ETNA
Electronic Transactions on Numerical Analysis.
Volume 20, pp. 86-103, 2005.
Copyright  2005, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
UNIFORM CONVERGENCE OF MONOTONE ITERATIVE METHODS FOR
SEMILINEAR SINGULARLY PERTURBED PROBLEMS OF ELLIPTIC AND
PARABOLIC TYPES
IGOR BOGLAEV
Abstract. This paper deals with discrete monotone iterative methods for solving semilinear singularly perturbed
problems of elliptic and parabolic types. The monotone iterative methods solve only linear discrete systems at each
iterative step of the iterative process. Uniform convergence of the monotone iterative methods are investigated and
rates of convergence are estimated. Numerical experiments complement the theoretical results.
Key words. singular perturbation, reaction-diffusion problem, convection-diffusion problem, discrete monotone
iterative method, uniform convergence
AMS subject classifications. 65M06, 65N06
1. Introduction. We are interested in monotone iterative methods for solving nonlinear
singularly perturbed problems of elliptic and parabolic types.
Firstly, introduce singularly perturbed problems which correspond to the reaction-diffusion
and the convection-diffusion problems of the elliptic type
(1.1)
"!#$%$
&')(*(+,
.- 0/0)$1$2&3((4&65*78or$
&95 ! (:
#;<>=?2@?9A? $CB ? ( D+EGFHIFKJML B D+ENF6FOJML
(1.2)
MPRQHS UT EVW#,>= ? B YXA.X9Z4PN\[,"]4[);
5 7 QH^ 7 T E<_5 ! QH^ ! T E
A`
/
on
S
on
?2
[a?2
^ 7.b
`
5 78b
! are constants. If , and ! are
where and are small positive parameters, and
sufficiently smooth, then under suitable continuity and compatibility conditions on the data,
a uniqued
solution
of (1.1) exists (see [10] for details). 9\f
ceJ
For
, the reaction-diffusion problem (1.1) with
is singularly perturbed and
characterized
by
the
boundary
layers
(i.e.,
regions
with
rapid
change
of the solution) of width
g #hjilk
h [a?
/Cc@J
near
(see
[2]
for
details).
For
,
the
convection-diffusion
problem (1.1)
mn with g
is singularly perturbed and characterized by the regular boundary layers of
o/Vhpiqk0/Vh rJ
CJ
width
at
and
(see [12] for details).
Secondly, introduce singularly perturbed problems which correspond to the reactiondiffusion and the convection-diffusion problems of the parabolic type
u
(1.3)
N&'as
t.,.
Received August 4, 2004. Accepted for publication February 25, 2005. Recommended by Y. Kuznetsov.
Institute of Fundamental Sciences, Massey University, Private Bag 11-222, Palmerston North, New Zealand,
E-mail: I.Boglaev@massey.ac.nz
86
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UNIFORM CONVERGENCE
#t>=dvH? B w E<x2yz{?9KD1ECF6dFOJ|L B D1ECF6}FKJML
87
P QHE<Wt.,>= v B jYX\8X6.
5*7YQH^7 T E<_5 ! QH^ !~T E
where
on
? from (1.2). The initial-boundary conditions are defined by
` 5*7.b
A`)>#t>=I[a? B EVx y@#;EA)€|<0>= ? ‚
€
If , , ! and
are sufficiently smooth, then under suitable continuity and compatibility
conditions on the data, a unique solution of (1.3) exists (see [11] for details).
ƒc„J
†f
For
, the reaction-diffusion problem (1.3)
g with
hpiqk2h [ais? singularly perturbed
near
(see [3] for details).
and characterized by the boundary layers of width
/Rc‡J
OFor
, the convection-diffusion problem (1.3) withg o/Vhpiqk0/Vh is singularly
HˆJ perturbed
3‰J and
characterized by the regular boundary layers of width
at
and
(see
[12] for details).
It is well-known that classical numerical methods for solving singularly perturbed problems are inefficient, since in order to resolve layers they require a fine mesh covering the
whole domain. For constructing efficient numerical algorithms to handle these problems,
there are two general approaches: the first one is based on layer-adapted meshes and the second is based on exponential fitting or on locally exact schemes. The basic property of the
efficient numerical methods is uniform convergence with respect to the perturbation parameter. The three books [9], [12] and [16] develop these approaches and give comprehensive
applications to wide classes of singularly perturbed problems.
