Applied Mathematics, 2011, 2, 981-986
doi:10.4236/am.2011.28135 Published Online August 2011 (http://www.SciRP.org/journal/am)
A Posteriori Error Estimate for Streamline Diffusion
Method in Solving a Hyperbolic Equation
Davood Rostamy, Fatemeh Zabihi
Department of Mat hematics, Imam Khomeini International University, Qazvin, Iran
Received December 23, 2010; revised May 30, 2011; accepted June 7, 2011
Abstract
In this article, we use streamline diffusion method for the linear second order hyperbolic initial-boundary
value problem. More specifically, we prove a posteriori error estimates for this method for the linear wave
equation. We observe that this error estimates make finite element method increasingly powerful rather than
other methods.
Keywords: Streamline Diffusion Method, Hyperbolic Problems, Wave Equations, Finite Element,
A Posteriori Error Estimate
1. Introduction
The wave equation based on rigorous a posteriori error
estimates is a largely subject, despite the importance of
these problems in the modeling of a number of physical
phenomena. A posteriori have made every method in-
creasingly powerful; such that there are various ap-
proaches to a posteriori error estimates and it has new
successfully applied to varied problems by several au-
thors (see Ainsworth and Tinsley Oden [1]; Asadzadeh
[2]; Gergouli [3]; Johnson [4] and [5]).
Gergouli et al. [3] and his teammates applied finite
element method for linear wave equation and obtained a
posteriori error estimates in L (L2) norm in Johnson
proved existence solution for second order hyperbolic
problems and used discontinuous Galerkin method for
them and obtained a priori and a posteriori error esti-
mates. In this paper, we do new work and use streamline
diffusion method (SD-method) for solving the linear
second order hyperbolic initial-boundary value problem.
Streamline diffusion methods (Asadzadeh [6]; Asadza-
deh and Kowalczyk [7]; Eriksson and Johnson [8];
Brenner [9]; Dubois [10]; Fuhrer [11] ) perform slightly
better than the standard finite element methods for
smooth solutions and non-smooth solutions hyperbolic
problems as a two-dimensional one which both is higher
order accurate and has good stability properties. Due to
the fact that artificial diffusion is added only in the char-
acteristic direction so that internal layers are not smeared
out, while the added diffusion removes oscillations near
boundary layers.
We consider the linear second order hyperbolic initial
boundary value problem (see Codina [12]; Haws, [13];
Gergoulus et al., [3]; Iraniparast [14]; Kalmenov [15]; in
Sobolov space Adams [16]; Shermenew [17]) as follows:


 

 

0
1
in 0,
,0on 0
,0,0on 0
,00on(0,]
tt
t
uauf T
uxu x
ux ux
ux T
 










(1)
Here, d

 is a bounded open polygonal domain
with boundary  and we have



12
00 1
,uH uL,
a is a scalar-value function in

C and




22
0, ;fL TL

.
For (1), we use one variable changing and obtain a
new problem. We apply SD-method for new problem
and obtain a posteriori error estimates. A posteriori error
bound provides a computable upper bound on the error in
some norm using the computed finite element solution
(see Ainsworth and Tinsley Oden [1]; Asadzadeh [2];
Burman [18]; Johnson and Szepessy [19]; Sandboge
[20]).
In order to make use of the theory of Semigroups we
write the system (1) in the following abstract form:

982 D. ROSTAMY ET AL.
 
  
 
2
2
0
2
in 0,
,0on0
0,0on(0, ]
t
wAwFT
wxw x
wt T
 






(2)
Here, we assume for and
t
vud x[0, ]tT

,
also:
  
T
,,,,wxtuxt vxt,

 

