Applied Mathematics, 2011, 2, 866-873
doi:10.4236/am.2011.27116 Published Online July 2011 (http://www.SciRP.org/journal/am)
Copyright © 2011 SciRes. AM
Analysis of an Il’in Scheme for a System of Singularly
Perturbed Convection-Diffusion Equations
Mohammad Ghorbanzadeh, Asghar Kerayechian
Department of Ap pl i e d M athematics, School of Mathematical Sciences, Ferdowsi University of
Mashhad, Mash had, Iran
E-mail: Ghorbanzadeh_imamreza@yahoo.com, krachian@math.um.ac.ir
Received May 8, 2011; revised May 22, 2011; accepted July 22, 2011
Abstract
In this paper, a numerical solution for a system of singularly perturbed convection-diffusion equations is
studied. The system is discretized by the Il’in scheme on a uniform mesh. It is proved that the numerical
scheme has first order accuracy, which is uniform with respect to the perturbation parameters. We show that
the condition number of the discrete linear system obtained from applying the Il’in scheme for a system of
singularly perturbed convection-diffusion equations is O(N) and the relevant coefficient matrix is well con-
ditioned in comparison with the matrices obtained from applying upwind finite difference schemes on this
problem. Numerical results confirm the theory of the method.
Keywords: Convection-Diffusion, Il’in Scheme, Uniform Convergence, Singular Perturbation, Condition
Number
1. Introduction
Consider the following system of coupled singularly
perturbed convection-diffusion equations: Find
such that
l
=Au
=u



2
1,,(0,1)0,1 l
l
uuC C
:=LEB
 
uuu
 
,
=0,1, 0= 1=0,x

f
uu (1.1)
With

12
=diag,,,l
E

, where ,
i
s
for
are known small positive diffusion co-efficients,
=1,,il


ij
=
A
ax ll
is an matrix, and
i
=
f
xf for
is a vector-valued right hand side.
Furthermore, we shall assume that is diagonal with
diagonal elements and define
=1,i,l

i
bx B

[0,1]
=>0 for =1,,.
min
kk
xbxk l
(1.2)
Some results for systems of singularly perturbed of
differential equations can be found in: Linss and Madden
[1], Madden and Stynes [2] and Gracia and Lisbona [3].
Bellew and O’Riordan [4], Cen [5], Amiraliyev [6]
and Andreev [7] used the finite difference method for a
coupled system of two singularly p erturbed convection –
diffusion equations.
T. Linss [8] considered an upwind finite difference
scheme on special layer adapted Shishkin and Bakhva-
lov meshes. He showed that the error in the discrete
maximum norm is bounded by and
1lnCN N
1
CN
for Shishkin and Bakhvalov meshes respectively, where
is independent of the perturbation parameters k
C
for , and is the number of mesh points
used. =1, ,klN
The discrete linear systems that arise from Shishkin or
Bakhvalov meshes do not have a good condition number.
H. G. Roos in [9] showed that the condition number of
the discrete linear system associated with the upwind
schemes on Shishkin meshes for a single equation is
22 2
ln
ONN
, which is not good when
is small.
In fact, if
is the coefficient matrix of the linear
system associated with the upwind schemes on Shishkin
meshes, then 2
22
ln
N
AC N
and 1
A
C
, where
Cis a constant independent of
and the norm is the
discrete maximum norm for matrices. Nevertheless, he
proposed a precondition which has reduced this condition
number to
21
ln
ON N
.
In this paper, we study the Il’in scheme (see [10]), for
problem (1.1). We show that for this method the error in
the discrete maximum norm is bounded by 1
CN
, where
is independent of the perturbation parameters k
C
=1,,kl
, and we prove that if
A
is the coefficients
M. GHORBANZADEH ET AL.867
matrix of the linear system associated with the Il’in
scheme on uniform meshes, then
A
CN and
1<.
A
C
So the condition number of the discrete
linear system associated with the Il’in scheme on
uniform meshes for single equation is , which is
better in comparison with the precondition of upwind
schemes on Shishkin meshes.