In the study of numerical methods for nonlinear singularly perturbed problems, the two
major points have to be developed: i) constructing parameter uniform difference schemes; ii)
obtaining reliable and efficient computing algorithms for computing nonlinear discrete problems. A fruitful method for the treatment of these nonlinear systems is the monotone method
(known as the method of lower and upper solutions, see [13] for details). The monotone
method leads to iterative algorithms which converge globally and solve only linear discrete
systems at each iterative step which is of great importance in practice. Since the initial iteration in the monotone iterative method is either an upper or a lower solution, which can be
constructed directly from the difference equation without any knowledge of the exact solution, this method eliminates the search for the initial iteration as is often needed in Newton’s
method. This elimination gives a practical advantage in the computation of numerical solutions.
In this paper, we investigate uniform convergence properties of the monotone iterative
methods constructed in [5]-[8].
The structure of the paper is as follows. In Section 1, we present differences schemes
which approximate the nonlinear problems (1.1) and (1.3). In Section 3, we construct a monotone iterative method for solving the nonlinear difference schemes which approximate the
nonlinear elliptic problems (1.1) and study convergence properties of the proposed method.
Section 4 is devoted to the construction and investigation of a monotone iterative method for
solving the nonlinear difference schemes which approximate the nonlinear parabolic problems (1.3). The final Section 5 presents results of numerical experiments.
2. Difference schemes.
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88
I. BOGLAEV
?
?>Š\
B (
schemes for solving (1.1). On introduce nonuniform mesh
?fŠ $ 2.1.
? Š Difference
:
? Š $ D*)‹ECŒ9fŒHŽ $ € \E<)’‘“J ”V$ ‹;•)‹l– 7 —a‹L
(2.1)
? Š( ™
˜:+šM’ECŒœ›RŒ•Ž(  € \EV ’ž rJ ” (ŸšY\+š – 7>—+š: 2‚
For approximation
Ar (1.1), we use the classical difference scheme for the reaction-diffusion
problem with
and the upwind difference scheme for the convection-diffusion prob'O lem with
: ¡
Š£¢ ¤“"&6¤0 ¢ OE<;¤…=? Š ¢ •` [V? Š on
¡
¡
¡
r "! ¥#¦N$! &3¦G(M! § ¢ Š ¢ ŠŠ ¢¢ 0/¨¥o¦ $! &'¦ (! § ¢ &6*5 7or¦}$ © ¢ 6
& 5 ! ¦}( © ¢ ‚
-
(2.2)
(2.3)
¦ $! ¢ ¦ ( ! ¢
,
¦}$ © ¢ }
¦ (© ¢
and
,
are the central difference and the backward difference approximations to the second and first derivatives, respectively,
7 « ¢ ‹q– 78b š ¢ ‹ š , ”a$ ‹w © 7 •
¦ $! ¢ ‹ š rj”Vª $ ‹w © ~
¢ ‹ š ¢ ‹ © 78b š " ”a$Mb ‹ © 7 © .7 ¬ ¦ (! ¢ ‹ š rj”Vª (8š © 7 « ¢ ‹ b š – 7 ¢ ‹ š , ”V(Ÿš © 7 •
¢ ‹ š ¢ ‹ b š © 7 , ”a(+b š © 7 © 7 ¬ ”ª $ ‹ ƒ­ © 7 ” |$ b ‹ © 7& ” $ ‹ ,„”ª (Ÿš
O­ © 7 ” 4( b š © 7& ” (8š*,
where
¦ $ © ¢ ‹ šY™ ” M$ b ‹ © 7* © 7 ¢ ‹ š ¢ ‹ © 78b š1,‡¦ ( © ¢ ‹ šY ” 4( b š © 7 © 7 ¢ ‹ š ¢ ‹ b š © 7*
¢ ‹ š ¢ w a‹ š ¤r)‹ š >=? Š
and
.
2.2. Difference schemes for solving (1.3). On
?fŠ
where
is defined in (2.1) and
v
introduce a rectangular mesh
? ® D%t¯¨\°£±V’ENŒA°²Œ•Ž ® ; Ž ® ±R\x¨L³‚
For approximation
of problem (1.3), we use the implicit difference scheme
¡
(2.4)
Š£¢ w¤0t& J ¢ ¤0t³ ¢ w¤0t—±Vy,r
¤0t. ¢ ±d´
¢ ¤0tH`,¤0t. ¤0t0=µ[V? Š B ? ® ¡
Š¢
where on each time level
is defined in (2.3) and
¢ ¤0E•)€:w¤“.¤™= ? Š ¢ ‹ ¯ š ¢ a‹ š t ¯ .
