Research Journal of Mathematics and Statistics 3(3): 97-106, 2011
ISSN: 2040-7505
© Maxwell Scientific Organization, 2011
Submitted: May 27, 2011
Accepted: August 05, 2011
Published: September 25, 2011
Numerical Studies for Singularly Perturbed Convection Reaction Diffusion
Problems in Two Dimensions
1
N.H. Sweilam, 2M.M. Khader and 3F.M. Atlan
Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
2
Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt
3
Department of Mathematics, Faculty of Science, Amran University, Sa'ada, Yemen
1
Abstract: In this study, we study the numerical solutions of singularly perturbed convection-reaction-diffusion
problem over a square. Two different approaches are used to study the problem namely: Finite Element Method
(FEM) based on a posteriori error estimate, and Variational Iteration Method (VIM). In this study, we use the
adaptive h-refinement based on the a posteriori error estimate. Numerical simulation for the solution of the
proposed problem using FEM and VIM is given. From this simulation we can found that the finite element
method gives the solution with excellent agreement compared with the exact solution. Also, it's found that VIM
is invalid for some values of the perturbed parameter g, but FEM overcome on this shortcoming.
Key words: A posteriori error estimates, AMS Subject Classification (2000), finite element method, mesh
refinement, stationary convection reaction diffusion problem, variational iteration method
|||.|||. It uses only the information which is available during
the solution process, such as the discrete solution itself
and the data of the problem. In the 1970s, (Babuska and
Rheinboldt, 1978) did pioneering work on the a posteriori
error estimator. Throughout the 1980s and into the 1990s,
more approaches for analyzing the a posteriori error
estimators were developed for many classes of partial
differential equations. One of the problems with these
techniques is the frequent presence of problem-dependent
constants in the diffusion convection reaction equations.
These constants may dominate the a posteriori error
estimate in such a way that the error estimator does not
give reliable information for the adaptive algorithm.
Recently, efforts have been made to make the constants
only weakly depend on the problem. Several a posteriori
error estimators of this type are described in (Kunert,
2003) and (Verfürth, 1998).
We introduce the a posteriori error estimators for
finite element discretizations standard Galerkin or
Streamline-upwind Petrov-Galerkin method (SUPG) of
problem (1), this estimator is based on the evaluation of
local residuals. The estimator yields global upper and
lower bounds on the error of the finite element
discretization measured in a norm that incorporates the
standard energy norm of problem (1) and a dual norm of
the convective derivative. We have used a Matlab code
ffw (http://openffwgooglecode.com/svn/homepage/
downloads.htm) to solve the diffusion convection reaction
equation on any polygonal domain, using the standard
Galerkin finite element method. The code is based on
continuous piecewise linear basis on triangle element.
INTRODUCTION
Singularly perturbed problems arise in various
models of fluid flow (Hirsch, 1988) and (Hirsch, 1990)
such as in the Navier-Stokes equations (Kreiss and
Lorenz, 1989), in the equations modeling oil extraction
from underground reservoirs (Ewing, 1983), flows in
chemical reactors (Alhumaizi, 2007) and convective heat
transport with large Peclet number (Jakob, 1959). In this
paper, we interest to study the numerical solutions of the
following stationary linear convection reaction diffusion
problem in two dimensions:
− ε∆ u + b. ∇ u + cu = f in S
u = 0,
on MS
(1)
in a polygonal domain SdR2 with Lipschitz boundary
MS = 'D. The data must satisfy the following conditions:
(A1): 0< g <<1;
(A2): b ∈ [W 1,∞ (Ω )] , c ∈ L∞ (Ω ) , b
2
L∞
+ c
L∞
= Ο (1);
1
2
(A3): c( X ) − ∇.b( X ) ≥ 1;
(A4):
Γ − = {x ∈ Γ D : b( X ). n( X ) < 0} ⊂ Γ D
The a posteriori error estimator is used for estimating
the local errors so that adaptive mesh refinement can be
controlled. It estimates the error of the computed discrete
solution uh and the unknown solution u of the diffusion
convection reaction problem in some prescribed norm
Corresponding Author: N.H. Sweilam, Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
97
Res. J. Math. Stat., 3(3): 97-106, 2011
C
Over the last decades several analytical and
approximate methods have been developed to solve the
linear and nonlinear differential equations. Among them
the Variational Iteration Method (VIM) (Biazar and
Ghazvini, 2007; Goha et al., 2010; He, 1997), which is
proposed as a modification of the general Lagrange
multiplier method. Also, VIM is based on the use of
restricted variations and correction functional which has
found a wide application for the solution of nonlinear
differential equations (He, 2000; Sweilam and Khader,
2007; Sweilam and Khader, 2010). This method does not
require the presence of small parameters in the differential
equation, and does not require that the nonlinearities are
differentiable with respect to the dependent variable and
its derivatives. This technique provides a sequence of
functions which converges to the exact solution of the
problem. It has been shown that this procedure is a
powerful tool for solving various kinds of problems, for
example, VIM is used for solving the delay differential
equations in (He, 1997). This technique solves the
problem without any need to discretization of the
variables, therefore, it is not affected by computation
round off errors and one is not faced with necessity of
large computer memory and time. The proposed scheme
provides the solution of the problem in a closed form
while the mesh point techniques, such as finite difference
method provides the approximation at mesh points only.
