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Electronic Transactions on Numerical Analysis.
Volume 23, pp. 288-303, 2006.
Copyright  2006, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
UNIFORMLY CONVERGENT DIFFERENCE SCHEME FOR SINGULARLY
PERTURBED PROBLEM OF MIXED TYPE
ILIYA A. BRAYANOV
Abstract. A one dimensional singularly perturbed elliptic problem with discontinuous coefficients is considered. The domain under consideration is partitioned into two subdomains. In the first subdomain a convectiondiffusion-reaction equation is posed. In the second one we have a pure reaction-diffusion equation. The problem is discretized using an inverse-monotone finite volume method on Shishkin meshes. We establish an almost
second-order global pointwise convergence that is uniform with respect to the perturbation parameter. Numerical
experiments that support the theoretical results are given.
Key words. convection-diffusion problems, singular perturbation, asymptotic analysis, finite volume methods,
modified upwind approximations, uniform convergence, Shishkin mesh
AMS subject classifications. 34A36,34E05, 34E15, 65L10, 65L12, 65L20, 65L50
1. Introduction. Let us consider the following one dimensional elliptic problem
!#"$&%(')*+,-"/.01"
39 2!45 !#"$&%('9 26
."871"
(1.2)
;:=<?>A@BC!D.BE,FGF.)H,I ,-" : <?>A@ ,-"
(1.3)
!,JICK!LJ"MFN7?$OKQP5"
(1.4)
,SRTURVRW7 and
where
,XRY LVZ D Z P "[,XRY L\Z D Z P]
(1.5)
.
The functions and could be discontinuous at the interface point . Note that the sign
(1.1)
pattern of the convection coefficients is essential for the behavior of the solution; see [7, 8]
for more detailed discussion for convection-diffusion problem with discontinuous convection
T^7 and interior layers
coefficient. The solution&of
this
problem
has a boundary layer at
W
.
'
. In
, where convection-diffusion problems are present, the
with different widths at
D` . Since
width of the layer is _
the convection coefficient is positive, the characteristics of
the reduced problem point toward the interface point and an interior layer appears on the right
' . In ' 2 , we have a reaction-diffusion problem. Now the reduced
part of the boundary of
a `
problem is of zero order and boundary layers of width _
on both boundaries of
' 2 . The character of the layers can be readily seen on Fig. 6.1 appear
below.
There is a vast literature dealing with convection-diffusion and reaction-diffusion problems with smooth coefficients and smooth right hand side (see [10, 11, 13, 17] for surveys),
but there are only a handful of papers dealing with problems with discontinuous coefficients. Such problems usually present interior layers. The one dimensional convectiondiffusion problem with discontinuous input data is discussed recently by several authors;
see [4, 7, 8, 9, 10, 15] and references there. The one dimensional reaction-diffusion problem
with discontinuous input data and concentrated source is discussed in [6, 12, 14].
Our objective in this paper is to derive global pointwise
convergence of almost second
order that is uniform with respect to the small parameter . We construct partially uniform
b
Received October 24, 2005. Accepted for publication March 13, 2006. Recommended by L. Reichel. This work
was supported by the National Science Fund, Sofia, Bulgaria, project HS-MI-106/2005.
Department of Applied Mathematics and Informatics, University of Rousse ”Angel Kanchev”, Studentska str.
8, 7017 Rousse, Bulgaria (brayanov@ru.acad.bg).
288
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SINGULARLY PERTURBED PROBLEM OF MIXED TYPE
' ' 2
and
. The resulting global mesh is condensed
Shishkin meshes in each subdomain
closely to the boundary and interior layers. To derive a uniform difference scheme we use an
inverse-monotone finite volume discretization on layer adapted meshes. To obtain a second
' we use a modification of the monotone Samarski’s scheme analogous
order scheme in
'
2
to [4] . In
, where a reaction-diffusion problem is present, we modify the scheme at the
interface point, similarly to [5]. To prove -uniform convergence we use two types of techniques. The first one uses the discrete Green function to obtain a hybrid
stability inequality
P norm
that shows that the maximal nodal error is bounded by a discrete
of its truncation
error. A result like this is proved by Andreev in [1, 2] and used later in [3] to prove second
order -uniform convergence in maximum norm for convection-diffusion problem. The second one uses the embedding inequality and estimates in negative norm to prove second order
uniform convergence for reaction-diffusion problem on non-uniform meshes; see [16]. This
technique is used in [3, 5] to prove almost second order -uniform convergence in maximum
norm, for the corresponding problems.
An outline of the paper is as follows. In Section 2, we describe some properties
of the
differential solution, construct Shishkin decomposition of the solution, and derive uniform
bounds for their derivatives. In Section 3, we construct Shishkin mesh condensed near the
boundary and interface layers and obtain a finite volume difference scheme. Some a priori
bounds of the discrete problem are given in Section 4. The uniform convergence of the
constructed scheme is proved in Section 5. In Section 6 we give some numerical experiments
that support the theoretical results.
2. Properties of analytical solution. In this section we establish some properties of
solution to problem (1.1)-(1.4) to be used in the analysis of difference scheme. We begin
with the existence of a solution of a problem (1.1)-(1.4).
L EMMA 2.1. Suppose that
c%&dVe?fg')F1"3"hi%&dVe?fg')kj'2I#"QlmCno integer ]
&%id P 'pqd e r' j' 2 .
Then, problem (1.1)-(1.4) has a solution
(2.1)
Proof. The construction of a solution is similar to that described in [4] for a convectionP
diffusion equation. Let s and s f be particular solutions to the differential problems
s PpW"$i%&' "[ 2 s f U"$i%&' 2 ]
Consider the function
WDK!L s P5r,/vIP5w xy s P`D.0/v f #"Nz%i' "
s Qut s P5DDW
DKQP{ s f N7?/vF|w xy s f D.0/v!}#"Nz%i' 2 "
sf
v$~ D!470"€n"/‚"Nƒ satisfy the problems
where the functions ,
v ~ O,‚"$F47"hn-"!vIP`,3„70"-vIP0D.0Qy,-"Iv f r,Qy,-"-v f D.0Q470"
32$v ~ O,-"$!y‚"Nƒ‚"$v!}0.0Q ,-"-v!}N7?Iy,-"IvF|.0Qy,‚"-vF|JN7?I„7 ]
v$~FD!