?>Š B ?®
,
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89
UNIFORM CONVERGENCE
2.3. The maximum principle. On
¶
ing canonical form
w¤“·w¤“
(2.5)
?0Š
, we represent a difference scheme in the follow-
¸
¤0Ÿ¤¨ÀÁV·Âz¤¨Àq;&'ÃRw¤“.¤™=? Š ¹º#»:¼V½l¹"¾+¿
·w¤“\·K€£¤“’¤…=[a? Š and suppose
¶ that
¶
w¤“ T EV # ¤0¤ À >Q•E<’SM¤“ w¤“ ¸
¤0¤ À T E<¤r=? Š ¿
¹ º »:¼ º ½l¹"¾4¿
À w¤“ Ä ¤“Å
D+¤NL , Ä w¤“ is a stencil of the difference scheme. Now, we formulate
where Ä
a discrete maximum principle and give an estimate on the solution to (2.5).
L EMMA 2.1. Let the positive property of the coefficients of the difference scheme (2.5)
be satisfied.
·¤“
(i) If ¶
satisfies the conditions
w¤“·¤“³
¸
¹ º »:¼V½l¹"¾ ¿
#¤0¤ À <·ew¤ À œÃR¤“>Q•EVŒ6E.¤™=? Š ·w¤“>QHEajŒ•E:;¤…=[a? Š then
·w¤“>QHEajŒHE:.¤r= ? Š
.
(ii) The following estimate on the solution to (2.5) holds true
Æ · Æ Ç<È ŒHÉCÊM˲ÌjÍÍ · € Í%Í Î Ç È  Æ Ã
]4S Æ8Ç ÈMÏ (2.6)
where
Æ · Æ Ç È †ÉRÊMÇ<Ë È h ·¤“%hÐ
¹»
ÉRÊ4Ç Ë 2
· € ¤“ Ò ‚
ÍÍ · V€ Í%Í Î Ç È Ñ
¹ »:Î È ÒÒ
Ò
The proof of the lemma can be found in [17].
3. Monotone iterative method for the elliptic problems.
3.1. Monotone convergence. For solving the nonlinear difference scheme (2.2), we
investigate uniform convergence of the monotone iterative methods constructed in [5] and
[7].
Additionally, we assume that from (1.1) satisfies the two-sided constraints
ECF•S Œ•4P}ŒHS _S ŸS (3.1)
We say that
¢ ¤“
const
‚
¡ upper solution of (2.2) if it satisfies the inequalities
is an
Š ¢ &9w¤0 ¢ >Q•E<’¤™=? Š ¢ ¤“
¢ Q9`
on
[a? Š ‚
Similarly,
is called a lower
if it satisfies all the reversed inequalities.
˜ ¢ ½lÓM¾ solution
The iterative sequence
is constructed using the following recurrence formulas
(3.2)
¢ ½ € ¾ ¤“
fixed
¢ ½ € ¾ ¤“A`w¤“.¤™=µ[a? Š ETNA
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90
¡
I. BOGLAEV
2Õ ½lÓ|¾ ¤“._¤…=? Š ¥ Š &'S §;Ô l½ Ó – 7 ¾ ¡
½
l
M
Ó
¾
Õ ¤“ ¢ ½lÓ|¾ &6IÖ*¤0 ¢ ½lÓM¾× Ô ½lÓ – 7 ¾ ¤“OE<’¤™=µ[a? Š ¢ ½lÓ – 7 ¾ ¤“f ¢ l½ Ó|¾ ¤“& Ô ½qÓ – 7 ¾ ¤“.¤…= ? Š ‚
The following proposition½ gives
the monotone property of the iterative method (3.2).
