American Journal of Computational Mathematics, 2011, 1, 264-270
doi:10.4236/ajcm.2011.14032 Published Online December 2011 (http://www.SciRP.org/journal/ajcm)
Copyright © 2011 SciRes. AJCM
An Upwind Finite Volume Element Method for Nonlinear
Convection Diffusion Problem
Fuzheng Gao, Yirang Yuan, Ning Du
School of Mathematics, Shandong University, Jinan, China
E-mail: fzgao@sdu.edu.cn
Received August 17, 2011; revised September 6, 2011; accepted Septembe r 15, 2011
Abstract
A class of upwind finite volume element method based on tetrahedron partition is put forward for a nonlinear
convection diffusion problem. Some techniques, such as calculus of variations, commutating operators and
the a priori estimate, are adopted. The a priori error estimate in L2-norm and H1-norm is derived to determine
the error between the approximate solution and the true solution.
Keywords: Nonlinear, Convection-Diffusion, Tetrahedron Partition, Error Estimates
1. Introduction
Consider the following nonlinear convection-diffusion
problem:

t
uu ugut
 Fxx xJ
(1)

0ut tJ xx (2)
 
0
0uu xxx
3
(3)
where is a bounded region with piecewise
smooth boundary .
R
is a small positive constant
and is a smooth
vector function on

ufFx
x
12
u

3
fufu x x
R ,
00Fx .
The finite volume element method (FVEM) is a dis-
crete technique for partial differential equations, espe-
cially for those arising from physical conservation laws,
including mass, momentum and energy. This method has
been introduced and analyzed by R. Li and his collabo-
rators since 1980s, see [1] for details. The FVEM uses a
volume integral formulation of the original problem and
a finite partitioning set of covolumes to discretize the
equations.The approximate solution is chosen out of a
finite element spaces [1-3] The FVEM is widely used in
computational fluid mechanics and heat transfer prob-
lems [2-5]. It possesses the important and crucial prop-
erty of inheriting the physical conservation laws of the
original problem locally. Thus it can be expected to cap-
ture shocks, or to study other physical phenomena more
effectively.
On the other hand, the convection-dominated diffusion
problem has strong hyperbolic characteristics, and there-
fore the numerical method is very difficult in mathemat-
ics and mechanics. when the central difference method,
though it has second-order accuracy, is used to solve the
convection-dominated diffusion problem, it produces
numerical diffusion and oscillation near the discontinu-
ous domain, making numerical simulation failure. The
case usually occurs when the finite element methods
(FEM) and FVEM are used for solve the convection-
dominated diffusion problem.
For the two-phase plane incompressible displacement
problem which is assumed to be -periodic, J. Douglas,
Jr., and T.F.Russell have published some articles on the
characteristic finite difference method and FEM to solve
the convection-dominated diffusion problems and to
overcome oscillation and faults likely to occur in the
traditional method [6]. Tabata and his collaborators have
been studying upwind schemes based triangulation for
convection-diffusion problem since 1977 [7-11]. Yuan,
starting from the practical exploration and development
of oil-gas resources, put forward the upwind finite dif-
ference fractional steps methods for the two-phase three-
dimensional compressible displacement problem [12].
Most of the papers known concern on the FVEM for
one- and two-dimensional linear partial differential equa-
tions [1-4,13,14]. In recent years, M. Feistauer [15,16],
by introducing lumping operator, constructed finite vo-
lume-finite element method for nonlinear convection-
diffusion problems. On the other hand, because the FEM
costs great expense to solve the three-dimensional prob-
lems, finite difference methods (FDM) are usually used
to approximate the problems [12]. These works inspire
F. Z. GAO ET AL.265
us to look into the subject how to use the upwind FVEM
to solve three-dimensional nonlinear convection-domi-
nated diffusion problems. In this article, we continue to
our work [17] and put forward an upwind FVEM for
three-dimensional nonlinear convection-dominated dif-
fusion problems based on tetrahedron partition and its
dual partition of . Some techniques, such as calculus
of variations, commutating operator and the a priori error
estimate, are adopted. The a priori error estimate in
-norm and
12
L
H
-norm is derived to determine the error
between the approximate solution and the true solution.
The remainder of this paper is organized as follows. In
Section 2, we put forward the upwind FVEM for prob-
lem (1). In this section, we introduce notations, construct
tetrahedron mesh partition h of and its dual parti-
tion. Some auxiliary lemmas and the a priori error esti-
mate in -norm and
T
1
2
L
H
-norm of the scheme are
shown In Section 3 and Section 4, respectively. In Sec-
tion 5, some concluding remarks are presented.
Throughout this paper we use C (without or with sub-
script) to denote a generic constant independent of dis-
crete parameters. We also adopt the standard notations of
Sobolev spaces and norms and semi-norms as in [18, 19].
2. The UFVE Method
Suppose problem (1) satisfy condition (A1):

1
C Continuity condition:
 
2
g
uL R x is
Lipschitz continuous w.r.t. the second variableu.

