Journal of Applied Mathematics and Physics, 2013, 1, 16-20
http://dx.doi.org/10.4236/jamp.2013.13004 Published Online August 2013 (http://www.scirp.org/journal/jamp)
Optimal Convergence Analysis for Convection Dominated
Diffusion Problems
M. A. Mohamed Ali
Mathematics Department, Faculty of Science, Suez Canal University, Ismailia, Egypt
Email: mohamedali1961@yahoo.com
Received July 6, 2013; revised August 9, 2013; accepted September 1, 2013
Copyright © 2013 M. A. Mohamed Ali. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
In classical mixed finite element method, the choice of the finite element approximating spaces is restricted by the im-
position of the LBB consistency cond ition. The method of H1-Galerkin mixed finite element method avoids completely
the imposition of such a condition on the approximating spaces. In this article, we discuss and analyze error estimates
for Convection- dominated diffusion problems using H1-Galerkin mixed finite element method, along with the method
of characteristics. Optimal order of convergence has been achieved for the error estimates of a two-step Euler backward
difference scheme.
Keywords: H1-Galerkin Mixed Finite Element Method; Characteristics Method; LBB Condition; Optimal Error
Estimates; and Euler Backward Difference Scheme
1. Introduction
The convection-dominated diffusion problems have been
treated heavily using finite element methods [1-4]. Mixed
finite element method has been proposed by Douglas et
al. [5]. But these methods need to satisfy the Ladyzhen-
skaya-Babuska-Brezzi (LBB), consistency condition [6-9] ,
on the approximating spaces which restrict the choice of
the finite element spaces. In this case it is a special case
of those defined by Raviart and Thomas [10]. Pani [11]
has proposed and analyzed an H1-Galerkin mixed finite
element method which is not restrictive in the sense that
the approximating finite element spaces don’t need to
satisfy the LBB condition. Further, piecewise linear poly-
nomials can be considered for the approximating spaces.
Recently, an H1-Galerkin mixed finite element method
has been discussed for a class of second order Schrödinger
equation by LIU et al. [12].
Moreover, in convection dominated problems, standard
upwind finite difference methods are used for problems
which artificially smear fronts with excessive numerical
dispersion and produce solutions that depend strongly on
the orientation of the difference grids relative to the
streamlines of flow. Other standard techniques without
unwinding produce unacceptable oscillations in the ap-
proximations. These difficulties can be reduced sub-
stantially by using the Modified Method of Characteris-
tics (MMOC). This procedure was introduced and ana-
lyzed for a single parabolic equation by Douglas [13]
using backward single-step in the direction of charac-
teristic. Also, this procedure using two-step in the direc-
tion of characteristic has been analyzed by Ewing and
Russell [14], and the n exte nde d by R ussel l [1 5] to nonli ne ar
coupled systems in two and three spatial d imensions. The
H1-Galerkin mixed finite element along with the method
of characteristics has been applied to the convection do-
minated diffusion problems with a single step backward
Euler in the characteristic direction by Mohame d Ali [16] .
In this paper, to utilize the above advantages for the
convection dominated diffusion problems, we shall pro-
pose an H1-Galerkin mixed finite element method com-
bined with the method of characteristics, and examine the
rate of convergence for a Two-Step Euler backward dif-
ference scheme.
2. Variational Formulation
2.1. Consider the Convection-Dominated
Diffusion Problem
   


 

0
,,
,0,1
0,1,0,,0, 0,1.
uu u
cxbxaxf xt
txxx
xt J
ututux uxx


 




 
(1)
Assume that and , and the
00aa

00cx c
C
opyright © 2013 SciRes. JAMP
M. A. M. ALI 17
coefficients are smooth, where
0,1J. The above
problem arises in many applications involving diffusion
processes such as heat flow in a moving material, trans-
port of pollutants in lakes or channels, subsurface hydro-
logy and miscible displacement in porous media.
Setting

x
vaxu

ux
, Equation (1) can be rewritten as
 
,,
x
fxt
t
vv

 (2)
where
 
x
bx axcx
and

1
x
cx
.
Let
 

1
 
11
0:0 0HI HI


11
0
JH H
. We now
state the weak formulation as follows: Find a pair
such that

,:uv



, ,
x
I
fz

 

0
1
,,,
,,,
xx x
txx xx
aauw vwwH
buzvzvzzHI


1
.
 