¢ € ¾ ¢ ½€ ¾
be upper
and lower solutions of problem (2.2) and let
P ROPOSITION 3.1. Let
¢ ½lÓ|¾*Ù
satisfy (3.1). Then the upper sequence Ø
generated by (3.2) converges monotonically
¢
from above to the unique solution of (2.2), the lower sequence
¢
converges monotonically from below to :
Ø
¢ ½lÓ|¾ Ù
– 7
¢ ½ € ¾ Œ ¢ ½lÓM¾ Œ ¢ ½qÓ – 7 ¾ Œ ¢ Œ ¢ ½lÓ ¾ Œ ¢ ½lÓM¾ Œ ¢ ½ € ¾ rJ œS ]4S%
generated by (3.2)
on
? Š and the sequences converge with the linear rate Ú
.
The proof of the proposition can be found in [5], [7].
R EMARK
3.2. Consider the following approach for constructing initial upper and lower
¢ ½€ ¾
¢ ½€ ¾
¤“
?>Š
solutions
and
. Suppose that a mesh function Û
is defined on
and satisfies
A
`
a
[
?
Š
¡
the boundary condition ¡ Û
on
. Introduce
the following difference problems
(3.3)
¥ Š &9S § Ô Ü ½ € ¾ ƒÝ¨h Û &6¤0 Û %h’¤™=? Š Ô Ü ½ € ¾ ¤“fAE<’¤™=I[a? Š ZÝRrJ|%¨J|‚
¢ ½€ ¾ & Ô ©½ € 7 ¾
& Ô 7½ € ¾ ¢ ½ € ¾ Û
Û
Then the functions
are upper and lower solutions,
respectively.
The proof of this result can be found in [5], [7].
R EMARK 3.3. Since the initial iteration in the monotone iterative method (3.2) is either
an upper or a lower solution, which can be constructed directly from the difference equation
without any knowledge of the solution as we have suggested in the previous remark, this
algorithm eliminates the search for the initial iteration as is often needed in Newton’s method.
This elimination gives a practical advantage in the computation of numerical solutions.
R EMARK 3.4. We can modify the iterative method (3.2) in the following way. ProposiS1
tion 3.1 still holds true if the coefficient in the difference equation from (3.2) is replaced
by
S l½ ÓM¾ w ¤“AÉRÊ4Ë 4P)¤0 ¢ . ¢ l½ ÓM¾ w ¤“0Œ ¢ w¤“0Œ ¢ l½ ÓM¾ ¤“’¤
fixed
‚
To perform the modified algorithm we have to compute two sequences of upper and lower
solutions simultaneously. But, on the other hand, this modification increases significantly the
rate of the convergence of the iterative method.
Without
`,¤“Þ
ßE loss of generality, we assume that the boundary condition in (1.1) is zero, i.e.
. This
can always be obtained ¢ via½ ¾ a change of variables. Let the
¾
¢ ½ € assumption
€ is the solution of the following
initial function
be chosen in the form of (3.3), i.e.
¡
difference problem
(3.4)
¥ Š &9S § ¢ ½ € ¾ Oݨh ¤0ŸE:*h_¤™=? Š ETNA
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91
UNIFORM CONVERGENCE
w¤“\E
¢ ½ € ¾ ¤“OE<अ=[a? Š ZÝRJ:*¨J|
¢ ½ € ¾ ¤“. ¢ ½ € ¾ w¤“
ÝCrJ
ÝR¨J
where Û
. Then the functions
corresponding to
and
are upper and lower solutions, respectively.
¢ ½€ ¾
T HEOREM 3.5. Suppose that the initial upper or lower solution
is chosen in the
form of (3.4). Then the monotone iterative method (3.2) converges uniformly in the perturba
/
tion parameters and :
ÍÍ ¢ q½ Ó – 7 ¾ ¢ l½ Ó|¾ ÍÍ Ç£È ŒHS Ó Æ ¤0ŸE: Æ Ç<È _S âá S &9S% €Ú
€
S S
(3.5)
Í
Í
J —S ]MS
where Ú
.