2
C The vector function has 1-order con-
tinuous partial derivative w.r.t. and u.
uFx
x
Suppose the true solution of problem (1) possess cer-
tain smooth and satisfy:

R Regular condition:


212
uW LHLLH
 

2
.
Before presenting the numerical scheme we introduce
some notations. For simplicity we assume
is the
domain

000
L
LL
X
YZ  . Firstly, Let us
consider a family of regular tetrahedron partition in
the domain
h
T
, which is a closure of . Let be
maximum diameter of cell of h
T. For a fixed tetrahe-
dron partition
h
h
TK, we define a closed tetrahedron
set

1
K
N
ii
K and node set
 
12
1
011
2
1
M
MM
hhi ii
iiM
PP P


 i
,
where 0 is inner nodes set of and h
boundary
nodes set on . Let be all edges
set.
i
1
hi
Ee M 
E
Definition 2.1. Suppose that is a
set of tetrahedron partition of , the set h is called
regular, if there exists a positive constant

0
0
h
TT hh
T
1
independ-
ent of , such that
h

10
max 0
hKK
KT hh

h
 
where
K
h and
K
are the diameter of and the
maximum diameter of circumscribing sphere of tetrahe-
dron , respectively.
K
K
Definition 2.2. The two tetrahedron cells are called face-
adjacent if they have one common face, while edge-ad-
jacent if they have one common edge.
Definition 2.3. The two nodes are called adjacent if
they form one edge which belongs to h. Denote by E
i
j
jP
is adjacent to .
ii jh
For a given tetrahedron partition h
T with nodes
PPP
i
Ph
and edges
i
eE
hP
h
, we construct two kind
of dual partitions. First, we will construct the circumcen-
ter dual partition of .
Tih
 , let
hi hi
PKKTP
 is a vertex of
K
. Let
j
Q be
a circumcenter of
hi
K
P . Connecting
j
Q of the
two face-adjacent tetrahedron cells which belong to
hi
P, then we can derive a polyhedron i
P
K
which
surrounds the node i. P
j
Q are vertices of the polyhe-
dron i
P
K
which is called circumcenter dual partition
corresponding to node i. i
hPih
is the
circumcenter dual partition of h
T. Denote by the
midpoint of and its adjacent node
P

TK

P
ij
P
i
P
j
P.
The other dual partition as follows. h
 , let
k
eE
hk h
eKKT
 and k is a edge of e
K
. Denote
by 1
k and 2
k the vertices of the edge k and P Pe
j
Q
the circumcenter of the

hk
K
e . Suppose k
e
K
is a
polyhedron whose vertices are
1
k
P2
k
P
j
Qs. k
e
K
is
called dual cell for edge .
k
e

1
E
k
M
hek
K
T
is the other
dual partition to .
h
T
Let h
be the node set of dual partition. For
h
Q
, let Q
K
be tetrahedron cell which includes .
Q
Let
P
K
and Q
K
be volumes of dual cell
P
K
and
tetrahedron cell Q
K
, respectively. Let be the di-
ameter of tetrahedron cell Q
h
K
. As follows, we assume
that the partition family h is regular, i.e., there exist
positive constants 12
T
CC
independent of , such that
the following condition (A2) satisfies:
h
33
12
33
12
h
p
Qh
Ch KCh P
ChKCh Q


(4)
Suppose that a trial function space
1
0
h
UH,
whose basis functions are


1
1
M
ii
P
possessing the
form
01 23
x
yz
 
  based on [15], and
h
T
0
iih
PP
. Test function space
2
h
VL is a
piecewise constant function space corresponding to the
dual partition h
T
, whose basis functions are
h
PP
 .
Copyright © 2011 SciRes. AJCM
F. Z. GAO ET AL.
266

