Maximum Principle and Error Estimate of the Upwind Finite …m-hikari.com/ams/ams-2012/ams-53-56-2012/aboodAMS53-56... · 2012. 3. 4. · total flow that enters a cell through its

Applied Mathematical Sciences, Vol. 6, 2012, no. 53, 2625 – 2647

Maximum Principle and Error Estimate of the

Upwind Finite Volumes Method for Nonlinear

Convection-Diffusion Equation

Hashim A. Kashkool and Shamma A. Kadhum

Mathematics Department, College of Education University of Basrah, Basrah, Iraq

[email protected] Abstract. In this paper we used the combined F.E.-F.V. method to solve an initial-boundary value problem, provided that the nonlinear convective term is approximated by U.F.V. method on the finite volume mesh dual to a triangular grid of weakly acute type and the diffusion term is approximated by standard Galerkin finite element method. We proved the

stability and the discrete maximum principle under the condition ,ˆ)(ˆ0 ii Mc Ω∂Ω≤≤τ Ji∈ .

Also proved the approximate solution is convergent with error of )(0 τ and the local

truncation with error of )(0 2τ .

Keywords: finite volumes, Maximum Principle, Error estimate, Convection-Diffusion problem 1. Introduction Convection-diffusion processes appear in many areas of science and technology, and environmental protection. These problems are peculiar in the sense that it is a combination of

2626 H. A. Kashkool and S. A. Kadhum two dissimilar phenomena, convection and diffusion. It is a well-known fact that the use of classical Galerkin method with continuous piecewise linear finite elements leads to spurious oscillations when the local Peclet number is large. This is the reason that the numerical solution of convection-diffusion problems attracts a number of specialists. From an extensive literature devoted to problems let us mention the papers [1], [6], [12], [11], [10], [3], and [5]. In the theory of weak solutions for partial differential equations in divergence from there are two roughly equivalent formulations in common use, namely, the functional formulation involving integration against smooth test functions versus the finite volume type over arbitrary control volumes. The former corresponds to energy methods and leads naturally to F.E discretizations for elliptic and parabolic, i.e., diffusive, problems. The latter corresponds in a natural way to the physical formulation of the basic laws of mass, momentum, and energy in fluid mechanics leading directly to the well-known FV methods. In [9] the work reported here we investigate the approach by trying to have the best of both worlds, i.e., the combination of finite volumes for in viscid conservation laws with finite elements for the diffusion. Computational fluid dynamics problems usually require discretization of the problem into a large number of cells/grid points (millions and more), therefore cost of the solution favors simpler, lower order approximation within each cell/grid point. This is especially true of " external flow " problems, like air flow around the car or airplane, or weather simulation in a large area. We view space as being broken down into a set of volumes each of which surrounds one point. In particular the volume associated with each point is the region of space that is closer to that point than to any other point. Now we consider how can be used these cells to solve fluid flow problem. The way to think about it is to consider the total flow that enters a cell through its boundaries. It is claimed that the net flow (sum of flow in and out) has to be zero. Intuitively this is because the fluid (or air) has nowhere to go. This condition can be written formally as ∫

Ω∂=

idsnu 0.rr

where iΩ∂ is the boundary of the thi cell, ur is the flow and nr

is the vector normal to the surface let us mention the papers [11], [10], and [9]. Our main goal is to develop a robust theoretically based numerical method for the solution of viscous compressible flow applied of unstructured meshes. In this paper we proposed numerical schemes for the solution of viscous gas flow based on the combination of the FV method for the discretization of inviscid convective terms and the FE method applied to the approximation of viscous terms. This paper consists

2. The Convection-Diffusion Equation We consider the two–dimensional nonlinear convection–diffusion equation [6].

Maximum principle and error estimate 2627

),()).(.( txfuubutu

=∇+Δ−∂∂ r

λ TQT =×Ω ),0( , (2.1)

,0),( =txu on Ω∂ , (2.2)

),()0,( xuxu o= on −

Ω (2.3)

where ),( 21 xxx = and 0>λ is a given constant, the vector 2R:)( →TQubr

is a convection coefficient, the functions R:),( →TQtxf ( ),( txff = for simplicity) and the function

R: →Ωou are given. The weak form is to find )(],0[: 10 Ω→ HTu , for simplicity we will

denote )(.,)( tutuu == :

),,())),()).((.(()),(()),(( vfvtutubvtuvtudtd

=∇+∇∇+r

λ )(10 Ω∈∀ Hv ,

0)0( uu = . (2.4)

Assumption 2.1. [11] a) ))(],,0[( ,1 Ω∈ qWTCf for some 2>q .

b) )(,10

0 Ω∈ pWu for some 2>p . c) a positive constants 1γ and 2γ such that

1)(

0 γ≤Ω∞L

u , 2)(γ≤∞

TQLf .

Now, we define the set { }MuuY ≤= : and the coefficients of equation (2.1) satisfy the

following conditions A (2.1) )()())(),(()( ,1,1

21 YWYWububub ∞∞ ×∈=v

.

A (2.2) )(ubv

is locally Lipschitz –continuous vuLvbub −≤− )()(rr

, Yvu ∈∀ ,

Where 10 << L is Lipchitz constant? A (2.3) ))(;,0())(;,0( 21

0 Ω∩Ω∈ ∞∞ HTLHTLu , ))(;,0( 2 Ω∈′ ∞ LTLu and );,0( ∗∞∈′′ VTLu ,

where u is the weak solution of equation (2.4) and ∗V is the dual space of V . Lemma 2.1. [5] Let kU and ku are the approximate solution and the exact solution Respectively, if ChuU

L

kk ≤−Ω)(2 , ( )Ttk ,0∈ then

,1)(2 CUL

k ≤∇Ω

where 1C is a constant independent of h and τ .

Lemma 2.2. [8] Let VH ∩Ω∈ )(2ϕ and ϕϕ hh p= such that \

2628 H. A. Kashkool and S. A. Kadhum

)(10 Ω

−Hhp ϕϕ ,

)(2 Ω≤

Hch ϕ ),0( ohh∈

where hp is the projection and the constant c is independent of ϕ and h .

