A Posteriori Error Estimate for Streamline Diffusion
Method in Solving a Hyperbolic Equation
Davood Rostamy, Fatemeh Zabihi
Department of Mathematics, Imam Khomeini International University, Qazvin, Iran E-mail: [email protected]
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:
01
in 0,
,0 on 0
,0 ,0 on 0
,0 0 on (0, ]
tt
t
u a u f T
u x u x
u x u x
u x T
(1)
Here, dis a bounded open polygonal domain
with boundary and we have 1
2
0 0 , 1
u H u L ,
a is a scalar-value function in C
and
2 0, ; 2
f L T L .
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]).
2
2 0
2
in 0,
,0 on 0
0, 0 on (0, ]
t
w Aw F T
w x w x
w t T
(2)
Here, we assume vut for and
d
x t[0, ]T ,
also:
T, , , ,
w x t u x t v x t ,w x tt
,
u x t v x tt
, , t , T
0
0
1
T0
, 0 w x u x u x
a
I A
and
T, 0, ,
F x t f x t
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
0 1 be a subdivision of the time
in-terval into intervals
[0, ]T
N
t t
[0, ]T
0 t T
, 1n n n
I t t
,N 1
, with time
steps n n1 n, and introduce the corresponding space-time slabs, i.e.:
k t t n0,1,
, : , 1n n
S x t x t t tn
(3)for . Further, for each n let be a finite element subspace of
0,1, , 1
n N n
W
n
1 1
n
H S H S , (see Ad-ams, [16]) and let:
| 0, 0 , for
n n
n
W w W w t tI (4) We can formulate SD-method on the slab for (2),
as follows: n
S
For n 0, ,N1, find n such that: w W
,
, ,
n n n
t t n
n
t n n
w Aw g g Ag w g
F g g Ag w 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):
,
T dn Sn
w g
w g x td,n d
, T , .
n n
w g w x t g x t x
, lims 0
,
lims 0w x t w x ts
, lims 0
,w x t w x ts
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 N01 n and find| , such that:
n
w
W ww
,
B w g L g
(6)For gw and where the bilinear form B
., . andthe linear form L
.1
N
define by:
0 1
0 1
, ,
[ ],
n n
t t n
n N
n n
n n
B w g w Aw g g Ag
w g w g
where, we define w
w w1, 2
T such that for i1, 2:
wi wi,wi,,
w
w1 , w2
T
and
1
0 0
0
, ,
N
n n
L g F g gt Ag w 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
0
h S
n h
T
n h
KT . We introduce:
1 1 :
for , 0, 0 for
n
h n n k k
n
h n
W w H S H S w P K P K
T w t t I
k
where, P Kk
denotes the set of polynomials in K ofdegree less than or equal k and:
1
0 N
n
h h
n
W W
Thus (6) can be formulated as follows: Find whWh such that:
h,
B w g L g (7)
for gWh. Moreover, we know that the exact solution
fo
,
B w g L g
rgwand by use (6) and (7), we have the Galerkin
ogon
orth ality relation:
, 0B e g (8)
where,
. An a Posteriori Error Estimate for the
this section, we shall consider the following simplified
(9) where,
h
e w w .
3
SD-Method
In
version of SD-method for (7) with = 0: Find whWh,
such that for n0,1, ,N1:
1 N
1 , , 0 1 1 0 0 0 , [ ], , , , Nn n n n
h t h h n h
n
n n
N
n n
w Aw g w g w g
F g w g
0 hgW and
mplicit take
0 , 0
h
w .
For si y, we w00 and . In order
to the error, we
in
e
(10)
and denotes the adjoint of the operator L defined in nd
0
F
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
L A
t t T
x T 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
w w w (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 1 1 0 0 , , * , , , L T N t n n N N Tt n n
n n
e e e e L
e A
e e A
(12)We have for by part integrating:
d
(13) We define and
0,1, , 1 n N
1 0 1 0, d d
, , d d
, , d ,
n
n
T
t n S t
N
T T
n n S t
n N
T
n n t
n n
e e x t
e x t x t x e x t
e x t x t x e
T1, 2
e e e
1, 2
T andta
ob-in for n0,1, ,N1:
1 2 2 1 2 11 2 2 1
2 1 2 1 2 1 2 1
, d d
0 ( )
d d
1 0
.( ) d d
( ( ) )d d
d d
( )
d d ,
n n n n n n n T S n T S S S S S T n S
e A e A x t
a
e x
a
e e x t
e a e x t
a e e x t
a e e x t
Ae x t Ae
T T d d t (14) We define:According to (9),
1 0 1 0 2 1 1 2 1 0 1 1 0 0 , , , , , , , , , , , , [ ], ,[ ] T n n n N N N N N N N n n n nJ e x t x t dx
e e e e e e e e e e e e
N
.,tN T
0 and since
0
e w0 0, we get:
1 N
0
[ h], n
n
J w
(15) Then in (12), by use (15) and (16), we have:
1 2
1 1 1
2 N N N
( )
0 0 0
, , [ ],
t n h n
L
n n n
e e Ae w
1 2, N
e e
1 2
1 1
( )
0 0 0
1
0
1 1
,
0 0
1
0
, [ ],
,
[ ], ,
[ ],
N N
t n h n
L
n n n
N
h t h
n n
N N
h n h t h n
n n
N
h n
n
Ae w
w w A w w
w F w Aw
w
(16) where, is an interpolant of The idea is
ate u
tes for the Dual
e consider our interpolant in (16)
and
ˆ h
W
. now
to estim in terms of −1
e sing a strong
sta-ˆ
es for
bility estimat solution of the dual problem.