Proof. Using ¡ the mean-value theorem and (3.2), we obtain
¥ Š &'S § Ô ½lÓ – 7 ¾ « S œ P ½lÓ|¾ ¤“ ¬ Ô ½lÓM¾ ¤“_¤™=? Š where
(3.1),
Ô l½ Ó – 7 ¾ ¤“OE<अ=[a? Š P l½ ÓM¾ w¤“2r P ÌФ0 ¢ ½lÓ © 7 ¾ ¤“"&•ã ½lÓM¾ w¤“ Ô ½lÓM¾ ¤“ Ï E Fã l½ ÓM¾ w ¤“UF™J
,
(3.6)
Applying (2.6) to (3.2) for ä
(3.7)
Estimating
ÍÍ Ô ½ 7 ¾ ÍÍ Ç È Œ J ÍÍ
S Í
Í
Í
¢ ½€ ¾
. By (2.6) and
ÍÍ Ô l½ Ó – 7 ¾ ÍÍ Ç<È Œ Ó ÍÍ Ô ½ 7 ¾ ÍÍ Ç<È ‚
Ú Í
Í
Í
Í
J
and taking into
¡ account (3.4), we have
J
ÍÍ Š ¢ ½ € ¾ ÍÍ Ç & J ÍÍ Ö ¤0 ¢ ½ € p¾ × ÍÍ Ç È ‚
Õ ½ € ¾ ÍÍ Ç È Œ
S Í
Í
Í È S Í
Í
from (3.4) by (2.6), we get
ÍÍ ¢ ½ € ¾ ÍÍ Ç È Œ J Æ w¤0E Æ Ç<È ‚
S
Í
Í
¡
From here and (3.4),
it follows that
ÍÍ Š¢ ½ € ¾ ÍÍ Ç È Œ•S ÍÍ ¢ ½ € ¾ ÍÍ Ç È & Æ ¤0E Æ Ç<È •
Œ ­ Æ w¤0E: Æ Ç<È ‚
Í
Í
Í
Í
¢ ½€ ¾
Using the mean-value theorem, (3.1) and the estimate on
, we conclude that
ÍÍ w¤0 ¢ ½ € ¾ ÍÍ Ç<È Œ Æ ¤0ŸE: Æ ÇÈ &'S ÍÍ ¢ ½ € ¾ ÍÍ ÇÈ Œmå,Jf& S Æ ¤0E Æ ÇÈ ‚
S æ
Í
Í
Í
Í
Ô ½ 7 ¾ in the form
Substituting the above estimates in (3.7), we estimate
ÍÍ Ô ½ 7 ¾ ÍÍ Ç<È Œ6S Æ ¤0ŸE: Æ Ç È €
Í
Í
S€
where
(3.5).
is defined in (3.5). Thus, from here and (3.6), we conclude the uniform estimate
3.2. Uniform convergence of the monotone iterative method (3.2). Here we analyze
a convergence rate of the monotone iterative method (3.2) defined on meshes of the general
type introduced in [15].
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92
I. BOGLAEV
3.2.1. Layer-adapted meshes. The reaction-diffusion problem (1.1). For the reactiondiffusion problem (1.1), a layer-adapted
? $ mesh
E<*from
J*y [15]
? ( is formed
EV*J*y in the followingEVmanç $ y
ner. We divide each of the intervals
´
´
and
into three parts ´
,
ç $ %J —ç $ y J —ç $ *J*y
E<Ÿç ( y ç ( %J —ç ( y J œç ( *J*y
´Ž $ Ž (
, ´
, and ´
, ´
, ´
, respectively. Assuming that
E<Ÿç $ y J —ç $ *J*y
EVç ( y J œç ( %Jy
by ( 4, in the parts ´
, ´
and ´
, ´
$ ]4èdivisible
$y
Ž are
&éJ
Ž ]+è²&éJ
ç $ *J we
ύ allocate
and
mesh points, respectively, and in the parts ´
ç ( *J œç ( y
Ž $ ]M­R&êJ
Ž ( ]M­C&J
and
and
mesh points, respectively. Points
ç $ ´ J œç $ $ weç ( allocate
J —ç ( ,
and , (
correspond to transition to the boundary layers. We con? Š
? Š
Ì  ‘*ëì aí  ‘%ëì Ï and Ì ’ž ëì |í ;ž ëì Ï but
sider meshes
and
which are equidistant in
Ì E<  ‘%ëì Ï Ì aí  ‘%ëì *J Ï and Ì E< ’ž ëì Ï , Ì :í ’ž ëì %J Ï . On Ì EV  ‘ëì Ï , Ì )í  ‘*ëì %J Ï
graded in
ÌîEV ’ž ëì Ï Ìï|í ’,ž ëì *J Ï
and
,
EGñE
jJ+]4èNˆJ let our mesh be given by a mesh generating function ð with