Lemma 2.3. [10] Let VH ∩Ω∈ )(2ϕ and ϕϕ hh p= such that

)(2

2)(2 ΩΩ≤−

HLh chp ϕϕϕ ),0( ohh∈ ,

where hp is the projection and the constant c is independent of ϕ and h .

3. Finite Elements Triangulations [11] The F.E. method, in its simplest form, is a specific process of constructing subspaces VVh ⊂ , which are called F.E. spaces. The most characteristic is discretization such as triangulation

hT established over the closure Ω∂∪Ω=Ω−

of .2R⊂Ω Let hΩ be a polygonal approximation of the domain Ω , thus denotes a partition of Ω into disjoint triangles e such that the following properties are satisfied

i) ,Uh

hTee

∈

−

=Ω

ii) If ,, 21 hTee ∈ then o/=∩ 02

01 ee or 1e and 2e have a common

side or 1e and 2e have a common vertex,

iii) hP−

Ω∈ for any vertex P of each .hTe∈

Then the triangulation hT is called a basic mesh.

Remark 3.1[11]. We associate the index set };{ hipjiJ−

Ω∈≠= and { }hi

o

pJiJ Ω∈∈= ; be

the set of the indices of all interior vertices. Let { }Jipih ∈= ;P be the set of all vertices of all

.hTe∈ Let iφ , Ji∈ be the continuous function in h

−

Ω such that )( piφ is a linear on each

hTe∈ and ijji p δφ =)( for all jp , Jji ∈, , where ijδ is Kronecker`s delta. The linear F.E.

spaces hV and hV0

φφφ ),(:{ hh CV−

Ω∈= is linear function for each }.hTe∈

0,:{0 =∈= φφφ hh VV on }hΩ∂ .


By hI we denote the operator of the interpolation hhh VCI 0)(: →Ω−

. Hence, if :v R→hP ,

then ,0hh VvI ∈ ),())(( iih pvpvI = hip P∈ .

4. The Dual Finite Volumes To construct the dual mesh { }Jiih ∈Ω=Ω ;ˆˆ over the basic mesh hT , the dual F.V. iΩ̂

associated with a vertex hiP P∈ is a closed polygon obtained in the following way [11] : By

joining the center of every triangle hTe∈ that contains the vertex iP with the center of

every side of e containing iP . If hhiP Ω∂∩∈P , then complete the obtained contour by the straight segments joining iP with the centers of boundary sides (i.e. sides which are subsets of

hΩ∂ ) that contain iP . In this way, one can get the boundary iΩ∂ ˆ of the finite volume iΩ̂ (see Figure 4.1). It is obvious that

UJi

ih∈

−

Ω=Ω ˆ .

The interiors of Jii ∈Ω ,ˆ , are mutually disjoint. If for two different F.V. iΩ̂ and jΩ̂ their

boundaries contain a common straight segment, which call neighbors, then

.

1

ˆˆjiji

ij

ijij Γ=Ω∂Ω∂=Γ=Γ=

IUl

α

α ,

where the set ijΓ consists of one or two straight segments αijΓ and αα

jiij Γ=Γ as shown in

Figure 4.1. We have

⎪⎪⎩

⎪⎪⎨

⎧

Ω∂ΩΩ

Ω⊂ΩΩ

=

.ˆtoadjacentareˆandˆfor1

ˆˆorˆfor2

hji

hji

ijl

If hhiP Ω∂∩∈ ˆP , then we denote by 1,1, −− =Γ ii lα for 2,1=α , the segments that form

hi Ω∂∩Ω∂ ˆˆ . Put

{ }

.ˆfor)(

ˆfor1)()(

⎪⎩

⎪⎨

⎧

Ω∩∈

Ω∂∩∈−∪=

hhi

hhi

Pis

PisiS

P

P


Figure 4.1: Basic triangular mesh in a domain hΩ̂ and the dual F.V.

For every hi Ω∈Ω ˆˆ we have

U UUl

)( 1)(.ˆ

iSj

ij

ijiSj

iji∈ =∈

Γ=Γ=Ω∂α

α (4.1)

the open segments obtained by removing the endpoints from αijΓ are mutually disjoint.

Remark 4.1. We introduce the following notations a) =e area of hTe∈ .

b) =Γ= ijijβ length of ijΓ , =αβij αijΓ .

c) =Ω iˆ area of hi Ω∈Ω ˆˆ , =Ω∂ i

ˆ length of iΩ∂ ˆ .

d) ( )== αααijijij nnn 21 , unit outer normal to iΩ∂ ˆ on α

ijΓ .

e) Let us consider a partition Ttttt kk =<<<<= +110 ...0 of the time interval ( )T,0

and set kk tt −= +1τ for ,...1,0=k .

Assumptions 4.1. [11] a) Triangulation family hT is of weakly acute type. This means all angles of all

hTe∈ , are less than or equal to 2π .

b) Assume that the domain Ω is polygonal and thus Ω=Ωhˆ .

c) Define eh = the length of the longest side of the triangle hTe∈ ,

Maximum principle and error estimate 2631 eθ =the magnitude of the smallest angle of the triangle hTe∈ ,

and put eh

hhTe∈

= max , eh

h θθTe∈

= min .

d) Triangulation family hT , ),0( 0hh∈ , be regular, i.e. there exists 00 >ϑ

such that, ≥hθ 0ϑ > 0 ).,0( 0hh∈∀ e) In view of Assumption (c) which implies the existence of a constant 0ˆ >c such that

ech ˆ2 ≤ , hTe∈ , ),0( 0hh∈ .

f) Introduce the inverse stability assumption )(0 τ=h it means that we consider the condition τch (≤

where a constant 0>c( independent of h and τ . 5. The Upwind Finite Volumes Scheme Let 0>τ be the time step, the U.F.V. scheme for [ )Ttk ,0∈ to find h

k VU 01 ∈+ such that

,),(),,(),(),( 111

hhk

hkk

hhk

hh

kk

vfvUBRvUvUU +++

=+∇∇+− r

λτ

,0hh Vv ∈∀

,00 uIU h= (5.1)

Where hI is the interpolation operator and ),,( vUBR kkr

is derived as follows: By using (4.1)

and the definition of the mass lamping operator hL for all hVvu ∈, we get [13],:

∫∇=∇Ω

vdxuubvuub )).(.()),).(.((rr

∑ ∫∇≈∈ ΩJi h

hvdxLuubˆ

)).(.(r

∑ ∫∇=∈ ΩJi i

i dxuubpvˆ

)).(.()(r

= ∑ ∫∈ Ω∂Ji i

i nudsubpvˆ

)()(r

= ∑ ∫∑ ∑∈ Γ∈ =Ji

ijiSj

ij

i nudsubpvαα

)()()( 1

rl

, (5.2)

replacing the function u on αijΓ by some convex combination of the nodal values of iu and

ju with parameter ijH

)),).(.(( vuubr

∇ ≈ ∑ ∑ ∫+∑∈ = Γ∈Ji

ij

ij

ijjjiiijisj

i dsnubuHuHpvl r

1α α

α

)(,)()()(

then we define


=),,( vUBR kkr

∑ ∑ +∑∈ =∈Ji

ijkij

kj

kji

ki

kij

isji UHUHpv

l

1

,

)(,)()(

α

αβ (5.3)

where )( ikk

i pUU = , )( ii pvv = and ∫=

Γα

ααβij

ijkk

ij dsnBr

, , ).( kk UbBrr

=

Where the function { }1,: =∈× =→ nnSSH 22 RR,R is called a numerical flux.

With ),,( ijkija φφλ ∇∇= Jji ∈,

where ,iφ jφ are the basis functions in )(10 ΩH .

5.1.The Properties of the Numerical Flux ( [11] , [7] ) We use the following assumptions

a) ),,( nzyHH = is locally Lipschitz-continuous with respect to y, z

for any 0≥M there exists a constant )(Mc > 0 such that

))((),,(),,( **** zzyyMcnzyHnzyH −+−≤− ].,[,,, ** MMzzyy −∈∀

b) H is consistent nuubnuuH )(),,(

r= , .Vu∈∀

c) H is conservative ),,(),,( nyzHnzyH −−= ., Vzy ∈∀ d) H is monotone in the following sense: For a given fixed number 0>M , the function ),,( nzyH is non-increasing with respect to the second variable z on the set { }.],[,);,,(ζ MMzynzyM −∈=

5.2. The Basic Properties of the Discrete Problem [11].

a) If Ji∈ and hTe∈ is a triangle with the vertex hip P∈ then

.31ˆ ee i =Ω∩

b) The approximation h(.,.) of the 2L -scalar product can be defined with the aid of

numerical integration using the vertices eee PPP 321 ,, of hTe∈ , as the integration points

∑ ∫=∑== Ω∈

3

1,))((3)()(),(

khhkk

heh dxvLuLPvPuvu ee

Te ),(, hCvu

−

Ω∈

,)(2)(2 ΩΩ

=LhL

uLu ).( hCu−

Ω∈


c) We have

⎪⎩

⎪⎨⎧Ω

=,0

,ˆ),( j

ihiφφ

jiji

≠=

{ }

∑ Ω==∩∈∈ hiph

iiihi pupuuPeTe

e;

),(ˆ)(31),( φ ,Ji∈ ),(1

0 Ω∈Hu

),,(ˆ),( kiihik tpff Ω=φ ,Ji∈ [ ]Ttk ,0∈ .

Remark 5.1. a) from assumption (4.1-b) we find h.).,(.).,( ∇∇=∇∇ . b) By c we denote a generic constant independent of ,...,, kh τ which attains in general different values in different places.

6. The Maximum Principle [14] The DMP plays an important role in computational partial differential equations by guaranteeing that approximation of physically nonnegative quantities.

Lemma 6.1. If 0>τ time step and ),0( 0hh∈ then

,ˆ)(ˆ0 ii Mc Ω∂Ω≤≤τ .)(, oMcJi >∈ (6.1)

Proof. From the property (5.2 –a)

ii ee Ω==Ω∩ ˆ31ˆ ⇒ iiie Ω∂Ω=Ω∂ ˆˆˆ

31 , (6.2)

from the assumption (4.1-e) we find

c

heˆ

2

≥ ⇒ ii che Ω∂=Ω∂ ˆˆ3ˆ31 2 he T∈ , ),0( 0hh∈ ,

then equation (6.2) becomes

iii ch Ω∂≥Ω∂Ω ˆˆ3ˆˆ 2 ,

since hi ≥Ω∂ ˆ , we get

≥≥Ω∂Ω chii ˆ3ˆˆ hc1̂= where cc ˆ31ˆ1 = , (6.3)

we will consider the stability condition(see [4]) hMcc 1

1 )(ˆ0 −≤<τ , (6.4)

2634 H. A. Kashkool and S. A. Kadhum then from equations (6.3) and (6.4) , we obtain (6.1). □

Lemma 6.2 [13]. let )(),(),( ijijij dDcCaA === be KN × matrices satisfying the

conditions

1. 0>∑=∑=∑∈∈∈ Jj

ijJj

ijJj

ij dca . for ,oJi∈

2. ,0≥ijc for ,o

Ji∈ ,Jj∈

3. ,0≥ijd for ,o

Ji∈ ,Jj∈

4. ,0≤ija for ,o

Ji∈ ,Jj∈ ,ij ≠

and assume that gDwCuA rrr τ+= ,

then each component )(o

i Jiu ∈ is estimated by .maxmaxmax jJjjJjjJjgwu

∈∈∈+≤ τ

Theorem 6.1. If 0>τ and ),0( 0hh∈ satisfy the condition (6.1) and if assumption (2.1-c) holds then the solution of equation (5.1) is bounded and estimated by

MfTuUTQLLL

k ≤+≤ ∞Ω∞Ω∞ )()(

0

)(.