. Interpolation Estima
4
Solution
shall now
W ˆ Wh
, name
to be the space-time L2-projection of ly if the
first, we define the L2-projections:
2
: ( )
Pn L Whn
2 0, 2
: :
is constant on ,
n n n n
n
L S w L S w x
I x
,.
in space and in space time, respectively, by:
n
,
T d T d ,n h
P w x w x w
W
0,1
| ., d ,
n
n n I
n
w S w t t w
k
nThen, we can define ˆ n
n h
S W
by letting:
ˆ n
n n n h
S P W
n nP
where, Sn . Further, if we introduceP and
de-: fined by
P Sn Pn
Sn
and
Snn
Sn
respectively, then we can let to
Now, we define residual of computed solutio by:
n ˆ
h
W
be:
ˆ
h
P P W
nwh
0 h,t h
R Fw Aw
1 , , ,on
n n
h h
R w w S
,2 ,on
n
n h
n n
P I w
R S
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
RL2 :
12 2
2
( ) ( )
, xx
L L
R P C h IP R (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 p I
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( )
t L 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 2
2L L a posteriori error estimates for the scheme (9).
Theorem 1: The error e w wh, where w
so-lution of the continuous problems (2) and w
is the that of (9), satisfies the following stability estimate: h
1 1
2 2 2
2
0
( ) ( )
x e
L L
e C h IP R C 1
1 1
2 2
1 ( )
2
2 ( ) 2 ( )
x
e n L
x x
e L e n L
k R
h R k R
(20) Proof: Using the notation introduce above, we write (17) as:
may
1
1 1
2 ( )
[ ]
N N
L
w e
2 0
0 0
ˆ ˆ
, h ,( )
n n
n
n n n
R k
k I II
Below we shall estimate the terms I and II separately.
Splitting the interpolation error by writing
and , we have:
ˆ ˆ P P
ˆ n nP
12 2 1 0 0 1 1 ( ) ˆ n n N N
I R P P
0 0 0 0 20 ( ) ( )
ˆ , , N n n xx L L
R P R P
C h I P R
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 n n
0 R
,
, d d nt n
t
x x t x t
so that:
, d
, d dn n n
t n
n I I I
k x
x t t
x t (21) where,
and ˆn ˆ .,
tn :
1 0 1 0 1 0 1 1 2 0[ ] ˆ ,
[w ] ˆ , ˆ , [ ] , : N h n
n n n
N
h
n n
n n n
N
h
n n
n n n
N
h n
n n n
w
II k
k
k P P
k
w
k P
k
w
k P II II
k
To estimate II1, we use (21) to get:
1 2 2 1 2 2 1 1 1 0 1 10 n n
n n
R k Pk
1 1 0 1 1 02 ( ) ( )
1 ( ) ( )
ˆ ,
ˆ
ˆ
, ., d ., d d
., d d
n n n
n n N n n n n N n N t
n n I I t
n
N t
I t n
n
n L t L
n L t L
II k R P
R k P t t P t
R P t
k R P
k R
As for the
n
2
II -terms we can write:
1 2 1 2 0 1 , , 0 1 , , 0 1 , 0 , . n n I n n P I k
1 , 02 ( )
[ ]
,
, ( ., d
., d d
, d
, ., d d
n n n n n N h n
n n n
n n
N
h h
n n
n n n
n n
N
n h h
n I n n t I t n n N n h n n N n h n I I n n n n L w
II k P
k
w w
P I k k
P w w
P I t t
k
t
P I w
t t
P I w
P I t t
k k R
12( ) 2 2 ( ) 2( )
xx L k Rn L t L 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.
6. References
[1] M. Ainsworth and J. Tinsley Oden, “A Posteriori Error Estimation in Finite Element Analysis,” Wiley-Intersci-ence, New York, 2000.
