volume scheme for the incompressible fluids 9
and for all t 6∈ [0, T ] we set uε(t) = ũε(t) = ũcε(t) = fε(t) = 0, pε(t) = 0. We
recall that the Fourier transform v̂ of a function v ∈ L1(R) is defined by
∀ τ ∈ R, v̂(τ) =
e−2iπτt v(t) dt. (19)
We have the following result.
Lemma 1 Let 0 < γ < 1
. There exists C > 0 such that for all ε > 0
|τ |2γ |̂̃uε(τ)|2 dτ ≤ C.
Proof. Let χI be the characteristic function of an interval I ⊂ R. We define
the application gε : R → P0 as follows. For all t 6∈ [t1, T ] we set gε(t) = 0.
For all t ∈ [t1, T ], gε(t) ∈ P0 is the solution of
∆̃hgε = ∆̃hũε + fε − b̃h
2uε − uε(t− k), ũε
with Pε = −∇hpε− 2 ũε(t−k)−uεk +
ũε(t−2 k)−uε(t−k)
χ[t2,T ]. We have omitted
most of the time dependancies for the sake of concision. Let us estimate gε.
We have
−(∆̃hgε,gε) = −(∆̃hũε,gε)− (fε,gε) (20)
2uε − uε(t− k), ũε,gε
+ (Pε,gε).
According to proposition 4 we have
−(∆̃hgε,gε) = ‖gε‖2h , −(∆̃hũε,gε) ≤ ‖ũε‖h ‖gε‖h.
Using the Cauchy-Schwarz inequality and (6) we have
−(fε,gε) ≤ |fε| |gε| ≤ C |fε| ‖gε‖h.
According to proposition 2 and theorem 1
2uε − uε(t− k), ũε,gε
≤ C |2uε − uε(t− k)| ‖ũε‖h ‖gε‖h ≤ C ‖ũε‖h ‖gε‖h.
Using (18) we have
Pε = −
χ[t3,T ]
∇hpε+
χ[t3,T ]∇hpε(t−k)−
χ[t3,T ] ∇hpε(t−2 k).
Using proposition 3 and the Cauchy-Schwarz inequality we get
|(Pε,gε)| ≤ C
|pε|+ χ[t3,T ] |pε(t− k)|+ χ[t3,T ] |pε(t− 2 k)|
|divh gε|.
Using proposition 5 we have
|(Pε,gε)| ≤ C
|pε|+ χ[t3,T ] |pε(t− k)|+ χ[t3,T ] |pε(t− 2 k)|
‖gε‖h.
10 Sébastien Zimmermann
Let us plug these estimates into (20). By simplifying by ‖gε‖h and integrating
from t = t1 to T we get
‖gε‖h dt ≤ C + C
|pε| dt+
|fε| dt+
‖ũε‖h dt
According to the Cauchy-Schwarz inequality and theorem 1
|pε(t)| dt ≤
|pε(t)|2 dt
|pnh|2
Thanks to the stability of ΠP0 for the L
2 norm we have
|fε(t)| dt = k
|ΠP0 f(tn)| ≤ k
|f(tn)| ≤ k
‖f‖C(0,T ;L2) ≤ C.
And thanks to the Cauchy-Schwarz inequality and theorem 1
‖ũε(t)‖h dt ≤
‖ũε(t)‖2h dt ≤ C k
‖ũnh‖2h ≤ C.
Thus, since gε(t) = 0 for t ∈ [0, t1], we get
‖gε(t)‖h =
‖gε(t)‖h dt ≤ C.