Proof. We prove the theorem by mathematical induction. Clearly the theorem is valid for 0=k

,)(

0

)(

0

)(

0 MuuIULLhL

≤≤=Ω∞Ω∞Ω∞

we assume that the theorem is valid for m ),0( km ≤≤

.)()(

0

)(MfmuU

TQLLL

m ≤+≤ ∞Ω∞Ω∞ τ

The matrix form of (5.1)given by

let ∑=∈Jj

jih pUtxUL φ)(),( and choose )(xv ih φ= , ,o

Ji∈ one can get

∑∈

+

Jjhiji

k pU ),()(1 φφ ∑−∈Jj

hijik pU ),()( φφ ∑ ∇∇+

∈

+

Jjhiji

k pU ),()(1 φφλτ

∑−=∈Jj

ijk

ik BRpU ),,()( φφτ

r∑+∈

+Jj

hijki tpf ),(),( 1 φφτ , o

Ji∈ ,

let ),( ijwW = )( ijsS = and )( kijbB = , are the matrices of mass, Stiffness and convection,

respectively which have the components:

hijijw ),( φφ= ∑ Ω==∩∈∈ ];{

ˆ31

heipheie

PT, hijijs ),( φφλ ∇∇= , ),,( ij

knij BRb φφ

v−= .

Maximum principle and error estimate 2635 Then the matrix form of equation (5.1)

[ ] [ ] ),,()()( 11

++ ++=+ kii

ki

k tpWfpUBWpUSW τττ .Ji∈ We have [13].

0>iiw 0=ijw ( )ji ≠ ; ,0=∑∈Jj

ijs 0≤ijs ( )ji ≠ ,

it is easy to see

0)( ≤+ ijij sw τ , ),,( jiJjJio

≠∈∈ and ∑ >+∈Jj

ijij sw 0)( τ , )(o

Ji∈ ,

),,( ijkk

ij BRb φφr

−=

[ ]∑ ∑ +−+−+=∈ =)(

,

1)()()(

isj

kij

ijki

kji

ki

kij

kj

kji

ki

kij

ki

kji

ki

kij UHUHUHUHUHUH α

αβ

l

in view of the property (5.1-b) of H and 1=+ kji

kij HH (see [13] ),

∑ +∑∈ =)(

,

1)(

isj

kij

ki

kji

ki

ijkij UHUH α

αβ

l

∑ −+∑=∈ =)(

,

1)(

isj

kij

ki

kij

ki

ki

ijkij UHUUH α

αβ

l

∑ ∑=∈ =)(

,

1isj

kij

ijkiU α

αβ

l

0ˆ

=∫=Ω∂

dsnUBi

ki

kr

,

then

kijb [ ]∑ ∑ +−+=

∈ =)(

,

1)()(

isj

kij

ijkj

kji

ki

kij

ki

kji

ki

kij UHUHUHUH α

αβ

l

,

hence, if [11]

ijH =⎪⎩

⎪⎨

⎧

∑−

+−+

=

ijkijk

ikj

kj

kji

ki

kij

ki

kji

ki

kij

UUUHUHUHUHl

1

,)()(0

α

αβ

kj

ki

kj

ki

UU

UU

≠

= ,

then in case kj

ki UU ≠ we get

=∑ +−+=

ijkij

kj

kji

ki

kij

ki

kji

ki

kij UHUHUHUH

l

1

,)]()[(α

αβ )( ki

kjij UU −H ,

can be written as ∑ −=∈ )(

)(isj

ki

kjij

kij UUb H ,

by the property (5.1–d) of H , 0≥ijH , ,o

Ji∈ )(isj∈ then 0≥kijb .

From the proof of [5, theorem 3.5.1 ], gives 0=∑∈

kij

Jjb ,

oJi∈ ,

now, we want to prove 0)( ≥+ kijij bw τ , by the property (5.1-a) of H

2636 H. A. Kashkool and S. A. Kadhum ))(()()( k

jki

ki

ki

kj

kji

ki

kij

ki

kji

ki

kij UUUUMcUHUHUHUH −+−≤+−+

kj

ki UUMc −= )( ,

then ijH α

αβ ,

1

)(kij

ij

ki

kj

kj

ki

UU

UUMc∑

−

−≤

=

l

∑≤=

ijkijMc

l

1

,)(α

αβ ,

so, ∑ Γ=≤≤=

ij

ijkijij McMc

l

1

, )()(0α

αβH ,

by using (4.1) we have, i

isjij Mc Ω∂≤∑≤

∈

ˆ)(0)(H ,

hence, the condition (6.1), gives

,ˆ

ˆ)(0τ

iiij Mc

Ω≤Ω∂≤∑≤

∈s(i)jH

oJi∈ ⇒ 0ˆ

)(≥∑−Ω

∈ isjiji Hτ ,

oJi∈ ,

it is easy to see that

0)]ˆ([)()()(

≥∑+∑−Ω=+∈∈

kj

isjij

ki

isjiji

kijij UUbw HH τττ , ),,,( jiJjJi

o

≠∈∈

then by lemma 6.2 we get,

11 maxmaxmax +

∈∈

+

∈+≤ k

jJj

kjJj

kjJj

fUU τ ,

Since 1)(

0

)(

0

)(

0 γ≤≤=Ω∞Ω∞Ω∞ LLhL

uuIU and 2)(γ≤∞

TQLf , by induction assumption we

get

)()(

0

)(

1 )1(TQLLL

k fkUU ∞Ω∞Ω∞

+ ++≤ τ . □

Lemma 6.3. If ∗c is a positive constant and 0)( >Mc such that

)(1

0),,(

Ω∗≤

Hkk vcvUBR

r, ),(Ω∩∈ ∞LVu h ),0(, 0hhVv h ∈∈

Proof. By using Theorem 6.1, the property (5.1-c) of H , the relations ( ααjiij Γ=Γ , αα ββ ,, k

jikij =

and ααjiij nn −= ), (5.3) and use the property (5.1-b) of H ,can be get


=),,( vUBR kkv )]()([)(21 ,

)( 1ji

kij

kj

kji

ki

kij

Ji isj

ijpvpvUHUH −+∑ ∑ ∑

∈ ∈ =

α

αβ

l

)]()([))0()0((21)(

21

)(

,

1

,

)( 1ji

Ji isj

kij

kkji

kkij

ijkij

kj

kji

ki

kij

Ji isj

ijpvpvUHUHUHUH −∑ ∑ +∑−+∑ ∑ ∑=

∈ ∈ =∈ ∈ =

α

α

α

αββ

ll

then by using the property 5.1-a of H we find

),,( vUBR kkv )]()([)( ,

)( 1ji

kij

k

Ji isj

ijpvpvUMc −∑ ∑ ∑≤

∈ ∈ =

α

αβ

l

,

since 3

, hkij ≤αβ and α

ijeji vhpvpv ∇≤− )()( then

),,( vUBR kkvα

α ijeJi isj

ijk vhUMc

∇∑ ∑ ∑≤∈ ∈ =−

Ω

2

)( 1max

3)( l

, (6.5)

from Assumption (4.1-e) and the fact that each he T∈ appears in the above sum as some αije

at most six times, theorem 6.1 and from Cauchy-Schwartz inequality we conclude that