(3)
Integrating by parts the first two terms of (3.b) we
have
 
 
10
,
1
x
vz
v z


 
,, ,
,1 0
tx xx
x
uz vzvz
fzv z


 
 0,
and hence with
 
1
x
ax
, we obtain



 
1
10
,,
,11 ,
tx x
x
vvzvz v


,
00
x
z
.
f
zvz zHI
 
 
 
 vz

0
(4)
Here, and 0. We shall use the
standard Galerkin method to solve equation , and
the characteristics method combined with the Galerkin
procedure to approximate (4).

11


3.a
With
 
22
xx

, let

x

denote
the characteristic direction associated with the operator
t
vv
x
, where





.
xx
x
xt




xx


Therefore, the term in (4) may be re-
placed by
,
tx
vvz


,
v
x
z

.
. Defining the bilinear form


 
1
,,,,
xx
A
vzv zvzzHI

 
Then Equation (4) can be rewritten as:
  

   
1
10
,,
,1100,
x
v
xzAvz
x
.
f
zvz vzzHI





 
In our error analysis we shall use standard Sobolev
spaces
m
H
I with norm m
. For simplicity, we
write

m
H
I as m
H
.
3. Second-Order Backward Euler Method
Let and be the finite dimensional subspaces of
h
Vh
W
1
H
I and
1
0
H
I
0r
, respectively. Assume that h and
satisfy the following approximation properties, for
and integers,
V
h
W0k
 


1,1,
11
0
1
1
1
inf, ;
inf ,
.
hh
rppp
hh
kk
hh
vV
r
hh WILIW I
wW
r
vvhvvChvvH H
wwhwwChw
wH

 
 
1
We shall approximate (5) using the second -order back-
ward Euler approximati on method, using the f orm ula
12
31
2
22
.
nn n
nuu u
u
tt



For the approximation along the characteristic direc-
tion, we define



, 2,
xx
x
xtxx
xx


t
 

111
,,
nnn
uuxtux


and

122
,.
nnn
uuxtux


Then, the time derivative along the characteristic di-
rection is approximated by
1
2
22
11
33
11
222
2
2
22
1
nn
nn nn
nvvvvv v
v
t
t
 


 






2
(6)
Then, the Galerkin procedure is to determine a pair
01
,:,,,
N
hh
UVt ttWV such that




  
2
1
31
22
10
,,,
2
,,
,1100,
nn
xxx h
n
nn n
nnn
x
h
aaUw VwwW
VV V
bzAVz
t
f
zVz VzzV
 








(7)
where

,,AA
,


. Here,
is cho-
sen large enough to ensure the coercivity of
A
, that is
there exists 00
such that

2
01
,A

Following Wheeler [17], we now define elliptic pro-
jections and
h
UW
h
VV
of and respect-
tively, through u v
Copyright © 2013 SciRes. JAMP
M. A. M. ALI
18




,0,
,0,
x
xx h
h
aauUw wW
bAvVz zV


(8)
Letting and
uU

vV
, it is now quite
standard to obtain the following estimates, for
,
1p



1
,1,
,1,
1
,1,,1,
,0,1;
,0,1.
rj
tt
jpr p
jpr p
kj
tt
jpk pjpkp
aChuu
bChvvj





 j
 
Following Pani and Anderssen [18], we have the fol-
lowing super converg ence results at the knot points
0,1x
 
2
1
10k
k
cCh

 .v (9)
4. A Priori Error Estimates for Double-Step
Backward Euler Method
Let

uU uUUU
  

and
 
vV vVVV

 
 .
Now, subtract (7.a) from (3.a) then use (8.a) to have
the following error equation in
 
,,,
nnn .
x
xx
aw wwW

 
h
(10)
Below we state the following lemma for our subse-
quent use (the proof of this lemma is given in Mohamed
Ali [16], p age 46) .
Lemma 4.1
Let be an approximate solution satisfying (7.a)
and be the elliptic projection of defined through
(8.a). Then satisfies the following estimates,
n
U
n
Uu
UU