1
0otherwise
P
PK
P


and

0h
PP
 
For the following analysis, we introduce two interpo-
lation operators. Suppose that and are inter-
polation operators from
h
h
1
0
H
to and , respec-
tively, satisfying
h
Uh
V

1
1
M
hi
i
uuPP
 
i
(5)

Ph
hKT
uuPP

 
(6)
Multiplying both sides of (1) by , integrating on
dual partition cell
v
i
P
K
P, using Green formula, and sum-
ming with respect to , we have
i
h



1
0
t
uvauvbuvgv vH 
d
(7)
where

d
PP
ii
ih KK
P
auvuvu vs



 

x
(8)

d
ih
PKK
PP
ii
buvv vs


  

Fx Fd
(9)
Converting into [1]
F
 
0d
u
uu
FxFxuu (10)
Let




0
0
max 0()d
max 0()d
u
ij ij
u
ij ij
uuu
uu



xFx
xFx
u
uu
(11)
where ij
is the unit outward normal vector of
ij i
P
K
 . For we introduce bilinear
form
hhhh
uUvV
 

 

i
ih
hhhhi ij
Pj
ijij hiijij hj
buvvP
Pu PPu P

 

 




(12)
where ij
is the area of .
ij
So far, we can obtain the semi-discrete upwind finite
volume element scheme: Find such that
hh
uU



,hthh hhh hhh
uvauvbuvg uv x
(13)
where

d
P
i
ih
hhh h
K
P
au vuvs



Let tTN , denote by

nn
tntuut 
n

12
nn
hh
uutn N,

11nnn
thh h
uuu t
 .
If approximate solution h
is known, then
can be found by the following full-discrete upwind finite
volume element scheme.
1n
h
uU
n
h
u


11
1
nn n
hthhhhhhh
n
hh
uv auv buv
gu v
 
1
 
x (14)
3. Auxiliary Lemmas
Define the discrete norm and the discrete semi-norm [1]
as follows.


22
2
00i
Ph
i
hhh hi
hKT
uu uP


P
K

(15)
 

2
2
21
11
E
k
M
hhkhkk
hk
uuPuPeK
e

(16)
22
101
hhh
hh
uuu
2
h

 (17)
obviously, the discrete norm and the discrete semi-norm
are equivalent to the continuous norm and the full-norm
on , respectively.
h
Lemma 1. Suppose all cells Q
U
K
of the partition h
satisfy conditions (A2), h
T
T
is a circumcenter dual parti-
tion. hh h
uu U
 , there exist positive constants 0
C
independent of h
such that
2
1
hhhhhh
auuuuU
 
(18)
011
hhh
hhhhh
auC uuU
uuu
 
(19)
h
hhhhhh hh
auu auuuU
u

  (20)
Remarks:
1) From Lemma 1, we know that is symmet-
rical and positive definite in .

a
h
U
2) Let

1
2
1
hhhh
uauu
 , then 1
is equiva-
lent to 1
in .
h
U
Lemma 2. Let
1
2
0
hhhh
uuu

 h
, 0
is equi-
valent to 0
in .
h
U
The proof of lemma 2 can be completed by computing
integral on cell Q
K
, directly.
Theorem 1. (Trace Theorem) [20]. Suppose that
has a piecewise Lipschitz boundary, and that is a real
number in range
p
1p
. Then there exists a constant
, such that
C
  

1
11 11
pp
p
pp
p
LLW
vCvv vW
 
 
Lemma 3. For h small enough, suppose P
is a
random point in dual partition cell i
P
K
,
ij
ij PP
K
K

 ,
Copyright © 2011 SciRes. AJCM
F. Z. GAO ET AL.267
then
 
2
12
dPP
ij ii
i
KK
j
uPusChuu

 
 x
(21)
Proof. From Hölder inequality, we can get that
 
 

1
2
2
d
d
i
ij
i
j
j
ij uPus
Chu Pus



x
x
Using Taylor expansion, trace theory in which we
choose and Hölder inequality, we can complete
the proof of lemma 3.
2p
Lemma 4. For hh h
uu U  there exists a positive
constant , such that
C

,
hhh hhh
uuuu
 
(22)

0.
hhhh h
uuCuu
 0
(23)
Proof. From the properties of the functions in , for
each partition cell
h
U
h
K
T, we know that h
K
u has
the following expression.