),,( vUBR kkv ∫ΩΩ∇≤

∞dxvUMcc

L

k

)()(ˆ2

)(10)(

21

))(meas)((ˆ2ΩΩ∞Ω=

HL

k vUMcc ,

with 21

))(meas()(ˆ2 Ω=∗ MMccc . □ 7. The Stability of Approximate Solution

Theorem 7.1. There exists a constant 0>δ independent of τ,h and m such that

δ≤Ω)(2L

mU

Proof. In view of theorem 6.1, Assumption 2.1-c and the condition (6.4). If we set 1+= kh Uv

in (5.1) then ,),(),,(),(),( 1111111

hkkkkk

hkk

hkkk UfUUBRUUUUU +++++++ =+∇∇+− ττλτ

r0>τ (7.1)

we will assume the relation

)(21),( 2

)(22

)(22

)(2 ΩΩΩ−+−=−

LLLh zyzyyzy ,


valid for ),(,−

Ω∈Czy we get

)(21),(

2

)(212

)(2

2

)(2111

Ω

+

ΩΩ

+++ −+−=−L

kk

L

k

L

kh

kkk UUUUUUU ,

then (7.1) becomes

2

)(10

12

)(212

)(2

2

)(21 2

Ω

+

Ω

+

ΩΩ

+ +−+−H

k

L

kk

L

k

L

k UUUUU τλ

,),,(2),(2 111h

kkkh

kk UUBRUf +++ −=r

ττ [ )Ttk ,0∈ (7.2)

in virtue of Lemma 6.3 and Young's inequality we have

),,(2 1+kkk UUBRr

( ) 2

)(10

12*

Ω

++≤H

kUc εε

, (7.3)

by the property (5.2-d) and Cauchy-Schwartz inequality we find

hk

hk

hhkk ULfLUf ),(),( 1111 ++++ =

)(

1

)(

122 Ω

+

Ω

+≤L

khL

kh ULfL , (7.4)

now, by using the definition of the mass lamping operator hL in [13] we get

)(2

1

Ω

+

L

kh fL

21

21 )(ˆ)( ⎟⎟

⎠

⎞⎜⎜⎝

⎛∫ ⎟

⎠⎞⎜

⎝⎛ ∑=

Ω

+

∈ppf ii

k

Jiμ

[ ] ))(,1;,0(

1

Ω

+≤ qWTC

kfc (7.5)

further, we use [ 13, lemma 2.2 ] and from the definition 1.2 in [2] we get ,~

)(10

1112)(2

1

Ω

+−

Ω

+ ≤H

k

L

kh UccUL (7.6)

from (7.5), (7.6) and Young's inequality, we have

),(2 11 ++ kk Uf [ ]

2

)(10

12

))(,1;,0(

121

22 εε/~

Ω

+

Ω

+− +≤H

kqWTC

k Ufcc , (7.7)

now, by substituting (7.3), (7.7) in (7.2), choosing 2λε = and using Remark (5.1-b), we get

2

)(10

12

)(212

)(2

2

)(21

Ω

+

Ω

+

ΩΩ

+ +−+−H

k

L

kk

L

k

L

k UUUUU τλ τc≤ [ )Ttk ,0∈ ,

where [ ]

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+=

Ω

+− λ~2 2*2

))(,1;,0(

121

22 cfccc

qWTC

k .Summation over 1,...,0 −= mk , ( ]Ttm ,0∈

then

2

)(201

0

2

)(10

11

0

2

)(212

)(2 Ω

−

= Ω

+−

= Ω

+

Ω+≤∑+∑ −+

L

m

k H

km

k L

kk

L

m UcTUUUU τλ , (7.8)

since 00 uIU h= and by using the definition 1.2 in [2] then


2

)(10

021

22

2

)(20 ~

Ω

−

Ω≤

HhLuIccU ,

then (7.8) becomes

δτλ ≤+≤∑+∑ −+Ω

−−

= Ω

+−

= Ω

+

Ω

2

)(10

021

22

1

0

2

)(10

11

0

2

)(212

)(2~

Hh

m

k H

km

k L

kk

L

m uIcccTUUUU , ( ]Ttm ,0∈

the last inequality is a consequence of the bounded

cuIHh ≤

Ω

2

)(10

0 , ),0( 0hh∈

and the second and third terms are nonnegative, , we obtain δ≤Ω)(2L

mU . □

8. The Local Truncation Error We suppose that the exact solution VTu →),0(: of (2.4) satisfies the condition A(3.3).

Hence ∩Ω∈ ))(];,0([ 2LTCu ∩∗)];,0([1 VTC )( TQL∞ and set ∞<= ∞ )( TQL

kuM .

Lemma 8.1. Under conditions A(3.3), for ),0[ Ttn ∈ we have

(a) ,)),((),()(1

0

21

1Ω+

+ ≤′−−Hk

kk vcvtuvuu ττ ,Vv∈

(b) )),).(.(()),).(.(( 11 vuubvuub kkkkrr

∇−∇ ++ )(1

0 Ω≤

Hvcτ , Vv∈ .