ON
The paper is organized as follows: in section 2, we
give some properties of the solution of (1.1) and in
section 3, we state the difference approximation Il’in
scheme. In section 4, we analyze the error of Il’in
scheme applied to (1.1), and some numerical examples
are presented in section 5.
2. Properties of the Exact Solutions
In this section, we analyze the exact solution of (1.1).
Assume that
0,1vL
01
=<<<x
and , where

=|0 =1
iN
xxx
=
i
x
ih and
=1hN. Consider the following norms:


,
(0,1)
1
11,
0=0
=,=
max max
=d,=|
i
xx
i
N
i
i
vvxv
vvxxvhv


,
,
v
N
where i
for Consider the fol-
lowing discrete norm:

=
i
vvx = 0,,.i
1,
1,1 =1
=,
N
hi
Wi
vvhDv
where the space 1,1 contains , such that
h
W1N
vR
1,1 <.
h
W
v Difference operators are defined as follows:
0
11
=,=,=
2
ii iiii
iii
uu uuuu
Du Du Du
hh


 11
.
h

Let
11
00
==
NN
N
RvRvv

=0, and consider the
space on We define the following norm on
as
0
1,1
h
W1
0.
N
R
0
1,1
h
W01,1=.
hi
WDv
=1
N
i
vh
Also for the dual of , which is denoted by
0
1,1
h
W1,
h
W
,
we define
01,1
1,
|< ,>|
=max ,
h
h
WW
vf
fv

where 1
=1
,= N
ii
i
vf hvf
. In [11], it has been shown
that
,
1, =
min
h
WcR
Dff c,
 (2.1)
or equivalently
,
1, :=
=
min
h
WFDFf
f.F
 (2.2)
For the vector-valued function
 
1
=,,
T
l
x
vx vx
v, consider the infinity norm as
=1, ,
=.
max i
il
xv
v
To estimate the error in our difference approximation,
we shall require some bounds for the derivatives of the
solution of (1.1), under the assumptions

=1 0, ()>0 and () 0
for , [0,1] and ,=1,,
l
ij iiij
j
ax axax
ijxij l


(2.3)
and strict inequality hold at least for one i.e.,
k

=1 >0
l
kj
j
ax
(2.4)
Lemma 2.1 If , in
 
1
=,,
T
l
yx yxy
0Ly
and
,10y

0xy00y then in
.
Proof. Let
i
y
x be minimum at i for
, i.e, t
=1,2, ,
il
ii
yt

=min i
xyx
 and also assuming
=
=1, ,
min
j
ji
il
t
yt yi
(2.5)
If
0
jj
yt the lemma is proved. So let
<0
jj
yt .
If
=
j
jkj
tyt y for , then it follows =1,2,,kl
that
=0
j
t
y and
0
j
t
 y. By (1.1)

 
:=
=.
j
jj
tj
jkj
LEtBtAt
At ytA

yyyy
y1

In this case according to (2.4) since , the
th component of

<0
kj
yt
k
j
A
ty is negative, which is a
contradiction to the assumption of the lemma. If there is
a with 1kkl
such that

<

j
jkj
ytyt then
 
 
 

 

 
=1
=1
=1 ,
=1, ,=1, =1
=
.
max
j
jjjjjjjmj mj
m
l
jjjjm jj j
m
l
jm jm jj j
mmj
ll
mj jjjmjjjjmj
ml mmjm
ytbtyta tyt
yta tyt
atyt yt
ytyta tyta t
 
 



 

l
If
=1, <0
l
jm j
mmj
at
, it is obvious that the right
hand side of the above inequality is negative. If
=1, =0
jm j
mmj
at
l
, then since we have >0
jj
a
=1 >0
jm j
at
l
m
so the right hand side is negative and
again we reach a contradiction. So the lemma is proved.
Copyright © 2011 SciRes. AM
M. GHORBANZADEH ET AL.
Copyright © 2011 SciRes. AM
868
3. Discretization Theorem 2.2 Suppose solves (1.1) and assume
that for satisfies (2.3) and (2.4), then
u
ij
a,=1, ,ij l
.uC

By theorem 2.2, T.Linss [10] has proved the following
lemma.
f
Lemma 2.3 Let be the solution of (1.1) and
suppose (1.2), (2.3) and (2.4) hold. Then for
u
0,1x
and
=0,1n

 
1exp
1
1 exp.
nk
kk
k
n
k
k
n
kk
k
k
x
Cb
ux x
Cb
k




 







 





,,,
By lemma (2.3) and the application of the technique
mentioned in [12], the following lemma can be proved in
a similar way.
Lemma 2.4 Let satisfy (1.1). If u
 