0000111 1
22223333
,,,,,,,, ,
,,,,,, ,
hiiiiiiii
K
iii iiii i
uxyzt uxyztuxyzt
uxyztuxyzt


 (24)
where

0123
1
6
lllll
axbyczdlii ii
Ve

and is the volume of tetrahedron i.e.,
Ve 0123
iiii
PPPP
00
0
11
1
22
2
33
3
1
1
1
1
6
1
ii
i
ii
i
ii
i
ii
i
x
yz
x
yz
Ve
x
yz
x
yz

0123
Pl iiii
l
i, whose coordinates are ll

l
iii
x
yz
PP ,
are four vertices of tetrahedron cell 0123
iiii
which
belongs to h. 012 3l are the volume coor-
dinates which are corresponding to tetrahedron cell
. For ,
PP
T
0123
iiii
PPPPl
liiii
0
i
11
11
22
0202
33
33
11
11
222
020
333
3
3
11
11
11
1
1
1
ii
ii
ii
iiii
ii
ii
ii
ii
iii
iii
iii
i
i
yz xz
ayzbxz
yz xz
1
2
i
i
x
yxy
cxydxy
z
z
x
yz
xy
 
 
Analogously, we can define the remaining coefficients
123lll l
abcd liii
 . Further,

0123
ddd
h
hhh hl
KTliiii h
KK
P
l
uuuP uxy

z

 
For simplifying numerical integral, we divide the po-
lyhedron integral domain 0
i
P
K
K
into six tetrahedron
integral domains
0010120123
0010130123
0020120123
002 023 0123
003 013 0123
1
2
3
4
5
tetrahedron
tetrahedron
tetrahedron
tetrahedron
tetrahedron
iiiiiiiiii
iii iiiiiii
iiiiiiiiii
iii iii iiii
iiiiiiiiii
VPPP
VPPP
VPPP
VPPP
VPPP
P
P
P
P
P





003023 0123
6tetrahedron iii iii iiii
VPPPP

where 01
ii is the midpoint of segment 01
ii
while
and 012 3
iiii are circumcenters of triangular surface
012
iii and tetrahedron 0123
iiii
, respectively.
Analogously, we can define the remaining points.
P
P
PP
012
iii
PPPPPPPP
Noting the Equality (24), we have that




00 1
22 33
ddd ddd
[
]ddd .
PP
ii
P
i
hhK
KK KK
ii ii
KK
ii ii
u xyzuxyz
uPtuPt
uPtuPt xyz
1





 


For simplicity, we will omit the variable in func-
tion
t
uxyzt

01
ii

. From volume coordinate formula, not-
ing 23
1
ii

, we can derive





 

0
00 11
22 33
012
6
1
ddd
ddd
7333
48
P
i
j
hK
KK
ii ii
V
j
ii ii
iii
uxyz
uP uP
uPuPxyz
KuPuP uPuP




3
i
  


Further,

7333
3733
33 73
48
333 7
h
hhh KT
K
uu



 


where


0123
hihihihi
uP uPuP uP

and


0123
T
hi hihi hi
uP uPuP uP


From the above equality, we can complete the proof of
Copyright © 2011 SciRes. AJCM
F. Z. GAO ET AL.
268
h
Lemma 4 easily.
4. Convergence Analysis
Now we consider the error estimates of the approximate
solution. Let

nn nnnnnn
hhhhh
uu uuuue
  
Choosing in (7), then we have
1n
tt



111 1nnnn
t
utvau vbuvguv

 x(25)
Subtracting (14) from (25), we obtain that

 



11
1
11
11
nn
hthhhh
nn
hhh
nn
hh hh
nn
hh
ev aev
rv av
bu vbu v
g
uguv
 


 
 

xx
n
t
(26)
where .

11nn
ht ht
ruu

 
1nn
Choosing hh in Equality (26), denote
by 12
and 1234
TT the left and right hand side
terms of Equality (26), respectively. We will analyze the
six terms successively.
h h
vee 
TTWW
For , from the definition of
1
W0
, we have that
2
1
100
1
2
nn
hh
Wee
t


2
(27)
Rewriting as
2
W





11
2
11
11
21 2223
nn nn
hhhhh
nn nn
hhhhhh
nn nn
hhhhhh
Wae eee
aeeae e
aeeaee
WW W

 
 




(28)
From (20) of Lemma 1, we can get the estimate to
as follows.
23
W
23 0W (29)
From (27)-(29), we have
22
12 01
22
11 1
01
1
22
1
24
nn
hh
nn n
hh h
t
WWe e
t
t
ee e
 