Proof. (a) We based on the following result [10]: If ∗→VT),0(:ω is such that );,0(, 1 ∗∈′ VTLωω and Vv∈ then ),0(, 1 TLv ∈⟩′⟨ω and

=⟩′⟨∫ dtvtt

t),(

2

1

ω ,),()( 12 ⟩−⟨ vtt ωω ].,0[, 21 Ttt ∈ Here ⟩⟨ v,ϕ denotes the value of a

functional ∗∈Vϕ at a point Vv∈ . A similar result holds, if ))(;,0(, 21 Ω∈′ LTLωω and the

duality ⟩⟨.,. is replaced by the 2L -scalar product (.,.) . Let Vv∈ , then

)),((),( 11 vtuvuu k

kk+

+ ′−− τ −−= + )),()(( 1 vtutu kk ( ) .),( 1

1dtvtu k

kt

kt+

+′∫

dtvtutu k

kt

kt⟩′−′⟨∫= +

+),()( 1

1 (8.1)

Since );.0( *VTLu ∞∈′′

⟩′−′⟨ + vtutu k ),()( 1 ωω dvut

kt⟩′′⟨∫=

+

),(1

, (8.2)

2640 H. A. Kashkool and S. A. Kadhum by substituting (8.2) in (8.1) and using Cauchy-Schwartz inequality, we get

)),((),( 11 vtuvuu k

kk+

+ ′−− τ dtdvukt

kt

t

kt∫ ⎟⎟

⎠

⎞⎜⎜⎝

⎛⟩′′⟨∫=

+

+

1

1

),( ωω

dtvtutukt

kt

k∫ ⟩′−′

⟨=+

+1

1 ,)()(τ

τ dtvukt

kt∫ ⟩′′⟨=+1

),(ωτ

),)()(( 12 vtutu kk

ττ

′−′= + )),((2 vtu ′′= τ

)(10

2Ω

≤H

vcτ .

(b) By using Green's theorem then

=∇−∇ ++ )),).(.(()),).(.(( 11 vuubvuub kkkkrr

),).((),).(( 11 vuubvuub kkkk ∇−∇++rr

adding and subtracting ),).(( 1 vuub kk ∇+r

, by using Cauchy-Schwartz inequality, Theorem 8.1 and the conditions A(3.1)–A(3.3), we get

)),).(.(()),).(.(( 11 vuubvuub kkkkrr

∇−∇ ++ ),)).()((()),).((( 111 vuububvuuub kkkkkk ∇−+∇−≤ +++rrr

)(1

0)(21

ΩΩ

+ −≤HL

kk vuuξ , (8.3)

since ))(;,0( 2 Ω∈′ ∞ LTLu

)(2

1

Ω

+ −L

kk uu ⎟⎠⎞⎜

⎝⎛ Ω∞′≤

)(2:,0 LTLuτ τc≤ ,

then (8.3) becomes

≤∇−∇ ++ )),).(.(()),).(.(( 11 vuubvuub kkkkrr

)(10 ΩH

vcτ , Vv∈ .

Theorem 8.2. If ku and kU are solutions of (2.4) and (5.1), respectively at time ktt = and

ku satisfies the conditions A(3.3) then L.T.E ≤ )(0 2τ , 0>τ .

Proof. The exact solution u satisfies (2.4) at 1+= ktt then

).,()),).(.((),()),(( 11111 vfvuubvuvtu kkkk

k++++

+ =∇+∇∇+′r

λ ,Vv∈∀

adding and subtracting ),(1

vuu kk

τ−+

and )),).(.(( vuub kkr

∇ with setting hh Vvv ∈= , gives

),( 1h

kk vuu −+ +∇∇+ + ),( 1h

k vuλτ )),).(.(( hkk vuub

r∇τ += + ),( 1

hk vfτ

)]),((),[( 11

hkhkk vtuvuu +

+ ′−− τ )]),).(.(()),).(.([( 11h

kkh

kk vuubvuub ++∇−∇+rr

τhh Vv ∈ , (8.4)

by subtracting equation (8.4) from the equation (5.1) and using Remark (5.1-a) such that


hhkk vUU ),( 1 −+ ),( 1

hkk vuu −− + )],(),[( 11

hk

hk vuvU ∇∇−∇∇+ ++ λλτ

)]),).(.((),,([ hkk

hkk vuubvUBR

rr∇−+τ )],(),[( 11

hk

hhk vfvf ++ −=τ

)]),((),[( 11

hkhkk vtuvuu +

+ ′−−− τ )]),).(.((),).(.([( 11h

kkh

kk vuubvuub ++∇−∇−rr

τ , hh Vv ∈∀

adding and subtracting ),( 1h

kk vUU −+ and )),).(.(( hkk vUUb

r∇τ , gives

),( 11h

kk vuU ++ − ),( hkk vuU −− )),(( 11

hkk vuU ∇−∇+ ++τλ

)]),).(.(()),).(.([( hkk

hkk vuubvUUb

rr∇−∇+τ 21 II += ,

hh Vv ∈∀ (8.5) where =1I )]),((),[( 1

1hkh

kk vtuvuu ++ ′−−− τ

)]),).(.((),).(.([( 11h

kkh

kk vuubvuub ++∇−∇−rr

τ ,

2I +−= ++ )],(),[( 11h

khh

k vfvfτ ]),(),[( 11hh

kh

k vUvU ++ −

]),(),[( hhk

hk vUvU −− )],,()),).(.([( h

kkh

kk vUBRvUUbrr

−∇+τ .

To estimate 1I , we use Lemma 8.1,

)),).(.((),).(.(()),((),( 111

11 h

kkh

kkhkh

kk vuubvuubvtuvuuI +++

+ ∇−∇+′−−≤rr

ττ

)(1

0

2Ω

≤Hhvcτ ),(0 2τ≤

hh Vv ∈ , 0>τ . (8.6)

We can write ∑==

4

122

iiII . The estimate 21I is a consequence of [ 12, Theorem 4.1.5 and the

proof of Theorem 4.1.6 ]

),(),( 1121 h

khh

k vfvfI ++ −= τ )(1

0 Ω≤

Hhvchτ , hh Vv ∈∀ . (8.7)

The estimate of 22I and 23I are a consequence of [ 12, Theorem 3.1.6 ] and using Lemma 2.1, we have

=22I hhk

hk vUvU ),(),( 11 ++ −

)(2

10 Ω

≤Hhvch , (8.8)

so we find

=23I hhk

hk vUvU ),(),( −

)(10

2Ω

≤Hhvch . (8.9)