In what follows we use the following notations and
definitions:
C
D u=
C
a p b ⇔ a ≤ cb, a ≅ b ⇔ a p b and b p a ,
where the constant c must be independent of any mesh
size h and of , (Adams, 1975).
FINITE ELEMENT DISCRETIZATION
In this section, we implement the finite element
method of Eq. (1) (Verfürth, 1998). For any bounded
subset T of S with polygonal boundary ( we denote by
Hk(T), k , N, L2(T) = H0(T), and L2(() the usual Sobolev
and Lebesgue spaces equipped with the standard norms.
.
and .
0,γ
= .
2
}
1
2 2
0
L2 (γ )
{
on H1(T)
Set H D1 (Ω ): = v ∈ H 1(Ω ): v = 0 on Γ D
}
Then the standard variational formulation of problem (1)
is: find u ∈ H D ( Ω) such that:
1
a(u, v ) = f (v ) ∀ v ∈ HD1 (Ω )
(2)
where,
a(u, v ) = ∫ ε∇ u. ∇ vdx − ∫ b. ∇ u vdx + ∫ cuvdx
Let m be a non-negative integer and let 1#p#4. The
Sobolev space Wm,p (S ) of order (m,p) is the linear
space of functions (or equivalence classes of
functions) in Lp (S ) whose distributional derivatives
D"u are in Lp(S ):
Ω
and f ( v ) =
Ω
∫
Ω
fv dx
Ω
W m, p ( Ω ) =
are the bilinear form and the linear functional defined for
{v ∈ L ( Ω), D v ∈ L ( Ω), for 0 ≤ α ≤ m}
C
H k (ω )
{
∂ x 1α 1 , ∂ x 2 α 2 ,..., ∂xnα n
α
= .
|||u|||ω = ε ∇ u 0 + u
∂αu
p
k ,ω
and (Adams, 1975). Similarly, (.,.)T and (.,.)( denote the
scalar products of L2(T) and L2((), respectively. If T = S
we will omit the index S. On the other hand, |||.|||T denotes
the canonical restriction of the energy norm
Let " = ("1,"2, …., "n) be an n-tuple of non-negative
integers and denote *"*=("1+"2+…+"n). Then by
D"u we shall mean the "th derivative of u defined by:
α
We also use the following convention:
all u, v ∈ H D ( Ω) . Problem (2) admits a unique solution
p
1
(Kunert, 2003). Moreover, assumptions (A1)-(A4) and
integration by parts imply that:
The Hilbert-space H(div, S) = {v 0 L2(S): div v 0
L2(S)} and the inner product is given by the bilinear
form:
1
a( v , v ) ≥||| v|||2 , v ∈ H D
( Ω)
a( u, v ) = ( u, v ) L2 ( Ω ) + ( div u, div v ) L2 ( Ω)
and
98
(3)
Res. J. Math. Stat., 3(3): 97-106, 2011
a(v , w) ≤ ||| v||| ||| w|||{1+ ||c||L∞ }
+ |||v||| || w||0 ε −1/ 2 ||b||L∞ ,
∀ v, w ∈
HD1
Finally, we introduce some useful notations: Eh
denotes the set of all n-1-faces inTh. It can be split in the
form Eh = Eh,S c Eh,D where Eh,S and Eh,D refer to interior
faces, and faces on the Dirichlet boundary 'D,
respectively. For E 0 Eh, hE is the diameter of E. The
shape regularity implies that hT – hE and hT – hT!