70"€n"h-"Nƒ
From (2.1) follows that the functions
l within their domains. By the maximum principle, ,
(2.2)
Z exist
Iv ~ Z and7)Dhave
$47derivatives
"hn-"/-"/ƒJ andup to order
v P "/v } Z ,-"[v f "hv } mY, ]
x
The second conjugation condition in (1.3) yields the following condition for :
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I. BRAYANOV
9
9
x !v f .0!Gv!} D.0…:A s 9 f D.0I s P D.0w K P s f 7†:‡v!| D.0
K L s P r,†:‡v P .0 s P 0. v f .0$ s f .0v } D.0 ]
x
By virtue of (2.2), is uniquely defined.
The uniqueness of the solution follows from Lemma 2.2 below.
%‰d L 'pqŠd f g' j
L EMMA 2.2. (the maximum principle). Suppose that a function ˆ
' 2 satisfies
9 ˆ D Z ,-"!&%i')Q"[Q2 ˆ D Z ,‚"Fz%‰'2"
<Œ>A@
ˆ ‹: mY,-" ˆ r, Z ,-" ˆ N7? Z ,-"
Z , for all z% ' .
then ˆ
‚L be a point at which ˆ attains its maximal value in ' . If ˆ L` Z , ,
Proof. Let
L5*Ž, , the proof is completed by
there is nothing to prove. Suppose therefore that ˆ
showing that this leads to contradiction. With the above assumption
value,
;L&%y' jG' 2 or ;L‰. . If ‚L*%y'p then ˆ D;L Z , , ˆ on ‚the
L5cboundary
, . Therefore
either
 ˆ D;LY, , which is a contradiction.
L O. . There are two possible cases.
The only possibility remaining is that
D
L
L is9 a point at which ˆ attains
1. The function ˆ
isL not differentiableL at . Since
Z
T
J
,
S
‘
m
,
O
J
,
,
: <?>’<?“ Ru, , which is the
its maximal value, then ˆ
,ˆ
and ˆ
required contradiction.
L
L z”, and there exists a
2. The function ˆ L L is differentiable at . Then ˆ
(
?
—
i
6
%
D
N
"
E
`
˜
1
/
"
U
˜
C

,
™
such that
subinterval •;–
ˆ D pC,-" and ˆ DR ˆ D L #"!š‰% ‚• – ]
P % •‚– . There exists f % •;– , such that
Let
P ! ˆ D L RC,-"
†
I
ˆ
f
AP{G‚L
ˆ
}›% • – , satisfying
and
f ! D‚L5 RC, ]
ˆ g } I ˆ f Gˆ L
} %&' 2 and 2 ˆ } T, , which is the required contradiction.
Then
]
'
Denote by œ œ8;žŸ ;¡ the maximum norm in . An immediate consequence of the maximum principle is the following
i%&d L 'stability
qd f gresult.
' j' 2 be a solution of problem (1.1)-(1.4), then
L EMMA 2.3. Let
(2.3)
and
(2.4)
(2.5)
œ œ#žpŸ ‚¡ ZT¢Š£5¤S¥o¦ K L¦ " ¦ K P`¦ " œ œ ;# L žŸ ‚¡ § "
¦ Ÿ e ¡ D -¦ Z (
d ¨/73He`©0Q"N`/ªI"$i% ' "
¦ Ÿ e ¡ D -¦ Z 4
d «736 Ae#¬hf © 2 Dw"0­S"$&% ' 2 "
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SINGULARLY PERTURBED PROBLEM OF MIXED TYPE
291
where
©QD"N0QO® ¤¯‰° D .o* …ŒL ± "I©2pDw"0Qy© 2P "N`w© 2f Dw"01"
L
L
© 2P "N`3O® ¤¯i° DŠGa .0 a ± "I© 2f D"N0QC® ¤¯&° 7pGa a ± "
d
and is independent of positive constant.
Dk³B´¶µ , where ´ ¢Š£5¤\· œ K L œ " œ K P œ ";¸†¹J¸…º ¾ “žp» ¼½ ¿
Proof.
Put ² 
P
²  %‰d ' , ²  ,J Z , , ²  N7? Z , and for each i%i' ,
. Clearly
 ²  D Z , ]
DŠmÀ, for all W% ' , which leads to the
It follows from the maximum
principle that ² 
desired bound (2.3) on . Now we consider the solution independently in each subdomain
'p and using the arguments in [4, 11], to obtain the remaining bounds in (2.4) and (2.5).
But for the numerical analysis below we shall need a decomposition of the solution into
regular and singular part. Using the results in Lemma 2.2 and Lemma 2.3 we obtain the
following estimates.
5"h
be sufficiently smooth in each subdomain
T HEOREM 2.4. Let the functions
' and
' 2 . Then the solution of problemand
(1.1)-(1.4) admits the representation
!DIyÁDw ˆ #"
Á and the singular part ˆ satisfy for all natural l , , Z l Z ƒ
where the regular part
the estimates
(2.6)
œ Á Ÿ e ¡ œ  ž Ÿ ‚¡ Z do"
and
¦ ˆ Ÿ e ¡ ¦-Z d)Ae12 P ¬/f?©QDw"01";i% ' "
¦ ˆ Ÿ e ¡ ¦-Z d) Ae#¬hf © 2 Dw"01";i% ' 2
(2.8)
d
for some positive constant Áindependent
of the small parameter .
D
Proof. The regular part
is sought in the form
}
"Ä&%i' "
ÁQ  ~ Á`~/6 |ŒÃ | #"IÁ`~N3 t ÁÁ ~2 1#"Ž
&%i' 2 "
~
~> L
(2.7)
Á ~ D are solutions to the problems
D#Á ~ Á ~ Å ~ D1"Ii%&')Q"$Á ~ r,I ² ~ "$! ,-"87"hn-"/-"
Å L {y!D1" ² L yK L "IÅ Æ DQ+DÁ Æ P " ² Æ O,‚"ÇU
70"hn-"/ ]
Á ~2 D satisfy
and the functions
Á 2 Á L 2 !DD "$Á ~2 ~ P "F$y,‚"87"hn"h ]
where the functions
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I. BRAYANOV
From the presentation it is clear that the regular component has -uniformly bounded derivatives.