=, for =1
kkk
uxvx zxkl
where


 


  

1
1
0exp 0
0
=1exp 11,
1
kk kk kk
k
k
kk kkk k
k
ubxb
b
vx ubxb
b


then, for
0,1x and
=0,1
n


.
n
k
zxC
Remark 2.5 By lemma 2.4 and direct computation,
without loss of generality, assume that k
bk
; hence,
we have
 
  

=1
1
=1
=
=00
l
kkkkk kmm
m
l
kkmmkkkkk
m
k
Lzbxzxazx
f
avxb bbxvx
gx
 
 
 

z
and for
=0,1
n


1exp, =1,,
nnk
kk
k
x
g
xC k








l
In this section, we deal with the discretization of
prob-em (1.1) by the Il’in scheme. We apply the Il’in
scheme : Find such that:
1
0
l
N
UR
0
;;
;;
=1
=,
kk kjkjkjkj
j
l
km jmjk j
m
LUD DUbDU
aU f


:=

(3.1)
for and , where
=1,,
kl=1, ,1jN
=coth
kk k
x
qx qx
with

=2
k
kk
hb x
qx
,
kj
U
is the approximate value for that is obtained by
the Il’in scheme, ;
bb and

kj
ux

j
x=
kj k
;=
km jkmj
aax.
Consider the diagonal matrices

1
=diag,,l
Qxqx qx
and

1
coth()= diagcoth,,coth.
l
Qxq xq x
We define
 
11
=coth= coth
22
hE BhE B
Qx Qx
.
We rewrite (1.1) in matrix form as follows:
0=
for =1,1,
j
jj jjj
EDDU BDUAUf
jN

 
j
(3.2)
where
1
=,,
T
jjlj
UU U,

1
=,,
T
j
jl
DU DUDU
 
j
lj
,

1
=,,
T
jj
D
UDUDU
 
00 0
1
=,,
T
and
j
jlj
D
UDU DU The corresponding coeffi-
cient matrix in (3.1) is:
11 121
21 222
12
=
l
l
h
ll ll
DD D
DD D
A
DD D






 
(3.3)
where blocks ij in Dh
A
are and
diagonal blocks for are tridiagonal
matrices as follws

1NN 
, ,l
1
kk
D=1k
;1;1 ;1
;1
22
;2 ;2;2;2 ;2
;2
22 2
;1 ;1;1;1
22
20
2
20
=22
2
02
kkkk k
kk
kk kkkkk k
kk
kk
kkN kNkkN kk N
b
ah
hh
bb
a
Dhh
hh h
ba
hhh
 
 
 
 











 


 
M. GHORBANZADEH ET AL.
Copyright © 2011 SciRes. AM
869
b
and for where are diagonal
matrix as
ij
D,=1, ,ij lij

;1;1
=diag,,
ijijij N
Daa
.
Remark 3.1 When is constant
and , the system (1.1) is reduced to
1
=diag,, l
Bb
0A
=.EB
 
uuf
Then its general solution for homogeneous equation,
which is also boundary layer function, is

 
=exp for >0 and
1
=exp for <0,
k
kk k
k
k
kk k
k
bx
ux Cb
bx
ux Cb






where for are arbitrary constants. The
difference operator (3.2) is exact for these boundary
layer functions. i.e.,
k
C=1,,kl
0=
for =1,,1,
j
jj jj j
EDDU BDUEB
jN
  
 
uu
j
(3.4)
where index j indicates the evaluation at point
j
x
,
and .

1; ;
=,,T
jjlj
uuu

;=
kjk j
uux
In what follows, we try to estimate the condition number
of the discrete linear system associated with the Il’in
scheme presented. Then we apply M-criterion [13], [14]
to obtain the condition number.
Theorem 3.2 (M-criterion). Let a matrix
satisfy
0
ij
a
for ij
. Then
is an M-matrix if and only
if there exists vector such that . Further-
more, we have >0e>0Ae

1.
min k
k
e
A
A
e
(3.5)
Theorem 3.3 The corresponding coefficient matrix in
(3.1) is an M-matrix and satisfies
h
A
CN
and

12
h
A
where is independent of and
CN k
for
and =1,,kl
=1, ,
minkl
k
=
Proof. To evaluate the maximum norm of the matrix
h
A
for the ith row of the th row of the block matrix k
h
A
, where =1,,1iN
and , we have
=1,,k
l