whenever, E d MT and T 1 T! … N. For any piecewise
continuous function N and any Eh, S, we denote by [N]E the
jump of N across E in an arbitrary but fixed direction nE
orthogonal to E. For any T 0 Th and E 0 Eh we set:
(4)
(Ω )
We denote by Th, h >0, a regular triangulation of S
into n-simplices, which satisfies the following two
properties:
C
C
Admissibility: any two elements are either disjoint or
share a complete smooth k- dimensional, 0#k# n-1,
sub-manifold of their boundaries;
ωT =
ω~T =
sup hT / ρ T p1
T ∈Th
Shape regularity: sup
h >0
Set
{
= v ∈ C( Ω): v|T ∈ Pk , ∀T ∈ Th
{v ∈ s ( Ω): v = 0,
1
h
on Γ D
T ′,
E ⊂∂ T ′
A RESIDUAL ERROR ESTIMATOR
In this section, we introduce the error estimators as
given in the following theorem.
}
1
sh1, D ( Ω) = sh1 ( Ω) I H D
( Ω) =
U
T I T ′≠ φ
T ′,
ωE = U T′
Here, hT and DT denote the diameter of T and the
diameter of the largest ball insribed into T.
For k ,N, we denote by pk the set of polynomials of
degree at most k and set:
sh1
U
φ ≠ T I T ′∈Eh
Theorem 1: (Verfürth, 1998)
Denote by u and uh the unique solutions of problems (2)
and (5), respectively. Let fh be arbitrary approximations of
f by piecewise polynomials of degree at most k with
respect to Th:
}
⎧
⎫
Set α S = min⎨ hsε 2 ,1⎬ , S ∈ Th U Eh ,
1
Now, we can consider the following discretization of
⎩
problem (1): Find u ∈ sh , D ( Ω) such that:
1
⎭
and
aδ ( uh , vh ) = f δ ( vh ) ∀ vh ∈ sh1, D (Ω )
ηR2 ,T = α T2 || f h + ε∇ uh − b. ∇ uh − cuh ||20,T
(5)
+
where,
aδ ( uh , vh ) = a( uh , vh ) +
∑
T ∈Th
δT ( − ε∆ uh + b. ∇ uh + cuh , b. ∇ uh ) T ,
fδ ( vh ) = f ( vh ) +
∑
T ∈Th
(8)
0, E
Then the following a posteriori error estimates are valid:
δT ( f , b. ∇ vh ) T
⎫
⎧⎪
2 ⎪
||| u − u h ||| p ⎨ ∑ η R ,T ⎬
⎪⎭
⎪⎩ T ∈Th
1/ 2
⎧⎪
⎫⎪
+ ⎨ ∑ α T || f h − f ||0,T ⎬
⎪⎩ T ∈Th
⎪⎭
et al., 1992), if δT > 0 ∀ T ∈ Th . From assumptions (A1)(A4) and a local inverse estimate it is follows that Eq.(5)
admits a unique solution if δ T p hT2 (Franca et al., 1992).
{(
η R ,T p 1+ || c|| L∞ (ω E )
In what follows we assume that:
∀T ∈ Th .
2
(6)
Equation (5) is the standard Galerkin approximation,
if δT = 0 ∀ T ∈ Th and the SUPG discretization of (Franca
δ T p hT ,
1
−1
∑
ε 2α E J ( uh )
∂
I
Ω
⊂
E
T
2
+ ε −1/ 2 || b||
(7)
99
(
L∞ ω E
1/ 2
,
)
⎫
⎭
) α T ⎬ α T || f h − f ||0,ω T
Res. J. Math. Stat., 3(3): 97-106, 2011
{(
{ε
η 2 R, T p 1+ ||c||L∞(ω E )
+ Tmax
∈Th
−1/ 2
)
The VIM gives the possibility to write the solution of
Eq.(10) with the aid of the correction functional:
}
||b||L∞(ω ) α T ⎫⎬ ×
E
⎭
⎧
|||u − uh |||ω T + ⎨ ∑ α T2 f h − f
⎩ T ∈Th
2
0, T
⎫
⎬
⎭
0
1/2
[
] , if E ∈ E
E
⎫
⎪
⎬
if E ∈ Eh , D ⎪⎭
h ,Ω
]
It is obvious that the successive approximations un,
n$0 (the subscript n denotes the nth order approximation),
can be established by determining the general Lagrange
multiplier, 8, which can be identified optimally via the
variational theory. The function u~n is a restricted
variation, which means δu~ = 0 . Therefore, we first
where
⎧
⎪ − ε∂nE uh
J ( uh ) = ⎨
⎪⎩ 0,
[
t
un +1 = un + ∫ λ (τ ) Lun + Ru~n + F ( u~n ) dτ , n ≥ 0 (11)
(9)
n
determine the Lagrange multiplier 8 that will be identified
optimally via integration by parts. The successive
approximations un,n $1, of the solution will be readily
obtained upon using the Lagrange multiplier obtained and
by using any selective function u0. The initial values of
the solution are usually used for selecting the zeroth
approximation u0. With 8 determined, then several
approximations u n , n $ 1, follow immediately.