Using definition (19) we obtain finally
∀ τ ∈ R , ‖ĝε(τ)‖h ≤ C. (21)
With this estimate we can now prove the result. Since the function ũcε is
piecewise C1 on R, and discontinous for t = 0 and t = T , equation (17) reads
dũcε
dũcε
(t− k) = ∆hgε +
(ũ0h δ0 − ũNh δT )−
(ũ1h δt1 − ũNh δT+k)
ũ1h − ũ0h
χ[0,t1] −
ũNh − ũ
χ[T,T+k]
where δ0, δt1 , δT and δT+k are Dirac distributions located respectively in 0,
t1, T and T + k. Let τ ∈ R. Applying the Fourier transform we get
−2iπτ
−2iπτk
̂̃uε(τ) = ∆hĝε(τ) +α
(ũ0h − ũNh e−2iπT )−
(ũ1h − ũNh e−2iπT )
e−2iπk − 1
ũ1h − ũ0h
ũNh − ũ
−2iπT
e−2iπk − 1
Convergence of a finite volume scheme for the incompressible fluids 11
Taking the scalar product with i ̂̃uε(τ) we get
e−2iπτk
|̂̃uε(τ)|2 = i
∆hĝε(τ), ̂̃uε(τ)
α, ̂̃uε(τ)
Let us bound the right-hand side. According to proposition 4 and (21)
∆hĝε(τ), ̂̃uε(τ)
)∣∣∣ ≤ ‖ĝε(τ)‖h ‖̂̃uε(τ)‖h ≤ C ‖̂̃uε(τ)‖h.
On the other hand, using theorem 1, one checks that α is bounded. Thus,
according to the Cauchy-Schwarz inequality and (6)
α, ̂̃uε(τ)
)∣∣∣ ≤ |α| |̂̃uε(τ)| ≤ C |α| ‖̂̃uε(τ)‖h ≤ C ‖̂̃uε(τ)‖h.
Hence we have
∀ τ ∈ R, |τ | |̂̃uε(τ)|2 ≤ C ‖̂̃uε(τ)‖h.
If τ 6= 0, multiplying this estimate by |τ |2 γ−1, we get |τ |2 γ |̂̃uε(τ)|2 ≤
C |τ |2 γ−1 ‖̂̃uε(τ)‖h. Using the Young inequality and integrating over {τ ∈
R ; |τ | > 1} we obtain
|τ |>1
|τ |2 γ |̂̃uε(τ)|2 dτ ≤
|τ |>1
|τ |4 γ−2 dτ + C
|τ |>1
‖̂̃uε(τ)‖2h dτ.
For |τ | ≤ 1 we have |τ |2 γ |̂̃uε(τ)|2 ≤ |̂̃uε(τ)|2 ≤ C ‖̂̃uε(τ)‖2h thanks to (6).
|τ |2 γ |̂̃uε(τ)|2 dτ ≤
|τ |>1
|τ |4 γ−2 dτ + C
‖̂̃uε(τ)‖2h dτ.
Since 4 γ−2 < −1 we have
|τ |>1
|τ |4 γ−2 dτ ≤ C. On the other hand, thanks
to the Parseval theorem and thorem 1
‖̂̃uε(τ)‖2h dτ ≤
‖̂̃uε(τ)‖2h dt ≤ k
‖ũnh‖2h ≤ C.
Hence the result.
We introduce the following spaces
H = {v ∈ L2 ; divv ∈ L2 et v · n|∂Ω = 0} , V = {v ∈ H10 ; divv = 0}.
We also set
((u,v)) =
(∇ui,∇vi) , b(u,v,w) = −
(vi,u · ∇wi)
for all u = (u1, u2) ∈ H1, v = (v1, v2) ∈ H1, w = (w1, w2) ∈ H1. We have
the following result.
12 Sébastien Zimmermann
Theorem 2 We assume that the initial values of the scheme fulfill hypothesis
(HI). We also assume that the space step h and the time step k are such
that h ≤ C kα with α > 1. Then we have uε → u in L2(0, T ;L2) with
u ∈ C(0, T ;H) ∩ L2(0, T ;V) ,
∈ L2(0, T ;L2). (22)
We also have u(0) = u0 and for all ψ ∈ C∞0 ([0, T ])
∀v ∈ V,
(u,v) + ((u,v)) + b(u,u,v) − (f ,v)
dt = 0. (23)
Proof. In what follows, sub-sequences of a sequence (vε)ε>0 will still be
noted (vε)ε>0 for the sake of convenience. All the limits are for ε → 0.
According to theorem 1 and hypothesis (HI) we have
‖uε‖2L2(0,T ;L2) = k (|u
h|2 + |u1h|2) + k
|unh|2 ≤ C.