 
 

1;;;
;; ;;
=1 =1
;;;;;;;;
2=1
;;; ;;
2=1
1 cothcoth1 coth
22
=1coth2coth2coth1 coth
=coth11coth =
lN l
kikiki
h
ijkikiki kmi
j m
l
kkikiki kiki kikikikmi
m
l
kki kikikikmi
m
bbb
aqq qa
hhh
qqqqqqq qa
h
qqqq a
h
  





 


;
;; ;
2=1
4coth .
l
kki kikmi
m
qq a
h
By

2;
;; ;
coth 1,
1
ki
ki kiki
q
qq C
q

See [14], we conclude that




2
1;
;; ;
22
=1=1 ;
2;;
2=1
;
22 2
;;
;;
2
2=1 =1
;
4coth111
1
22
24 2
=.
2
22
lN lki
hkk
ijkik ikm i
jm
ki
l
ki kkm i
m
ki k
ll
kii kkkik
km ikm i
mm
ki k
q
aqq aC
q
hh
b
Ca
h
bh
bh bhbh
CaC
h
bh h

 










 

 




a
Since in singularly perturbed equations ,h
is considered, hence
M. GHORBANZADEH ET AL.
870

1
=1
21=.
2
lN i
ij
j
b
aCCCN
hh




Therefore,
1
=1
=.
max
N
hij
ij
A
aC





N
kx
If we choose where
1
=,,T
l
ee

e=1
ii
e
for
and for k, then

<0
k
bx =1
k
ii
ex

bx>0 h
A
e
h
for the th row of the th row of the block matrix
i k
A
for case we have

bx<0
k




;;;
1
;;
;
;;;
1
;;
=1,
=coth 1
222
coth1
2
coth1
222
>
kiki ki
hi
ik
kiki kk ii
k
kiki kii
k
lm
km iik i
mmk
bhbb
Ax
hh
bhb
ax
h
bhbb
x
hh
ae b

















 






e
>
k
and from (3.1), for we have . Hence the
M-criterion yields that
ijh0
h
ij
a
A
is an M-matrix and
 
12.
min
hh
k
k
e
AAe

A similar proof is for the case .

>0
k
bx
Notation. Throughout the above theorem we let
denote positive constant that may take different values in
different formulas but that are always independent of
both the perturbation parameters
C
k
and of , the
number of mesh intervals. By theorem 3.3, the condition
number of
N
is of order
ON. We recall that, in
theorem 3.2 and theorem 3.3,
A
is the usual
maximum norm of matrices. This shows that the matrices
arising from the Il’in scheme for discretization of a
singularly perturbed differential equation are well
conditioned respect to the upwind finite difference
method applied on Shiskin or Bakhvalov meshes.
For arbitrary , we define

1
10
=,,l
N
l
UU UR
,,
=1, ,
=.
max i
il
UU
A single equation of (3.1) can be written as follows:

0
;
;; ;
=1,
:=
=.
kkkkjkjkjkj
j
l
kk jkjk jkmjmj
mmk
LUqD DUbDU
aU faU





(3.6)
Suppose solves (3.1). We propose the following
lemma to obtain a bound for
U.U
Lemma 3.4 Suppose
A
is a matrix such that
and >0
ii
a
0
ij
a
for )( ji
and Also assume
,=1, ,.ijn
that for Then for every arbi-
trary vector we have
=1 1
jk
ka
l=1,,.j

,T
n
n
1
=,

,,
A


Proof. Suppose for the element of j
, ,=.
j

Without lose of generality, let =
j
j

(otherwise we
consider =
A
A
).

=1 =1,
=1 ,
==
,
nn
j
kk jjjjkk
jkkkj
n
jj jjkk
kkj
Aaa a
aa




(since 0
jk
a
for jk
). Therefore


=1 ,
=1, =1
=1,
=0
n
jjjj jkkj
jkkj
nn
.
j
jjjk kjkj
kkjk
n
jk kj
kkj
Aa a
aaa
a
 


 
 


So
>0.
j
j
A

Hence
 
,,
=1, ,
==
max j
kj
kn
AAA



Similar to theorem 2.2, we have the following theorem
in the discrete case.
Theorem 3.5 Suppose solves (3.1). Assume
and satisfy (1.2), (2.3) and (2.4). Then
Uk
b
ik
a
,,
.UCf