œ DÁ ~ Ÿ e ¡ œ ;žBŸ È;¡ Z ´("I, Z l Z ƒ‚"!!y,-"Œ70"hn-"/ ]
} > ~ P€Ë ~ Ë ~ \
Ê
É
~ L ˆ
ˆ f , where the functions
The singular part is represented as ˆ
€
P
Ë
$

y
‚
,
8
"
0
7
€
"
n
h
"

~
ˆ ,
are solutions to the problems
Q2 ˆ P€Ë ~w ,-"¶
%i'2"
9 &
<?>A@
ˆ P1Ë ~hD.0I
Á`~: " ˆ P1Ë L N7?IyK P *Á ~2 71" ˆ P1Ë ~/N7?I
pÁ ~2 N7?#"Qm(7 ]
P€Ë ~ +, in ' . Similarly as in [11] we can prove that ˆ P1Ë ~ satisfy the estimates (2.8).
and ˆ
Fy,‚"87"hn"h the problems
Ë~
The second term ˆ f satisfy for
(2.9)
9  ˆ f Ë ~wO,‚"Mi9 %&'  "
9
<?>A@
<Œ>A@
<Œ>’@
ˆ f Ë ~D: y,‚" ˆ f Ë ~ : Á~ ˆ P1Ë ~ : "
ˆ f Ë ~h,JIy,‚" ˆ f Ë ~/N7?I , ]
Ë~
We shall prove that the functions ˆ f satisfy the bounds
P
i%i' "
Ÿ e Ë ~ ¡ ¦Z t ´W Ÿ e12 P ¬/f ¡ ® ¤¯ Np L .›*/Ì?0#"
¦
´W Ÿ e12 ¡ ¬hf ® ¤¯ N a L Í*.0hÌ0a `1"Îi%i' 2 ]
ˆ f
(2.10)
lŠO, and $y,‚"87"hn"h :
Define the following barrier function for
B´ a F® ¤¯ p8LJD.›G/Ì0µ f Ë ~ "
i%&' "
² ~ Ë  D{ t B´
a F® ¤¯ a #LJDÍG.0/Ìa 0ˆ µ ˆ f Ë ~ "Ïi%&' 2 ]
P1Ë ~ we have
Á0~
Then using the estimates for and ˆ
² ~ Ë  a ´ ® ¤¯ Np L .›*/Ì?0 ¨ Lf G L DI a `D ª\Z ,‚"i%i' "
Q2 ² ~ Ë  y´ a !® ¤¯ a L Dk*.0hÌ a 0A L G/ Z ,‚"
9
´( L 9 Á : <?>A@ mT,‚"
Ë
L
~
:
y
‚
,
U
"
Ð
´
?
<
A
>
@
a
Ë
~
?
<
A
>
@

a ÓÒ ~ ˆ P€Ë ~
²
² FÑ
² ~ Ë  ,JI4B´ a F® ¤¯ p L .Ì?0 Z ,-"
a
a
² ~ Ë  7I4B´ F® ¤¯ a L 7pG.0/Ì 0 Z , ]
l‰4, follow by the maximum principle. The estimates for l*m
7
Now estimates (2.10) for
P1Ë ~ and bounds
are derived by induction, similarly to [4]. Finally from the estimates for ˆ
à |and
(2.10) we obtain (2.7)
(2.8).
The remainder
solves the problem
9
9
Ã
 | DÁ }  N g"&%i'  " à | : <Œ>A@ y,-" à | : <?>A@ ,-" à | ,JIy,‚" à | 7?I , ]
Applying estimates (2.4),(2.5) in Lemma 2.3 we obtain
(2.11)
| à 
|
œ | Ÿ l‚€™  ž Ÿ È ¡ Z ´W e Z ´("I, Z l Z ƒ ]
The bound (2.6) follows from representation (2.8) and the estimates (2.9), (2.11).
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3. Numerical approximation.
293
3.1. Grid and grid functions. To obtain -uniform convergent difference scheme we
Ô Ö condensed near to boundary
shall
a special partially uniform
(Shishkin) mesh ÕQ
47 construct
O
.
and around interface point
Õ!Ô Ö 4—Œ ~ "/ ~ y ~ PIT× ~ "/F47"hn-" ]8]Œ] "NØu6Ù&OÚi"/‚Lo ,-"/;ÛWO."NÜ(
7`™"
where
@
× P f Ÿ Û –Ná ¡ "
$„70" ]8]Œ] "NØ&Ì`n-"
× f f Û –Ná "
$CØ&Ì`ny70" ]Œ]8] "/Ø
× ~ Mß ÞÞÞÝ × } | –…â " @
$CØuy70" ]8]Œ] "NØäÙw̃;"
ÞÞ × | f 㠟 P f – â ¡ "Ï$CØuÙw̃By70" ]8]8] "NØäE`ÙwÌ5ƒ‚"
} | ã
$CØuE`ÙwÌ5ƒpy70" ]8]Œ] "/Ú&"
ÞÞÞ × ã – â "
˜ P ¢SÞÞà åçæ — • P !è æ ØiÌ5 L "/.Ì`n™"M˜ f ¢Såçæ — • f a !è æ ÙwÌ a L "Œ7pG.0/Ì5ƒ-™ ]
The structure of the mesh follows the pattern of the boundary and
the interior layers. Near
‡0 in ' and _ ga 0 in ' 2 . The parameters • P and
the layers it is _
• f play essential role
in the proof of the uniform convergence below. In section 2 we obtained that the singular
rÚ é?ê #"hlk+7"hn . Thus the error due
part is small outside of theÚ layers.1"€More
exactly
it is _
z
l
ë
0
7
€
"
n
é
ê
to singular part will be _
at the points where condensed and coarse mesh
• % P`" • f mYn ; see [3, 4].
meet. Therefore
to
obtain
second
order
of
convergence
we should have
F
Á
D
~
~
~
Õ$Ô Ö . Let in addition
Let
and
be mesh functions of the discrete argument
ì be a partially continuous function with possible discontinuity at the mesh point ’ÛëÀ. .
~ ì D ~ and ì Û  ì DÛyµY, . We shall also use the following notations for the
Denote ì
grid,í
í
ׂ~w ׂ~En ×;~ 2 P " ì Ô Û × Û 2 P ì Û2 T×
n×Û
í
í*
Á
{
P
~
îÍ2!Á`~w î‰Á`~ 2 P " îk2H 2 Á ~ "
×;~
Û ì Û I" î Á`~w `Á ~*Á`~ P 6
Á`~*í Á`~ P
"
ïî
×~
×;~
í
í Gî Á ~
î
Á
{
P
~
2
]
îÍ2$î‰wÁ`~w
×;~
Further we shall use the discrete maximum norm
œ œŒð ‹Ë Jò ñ ó Œ< ¢Šô…õ £5Jò ñ ¤ ó ¦ ~ ¦ö]
3.2. Finite difference scheme. To obtain a numerical approximation we shall use the
~ P y;~/Bׂ~†Ì`n , ~ P ¬hf C!D ~ P ¬hf .