We also deduce from (6), hypothesis (HI) and theorem 1
‖ũε‖2L2(0,T ;L2) = k |u
h|2 + k
|ũnh|2 ≤ C + C k
‖ũnh‖2h ≤ C.
A simple computation shows that there exists C > 0 such that
‖ũcε‖L2(0,T ;L2) ≤ C ‖ũε‖L2(0,T ;L2) ≤ C.
Thus the sequences (uε)ε>0, (ũε)ε>0 and (ũ
ε)ε>0 are bounded in L
2(0, T ;L2).
Therefore there exists u ∈ L2(0, T ;L2), ũ ∈ L2(0, T ;L2) and ũc ∈ L2(0, T ;L2)
such that, up to a sub-sequence, we have
uε ⇀ u , ũε ⇀ ũ , ũ
ε ⇀ ũ
c weakly in L2(0, T ;L2).
We claim that the limits u, ũ, ũc are the same. Indeed, let us consider uε−ũε.
Since unh − ũ
h = (u
h − u
h ) + (u
h − ũ
h ) we have
‖uε − ũε‖2L2(0,T ;L2) ≤ 2 k
|unh − un+1h |
2 + 2 k
|un+1h − ũ
According to theorem 1 we have k
n=0 |unh−u
h |2 ≤ C
n=0 k
3 ≤ C k2.
Thanks to (18) we also have
|un+1h − ũ
|∇h(pn+1h − p
h)|2 ≤ C
3 ≤ C k2.
Thus ‖uε − ũε‖L2(0,T ;L2) → 0 and u = ũ. One checks in a simililar way that
ũ = ũc. Now, using the Fourier transform, we prove the strong convergence
Convergence of a finite volume scheme for the incompressible fluids 13
of the sequence (uε)ε>0 in L
2(0, T ;L2). We set vε = uε −u. Let M > 0. We
use the splitting
|v̂ε(τ)|2 dτ =
|τ |≤M
|v̂ε(τ)|2 dτ +
|τ |>M
|v̂ε(τ)|2 dτ = IMε + JMε .
Let us estimate JMε . Since |v̂ε(τ)|2 ≤ 2 |ûε(τ)|2 + 2 |û(τ)|2 we have
JMε ≤ 2
|τ |>M
|ûε(τ)|2 dτ + 2
|τ |>M
|û(τ)|2 dτ.
According to lemma 1 we have
|τ |>M
|ûε(τ)|2 dτ ≤
|τ |>M
|τ |2γ |ûε(τ)|2 dτ ≤
|τ |>M
|û(τ)|2 dτ.
Therefore, for all ε > 0, we have JMε → 0 when M → ∞. We now consider
IMε . Let τ ∈ R. Since uε ⇀ u in L2(0, T ;L2), we deduce from definition (19)
̂̃uε(τ)⇀ û(τ) weakly in L2.
For all t ∈ R we have ũε(t) ∈ P0. From definition (19) we infer that ̂̃uε(τ) ∈
P0. Now, prolonging ̂̃uε(τ) by 0 outside Ω, we deduce from lemma 4 in [7]
that there exists a constant C > 0 such that
∀η∈ R2 , |̂̃uε(τ)(· + η)− ̂̃uε(τ)|2 ≤ ‖̂̃uε(τ)‖2h |η| (|η|+ C h).
Using definition (19), the Cauchy-Schwarz inequality and theorem 1, we have
‖̂̃uε(τ)‖2h ≤ C
‖ũε(t)‖2h dt ≤ C k
‖ũnh‖2h ≤ C.
Thus, using the compactness criterium given by theorem 1 in [7], we get
̂̃uε(τ) → û(τ) in L2. Thus ̂̃vε(τ) = ̂̃uε(τ)− û(τ) → 0 in L2. Therefore for all
M > 0 we have IMε → 0. Using the Parseval inequality, and gathering the
limits for IMε and J
ε , we get
|v̂ε(τ)|2 dτ =
|vε|2 dt =
|uε − u|2 dτ → 0.
We have proven that uε → u in L2(0, T ;L2).