T
,,,,
ttt
wxtuxtvxt
  

T
001
0,
0wx uxux a






I
A
and
  
T
,0,,
F
xtfxt
where, I is identity matrix.
The rest of this study is organized as follows. In Sec-
tion 2, we define slabs for space-time domain and obtain
SD-method for (2) this slabs. In Section 3, we consider a
posteriori error estimates for SD-method form of Section
2 and obtain dual problem. In Section 4, we define in-
terpolation estimates for dual problem. In Section 5, we
complete proof for a posteriori error estimates by using
definitions in Section 4.
2. The Streamline Diffusion Method
In this section, we consider the SD-method for solving (2)
that is based on using finite element over the space-time
domain . To define this method, let
01 be a subdivision of the time in-
terval into intervals
[0,]T
N
tt
[0,]T
0t T 

1
,
nnn

I
tt


, 1N
, with time
steps 1nn n , and introduce the
corresponding space-time slabs, i.e.:
kt t

 0,1,n
 
1
,: ,
nn
Sxtxttt



n
(3)
for . Further, for each n let be a
finite element subspace of
0,1, ,1nNn
W

n
 
11
n
H
SHS, (see Ad-
ams, [16]) and let:


|0, 0 , for
nn n
WwWwt tI 
 (4)
We can formulate SD-method on the slab for (2),
as follows:
n
S
For , find such that:
0, ,1nN n wW


 
,
,,
nn n
tt
n
n
tn
n
wAwg gAgwg
FggAgw g





 
,
n

(5)
where, Ch  with C is a suitable chosen (suffi-
ciently small, see Johnson, [18]) positive constant and
parameter h is defined in the following. Further, we de-
fine the following notations for (6):


,d
T
nSn
wgwgxt
d
,d
n

,,.
T
nn
wgwxtgxtx





00
, lim,lim
ss
wxt wxts




0
,lim ,
s
wxt wxts




The terms including ,

 in the above formula is a
jump conditions which imposes a weakly enforced con-
tinuity condition across the slab interfaces, at tn and is the
mechanism by which information is propagated from one
slab to another. For more concisely, after summing over
n, we may rewrite (5) as follow:
We assume and find|, such that:
1
0
Nn
n
w



Www


,Bwg Lg

(6)
For
g
w

 and where the bilinear form


., .B and
the linear form


.L
1N
define by:
 

0
1
0
1
,,
[],
nn
tt
n
n
Nnn
n
n
Bw gwAwggAg
wg wg



 





where, we define such that for

T
12
,www

1, 2i

:
  

,, 12
,,
iii
wwww ww

 T
and
  
1
00
0
,,
N
n
n
LgFggtAgw g 



 

For ,we define such that be a triangulation
of the slab n into triangles K satisfying as usual the
minimum angle condition (Ciarlet [21]) and assume that
the parameter h is represented with the maximum di-
ameter of the triangles
0hS
n
h
T
n
h
K
T. We introduce:
   



11
:
for,0,0for
n
hnnk
k
n
hn
WwHSHSwPK PK
Tw ttI
 

k
where,


k
PK denotes the set of polynomials in K of
degree less than or equal k and:
1
0
Nn
hh
n
WW



Thus (6) can be formulated as follows:
Find h
wW
h

such that:

 
,
h
Bw gLg (7)
for h
g
W

. Moreover, we know that the exact solution
of (6) satisfies:

D. ROSTAMY ET AL.
983

fo

,BwgLg
r
g
wand by use (6) and (7), we have the Galerkin
ogonorthality relation:

,0Beg

(8)
where,
. An a Posteriori Error Estimate for the
this section, we shall consider the following simplified

(9)
where,
h
eww .
3SD-Method
In
version of SD-method for (7) with  = 0: Find whWh,
such that for 0,1, ,1nN :
1N


1
,,
01
1
00
0
,[], ,
,,
N
nn n n
ht hhnh
n
nn
N
n
n
wAwgwg wg
Fgw g









0
h
g
W and
mplicittake
0
,0 h
w.
For siy, we 00w

and . In order
to the error, we
ine
(10)
and denotes the adjoint of the operator L defined in
nd
0F
con obtain a representation of sider the
following auxiliary problem, referred to as the linearized
dual problem: Find  such that:
 