Proof. Dividing (3.6) by , we have
;kkj
a

;
;;
;;
=1, ;
=:=
.
kj
kkkkkkj
j
jkkj
lkmjmj
mmkkk j
f
LULUaa
aU
a


(3.7)
The matrix associated with operator is a matrix
that satisfy lemma 3.4. We have k
L
,,,
,
=1 ,
.
k
kkkkk
lkm m
mmkkk
f
ULUa
aU
a

Therefore
Copyright © 2011 SciRes. AM
M. GHORBANZADEH ET AL.
Copyright © 2011 SciRes. AM
871
1
=
==1,,=1,
N
ij i
j
Ghgi ljN

,,
=1, .
lk
kkmm
mmkkk
f
UU
a



1, (4.4)
where
i
g
x for is defined in remark 2.5.
By (4.1)-(4.4), we can establish the following theorem.
=1,,il
By (2.3) and after some manipulation, we obtain
,,
.UCf

Theorem 4.1 Let be the solution of (1.1) and
be the solution obtained by the Il’in scheme (3.1).
Suppose the data
uU
1
,, ,0,1
kkkmk
a fCgb for
satisfy (1.2), (2.3) and (2.4). Then ,=km
1, ,l
Corollary. According to theorem 3.5, we have
,,.UCLU
 uu
(3.8)
,.U
uCh (4.5)
4. Error Analysis Proof. From lemma 2.4, we can split
x
u as
=
x
xuvzx, where

x
v is a boundary layer
function. Hence,
For the error analysis of Il’in scheme applied to the
system of singularly perturbed differential equations
(1.1), suppose that for and
satisfies (1.1). Similar to lemma 2.4, we split

>0
i
bx =1, ,ilu
=
x
u
 
x
xvz, where 1

=,


,
T
l
vx ,()vxv=
x
z
1 So we have ((), ,
l
zx z()).
T
x
 
k
x=
kk
zxux ,
and for v
= 0
n,1
,,,
=.UZVZV
,


uzvzv
First, consider ,
Z
z. From (4.1)-(4.4), we have

=(),=0,1
iiii
Lgon


zz
and

()
,
n
k
zx C
=and=
for= 0,,.
iiij ij
jj
LZDAZgD G
jN


and
 
 

1
0
=exp0
0
kk
kk
k
u
vxb x
b
.
k
d
,
Thus, by
=
ii
Lxgxz and

=,
ii
j
LZ gj


=, =0, ,
ii
iij
j
x
AZ GjN
0,1 andxx

z (4.6)
First we analyze the truncation error in a single equation.
We introduce the co ntinuous and discrete operators
with constants
and
.

 

 
1
1
=1
=
d0,1,=1,
iiiiiii
x
l
im m
xm
x
vxbvbsvss
avssxil



v
(4.1) From (2.1),
,
1, =().
min
h
ii
WcR
LZ AZc

zz
Taking =c
, from (4.6) we get
and
,
)(
~
=][
;
1=
1
=
;1
1
=
;


mim
l
m
N
j
ii
N
j
ijjiijiji
VahbDhV
VbxVDVA


(4.2)
 
,
1, .
h
iiiii
i
W
LZAxG x


zzz
Furthermore,





1
;
=
1
;1;
=
=
dd
N
iiiiiiii
jjj
N
xx
NN
iiiii
xx
jj
j
AG Dzzh
where
1
=ii
ivv
Dv h
and


=.
coth 1
ii
h
hqq
,
s
shzDb bzss


 
 

zz
where
We note that . We introduce the continuous and
discrete functions >hh
=1
=.
l
iimm
m
az g
i
 
1
=d=1,
ii
x,
x
gs sil
(4.3) Applying Taylor expansion with the integral remainder
form and using lemma 2.4 and remark 2.5, we have
and
 


  
11 11
;;1 ;
111 1
d=dd|() |d|d|
=dd|dd,
xxt xtx
iiiiii ii
xxxxxx
xxx x
ii ii
xtx t
htt sstsdstChhzDbbz
btzs stbtzs stCh
 

 
 

 
 
 


 