[2] M. Asadzadeh, “A Posteriori Error Estimates for the Fok-ker-Planck and Fermi Pencil Beam Equations,” Mathe-matical Methods in the Applied Sciences, Vol. 10, No. 3,
2000, pp. 737-769. doi:10.1142/S0218202500000380
[3]
[4]
Element Method,” Cambridge
“Discontinous Galerkin Finite Element Hyperbolic Problems,”
Com-E. H. Gergouli, O. Lakkis and C. Makridakis, “A Poste-riori L1(L2)-Error Bounds in Finite Element
Approxima-tion of the Wave EquaApproxima-tion,” arXiv:1003.3641v1[math. NA], 2010, pp. 1-17.
C. Johnson, “Numerical Solutions of Partial Differential Equations by the Finite
University, Cambridge, 1987. [5] C. Johnson,
Methods for Second Order
puter Methods in Applied Mechanics and Engineering,
Vol. 107, No. 5, 1993, pp. 117-129.
doi:10.1016/0045-7825(93)90170-3
[6] M. Asadzadeh, “Streamline Diffusion Methods for the Vlasov-Poisson Equations. RAIRO Math,” Modelling and Numerical Analysis, Vol. 24, No. 3, 1990, pp. 177-
196.
4
Streamline Diffusion Methods for the Vlasov-Poisson- Fokker-Planck System,” Numerical Methods for Partial Differential Equations, Vol. 21, No. 2, 2005, pp. 472-495.
doi:10.1002/num.2004
tics of Computation, Vol.
[8] K. Eriksson and C. Johnson, “Adaptive Streamline Diffu-sion Finite Element Methods for Stationary Convection- Diffusion Problems,”Mathema
60, No. 4, 1993, pp. 167-188.
doi:10.1090/S0025-5718-1993-1149289-9
[9] S. C. Brenner and L. R. Scott, “The Mathematical Theory of Finite Element Method,”
Springer-1994.
Verlag, New York,
-122.
ent Approximation of the Hy-[10] F. Dubois and P. G. Le Floch, “Boundary Conditions for
Nonlinear Hyperbolic Systems of Conservation Laws,”
Journal of Differential Equations, Vol. 71, No. 3, 2001,
pp. 93
[11] C. Fuhrer and R. Rannacher, “An Adaptive Streamline Diffusion Finite Element Method for Hyperbolic Con-servation Laws,” East-West Journal of Numerical Mathe- matics, Vol. 5, No. 2, 1997, pp. 145-162.
[12] R. Codina,” Finite Elem
perbolic Wave Equation in Mixed Form,” Computer Methods in Applied Mechanics and Engineering, Vol.
197, No. 6, 2008, pp. 1305-1322.
doi:10.1016/j.cma.2007.11.006
[13] L. Haws, “Symmetric Greens Functions for Certain H
1016/0898-1221(91)90216-Q
y-perbolic Problems,” Computers & Mathematics with Ap-plications, Vol. 21, 1991, pp. 65-78.
doi:10.
59
[14] N, Iraniparast, “A Boundary Value Problem for the Wave
Equation,” International Journal of Mathematics and Mathematical Sciences, Vol. 22, No. 4, 1999, pp. 835-
845. doi:10.1155/S01611712992283
of a Selfadjoint Problem
demic Press, New [15] T. Kalmenov, “On the Spectrum
for the Wave Equation,” Akad. Nauk. Kazakh. SSR, Vest-nik, Vol. 1, No. 3, 1983, pp. 63-66.
[16] R. A. Adams, “Sobolev Spaces,” Aca York, 1975.
[17] A. Shermenew, “Nonlinear Wave Equation in Special Coordinates,” Journal of Nonlinear Mathematical Phys-ics, Vol. 11, No. 2, 2004, pp. 110-115.
doi:10.2991/jnmp.2004.11.s1.14
[18] E. Burman, “Adaptive Finite Element Methods for Com-pressible Two-Phase Flow,” Mathematical Methods in the Applied Sciences, Vol. 10, No. 2, 2000, pp. 963-989.
doi:10.1016/S0218-2025(00)00049-5
[19] C. Johnson and A. Szepessy, “Adaptive Finite Element
99-234. Methods for Conservation Laws Based on a Posteriori Error Estimates,” Communications on Pure and Applied Mathematics, Vol. 48, No. 3, 1995, pp. 1
doi:10.1002/cpa.3160480302
[20] R. Sandboge, “Adaptive Finite Element Methods for Sys- tems of Reaction-Diffusion Equations,” Computer Meth-ods in Applied Mechanics and Engineering, Vol. 166, No. 3, 1998, pp. 309-328.
doi:10.1016/S0045-7825(98)00093-0