Consequently, the exact solution may be obtained by
using:
Remark 1: In the upper bounds for|||u-uh||| one may
replace fh by f. The second term on the right-hand side of
the first estimate of Theorem 1 then of course vanishes.
The main idea is to generate a sequence of solutions
on successively finer mesh, at each stage selecting and
refining those elements that are judged to contribute most
to the error. The process is terminated either when a
maximum number of elements T is exceeded, or when
each triangle contributes less than a preset tolerance, v. In
the end of this section, we present an algorithm for a
posteriori error estimate based adaptive mesh refinement.
u = lim un
SOLUTION PROCEDURE USING VIM
The algorithm is given by the following steps:
C choose an initial mesh
C while the number of elements are not too many do
C compute uh
C for all elementsT do
C compute the element indicator 0R,T defined by (8),
C if 0R,T > v then
C refine element T
C end if
C end for
C end while
In this section, we implement VIM to obtain the
approximate solutions of the same problem (1). Now, to
illustrate how to find the value of the Lagrange multiplier
8, we will consider the following case, which dependent
on the order of the operator L in Eq.(10), we study the
case of the operator:
L=
ANALYSIS OF THE VARIATIONAL
ITERATION METHOD
∂2
(Without lose of generality).
∂ x2
We rewrite Eq.(1) in the following form:
uxx + uyy − (1 / ε )(b. ∇ u + cu − f ) = 0
To illustrate the analysis of VIM, we limit ourselves
to consider the following nonlinear differential equation
in the type:
Lu+Ru+F(u) = 0
(12)
n→∞
(13)
Making the correction functional stationary, and noticing
~
that δun = 0 we obtain:
(10)
with specified initial and boundary conditions, where L
and R are linear bounded operators i.e., it is possible to
find numbers m1,m2>0 such that ||Lu|| # m1 ||u||, ||Ru|| # m2
||u||. The nonlinear term F(u) is Lipschitz continuous with
δun+1 = δun + δ ∫ λ (τ )
x
[
| F (u) − F (v )|≤ m| u − v|, for some constant m > 0.
100
0
unττ +u~nyy −(1/ ε )( b.∇u~n +cu~n − f
) ] dτ
Res. J. Math. Stat., 3(3): 97-106, 2011
[
= δun + − λ&(τ )δun + λ (τ )δu&&n
]
⎛ πx ⎞
− εy sin⎜ ⎟ (1 − y )
⎝ 2⎠
τ =x
x
+ ∫ λ&&(τ )[δun ]dτ = 0
⎛ πx ⎞
⎛ πx ⎞
+ πε 2 y cos⎜ ⎟ (1 − y ) + 2ε ( x − 1) sin⎜ ⎟
⎝ 2⎠
⎝ 2⎠
0
−
~
where, . denotes the differential w.r.t. x and δun = 0 is
~
considered as a restricted variation i.e., δun = 0 yields the
The exact solution of this problem is given by:
following stationary conditions:
λ&&(τ ) = 0, 1 − λ&(τ )
τ=x
1 2
⎛ πx ⎞
επ y sin⎜ ⎟ (1 − x )(1 − y )
⎝ 2⎠
4
= 0, λ (τ )
τ=x
=0
⎛ πx ⎞
u( x , y ) = εy sin⎜ ⎟ (1 − x )(1 − y )
⎝ 2⎠
(14)
Equation (14) is called Lagrange-Euler equation with its
natural boundary conditions. The Lagrange multiplier,
therefore, can be readily identified: λ (τ ) = τ − X .