 
 
tt
M. GHORBANZADEH ET AL.
Copyright © 2011 SciRes. AM
872
and





1
1
=d
dd.
xx
i
iii i
xt
j
xx
ii
ii
xt
Dzzhhzz sst
h
hhz zsst
hh













d
On the other h a nd, we have




=1 coth1
=.
2
iii i
i
ii
hhq q
hbh
qCh
 

Hence

1=1
1dd .
iii
j
l
xt
iii imm
xx m
Dz z
Chgb zazstCh
h



 



Thus
,.
ii ii
A
GC

 zz h
In [15] it has been shown that
 
,1,
2.
h
i
ii
ZLZ

 zz
Hence we have
,.
Z
Ch
z (4.7)
To bound the boundary layer function
=
x
v
,v
, a direct computation gives
 
T
1,,
l
vx vx
1
LE
v
 
=00BBBxA


v
and, in the grid points we can write


 

 

1
2
=si
sinh
sinh0 .
LVQ xB xQ
h
QxQV AV

nh0
By a technique used in [9], we can show that
,.VC
vh (4.8)
So by (4.7) a n d (4.8) we have
,.UC
uh
This completes the proof of the theorem.
5. Numerical Experiments
In this section, we compare the Il’in scheme with the
upwind finite difference scheme (see [8,15]) for the
following two examples.
Example 1. Consider
 

11 11211
222 1222
2= 0=1=0,
24=cos 0=1=0,
x
uuuueuu
uuuuxu u
 

 

In this example, we expect two layers at which
behave like
=0x
1
exp
x


for and like
1=1b
2
2
exp
x



for 22b
. Let 8
1=10
and , which are
sufficiently small values to bring out the singularly
perturbed nature of the problem. The exact solution to
the test problem is not available, so we estimate the
accuracy of the numerical solution by comparing it to the
numerical solution computed on the finer mesh. Let
6
2=10
N
U
be a numerical solution in grid point. We estimate
the error by N
2
,
NN
UU
The rates of convergence rN are computed using the
following formula:
2
,
224
,
log .
NN
N
NN
UU
rUU




For the upwind scheme we use Shishkin mesh with
1=m
and 22
1
=min, log
2N

. We divide the inter-
vals
1
0,
and
12
,
into 4N subintervals and
2,1
into 2N subintervals of equal length. For Il’in scheme,
we use the uniform mesh and divide the interval
0,1
into subintervals of equal length. Numerical results
are contained in Table 1. From this Table, we observe
that the Il’in scheme is a first order uniformly convergent
method.
N
Example 2. Let


111 1 2
222 2 3
2
3313 23
3=
0.5= cos
52=1
x
uuuue
uuuu x
uxuuu.x


 
 
 

In this example, we expect layers
1
exp 3,x
Table 1. Numerical results for example 1.
upwind
scheme Il’in
scheme
N Error
Rate
N
r Error Rate
N
r
128 4.62e-2 0.8019 5.00e - 3 1
256 2.65e-2 0.8461 2.50e - 3 1
512 1.48e-2 0.8945 1.25e - 3 1
1024 7.90e-3 0.9462 6.30e - 4 1
2048 4.10e-3 - - -
M. GHORBANZADEH ET AL.873
Table 2. Numerical results for example 2.
Upwind
scheme Il’in
scheme
N Error Rate Error Rate
32 1.03e - 2 1.0429 9.6e - 3 0.9652
64 5.0e - 2 0.9685 4.9e - 3 0.9826
128 2.55e - 2 0.9439 2.5e - 3 0.9913
256 1.33e - 2 0.9382 1.3e - 3 0.9957
512 6.9e - 3 0.9384 6.0e - 4 1
1024 3.6e - 3 - 3.0e - 4 -

23
0.5 15
exp,exp.
x
x





87
Let ,
1=10
2=10
and We use the upwind finite difference
scheme on a Shishkin mesh(See [11]). Assume that,
6
3=10.
312
12
3
3
3
=min,ln, =min,1ln,
234 0.5
1
=min, ln,
25
N
N
N










with
11
=min,3

and
31
=max, 3

. Then the
mesh is obtained by dividing each of the intervals
112 23
, ,,,0,
 
and
3,1
into 4
N
subintervals.
Numerical results are shown in Table 2.
From Table 1 and Table 2 we see that the rate of
convergence for Il’in scheme is close to 1, which agree
with the convergence estimate of theorem 4.1. Numerical
results confirm the theoretical results.
11. References
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