 



 


2
1
n
h
e
(30)
For each terms of the right hand side of (26). Using
interpolation theory, triangulation inequality and lemma
4, we know that

222 2
2
11
1000 2
nnnn
hhtt t
TCeetu hu


Similarly, we can bound as
2
T
11
221
nnn
hh
TChuee


Further, making use of triangulation inequality and
important inequality, we have that
22
12
211
nn n
hh
TCee hu


2
1
2
(32)
From the Lipschitz property of
g
ux in condition
2
C, making use of triangle inequality, important ine-
quality and Lemma 4, we have
22
41 1
420
nnn
hh
TChue e
 2
0

(33)
Combining (34),(35) with (36), we know that
22 2 2
1 21412
31121
nn nn
hh
TCeehu hu h


(34)
Combining (31), (32), (33) with (34) and applying
Sobolev space embedding theory, we know that the
of (26) satisfies
RHS

22 2
11
00
2
22 121
221
nn n
hh tt
nn
t
RHSCeetu
hhu u



2
0

(35)
From (30) and (35), using inverse estimate we know

22 22
11
01 01
222 2
11
00
1
2
22 121
22
1
22 2
1
4
1
nnn n
hhh h
nnn nn
hhh htt
nn
t
tt
eee e
t
ee Ceetu
hhu u



2
1
0





 
 
Further, we get that

22 2
11
010
22
11
10
2
222
22 1
11
2
02
22
2
1
nnnn
hhh h
nnn n
hhh h
n
nn
tt t
tt
eee e
tee Ctee
tu hhuu



2
1
2
0







(36)
Summing from 1 to with respect to in the
above inequality, we can obtain that
Nn
41
(31)


22 22
11
01 01
222
11
001
11
2
222
221
11
2
02
11
22
2
1
NNN N
hhh h
NN
nn nn
hh
hh
nn
NN
n
nn
tt t
nn
tt
eee e
teeCt ee
CttuCthh uu




 







 


(37)
Noting the equivalence of 0
and 1
with 0
Copyright © 2011 SciRes. AJCM
F. Z. GAO ET AL.269
and 1, respectively. Using the inverse estimate, we
have that there exist three positive constants
012

such that
22 22
2
021
0
01
01 h

 
Further, (37) may be rewritten as



22
2
01 2
01
0
2
2
21
00
00
2
22 121
22
0
22
1
N
Nn
hh
n
NN
n
n
htt
nn
Nnn
t
n
thete
CteCtt u
Cth h uu
 

 
 
 

(38)
Choosing in such way that
th
2
00 1
20th


,
further, (38) can be rewritten as


n
t
t
22
01
0
2
22
1
00
00
2
22 12 1
22
0
1
N
Nn
hh
n
NN
n
h
nn
Nnn
t
n
ete
CteCttu
Cth h uu


 


(42)
where 0
12
. Using discrete Gronwall’s lemma, we
know that

1
 

22
01
0
2
222
22 1
11
2
02
0
N
Nn
hh
n
Nn
nn
tt t
n
ete
Cttuh huu


 
(43)
Noting that , combining finite element space
interpolation theory, we can obtain the resulting error
estimates to the approximate solution as follows.
Nt T




1
0

2
2
0
hh
TH TH
LL
uu uu
Oh t
 

 (44)
where,


2
00
12
0
sup
sup .
n
TX TX
LL
X
ntT
Nn
X
NtTn
vvv
vt
 







Therefore we have the following theory.
Theorem 2. Suppose that the solution to the problem
(1) is sufficiently smooth. When and are small
enough and satisfy the relationship . The ini-
tial value is chosen as interpolation of , then the
Equation (44) holds.
h t
u
h
0
tO
0
h
u
5. Conclusions
In this paper, we continued our work [17] and presented
a class of upwind FVEM based on tetrahedron partition
for a three dimensional nonlinear convection diffusion
equation, analyzed and derived error estimate in -
norm and
2
L
1
H
-norm for the method. In the ongoing work,
we will discuss how to derive optimal error estimate in
-norm and how to code and present numerical results
to demonstrate the performance.
2
L
6. Acknowledgements
The research was partially supported by the Scientific
Research Award Fund for Excellent Middle-Aged and
Young Scientists of Shandong Province (grant no. BS-
2009HZ015), and NSFC (grant no. 10801092).
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