Now, the following variational iteration formula can
be obtained:
We start with an initial approximation, and by using the
iteration formula (15), we can obtain directly the
components of the solution as follows:
un +1( x , y ) = un ( x , y ) +
u1( x , y ) =
X
∫
0
(τ − x)[unττ + unyy − (1 / ε )(b.∇ un + cun − f ) ]dτ .
u0 ( x , y ) =
(15)
(
−
− ε∆ u − (2 + x )ux − 3 + y uy + u = f
(
)
(4ε (32 − 2π 2 x + − 32 + 2π 2 x( x − 1)
(
))
(
1
ε 256 − 16π 2 x + − 256 − π 2 x ( x − 1
2π 3
⎛
⎛ πx ⎞
× ⎜ − 8 + π 2 y( y − 1) cos⎜ ⎟
⎝ 2⎠
⎝
+
where u = 0 on 'D = MS and
(
1
π3
1
⎛ πx ⎞
πεxy(1 − x )(1 − y ) cos⎜ ⎟
⎝ 2⎠
2
⎛ πx ⎞
⎛ πx ⎞ ⎞ ⎞
cos⎜ ⎟ − 2π (5x − 3) sin⎜ ⎟ ⎟ ⎟⎟
⎝ 2⎠
⎝ 2 ⎠⎠⎠
(16)
in Ω = (0,1) × (0,1)
))
(
u2 ( x , y ) =
)
))
+ 2π 24 − π 2 y( y − 1) + x ( − 40 + π 2 y ( y − 1
Example 1: Consider the second order singularly
perturbed boundary value problem in two dimensions:
⎛ πx ⎞
f = εy sin⎜ ⎟ (1 − x )(1 − y ) − 3 + y 2
⎝ 2⎠
(
⎛
⎛ πx ⎞
⎜ − 8 + π 2 y( y − 1) cos⎜ ⎟
⎝ 2⎠
⎝
NUMERICAL EXAMPLES
(
1
1
πx
πεxy(1 − x )(1 − y ) cos +
2
2 2π 3
ε 256 − 16π 2 x + − 256 − π 2 x( x − 1) ×
We start with an initial approximation, and by using the
iteration formula (15), we can obtain directly the other
components of the solution u(x,y).
2
πx
1
πεxy(1 − x )(1 − y ) cos ,
2
2
)
(
))
+ 2π 24 − π 2 y ( y − 1
⎛
⎞
⎛ πx ⎞
⎜ ε sin⎜ ⎟ (1 − x )(1 − y )⎟
⎝ 2⎠
⎝
⎠
⎛
⎛ πx ⎞ ⎞ ⎞
+ x⎜ − 40 + π 2 y ( y − 1)) sin⎜ ⎟ ⎟ ⎟ ,
⎝ 2 ⎠⎠⎠
⎝
⎛ πx ⎞
− εy sin⎜ ⎟ (1 − x ) − (2 + x )
⎝ 2⎠
πx
1
u3 ( x, y ) = πεxy(1 − x )(1 − y ) cos
2
2
1
− 3 4ε 32 − 2π 2 x + − 32 + 2π 2 x( x − 1)
π
⎛
⎞
⎛ πx ⎞
⎜ 1 / 2πεy cos⎜ ⎟ (1 − x )(1 − y )⎟
⎝ 2⎠
⎝
⎠
((
101
(
)
Res. J. Math. Stat., 3(3): 97-106, 2011
by u(x,y) = gxy(1-x)(1-y). We start with an initial
approximation, and by using the iteration formula (15),
we can obtain directly the components of the solution as
follows:
⎛ πx ⎞
⎛ πx ⎞ ⎞ ⎞
cos⎜ ⎟ − 2π (5x − 3) sin⎜ ⎟ ⎟ ⎟⎟
⎝ 2⎠
⎝ 2 ⎠⎠⎠
+
1
2π 3
(
(
))
ε 256 − 16π 2 x + − 256 − π 2 x ( x − 1 ×
u0 ( x , y ) = εxy(1 − y ) ,
⎛
⎛ πx ⎞
⎜ − 8 + π 2 y( y − 1) cos⎜ ⎟
⎝ 2⎠
⎝
u1 ( x , y ) = εxy(1 − y ) +
(
+ 2π 24 − π 2 y( y − 1)
+ x 2 (ε + y − y 2 )),
⎛ πx ⎞ ⎞ ⎞
+ x − 40 + π y( y − 1) sin⎜ ⎟ ⎟ ⎟⎟
⎝ 2 ⎠⎠⎠
(
2
)
u2 ( x , y ) = εxy(1 − y ) +
1 2
x (6εy − 1)
6
+ x 2 (ε + y − y 2 )) − ( x 4 ( x 2 ( − 2ε + y − y 2 )
by the same way we can obtain other components of the
solution.