balance
equation for problem (1.1)-(1.4). Denote by ¬hf
í
D
N
"
P
~
~
¬hf 2 P ¬hf , F
70" ]8]8] "NØu67 and dividing by
Integrating equation (1.1) on the interval
× ~ we get
< ôçø
(3.1)
í
í
D ~ 2 P ¬hf G ~ P ¬hf 3 7 < ‡ô ú ágù†â r!I*D… DIGFNNû0 ]
×~
× F~ ÷ gá ù†â
After approximation of the integrals in (3.1) we obtain
(3.2)
where
í
~ 2 P ¬hf 7p à ~Ö 2 P F* ~ P ¬/f 73 à ~Ö …ýE ~ ~ y ~ "
× ~cü
Ö
~Ö y
;~G, ] þ ׂ~g1" à ~Ö × n5~ ~ ]
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Now we approximate
(3.3)
~ P ¬hfBÿ î ~ "$ ~ 2 P ¬hfBÿ î 2 ~ ]
and use that
à Ö
à Ö
7p à ~Ö 73 7 Ã Ö 73y à ~ Ö P "p73 à ~Ö 73 7 Ã Ö Tn à ~Ö 73 à ~ Ö P ]
~
~ ~
~ 5 in (1.1) we replace the first derivative by r‰ThÌ to obtain an
Ignoring the term
approximation
í
í
í
à ~Ö î ~
Öî ~
7
×
7
~ H ~ ~ ~
~
îÍ2 ° 73 Ã Ö P ± n îk2 ° 73y Ã Ö P ± ÿ n îÍ2 ° × 73~ gy
à ~Ö P ±
~ ~ í
í
í
7
×;~g`~
7
ׂ~r8~r~
7
×;~†5~
n îÍí 2 ° 73 à ~Ö P ± n îÍ2 ° 73 à ~Ö í P ± n îÍ2 ° í 7I à ~Ö P ± Œ~ 2 P × f~ 2 P Ì ×‚~ îÍ2!
7 îÍ2C°
ׂ~r8~
~
~ ÿ n7 îk2Y° 73yí ׂ Ã~†5Ö ~ P ± Ã
P
Ö
n í
73 ~ ±
n‚73 à ~Ö 2 í P P ~ × f~ 2 P Ì ×‚~
× f~ 2 P Ì ×‚~
7î 2 °
ׂ~r8~
8~ 2 P `~ 8~ 2 P 8~
~
Ã
Ã
P
P
Ö
Ö
n
73 ~ ±
n‚73 ~ 2 P ~ 2 P
n‚73 à ~Ö 2 P P ~ 2 P ]
í
Thus from (3.2) we get on the left the so called modified Samarskii scheme, see [2].
Ö ~ 4 îÍ2 ~Ö î‰ ~ ’ ~Ö î‰ï ~ E ~ Ö ~ y ~ Ö "MF
70" ]8]8] "N؏Y7 ]
(3.4)
í
where
í
× f~ 2 P Ì × ~
× ~ ~
7
~ P# ~
~Ö 73 à ~Ö € P "- ~ Ö yŒ~; îÍ2 °
Ã
P
Ö
í ± n‚7Qy à ~Ö P P Œ2 ~ 2 P "
n
3
7
~
í
2
×
Ì
×
~
P
7
×;~r`~
f
Œ
~
5
~
P
~
Ö ~ W ~ î 2C°
2
2
n
73 à ~Ö P ± n-73 à ~Ö 2 P P Œ~ 2 P ]
í
D ~ P ¬hf "N ~ 2 P ¬hf , B‘ØÊ(70" ]8]8] "/ÚÄy7 and
Integrating equation (1.2) on the interval
‚
×
~
dividing by we get
í
í 7 < ô‹ø ágù†â
D
G
I
r!!HD…!DNû ]
~ P ¬hf
(3.5)
× ~ ~ 2 P ¬hf
× ~w÷ < ô‡ú ágù†â
After approximation of the integrals in (3.5), applying (3.3), we obtain the following finite
í
difference scheme on the right of the interface
Ö ~
îÍ2!î‰ ~Œ~ ~ 5~N"M!OØuy70" ]Œ]8] "hÚÓT7 ]
(3.6)
To obtain numerical approximation on the interface we integrate equation (1.1) on the inD Û P ¬/f "NÛzŠ,J , equation (1.2) on the interval Ûik,‚"N Û 2 P ¬hf , sum the integrands
terval
and use the interface conditions (1.3) to get
< L
D Û 2 P /¬ f * Û P /¬ f Q
g!D$*D… D$HDF/û
×Û
× Û ÷ < <! ú ágù†â
wø
í
7 < ágù†â g!D$G…!DNNû0 ]
;× Û ÷ 2 L
í
(3.7)
í7
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SINGULARLY PERTURBED PROBLEM OF MIXED TYPE
After approximation of
í the integrals in (3.7) we get
Û 2 P h¬ f GÛ P h¬ f N7{ à ÛÖ †ýB Œ Ô ÛBÛ 5Ô Û ]
×‚Û ü
Now using the equation (1.2) and the interface conditions (1.3) Ã we approximate
í
×Û
ÛÖ
í
×Û
í
×Û
í
ׂÛ
í
Ö
73 à ÛÖ Û P ¬hf ° 73 7 Ã Ö Tn à ÛÖ 73 Ã Û Ö P ± Û P ¬hf ÿ
Û
Û ×Û
í
×;Û
î IÛ ÛÖ ïî IÛ g Û L H Û L$Û 1"
n × Û 73 Ã ÛÖ P í
í
Û 2 P ¬hfBÿ Û 2 P ¬hf 0× fÛ 2 P ۟ } ¡2 L ×Û
× Û
íÛ P
×
f
2 rÛ 2 L H5Û 2 L Û G Û 2 L Û 2 L Û 2 P ¬hf ÿ
× Û
í
î 2 ÛT × fÛ 2 P ¨gî 2 5Û ÛBî 2 ŒÛyH8Û 2 LŒî 2 ۛª ]
×;Û í
0× fÛ 2 P ۟ } ¡2 L Ì- × Û is added to ensure the second order of convergence (see
The term
the proof of Theorem 5.1 below). Thus we obtain the following difference scheme on the
interface
í
í
(3.8)
where
Ö ÛWÏ° ÛÖ îÍ2 ÛTÊ° × Û ÛÖ H
ÛÖ îi ÛTE Ö (
Û Û ÛÖ "
±
;× Û ±
×;Û
× fÛ í 2 P î 2 ŒÛc"
8 Û 2 5L í × fÛ 2 P "I Ö 8ÛC í ׂÛo8Û L
ÛÖ Û
Ã
P
Ô
n ׂU
Û 7{ ÛÖ ×;Û
ׂÛ
í ‚× Û›5Û L
× fÛ í 2 P î 2 Ûc]
ÛÖ Ô Û Ã
n × Û N37 y ÛÖ P × Û
Setting the boundary conditions
(3.9)
weÖ obtain the discrete problem ( ( ) are such that
(3.10)
Since
(3.11)
Ö
!L yK L " $Ü CK P "
): (3.4), (3.6), (3.8), (3.9). By (1.5) the coefficients in
Ö C,-"- ~ Ö C
m L Ì`n" ~Ö E
m L "-F
70" ]8]Œ] "NØG"I8~$mY L "!