and ð
which is supposed to be continuous, monotonically increasing,
ð
and piecewise continuously differentiable. Then our mesh is defined by
çV$ õ ‹ a‹;_ó ôò  ” $Vð J —ça$2J õ ‹ A].Ž$<;³AE<%‚*‚%‚ŽG$]+è 
’ OŽ$:]4èY&AJ:*‚%‚*‚ Ž$]+è~6J 
á $’ Ž$]+èY&\J|*‚%‚*‚ŸŽ$
# õ ‹ "ˆõ ‹ r; á Ž$:]4è<]Ž
á
ð
ça( wõ8š1
+šY ó ôò › ” (ð J œç ( J õ.šY9›].Ž(£,›G\E<%‚*‚*‚*Ž(|]4è 
› \ŽG(|]+è
&\J|*‚%‚*‚ á ŽG(4]4è¨6J 
N
õ š ˆõ š rq›“ á Ž ( ]+è£<].Ž ( ›N á Ž ( ]+èY&\J|*‚%‚*‚ŸŽ ( We also assume that ð
ð
” $O­ jJ œ­|çV$:<Ž $ © 7 º
” (¨\­ J œ­|çV(MŽ ( © 7 ‚
does not decrease. This condition implies that
” $ ‹ Œ ” $Mb ‹q– 74rJ|%‚*‚%‚ŽG$:]+è¨HJ|
” $ ‹ Q ” $|b ‹q– 7+’’ Ž$]+è
&AJ:*‚%‚*‚ŸŽ$~HJ|
á
”V(8š Œ ”V(+b š – 7 ,›GrJ|*‚%‚*‚ŸŽ ( ]+è¨HJ|
”a(Ÿš Q ”a(+b š – 7 )›G Ž ( ]+èY&\J|*‚%‚*‚ŸŽ ( 6J:‚
á
The convection-diffusion problem (1.1). For the convection-diffusion problem (1.1),
a layer-adapted
in the following
each
$ y J —ç We
$ %Jy|divide
( y:
? $ mesh
EV*Jy from
? ( [15]
E<is%Jformed
y
EV*J0—ç manner.
E<%J —ofç the
´
´
intervals
and
into two parts ´
,´
and ´
Ž $ ŸŽ (
Ž $ ]|­>&•J
J —ç ( *Jy
´ Ž ( ]M­C&êrespectively.
Assuming that
J
are even, in each part we allocate
jJ œ
ç $ (
and
mesh points in the - and -directions, respectively. Points
Š
jJ œç ( ? Š$
? and
correspond to transition to
the
boundary
layers.
We
consider
meshes
and
ÌîEV  ‘ë ! Ï
ÌîE< ;ž ë ! Ï
ÌÐ  ‘*ë ! %J Ï
ÌÐ ’ž ë ! %J Ï
which are equidistant in
and
but graded in
and
.
 ‘%ë ! *J Ï
’ž ë ! %J Ï
#õ|
Ì
Ì
On
and
with
E“mJ
J+]M­:UâE let our mesh be given by a mesh generating function ö
and ö
which is supposed to be continuous, monotonically decreasing,
ö
and piecewise continuously differentiable. Then our mesh is defined by
$ a‹;é  J ”a—
ça$
³AEV*J|%‚*‚%‚Ž $ ]M­ 
#õ ‹ ˆõ ‹ rœŽ$£]M­|<]Ž$’\Ž$£]M­ &AJ:*‚%‚*‚ŸŽ$<
ö
(:
+š
› J ” œ
çV(
›GOE<%J|%‚*‚*‚.ŸŽ(|]|­ 
#õ8š4ˆõ8šYrq›“œŽ(M]|­|<].ŽG(£›NAŽG(|]M­ &AJ:*‚%‚*‚ŸŽ(£
ö
”a$ O­2jJ œç $ < Ž $ © 7 ”V( \­ J —ç (  Ž ( © 7 ‚
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We also assume that ö
º
93
UNIFORM CONVERGENCE
does not decrease. This condition implies that
” $ ‹ Q ” $|b ‹l– 7+àOŽ$]M­2&\J|*‚%‚*‚ŸŽ$~HJ|
” (Ÿš¨Q ” (+b š – 7M÷›N\ŽG(|]M­ &\J|*‚%‚*‚ŸŽ(Y6J:‚
3.2.2. Shishkin-type mesh. The reaction-diffusion problem (1.1). For the reactionç $ jJ œç $ ç ( J —ç ( diffusion problem (1.1), we choose the transition points ,
and ,
as in
[12]:
ç $ •ÉCøqk ˜ è © 7 % jJ4]|ù S £~ilk2Ž $ _ç ( \ÉCølk ˜ è © 7 1 jJ+]:ù S Uiqk2Ž ( ‚