We now check the properties of u. First, proceeding as in [7], one checks
easily that u ∈ L2(0, T ;H10). Now let q ∈ L2(0, T ; C∞0 ). According to (12) we
have ∇h(ΠPnc
q) → ∇q in L2(0, T ;L2). Since uε → u in L2(0, T ;L2) we get
∇h(ΠPnc
q),uε
→ (∇q,u) = −(q, divu).
14 Sébastien Zimmermann
On the other hand, according to propositions 1 and 3, we have for all ε > 0
∇h(ΠPnc
q),uε
= −(ΠPnc
q, divh uε) = 0.
Thus we have
q divu dt = 0 for all q ∈ L2(0, T ; C∞0 ). Since the space C∞0
is dense in L2, we get divu = 0. Hence u ∈ L2(0, T ;V). Let us now check
the regularity of
. Using hypothesis (HI), (6) and theorem 1, we have
dũcε
L2(0,T ;L2)
|u1h − u0h|2 + k
|δũnh|2
≤ C + C k
‖δũnh‖2h
Thus the sequence
dũcε
is bounded in L2(0, T ;L2). Since uε → u in
L2(0, T ;L2) with u ∈ L2(0, T ;H1), proceeding as in, we get
dũcε
weakly ∈ L2(0, T ;L2)
and u ∈ C(0, T ;H).
Let us now prove that u satisfies (23). For the sake of simplicity, we omit to
note some time dependencies. According to (17) we have for all t ∈ [t1, T ]
dũcε
dũcε
(t− k)− 1
∆hũε + b̃h
2uε − uε(t− k), ũε
χ[t3,T ]
∇hpε +
χ[t3,T ] ∇hpε(t− k)−
χ[t3,T ] ∇hpε(t− 2 k).
Let v ∈ V ∩ (C∞0 )2 and ψ ∈ C∞([0, T ]) with ψ(T ) = 0. We set vh = Π̃P0v.
Multiplying the former equation by ψ vh and integrating over [t1, T ] we get
dũcε
dũcε
(t− k),vh
dt− 1
ψ (∆̃hũε,vh) dt
2uε − uε(t− k), ũε,vh
ψ (fε,vh) dt
χψ (∇hpε,vh) dt (24)
with χ = −χ[t3,T−2 k] + 13 χ[t2,t3] −
χ[t1,t2] − 73 χ[T−k,T ] −
χ[T−2 k,T−k]. We
now check the limits of the terms in this equation. First, according to (9), we
have vh → v in L2. We will use this limit in the computations below without
mentioning it. Since ψ(T ) = 0 we obtain by integrating by parts
dũcε
dt− ψ(t1) (ũ1h,vh)−
ψ′ (ũcε,vh) dt
dũcε
(t− k),vh
dt = −ψ(0) (ũ0h,vh)−
∫ T−k
′(t+ k) (ũcε,vh) dt.
Convergence of a finite volume scheme for the incompressible fluids 15
According to hypothesis (HI) we have ũ0h = u
h → u0 in L2 and ũ1h =
u1h → u0 in L2. It implies that (u0h,vh) → (u0,v) and ψ(t1) (ũ1h,vh) =
ψ(k) (ũ1h,vh) → ψ(0) (u0,v). On the other hand
ψ′ (ũcε,vh) dt =
χ[t1,T ] ψ
′ (ũcε,vh) dt→
ψ′ (u,v) dt
and since χ[0,T−k] ψ
′(·+ k) → ψ′ in L∞(0, T )
∫ T−k
′(t+k) (ũcε,vh) dt =
χ[0,T−k] ψ
′(t+k) (ũcε,vh) dt→
′ (u,v) dt.