To estimate 24I , by adding and subtracting )),).(.(( hhkk vLUUb

r∇τ then

24I ),,()),).(.(( hkk

hkk vUBRvUUb

rr−∇= τ


),,()),).(.(()),).(.(()),).(.(( hkk

hhkk

hhkk

hkk vUBRvLUUbvLUUbvUUb

rrrr−∇+∇−∇≤ ττ

224

124 II +=

by Cauchy-Schwartz inequality, Lemma 2.1, Theorem 8.1, [ 13, Lemma 2.1 ] and A(3.1) then

124I )),).(.(()),).(.(( hh

kkh

kk vLUUbvUUbrr

∇−∇≤τ )),).(.(( hhhkk vLvUUb −∇≤

rτ

)(10 Ω

≤Hhvhcτ . (8.10)

And 224I ),,()),).(.(( h

kkhh

kk vUBRvLUUbrr

−∇=τ

∑ ∑ −+−∫∑=∈ = Γ∈Ji

ij

jhihkij

kj

kji

ki

kij

ij

ijkk

isjpvpvUHUHdsnUUb

l r

1

,

)(,)()()()(

21

α

α

α

α βτ

If Ji∈ and )(isj∈ ,then the segment ji pp a side of triangles hije T∈α such that ααijij e⊂Γ then.

hpp ji ≤− , hpx i ≤− for αijx Γ∈ ,

3, hk

ij ≤αβ , αj )()(ije

kki

k uhpupu ∇≤− ,

α)()(ije

ki

kk uhpuxu ∇≤− for αijx Γ∈ , α

ijehjhih vhpvpv ∇≤− )()( , (8.11)

by the properties (5.1-a,b) of H and (8.11), we obtain

≤+−∫Γ

α

α

α β ,)()( kij

kj

kji

ki

kij

ij

ijkk UHUHdsnUUb

rα

α

α β ,)()())(( kij

kj

kji

ki

kij

ij

ijik

ik UHUHdsnpUpUb +−∫

Γ

r

αα

αββ ,, )()(max)(2 k

ijkj

ki

kij

ki

k

ijxUUMcUxUMc −+−≤

Γ∈ α

ije

kUMch ∇≤ )(2 ,

then we have

),,()),).(.(( hkk

hhkk vUBRvLUub

rr−∇τ αα

ατ

ijehije

k

Ji

ij

isjvUMch ∇∇∑ ∑∑≤

∈ =∈)(

21 3

1)(

l

, (8.12)

taking into account that each triangle he T∈ appears in the sum in (8.12) as some αije at most

six times, by using Assumption (4.1-e), Cauchy-Schwartz inequality and Lemma 2.1 we conclude that

),,()),).(.(( hkk

hhkk vUBRvLUUb

rr−∇τ h

k

hevUehMcc ∇∇∑≤

∈Tτ)(ˆ

)(10 Ω

≤Hhvhcτ , (8.13)

now from (8.10) and (8.13) we get )(1

024 Ω≤

HhvhcI τ , (8.14)

Maximum principle and error estimate 2643 by using Assumption (4.1-f) and from (8.7)- (8.9), (8.14) we get )(0 2

2 τ≤I , (8.15)

and from (8.6) and (8.15) the proof is complete.

9. The Error Estimate We define the projection hh VVp →: as follows: If V∈ϕ such

that ,hh Vp ∈ϕ

,)(1

0)(10 ΩΩ

≤HHhp ϕϕ .V∈ϕ (9.1)

Lemma 9.1. For any 0>M there exists a constant 0>ξ such that

)(1

0)(2212211 ξ)),).(.(()),).(.(((ΩΩ

−≤∇−∇HL

vzzvzzbvzzbrr

,

),()(, 1021 Ω∩Ω∈ ∞LHzz Mzz

LL≤

Ω∞Ω∞ )(2)(1 , .

Proof. Is similar to proof Lemma 8.1-b.

Theorem 9.1. If ku and kU are solutions of (2.4) and (5.1) respectively and satisfy the

conditions A(3.3) and τ be the time step satisfy ,ˆ)(ˆ0 ii Mc Ω∂Ω≤≤τ Ji∈ then

[ ]

)(0max)(2,0

τ≤Ω∈ L

k

Tkte , (9.2)

where a constant 0>c is independent of λτ ,,h .

Proof. Let ),0( ohh∈ and 0>τ satisfy the condition (7.4) and Assumption (4.1-f), From

(8.5), let the error of the method at time ktt = is defined by

kkk uUe −= , (9.3) let 1+= k

hh epv , adding and subtracting ),( 11 ++ kk ee , ),( 1+kk ee and ),( 11 ++ ∇∇ kk eeλτ , gives

),( 11 ++ kk ee ),( 1+− kk ee =∇∇+ ++ ),( 11 kk eeτλ

)]),).(.(()),).(.([( 11 ++ ∇−∇− kh

kkkh

kk epuubepUUbrr

τ 21 II ++ ),( 11 +++ kk ee

),( 11 ++− kh

k epe +− + ),( 1kk ee ++ ),( 1kh

k epe ),( 11 ++ ∇∇ kk eeτλ ),( 11 ++ ∇∇− kh

k epeτλ

denoting by VVI →: the identity operator ϕϕ =I( for )V∈ϕ then


),( 11 ++ − kkk eee =∇∇+ ++ ),( 11 kk eeτλ )]),).(.(()),).(.([(τ 11 ++ ∇−∇− kh

kkkh

kk epuubepUUbrr

21 II ++ ))(,( 11 ++ −+ kh

k epIe +−− + ))(,( 1kh

k epIe ))(,( 11 ++ −∇∇ kh

k epIeτλ , (9.4)

from (9.3) it follows that =− +1)( k

h epI 1111 ++++ +−− kh

kh

kk upUpuU 11 ++ −= kkh uup , (9.5)

hence, by using Lemma 2.2, Lemma 2.3 and (9.5) then

)(212

)(21)(

Ω

+

Ω

+ ≤−H

k

L

kh uchepI (9.6)

)(2

1

)(10

1)(Ω

+

Ω

+ ≤−H

k

H

kh uchepI (9.7)

by using the relation [ ]22 )()(41. BABABA −−+= and Cauchy-Schwartz inequality, we find