70" ]8]8] "/ÚÓT7 ]
× Û 2 P ,‚†a ?Ù P è æ Ù
then for
í sufficiently large
ÛÖ m LVC, ]
×Û
Ù
independent of holds
Therefore this difference scheme satisfies theÖ discrete maximum principle.
$L mY,‚" $Ü mY, and ~$mT, , then mC, on Õ Ô Ö .
L EMMA 3.1. If
An immediate consequence of this discrete maximum principle is the following inequality,
Ö satisfying zero boundary
L EMMA 3.2. Let be a solution to the discrete problem conditions. Then
n œ Ö œ ð Ë ò ó ]
‹
Ë
ñ
Z
ò
ó
# L
œ œð
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296
I. BRAYANOV
í
4. A priori estimates. Let Ö Ë
),
y ~ Ö "-F
7`؏Y70"
E Û Ö Ë Û Û Ö Ë "
be a solution to the discrete problem (
Ö Ë ~ 4 î 2 ~Ö î ~ ~Ö î ï ~ E ~ Ö
y,‚"M Ö Ë Û × n Û ÛÖ î‰ Û En` ÛÖ î‰
L
~
Û
where
Û Ö Ë Û L 73 8 Û Ã Ö L P "; Û Ö Ë W Û L 73 ` Û Ã
Û ^—Œ;L" ]8]8] "N;ÛU™
For the grid functions defined on the mesh Õ Ô Ö
define the scalar product
Û P í
ú Â ×;~ ~rÁ`~; × Û Û Á Û
N
"
Á
:
Œ
L
Ë
n s
(4.1)
s
òó
~> P s
Ö L P ]
Û and vanishing for
;L ,
and the norms
ú 4 ¦ Á ¦ "Œ7#: 8L Ë ú " Á ˋñ ú ¢Š£`¤ ú ¦ Á`~ ¦ö]
ò ó œ œ ð ò ó <Œô†õ ò ñ ó
Ë
Ö Ë
) is adjoint to (
) in the sense of the scalar product (4.1),
í
~Ö î ~ ! î 2 ~Ö ~ w ~Ö ~ y ~ Ö "-F
7`؏Y70"
Ö
ׂnÛ ÛÖ î ‘Û n5×; Û Û ‘Û E Û Ö Ë ‘Û y Û Ö Ë ]
;~/" Æ of problem ( Ö Ë ). As a function of
Now, we consider the Green function ;~ , with Æ held constant, it is defined by the relations
Ö Ë 3D~/" Æ I ˜ Ö ;~/" Æ #"-!
7 `Ø*" ,‚" Æ Qy, ]
(4.2)
;~
D~/" Æ satisfies the equations
As a function of Æ , with held constant, Green function Ö Ë { ~ " Æ 3O˜ Ö D ~ " Æ 1"ÇS„
7 `ØG" 3 ~ "/,J! , ]
(4.3)
í
where
Æ"
˜ Ö D~/" Æ I t ,‚×;"~" if ~w~ Æ]
if
Ö Ë ~ Ö for $„70" ]8]Œ] "NØëT7 . It is obvious that the solution to problem ( Ö Ë ) is
Denote
œ Á: ¦ P1Ë ò
Ö
The following problem (
í
Ë
Ö ~ „ î 2 L O,-"[ Ö Ë ‘Û ó
expressed in terms of Green function as
(4.4)
whereas the solution to (
Ö Ë
3D~g34 D~N" Æ 1 "€ ~ Ö Ë : LŒË ò ú "
ó
) is written as
3D~†I+ {
Æ "/;~†1"€ Æ Ö : L8Ë ò ú ]
ó
L EMMA 4.1. If conditions (3.10) are fulfilled, the Green function ative and -uniformly bounded:
(4.5)
, Z {;~/" Æ Z L P ]
~ " Æ is nonneg-
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SINGULARLY PERTURBED PROBLEM OF MIXED TYPE
Moreover, the solution to problem ( Ö Ë
297
) satisfies the estimate
P ÖË
ú
ú
œ œ ð ‹Ë ò ñ ó Z L œ A : ¦ P1Ë ò ó ]
(4.6)
~ %
Õ Ö . The fact that the Green function isÖ Ë nonnegative obviously follows
Proof. Fix
under conditions (3.10). To
from (4.2) and the validity of the maximum principle
Æ “ % Ö for
Ԉ such that
derive the estimate in (4.5), consider a point
(4.7)
×
í
Multiplying (4.3) by Æ for
Ç
Ç?L to Ø
with respect to from
í
(4.8)
3D~N" Æ “ I ¢Šõ £5¤ ú 3D~/" Æ ]
!#" $ñ ò ó Ǐ7%‚ØÓO7 , by ×;ÛVÌ`n
one obtains
for
Ǒn
Û P
Â × Æ Ö Ë {;~/" Æ w × Û Ö Ë {D;~/" Û
n
Æ> Ɠ
í
Û P
Æ Ö “ 3 ;~h" Æ “ w Â × Æ Æ Ö {;~/" Æ × n Û ÛÖ
Æ> Ɠ
í
Û P
Â × Æ ˜ Ö D ~ " Æ ’ × Û ˜ Ö ~ "0ۛ Z 7 ]
n
Æ> Ɠ
and summing up the results
’
Ë {;~/" Û Q
Since the Green’s function is nonnegative and (4.7) holds, then all terms in (4.8) are nonnegative. Thus (3.10) implies (4.5). The estimate (4.6) follows directly from (4.5) and the
representation (4.4). 2 2 ~ í Û 2 2 ~ be a solution to the discrete problem ( Ö Ë 2 ):
Let now , Ö Ë 2 ~ 2 îÍ2$î‰ ~ 2 E
8~ ~ 2 y
5~/"MFyØäy70" ]Œ]8] "hÚ
2 ,-"M Ö Ë 2 Û 2 Ï° ×‚Û 2 P# 8Û 2 L × Û n5 P ± î 2 ÛYE Û Ö
Ü
2
T7"
Ë 2 Û 2 y Û Ö Ë 2 ]
where
Û Ö Ë 2 8 Û 2 L3 × Û  2 P î 2 ŒÛU"I Û Ö Ë 2 y
5Û 2 L{ × Û  2 P î 2 5Û ]
~ Ö Ë 2 À5~ and ~ Ö Ë 2 uŒ~ for )äØë„7" ]8Œ] ] "/ÚÄ 7 . The problem ( Ö Ë 2
Denote
written in operator form
x 2 ~ 2 Ö Ë 2 "N€"$FyØ*" ]8]Œ] "/ÚÓY70" Ü(O, ]
2 „—Œ Û " ]8]8] "N Ü ™ and vanishing for Ü
For the grid functions defined on the mesh Õ Ô Ö
the scalar product
Ü P í
Ü
9
ø
ø
Â
Â
×;& Û s ÛBÁ`ÛU" s "NÁ: Ë ò × ~ ~Á ~]
s "/Á L8Ë ò ó ~ > Û P × ~ s ~ Á ~ (4.10)
ó ~> Û P s
2
2
) can be
(4.9)
where
× & Û B6×;Û Û
2 /P Z'#L × Û 2 P]
2 L × Ûf 2 P
, define
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298
I. BRAYANOV
'L
is independent of constant, given in (3.11).