Thus we have
dũcε
dũcε
(t− k),vh
dt→ −ψ(0) (u,v)−
ψ′ (u,v) dt. (25)
Let us now consider the discrete laplacian. Using proposition 6 and the split-
ting ∆̃hvh =
∆̃hvh −ΠP0(∆̃v)
+ΠP0(∆̃v) we have
∆̃hũε,vh) dt = Aε +Bε
with Aε =
ũε, ∆̃h(Π̃P0v)−ΠP0(∆̃v)
dt,Bε =
ũε, ΠP0(∆v)
Since
|Aε| ≤ ‖∆̃h(Π̃P0v) −ΠP0(∆̃v)‖−1,h
ψ ‖ũε‖h dt ,
using proposition 7 and the Cauchy-Schwarz inequality, we get
|Aε| ≤ C h ‖v‖2
ψ ‖ũε‖h dt ≤ C h
ψ2 dt
)1/2(∫ T
‖ũε‖2h dt
Therefore, using theorem 1: |Aε| ≤ C h
n=1 ‖ũnh‖2h
≤ C h. Hence
Aε → 0. On the other hand, using an integration by parts, we have
ψ (ũε, ∆v) dt →
ψ (u, ∆v) dt = −
ψ ((u,v)) dt.
By gathering the limits for Aε and Bε we get
ψ (∆̃hũε,vh) dt → −
ψ ((u,v)) dt.
Let us now consider the pressure. We use the splitting
(∇hpε,vh) = (∇hpε,vh − v) + (∇hpε,v −ΠRT0v) + (∇hpε, ΠRT0v). (26)
16 Sébastien Zimmermann
First, integrating by parts, we have
(∇hpε, ΠRT0v) = −
pε, div (ΠRT0v)
pε (ΠRT0v · nK,σ).
Since divv = 0, using the divergence formula and definition (13), one checks
that div (ΠRT0v) = 0. Thus −
pε, div (ΠRT0v)
= 0. On the other hand
pε (ΠRT0v ·nK,σ) =
σ∈Eint
(ΠRT0v)σ ·nKσ ,σ)
(pε|Lσ −pε|Kσ ) dσ
and since pε ∈ Pnc1 we get
pε (ΠRT0v · nK,σ) = 0. Thus the last
term in (26) vanishes. To bound the other terms, we use the Cauchy-Schwartz
inequality together with estimates (9), (14) and theorem 1. We get
|(∇hpε,v − vh)|+ |(∇hpε,v −ΠRT0v)| ≤ C h |∇hpε| ‖v‖2 ≤ C
‖v‖2.
Plugging these estimates into (26) we get
|χψ (∇hpε,vh)| dt ≤ C hk . By
hypothesis we have h
≤ kα−1 with α− 1 > 0. Thus for ε = max(h, k) → 0
χψ (∇hpε,vh) dt→ 0.
Let us now consider the convection term. We set uε = 2uε − uε(t − k)
and want to find the limit of
ψ bh(uε, ũε,vh) dt. We use the splitting
−bh(uε, ũε,vh) + b(u,u,v) = Aε1 +Aε2 +Aε3 with
Aε1 = b(u− uε,u,v) , Aε2 = b(uε,u,v) −
div(ui uε), v
Aε3 =
div(ui uε), v
− bh(uε,uε,vh).
By definition Aε1 = −
ui, (u − uε) · ∇vi
. Using the Cauchy-Schwarz
inequality we get
ψ |Aε1| dt ≤ ‖ψ‖∞ ‖v‖W1,∞ ‖u‖L2(0,T ;L2) ‖u− uε‖L2(0,T ;L2).
Since uε → u in L2(0, T ;L2) we also have ‖u − uε‖L2(0,T ;L2) → 0. Thus∫ T
ψAε1 dt → 0. Let us now consider Aε2. Since uε · n|∂Ω = 0 we obtain by
integrating by parts b(uε,u,v) =
vi, div(ui uε)
. Thus
Aε2 =
vi − vih, div(ui uε)
vi − vih,uε · ∇ui
Convergence of a finite volume scheme for the incompressible fluids 17
Using the Cauchy-Schwarz inequality we get
ψ |Aε2| dt ≤ C ‖ψ‖L∞ ‖v − vh‖L∞ ‖uε‖L2(0,T ;L2) ‖u‖L2(0,T ;H1).
Using a Taylor expansion one checks that ‖v − vh‖L∞ ≤ ‖v‖W1,∞ h. We
recall also that ‖uε‖L2(0,T ;L2) ≤ C. Therefore
ψAε2 dt → 0. Let us now
bound Aε3. For all triangle K ∈ Th and all edge σ ∈ EK ∩ E inth , we set
ũεK,Lσ =
ũε|K if (uε · nK,σ) ≥ 0
ũε|Lσ if (uε · nK,σ) < 0
Using the divergence formula one checks that
Aε3 =
σ∈EK∩E
(u− ũεK,Lσ ) (uε · nK,σ) dσ.