1
*
0,0, 0,
,0,
T
t
LA
ttT
xT x

 




*L
(2) a is a positive weight function. Note that this
problem is computed “backward”, but there is a corre-
sponding change in sign. Further, we shall first introduce
the following notation:

2
1/2
() , L
www

  (11)
Multiplying (10) by e and integrating by parts and
summing over n, we obtain the following error represen-
tation formula:




1
2
1
()
1
0
11
00
,,*
,
,,
L
T
Ntn
n
NN
T
tnn
nn
eeeeL
eA
eeA
 






 
 



 
(12)
We have for by part integrating:
d
(13)
We define and
0,1, ,1nN





 
1
0
1
0
,dd
,,d d
,,d,
n
n
T
tt
S n
NTT
nn t
S
n
NTnnt
n
n
eext
extxt xext
extxtx e







 
 



T
12
,eee

T
12
,

 and
ta
ob-
in for 0,1nN, ,1

 :





 
1
2
2
12
1
1221
21 21
21 21
,dd
0()
dd
10
.() dd
(())dd
dd
()
dd ,
n
n
n
n
n
n
n
T
S n
T
S
S
S
S
S
T
n
S
eAeAxt
a
ex
a
ee xt
eae xt
aeext
aeext
Aex tAe







 

 

 







 
 














TT
dd
t
(14)
We define:
According to (9),





1
0
10
21
12
1
0
11
00
,,
,,
,,
,,
,,
,,
[],,[ ]
Tnn
n
NN
NN
N
NN
nn
nn
Jextxtdx
ee
ee
ee
ee
ee
ee



 
 
  
 
 




  
 
 
  

 



 




N


., 0
N
tT   and since


0
e
00w

, we get:
1N
0
[],
hn
n
Jw




 (15)
Then in (12), by use (15) and (16), we have:

 1
2
11 1
2NN N 
() 00 0
,,[],
th n
n Lnn n
eeAew
 
 

 
 
So that recalling (9) and using the Galerkin orthogo-
nality (8), we obtain:

984 D. ROSTAMY ET AL.

1
2,
N
ee








1
2
1 1
() 00 0
1
0
11
,
00
1
0
,[],
,
[], ,
[],
N N
thn
n Lnn n
N
hh
t
nn
NN
hn hth
n
nn
N
hn
n
Aew
ww Aww
wFwAw
w
 
 

 








 

 
 
 



(16)
where, is an interpolant of The idea is
ate u
tes for the Dual
e consider our interpolant in (16)
and
ˆh
W . now
to estim in terms of −1esing a strong sta-
ˆ

es for bility estimatsolution  of the dual problem.
. Interpolation Es tim a 4Solution
shall nowWˆh W
, nameto be the space-time L2-projection of ly if the
first, we define the L2-projections:
2
:()PL Wn
nh
 



20,2
::
is constant on ,
nn nn
n
LSwLS wx
Ix
 


 ,.
in space and in space time, respectively, by:
n

,d d,
TT
nh
P
wxwx w 
 W


0,
1
|.,d,
n
nn I
n
wSw t tw
k
 

 n

Then, we can defineˆn
nh
SW by letting:
ˆn
nnnh
SP W 
n n
P  
where, n S . Further, if we introduce P and  de-
: fined by
 
nn n
P
SP S
and
 
nn n
SS  
respectively, then we can let to
Now, we define residual of computed solutioby:
n
ˆh
W be:
ˆh
PPW     
nh w
0h,th
RFwAw 
1, ,
,on
nn
hh
Rw wS




,
2,on
n
nh
n
n
PIw
RS
k



where, I is the identity operator.
In the end of this section, we shall give a lemma for
projection operators P,
le
interpolation estimates by the
aving the overall of I and II to next section.
Lemma 1: There is a constant C such that for residual