x 2 from (4.9) is selfadjoint and positive definite in the scalar
9 4.2.
L EMMA
The operator
ø
]
]
"
L8Ë ò ó defined in (4.10). For arbitrary discrete functions s "NÁ defined on Õ Ö2 and
product
ÜWOÁ`Ü(O, holds
satisfying s
and
9
(4.11)
9
9
"s NÁ)( ø x 2 s "/Á L8Ë ò ø Y
-rî s "/î 0Á : Ë ò ø Ö Ë 2 s "NÁ LŒË ò ø ]
ó
ó
ó
Proof. The operator
x 2
Ü P í í
 ׂ/~ îÍ2$î‰ g~ Á`~;
s
~> Û 2 P
"NÁ LŒË ò ø Y- î s "hî Á: Ë ò
ó
satisfies
9
9 ÖË
2 s "NÁ LŒË ò ø U"
ó
9 ÖË
ø 9 Ö Ë 2 "NÁ LŒË ø ]
2 s
s
òó
ó
Ö Ë2
~
Now the statement of the lemma follows from the positivity of
x 2 is selfadjoint and positive definite operator, then x . 2 P is also selfadjoint
Since
ø -îÍ2 Û
s "/Á ( s
Ü
 `×;~î ~î Á`~;
s
~> P
Á Û and positive definite operator. So we can define the energy norms
9
9
9
9
¦Á œ( ø +
* x 2 Á;"/Á L8Ë ò ø " ¦ œ Ÿ ( ø ¡ ú á +* x 2 P Á"NÁ LŒË ò ø ]
ó
ó
Ë
Ö
2 be a solution to problem ( 2 ) then
L EMMA 4.3. Let d 9 ÖË
ø
ø ú
œ 2 œ ð ˋò ñ ó Z a ¦ 2 œ Ÿ ( ¡ á ]
(4.13)
d Ë
.
for some positive constant independent
of the smallP parameter
Ö 2
Ö Ë2
x
W
ä
r
x
2
2
2
2
Proof. Since
then . Using the embedding inequality
(4.12)
(see [16])
ø
ø
œ 2 œ ð ‹Ë ò ñ ó Z dXî‰ 2"/îi F2 : Ë ò ó
and estimate (4.11) we have
ø d rî 2 "/î 2 :
œ 2 œ ð ‹Ë ò ñ ó Z X
9 ÖË
d
a 2 "?x 2 P d
Ë ò óø Z a
Ö Ë 2 L8Ë ø
òó
9
x 2 2 "
9
ad ¦ Ö Ë2
Ø
Ù ÿ ÚÍÌ`n . Let ,
Suppose that ÿ
5. Uniform convergence.
Ö
the discrete problem ( ). Then , satisfies
2 8L Ë ò ø
ó
ø ú
œ Ÿ( ¡ á ]
G
be the error of
Ö , g ~ Ö 6
5~gFC Ö ~AHI~rIOK!~/"M;~I% Õ Ö " , L , Ü O, ]
The next Theorem gives the main result in this paper.
T HEOREMP 5.1. Let the conditions in Theorem 2.4 be fulfilled. If the parameters of the
" m„n , Ù ÿ Ø ÿ ÚÍÌ`n , and Ú is sufficiently large (3.11) to hold, then the
mesh satisfy • • f
Ö solution of the discrete problem ( ) is -uniformly convergent to the solution to the
continuous problem (1.1)-(1.4) in discrete maximum norm and the following estimates hold
(5.1)
œ
* œ ð Ë ò ó Z d›ÚHfAè æ f Úi"
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299
SINGULARLY PERTURBED PROBLEM OF MIXED TYPE
d
for some positive constant independent of the parameters of the mesh and the small pa
rameter .