By writing this sum as a sum on the edges we get
Aε3 =
σ∈Eint
(vKσ − vLσ) ·
(u− ũεKσ ,Lσ) (uε · nKσ,σ) dσ.
Thus, using definition (11) and a quadrature formula
σ∈Eint
(vKσ − vLσ )
(ΠPnc
u− ũεKσ,Lσ) (uε · nKσ,σ) dσ
σ∈Eint
(vKσ − vLσ ) |σ|
(ΠPnc
u)(xσ)− ũεKσ,Lσ
(uε · nKσ,σ)σ.
We have |σ| ≤ h and, using a Taylor expansion, one checks that |vKσ−vLσ | ≤
h ‖v‖W1,∞ . Thus, thanks to the Cauchy-Schwarz inequality, we get
|Aε3| ≤ C h2
σ∈Eint
|uε(xσ)|2
σ∈Eint
|(ΠPnc
u)(xσ)− ũεKσ ,Lσ |
Using (4) we get
|Aε3| ≤ C
σ∈EK∩E
|uε(xσ)|2
σ∈EK∩E
|(ΠPnc
u)(xσ)− ũε|K |2
Therefore, using a quadrature formula
|Aε3| ≤ C |uε| |ΠPnc1 u− uε| ≤ C |uε| |ΠPnc1 u− uε|.
18 Sébastien Zimmermann
By writing ΠPnc
u − uε = (ΠPnc
u − u) + (u − uε) and using (12), we get
|Aε3| ≤ C |uε| (h ‖u‖1+ |u−uε|). Thus, using the Cauchy-Schwarz inequality,
we have
|ψ| |Aε3| dt ≤ C ‖uε‖L2(0,T ;L2) (h ‖u‖L2(0,T ;H1) + ‖u− uε‖L2(0,T ;L2)).
ψAε3 dt → 0. By gathering the limits for Aε1, Aε2, Aε3 we obtain∫ T
bh(uε,uε,vh)− b(u,u,v)
dt→ 0. Since
b(u,u,v) dt → 0, we get
bh(uε,uε,vh), dt →
ψ b(u,u,v) dt.
Finally, since vh ∈ P0, we have: (fε,vh) = (ΠP0 f ,vh) = (f ,vh). Therefore
ψ (fε,vh) dt =
χ[t1,T ] ψ (f ,vh) dt→
ψ (f ,v) dt.
We now gather the limits we have obtained into (24). The space V∩(C∞0 )2 is
dense in V. Hence we obtain for all v ∈ V and ψ ∈ C∞([0, T ]) with ψ(T ) = 0
−ψ(0) (u0,v)−
′ (u,v) dt+
((u,v)) +b(u,u,v)− (f ,v)
dt = 0.
Taking ψ = φ ∈ C∞0 ([0, T ]), we have φ(0) = 0 and from the definition of the
derivative in the distributional sense
φ′ (u,v) dt = −
(u,v) dt.
Thus we have proven (23). At last, let us show that the initial condition
holds. We have proven before that
dũcε
weakly in L2(0, T ;L2).
Let v ∈ V ∩ (C∞0 )2 and ψ ∈ C∞([0, T ]) such that ψ(T ) = 0. We have
dũcε
dũcε
(t− k),vh
Integrating by parts the limit we get
dũcε
dũcε
(t− k),vh
dt → −ψ(0)
u(0),v
ψ′ (u,v) dt.
By comparing this limit with (25), we get ψ(0) (u(0) − u0,v) = 0 for all
ψ ∈ C∞([0, T ]) with ψ(T ) = 0. Therefore u(0) = u0. At last, note that we
have proven so far the convergence of a sub-sequence of (uε)ε>0 towards u.
But the application u such that (22), (23) and u(0) = u0 hold is unique ([13],
p. 254). Thus the whole sequence (uε)ε>0 converges towards u.
Convergence of a finite volume scheme for the incompressible fluids 19
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