1
1
1.
nk
k
Chv




n
In the remaining part of this section we shall assume
that the coefficients
and
are bounded, and
ax
is bounded below by a positive constant, then




d.
d
xx
K
xxx






(11)
Now define the elliptic projection of through
(8.b) and the estimates (9.b) and (9.c) are hold.
V
v
The starting procedure for (7.b) will be required to
satisfy
 
1
22
111
1
k
tKh

 
This can be achieved by using a first-order method on
a time step smaller than t
, followed by several uses of
a multi-step procedure on somewhat larger steps until
time t
is reached, for details, see Ewing and Russell
[14].
We now state and proof the following theorem.
Theorem 4.1
Suppose that (12) hold. Then the error
of proce-
dure (7.b) satisfies
 


121
22
1
0, ;0, ;
0
32
3
0, ;
max
.
k
nk
t
LTHLTH
nN
LTL
K
vv
v
Kt
 h






Proof
Choose n
z
in (7.b) and note that 00
. Then,
the error equation in
can be written as


 

 

2
1
31
22
2
1
31
22
2
1
31
22
1
0
2,,
2
,,
2
11 1,
00 0
n
nn nnn
n
nn
n
nnn n
n
nn
nn
nn
A
t
vvv
v
t
zt
z
 

 
 
n










  












So that


12
31
22
2
1
31
22
2
11 2
111
2
2,,
2
,,
1
2, ,
2
2,2,
,
2
nn n
nnn
n
nn
n
nnn n
n
nn n
nn
nnn n
nn
nn n
A
t
vvv
v
t
tt
tt
t
 

 
 

 
 













  




















 

 

 
22
10
9
1
12,
2
1110 00
,,,
nnn
nnn n
nnn
i
i
t
zz
FF


 








(13)
t
(12)
Copyright © 2013 SciRes. JAMP
M. A. M. ALI 19
To treat the left hand side of (13), we use the following
stability lemma.
Lemma 4.2
If (12) and (13) hold, then for ,
2lN
 


11
2
4
122
2
41
,,,
33
4,2
3
l
llnnl l
n
lnll k
n
At
F
tKh t
 





(14)
For proof of this lemma, refer to R.E Ewing and
Russell [14].
By the lemma 4.2, theorem 4.1, will be proved if we
can handle the terms on the right-hand side of (14). The
first term can be ignored because of the coefficient 1
3; a
recursion argument would sho w that it could be removed
if the other terms were multiplied by 23
11
13
3
 2
.
For the second term, expressions of the forms
2
1
nt

and 2
n
K
t

can be respectively
hidden on the left-hand side of (13) and eliminated by the
discrete Gronwall’s lemma. Hence, using (9.b), the first
term on the right hand side o (13) can be handled as:

22
22
1
1
,
nnn n k
k
Kh v
 
 .
I.e.,
1
22
22
10, ;
2
k
lnk
LTH
n
FtKhv


(15)
and as in Ewing and Russell [14], the following terms on
the right- h and side o f (13) is bounded by


22
2
3
24
23
20, ;
ln
nLTL
v
F
tK t

, (16)
2
1
31
22
ll
nn
nn
2
F
t




t
, (17)
2
2
4
22
ll
nn
nn
2
F
t


 

t, (18)
2
22
22
50, ;
2
k
lnk
tLTH
n
FtKhu

, (19)
1
22
22
60, ;
2
k
lnk
LTH
n
FtKhv


, (20)
2
22
22
70, ;
2
k
lnk
tLTH
n
FtKhu

, (21)
1
22
22
80, ;
2
k
lnk
LTH
n
FtKhv


, (22)
and as in Mohamed Ali [16], the last term is bounded by

(23)
1
22
4
90, ;
2
k
lnk
LTH
n
FtKhv


Inserting these results in (14) completes the proof of
the theorem.
5. Conclusion
The computational process of the classical mixed finite
element methods faces some difficulties while choosing
the finite element spaces due to the restrictive LBB con-
dition which is a must for such methods. In this paper,
the application of this condition has been avoided by
using the H1-Galerkin mixed finite element method. For
more accurate and fast results, we have used the mo-
dified characteristics method for two-step backward euler
time discretization. This allows choosing the finite ele-
ment spaces freely and maintaining the optimality of the
order of convergence of the analysis. This problem has a
wide range of applications in the real life such as in the
transport of air and water pollutants, in oil reservoir flow,
in the modeling of semiconductor and so forth.
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