Á
Proof. Using the fact that the solution can be decomposed into regular part and
singular part ˆ , that satisfy the estimates (2.6)-(2.8) in Theorem 2.4 we can write the approxKI~
imation error in the form
K ~ CKQP1Ë ~ K f Ë ~
where
K 1P Ë ~w ~ Ö H`~AYr Ö Á`~AHIÁ`~†1"MK f Ë ~ Ö ˆ ~AH ˆ ~ ]
9
9
9
2
P
P
P
L
p
~
~
/
"
;
~
:
;
~
N
"
~
:
;~ 2 L "N~ 2 P : . We begin with the
~
~
2
Denote ,and internal points in Õ Ö . The
í approximation error has í the form, see [4]
K!~O î 2 î ~A*5 ~ /
. n7 D ~Ö 2 P î 2 ~‚ ~Ö î ï ~g!*?~ ~10
í
î^° n‚73 í × Ã ~ Ö P ~Ö î ~ E ~ ~ H ~ ±
~ n-N73× yf~ 2 P Ã Ì Ö ×‚ ~ P 8~~ 2 PP p?~ 2 P î 2 ~GŒ~~;T5~r
~ í 2
à ~Ö P ­ 0
Ì
×
7
y
«
~
Similarly to [2, 4], using that
is -uniformly bounded, we can prove
í }
í |
¦ K ~ ¦Z 3
d 243‡ ¦ × ~ 2 P{6× ~ ¦ ׂf~  <`¢Xõ£`5$¤ ô ¦ Ÿ Æ ¡ D ¦ 6 ׂf~  <0¢Šõ£55$¤ ô ¦ Ÿ Æ ¡ ¦
Æ> P
Æ> P
í
 f <`¢Šõ£`5$¤ ô ¦ Ÿ Æ ¡ D ¦ 67‘]
¦
P
¦
Ï
;
×
~
6
‚
×
~
‚
×
f
~
2
(5.2)
Æ> P
On the interface we present the approximation error in the form
í
í
í
K Û × Û rK P€Ë Û 6K f Ë Û 0 × Û 2 P 9 1P Ë Û 2 P ;: Û ’ × Û 2 P . P1Ë Û . f ËÛ ]
n ׂÛ
n ;× Û8×;& Û
n ×;Û
where
¦ KIP1Ë Û ¦ =
<< × nÛ <<
73
(5.3)
7
à ÛÖ
Z d324 `×;Û
¦ K f Ë ÛX¦ =
<< × nÛ <<
73
× Û JÁ Hî‰Á`Û L Ð ŒÛ L?ÁJ G Ö î‰Á`Û
0
Û L Û
n Û L
Ö
P ÐDÁ Û L * Û î Á Û L Û L Á Û L Ñ <<<
<
f
}
Â
Æ
<0¢Šõ 5 £5¤ ú ¦ Á Ÿ ¡ D ¦ T×;Û > P <`¢Šõ5 £`¤ ú ¦ Á Ÿ ¡ ¦ 67
Æ
×;Û
Ö
L n ˆ Û L Gîi ˆ Û L 0 Ð Û L ˆ Û L * Û î‰
Ö
P Ð ˆ Û L G Û î ˆ Û L{6?Û L ˆ Û L Ñ <<<
.‡ÁJÛ L . Û
ˆ
7
à ÛÖ
<
LÑ
ˆ Û LÑ
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300
I. BRAYANOV
} ¡ D ¦ T× Û Â f ¢Š£`¤ ú ¦ Ÿ Æ ¡ D ¦ 67
Ÿ
Z
24 0× u
Û
Š
¢
`
£
¤
¦
d
ú
<
`
õ
5
<`õ5 ˆ
(5.4)
ˆ
Æ> P
¦ P€Ë Û 2 0P ¦ ׂn Û ×;& Û P << î 2 Á Û 2 L Á Û 2 P ¬hf × fÛ > 2 P Á Û Ÿ } ¡ P <<
2 ¬/f <
2 <<
}
<
Ÿ
¡
Z dV× fÛ 2 P <0¢Šõ 5 £5¤ ø ¦ Á D ¦
(5.5)
¦ :’Û ¦ × n Û × & Û P << ÁJÛ 2 L × Û n 2 P ÁJÛ 2 L × fÛ 2 P Á Û Ÿ } 2¡ L *ÁÛ 2 P ¬hf × fÛ > 2 P Á
2 <<
} ¡ D ¦
Ÿ
Z
Š
¢
5
£
¤
¦
V
d
‚
×
f
Á
Û
P
ø
0
<
õ
5
2
(5.6)
¦ . P€Ë ÛX¦ × Û  2 P < Û 2 L H5Û 2 L Û H Û 2 L Û L GîÍ2I Û 2 L 6 Û îk2$ Û
<
Û
L
u
î
2 Û L < Z d‘°;× fÛ 2 P T× Û 2 P × Ûu<`¢Xõ5 £`¤ ú ¦ Ÿ f ¡ D ¦ ±
(5.7)
n
(5.8)
¦ . f Ë Û ¦ × Û P << îÍ2 ˆ
2 <
Z d)`×;fÛ 2 < P 0< ¢Šõ5 £5¤ ø
<
Û 2 {L ˆ Û 2 L × Û n 2 P ˆ Û 2 L × fÛ 2 P ˆ
¦ ˆ Ÿ | ¡ D ¦
}
Û Ÿ 2¡ P h¬ f <<
<<
2 L
}
Û Ÿ 2¡ L <<
<<
D ."Β7? we present the approximation error in the form
K ~ C î 2 J P1Ë ~ .?P1Ë ~ 6. f Ë ~ "
At the interior points on the interval
where
(5.9)
(5.10)
¦ P1Ë ~ ¦ << î‰Á`~;Á~ L? @ × > f~ Á ~ Ÿ } ¡ L ? @ <<
<<
<<
í7
¦ .?P1Ë ~ ¦ C
°Á ~ L ? @ *Á ~ L? @ << Á ~ ׂ~ 2
<<
Z d)0×;f~ <0¢Šõ £55$¤ ô ¦ Á Ÿ | ¡ ¦
í
¦ . f Ë ~ ¦ < îÍ2$î‰ ˆ ~’* ˆ ~ < Z d) ¢Xå‹æ
<<
(5.11)
Z V
d ×;f~ 0< ¢Šõ5£`¤ ôú ¦ Á Ÿ } ¡ ¦
× f~ 2 P Á Ÿ } ¡ × f~ Á Ÿ } ¡
> ~ 2 L ? @
> ~ L ?@
t <`¢Xõ£`5 ¤ ô ¦ ˆ
¦ × ~ 2 PH× ~ ¦€<`¢Šõ£`5$¤ ô ¦ ˆ Ÿ } ¡ D ¦ E× f~ <0¢Šõ£55$¤ ô ¦ ˆ
P f
<<
± <<<
<
Ÿ f ¡ D ¦ "
Ÿ | ¡ D ¦A]
, } . The first term , P
Ö Ë , ~ P OKQP1Ë ~ 6K f Ë ~ "$47" ]Œ]8] "/Ø*" , L P y, ]
Now we decompose the error
solution to the problem
,
in the form
,
,
,
Using Theorem 2.4 and estimates (5.2)-(5.4), similarly to [2, 4] we obtain
ú
ú
œ K P : ¦ 1P Ë ò ó _ rÚHfè æ f Úz1" œ K f : ¦ P1Ë ò ó _ a Ú6fAè æ f Úz ]
Applying the a priori estimates (4.6) we get
(5.12)
P
ú
ú
ú
œ, œ ð Ë ò ó Z d
« œ KIP€: ¦ P1Ë ò ó œ K f : ¦ P1Ë ò ó ­ Z d›Ú f è æ f Ú ]
is a
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SINGULARLY PERTURBED PROBLEM OF MIXED TYPE
The second term ,
f
301
is a solution
toí the problem
í
Ö Ë 2 , ~f O
« îÍ2JP1Ë ~ Í
î 2 f Ë ~r­ C
B ~ "NFyØäy70" ]8]8] "/Ú Y70"
Ö Ë 2 , Û f P1Ë Û 2 P D f Ë Û 2 P QE
B Û " , Ü2 , ]
×& Û
where
f Ë ~ :’Ûc"$$CØÀO7" ]8]Œ] "/Ú ]
Using Theorem 2.4 and estimates (5.5),(5.6), (5.9) we obtain
_ Ú f #" f Ë ~ _ rÚ f 1"$$CØÀO7" ]8]Œ] "/Ú ]
4rx 2 P B and ~wC P€Ë ~;D f Ë ~ then
Let F
Ü P í í
9 ø ú
9 ø
9
¦ B œ fŸ ( ¡ á ¦ F‚œ (f ¦ F " B L8Ë ò ø O F ÛG0Û 2 PQH Â × ~ F ~ îÍ2 ~
ó
~> Û 2 P
9 ø
ø
Z -î‰ F " J: Ë ò Z d a 5ÚHf ¦ F;œ ( ]
ó
P€Ë ~ Therefore
9
¦ B œ fŸ ( ø ¡ ú á Z d a `ÚHf ]
Applying the a priori estimates (4.13) we get
(5.13)
The last term ,
}
ø Z d ¦ 9 B ( ø ú Z d›ÚHf ]
f
Ë
œ , œ ð ò ó
a œŸ ¡ á
is a solution to the problem
Ö Ë 2 , ~} y
. P1Ë ~;6. f Ë ~/"$!OØ*" ]8]8] "/ÚÓY70" , Ü } , ]
Using Theorem 2.4 and estimates (5.7),(5.8), (5.10), (5.11), similarly to [2, 4] we obtain
. 1P Ë Û _ rÚHf’è æ f Úz1"$. P€Ë ~w _ ‡ `ÚHf’è æ f z
Ú #"-FyØäO7" ]Œ]8] "hÚëY70"
. f Ë ~w _ r Ú f è æ f z
Ú 1"$$OØG" ]Œ]8] h" Ú ]
Ö Ë 2 we find
Now from the comparison principle for operator
}
ø Z d . P 6.