+ 15ε (ε + y − y 2 ))) / 90ε ,
u3 ( x , y ) = εxy(1 − y )
Numerical simulation of example 1: The obtained
numerical and approximate solutions using the two
proposed methods, FEM and VIM of this example are
presented in Table 1 and Fig. 1-3. In Table 1, we
introduced the H1-semi-norm of the FEM with different
values of N and g. Figure 1 and 2 represent the behavior
of the approximate and exact solution at g = 5×10G4 and
g = 5×10G6, respectively, using VIM. Figure 3, presents
the numerical solution using FEM at g = 5×10G4 (left) and
g = 5×10G6(right).
1 2
x (6εy − 1) + x 2 (ε + y − y 2 ))
6
− ( x 4 ( x 2 ( − 2ε + y − y 2 ) + 15ε (ε + y − y 2 )))
+
/ 90ε + ( x 6 (28ε ( − 2ε + y − y 2 )
+ x 2ε (3ε + y − y 2 ))) / 2520ε
Table 1: The H1-semi-norm at different values of N and g
g
N
||| Lu-uh |||
5×10G1
202
5.6eG3
818
2.8eG3
5×10G2
62
1.0eG4
198
2.0eG4
5×10G4
70
1.1eG6
533
1.7eG7
5×10G6
71
2.5eG8
552
1.7eG9
Example 2: Consider the second order singularly
perturbed boundary value problem in two dimensions:
− ε∆ u + 2u = f , in Ω = (0,1) × (0,1).
1 2
x ( 6εy − 1)
6
(17)
where, u=0 on 'D = MS, and f = 2gxy(1-x)(1-y)+2g2(2x(1x) + y(1-y)). The exact solution of this example is given
(a)
102
Res. J. Math. Stat., 3(3): 97-106, 2011
(b)
Fig. 1: (a) The approximate solution and (b) the exact solution with g = 5×10G4 using VIM
(a)
(b)
Fig. 2: (a) The approximate solution, (b) the exact solution with g = 5×10G6 using VIM
103
Res. J. Math. Stat., 3(3): 97-106, 2011
Fig. 3: The adaptive solution with g = 5×10G4 (left) and g = 5×10G6 (right) using FEM
Fig. 4: The approximate solution (left) and the exact solution (right) with g = 5×10G2 using VIM
Fig. 5: The approximate solution (left) and the exact solution (right) with g = 5×10G4 using VIM
104
Res. J. Math. Stat., 3(3): 97-106, 2011
(a)
(b)
Fig. 6: The adaptive solution with (a) g = 5×10G2 and (b) g = 5×10G4 using FEM
by the same way we can obtain other components of the
solution u(x, y).
the numerical solution using FEM at (a) g = 5×10G2 and
(b) g = 5×10G4.
Numerical simulation of example 2: The obtained
numerical and approximate solutions using the two
proposed methods, FEM and VIM of this example are
presented in Table 2 and Fig. 4-6. In Table 2, we
introduced the H1semi-norm of the FEM with different
values of N and g. Figure 4 and 5 represent the behavior
of the approximate and exact solution at g = 5×10G2 and
g = 5×10G4, respectively, using VIM. Figure 6 presents
Table 2: The H1-semi-norm at different values of N and ,
g
N
||| Lu - uh |||
5×10G1
25
1.4eG2
947
2.0eG3
5×10G2
25
2.4eG4
973
1.9eG5
5×10G4
59
6.9eG7
685
7.9eG8
5×10G6
21
1.7eG8
595
1.5eG9
105
Res. J. Math. Stat., 3(3): 97-106, 2011
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CONCLUSION
In this paper, we implemented the finite element
method based on a posteriori error estimate and VIM to
study the numerical solutions of singularly perturbed
convection-reaction diffusion problems in two
dimensions. An algorithm of the proposed method (FEM)
is introduced and implemented it to obtain our results
using Matlab programming. From the given results we
can conclude that our methods are in excellent agreement
with the exact solution. But, we found that the VIM is
invalid at some values of the perturbed parameter,, say,
as in Fig. 5 in example 2.
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