RL
2

:

1
22
2
() ()
,xx
LL
RP ChIPR
 

  (17)
Proof: (see Johnson and Szepessy [19] and Sa
[20]).
Completion of the Proof of a
n osteriori error esti-
ate by estimating of the terms I and II in the error rep-
ndboge
5. The
Posteriori Error Estimates
this section we state and prove a pI
m
resentation formula (16). To this approach we introduce
the stability factors (see Burman [18]) associated with
discretization in time and space, defined by:
2()
tL
t




1
2()
e
L
e 
(18)
and
2
1
2
()
()
xx L
x
e
L
e






 (19)
respectively. We now apply the result of the previous
sections; using Catchy-Schwartz inequality in (16) cou-
pled with the interpolation estimate (17) and the strong
stability factors (18) and (19), to derive the


22
LL a
posteriori error estimates for the scheme (9).
Theorem 1: The error h eww , where w so-
lution of the continuous problems (2) and wis the that of (9),
sa
h
tisfies the following stabte: ility estima
 11
222
2
0
() ()
x
e
LL
eChIPR C 
 

 1
11
22
1
()
2
22
() ()
x
en
L
xx
een
LL
kR
hR kR



 



(20)
Proof: Using the notation introduce above, we
write (17) as:
may
 1
11
2
()
[]
NN
L
w
e



20
00
ˆˆ
,, ()
h
nn
n
nn
n
Rk
k
III



 


Below we shall estimate the terms I and II separately.


D. ROSTAMY ET AL.
985
Splitting the interpolation error by writing
and , we have:
ˆˆ
PP  ˆnn
P   



1
22
1
0
0
11
()
ˆ
n
n
NN
IRPP




00
00
2
0() ()
ˆ
,,
N
n
n
xx
LL
RP RP
Ch I PR 




 
 

where we have used the fact that is constant in the
time, (making the first integral zero) and then using in-
terpolation estimate (17) in the second integral. It re-
mains to estimate the terms II, to this end, we need the
following notation:
n
nn
 

0
R
 
,,dd
n
t
n
t
x
xtx t








so that:
 
,d,dd
nnn
t
n
nIII
kx xttx

t



 
 (21)
where,




 and

ˆˆ
.,
nn
t :




1
0
1
0
1
0
1
12
0
[]ˆ
,
[]wˆ
,
ˆ
,
[]
,:
Nh
n
nnn
Nh
nn
nnn
Nh
nn
nnn
Nh
n
nnn
w
IIk k
kPP
k
w
kP
k
w
kPII II
k



















To estimate 1
I
I, we use (21) to get:

 

1
22
1
22
1
11
0
1
1
0
nn
nn
RkPk



1
1
0
1
1
0
2() ()
1() ()
ˆ
,
ˆ
ˆ
,.,d.,dd
.,d d
nnn
nn
N
nn n
n
N
n
Nt
nn IIt
n
Nt
It n
n
nt
LL
nt
LL
IIkRP
RkPt tPt
RP t
kR P
kR




 
 




















As for the
n
2
I
I-terms we can write:











1
2
1
2
0
1,,
0
1,,
0
1,
0
,.
nn
I
nn
PI
k



1,
0
2()
[]
,
,(.,d
.,d d
,d
,.,dd
n
nn
nn
Nh
n
nnn
nn
Nhh
nn
nnn
nn
Nnhh nI
nn
t
It n
n
Nnh
n
n
Nnh
n
II
nnn
nL
w
IIk P
k
wwPIk
k
Pw wPI tt
k
t
PIw t t
PIw PIt t
k
kR
































1
222
2
()() ()
xx nt
LLL
kR



The a posteriori error estimate now follows immedi-
ately after collecting the terms and using the definition of
the stability factors (18) and (19).
For 0


in (7), we can obtain a posteriori error es-
timates with similar way.
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