ø
f œ ð Ë ò ó Z d›ÚHf’è æ f Ú ]
(5.14)
œ , œ ð Ë ò ó
œ
The theorem follows now from estimates (5.12)-(5.14)
Ù
Ø
ڊÌ0n
ÿ
R EMARK 5.2. The requirements ÿ
are a technicality. In the general
case
DØ f è æ f ØÄ
it is clear
from
the
analysis
above
that
the
order
of
convergence
will
be
_
Ù f è æ f ÙÍØٛè æ Ø è æ Ù .
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I. BRAYANOV
6. Numerical results. Consider the problem
5 yN73DH IKJŒLA/- 73MJ åçæ 9LAÌ`nN-473MJ åçæ 9LANH IKJ?LA/1" i%Hr,-"h, ] þ 1"
9 5 yƒ) ' 9 OP 9LAÌ0n0N-yBTnJ å‹æ 9LAÌ0n0NH IKJ?LAÌ0n0#"Mi%Hr, ] þ "Œ7?1"
;: <?> L ? @ O,-" : <Œ> L ? @ ,-"$!,$yFN7?Iy, ]
The solution of this problem exhibits typical boundary and interface layer behavior (see Fig.
Ø ÙÊ ÚÍÌ`n . The exact solution of this problem is not
6.1). For our tests we take
known. To investigate the convergence rate we compare our numerical results with the linear
ڔä7  > ƒ . Table 6.1 displays the results of our numerical
interpolation of the solution
for
Ú
we observe almost second-order -uniform convergence. The
experiments. For large
convergence rate is taken to be
Ü
Ü OèQIR f Ë
œ S Ü œ ð Ë $ Ì œ S f Ü œ ð Ë $ "
Ú
PL
where œ S
œ ð $ is the maximum error norm error for the corresponding
Ú ^7n > value
&ofÊn . Figure
6.1 shows the approximate solution and the maximal error for
and
. It
illustrates very well the boundary and interior layers behavior of the solution. Thus the numerical results support the theoretical ones and show the effectiveness of the special meshes.
TABLE 6.1
Error of the solution on Shishkin’s meshes
TVUVW
T b]
cd
TbeYfg
cd
TbeYfNh
cd
TbeYfNi 2
cd
TbeY fNj
cd
TbkY fml9n
cd
TbkY fmlg
cd
TbkYfml9h
cd
TbkY fml9i
cd
TbkY fml9j
cd
XVY
Z\[
]YV^
YV_VZ
_]Y
]`VY\[
YV`\[a^
6.53e-5
2.01
2.34e-4
2.10
2.01e-3
1.83
2.74e-3
1.55
1.04e-2
1.91
1.80e-2
1.18
1.80e-2
1.18
1.80e-2
1.18
1.80e-2
1.17
1.80e-2
1.17
1.62e-5
2.01
5.46e-5
2.06
5.67e-4
1.93
9.34e-4
1.28
2.77e-3
1.98
7.94e-3
1.51
7.96e-3
1.40
7.98e-3
1.40
7.98e-3
1.40
7.99e-3
1.40
4.04e-6
2.00
1.31e-5
2.03
1.49e-4
1.96
3.84e-4
1.43
7.03e-4
1.99
2.79e-3
1.98
3.02e-3
1.54
3.02e-3
1.54
3.03e-3
1.54
3.03e-3
1.54
1.01e-6
2.00
3.19e-6
2.02
3.81e-5
1.98
1.42e-4
1.53
1.77e-4
2.00
7.09e-4
1.99
1.04e-3
1.60
1.04e-3
1.61
1.04e-3
1.61
1.04e-3
1.61
2.52e-7
2.00
7.87e-7
2.01
9.65e-6
1.99
4.93e-5
1.61
4.41e-5
2.00
1.78e-4
2.00
3.41e-4
1.68
3.41e-4
1.68
3.41e-4
1.68
3.42e-4
1.68
6.28e-8
2.01
1.95e-7
2.02
2.42e-6
2.01
1.61e-5
1.68
1.10e-5
2.01
4.44e-5
2.01
1.06e-4
1.74
1.06e-4
1.73
1.07e-4
1.73
1.07e-4
1.73
1.56e-8
2.04
4.80e-8
2.07
6.00e-7
2.07
5.03e-6
1.78
2.72e-6
1.95
1.10e-5
2.07
3.19e-5
1.77
3.20e-5
1.80
3.20e-5
1.80
3.20e-5
1.80
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ETNA
Kent State University
etna@mcs.kent.edu
303
SINGULARLY PERTURBED PROBLEM OF MIXED TYPE
approximate solution
0.9
absolute error
−3
3
x 10
0.8
2.5
0.7
0.6
2
0.5
1.5
0.4
0.3
1
0.2
0.5
0.1
0
0
0.2
0.4
0.6
0.8
1
0
0
0.2
0.4
F IG . 6.1. Approximate solution and error on Shishkin mesh,
0.6
0.8
1
TbkY fml9no WMb]YV^
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