),( 11 ++ − kkk eee ⎟⎠⎞⎜

⎝⎛ −−+=

ΩΩ

+

Ω

+ 2

)(2

2

)(212

)(21

21

L

k

L

kk

L

k eeee . (9.8)

And 2

)(10

111 ),(Ω

+++ =∇∇H

kkk eee , (9.9)

it follows from Lemma 9.1, (9.1) and (9.3) that

)),)(.(()),)(.(( 11 ++ ∇−∇ kh

kkkh

kk epuubepUUbrr

)(10

1

)(2 Ω

+

Ω≤

H

k

L

k eec , (9.10)

and since

)(10

121 Ω

+≤H

kecI τ , (9.11)

)(1

0

12

)(10

12 Ω

+

Ω

+ +≤H

k

H

k echehcI τ (9.12)

furthermore, by using Cauchy-Schwartz inequality and (9.6), we have

))(,())(,( 111 +++ −−− kh

kkh

k epIeepIe )(2

1

)(212

Ω

+

Ω

+ −≤H

k

L

kk ueech , (9.13)

so, by using Cauchy-Schwartz inequality, from the definition 1.2 in [2] and (9.6), we have

11 )(, ++ −∇∇ kh

k epIe )(2

1

)(10

1

Ω

+

Ω

+≤H

k

H

k uech , (9.14)

now, by substituting (9.8)–(9.14) in (9.4). Taking into account conditions A(3.3) we obtain the estimate

≤+−−+Ω

+

ΩΩ

+

Ω

+ 2

)(10

12

)(2

2

)(212

)(21 2

H

k

L

k

L

kk

L

k eeeee τλ)(1

0

1

)(2 Ω

+

Ω H

k

L

k eecτ


)(10

12

Ω

++H

kecτ)(1

0

12

)(10

1

Ω

+

Ω

+ ++H

k

H

k echechτ)(2

122Ω

+ −+L

kk eech)(1

0

1

Ω

++H

kechτλ ,

We use (6.4) and assumption (4.1-f) then

≤+−−+Ω

+

ΩΩ

+

Ω

+ 2

)(10

12

)(2

2

)(212

)(21 2

H

k

L

k

L

kk

L

k eeeee τλ)(1

0

1

)(2 Ω

+

Ω H

k

L

k eecτ

)(10

1

Ω

++H

kehcτ)(1

0

1

)(10

1

Ω

+

Ω

+ ++H

k

H

k ehcehc ττ)(2

122Ω

+ −+L

kk eech)(1

0

1

Ω

++H

kechτλ ,

by using Young's inequalities we find that

≤+−−+Ω

+

ΩΩ

+

Ω

+ 2

)(10

12

)(2

2

)(212

)(21 2

H

k

L

k

L

kk

L

k eeeee τλ 2

)(2 ΩL

kecλτ 2

)(10

1

4 Ω

++H

keτλ

2

)(10

1

4 Ω

++H

keτλ 2

)(10

12

4 Ω

+++H

kehc τλλτ 2

)(214

Ω

+ −++L

kk eech2

)(10

12

4 Ω

+++H

kech τλτλ ,

and by using Assumption (4.1-f) , since the third term is nonnegative we have

≤Ω

+ 2

)(2

1

L

ke BeAL

k τ+Ω

2

)(2, [ ),,0 Ttk ∈ (9.15)

where

,1λτcA += ⎥

⎦

⎤⎢⎣

⎡++= 32

2

)1( hhcB λλ

,

by induction over ,...,2,1,0=k from (9.15) we easily deduce that

≤Ω

2

)(2L

ke BAAeA kk

L

k )1...( 212

)(2

0 ++++ −−

Ω

τ ,

multiplying and dividing the second term by )1( −A

≤Ω

2

)(2L

ke 112

)(2

0

−−

+Ω A

ABeAk

L

k τ , [ )Ttk ,0∈ ,

since ),exp(λτcA ≤ then

≤Ω

2

)(2L

ke λλ

λλλ

⎟⎠⎞

⎜⎝⎛ −⎥⎦

⎤⎢⎣

⎡+++

Ω

1)exp()1()exp( 3222

)(2

0 k

L

k cthhce

ct ,

taking into account that ),()( ,10

10 Ω⊂Ω∈ po WHu by virtue of [ 8, Theorem 2.27 ] and the

conditions A(3.3) we have

2

)(2022

)(20

ΩΩ≤

HLuche ,

also we obtain the estimate

2

1

23

212

)(2 1)exp()1()2

exp( ⎟⎠⎞

⎜⎝⎛ −⎥⎦⎤

⎢⎣⎡ +++≤

Ω λλλ

λkk

L

k cthhch

ctce , [ ]Ttk ,0∈ ,


now [ ] )(2,0

maxΩ∈ L

k

Tkte )exp()exp()1()exp( 2

3

λλ

λλ

λcThchcTccTc +⎟

⎠⎞

⎜⎝⎛ ++≤ ,

finally, by using Assumption (4.1-f) , which immediately implies (9.2). □ Conclusions In this paper we have analyzed the U.F.V. method for solving nonlinear convection-diffusion equation. This method satisfies the convergence, the discrete maximum principle and the discrete conservation law. We derive the U.F.V. method for the convection term )).(.( uub

r∇

which is discretized on the barysentric finite volume mesh dual to a triangular grid of weakly acute type under some assumptions on the regularity of the exact solution of the continuous problem. The discrete maximum principle implying the −∞L estimate and the stability of approximate solutions are the basic tools used in the investigation of error estimate are proved

under the condition ,ˆ)(ˆ0 ii Mc Ω∂Ω≤≤τ Ji∈ .

References

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[2] A. Quarteroni and A. Valli, " Numerical approximation of partial differential equations ", ISBN 3-540-57111-6. Springer-Verlag, Berlin Heidelberg, New York (1997).

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Maximum Principle and Error Estimate of the Upwind Finite …m-hikari.com/ams/ams-2012/ams-53-56-2012/aboodAMS53-56... · 2012. 3. 4. · total flow that enters a cell through its