ç $|b ( J+]+è
Ž $|© b ( 7
If
, then
are very small relative to . In this case, the difference scheme (2.2)
can be analyzed using standard techniques. We therefore assume that
ç $ rJ+]:ù S £~iqk2Ž $ _ç ( ™jJ4]|ù S £~ilk2Ž ( ‚
Consider the mesh generating function ð in the form
(3.8)
In this case the meshes
?fŠ $
and
?fŠ (
ð
õ|•è:õ‚
are piecewise equidistant with the step sizes
Ž $ © 7 F ” $NFA­MŽ $ © 7 ” $%R•è
J+] ù S £"Ž $ © 7 qi k2Ž$
Ž ( © 7 F ”a( FA­MŽ ( © 7 ”a( •è
J+]|ù S £"Ž ( © 7 ilk2Ž ( ‚
The difference scheme (2.2) on the piecewise uniform mesh (3.8) converges -uniformly
to the solution of (1.1):
(3.9)
Æ ¢ — Æ Ç<È Œ•ú~Ž © ! iqk ! ŽI_Žˆ\ÉNøqk~D%Ž $ Ž ( L
ú
/
where (sometimes subscripted) denotes a generic constant that is independent of or and
Ž
. The proof of this result can be found in [12].
The convection-diffusion problem (1.1). For the convection-diffusion problem (1.1),
J —ç $ J —ç ( we choose the transition points
and
as in [12]:
ç $ AÉCøqk˜:­ © 7 % w­]+^ 7 /ilk2Ž $ _ç ( \ÉCølk˜­ © 7 1 w­:]4^ ! /’iqk Ž ( 2‚
ç $|b ( rJ+]|­
Ž $M© b ( 7
/
If
, then
are very small relative to . In this case, the difference scheme (2.2)
can be analyzed using standard techniques. We therefore assume that
ç $ rz­:]4^ 7 £/’iqk Ž $ _ç ( ™w­]+^ ! /ilk2Ž ( ‚
Consider the mesh generating function ö in the form
(3.10)
ö
õ|J 3­4õ‚
? Š$
? Š(
In this case the meshes
and
are piecewise equidistant with the step sizes
Ž $ © 7 F a” $ FH­|Ž $ © 7 ”V$%- ™#è]+^ 7 /MŽ $ © 7 ilk2Ž $ ETNA
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94
I. BOGLAEV
Ž ( © 7 F ” (FA­MŽ ( © 7 ” (.- ™#è]+^ ! /MŽ ( © 7 ilk2ŽG(:‚
/
The upwind difference scheme (2.2) on the piecewise uniform mesh converges -uniformly
to the solution of (1.1):
Æ ¢ Þ Æ Ç<È Œ•ú~Ž © 7 iqk ! ŽI_Žˆ\ÉCølk~D+Ž $ ŸŽ ( L
(3.11)
ú
/
Ž
in [12].
where constant is independent of and . The proof of this result can
¢ ½ be
€ ¾ found
T HEOREM 3.6. Suppose that the initial upper or lower solution
is chosen in the
form of (3.4). Then the monotone iterative method (3.2) on the piecewise uniform meshes
(3.8) and (3.10) converges parameter-uniformly to the solution of problem (1.1):
ÍÍ ¢ ½lÓM¾ Þ ÍÍ ÇÈ Œ… ú ¥ Ž © !7 iqk
ú ¥ Ž © qi k
Í
Í
J —S ]4S%
ú
where Ú
and constant
Proof. Using (3.6), we obtain
! Žé& Ó § ! Žé& Ú Ó § Ú
/
is independent of
Ó –û
ÍÍ ¢ l½ Ó )– û ¾ ¢ l½ Ó|¾ ÍÍ <Ç È Œ ¸ ©
Í
Í
l‹ ü Ó
Œ Ú
J S
'\f)
for
'\-4
for
or and
7
ÍÍ ¢ ½ ‹l– 7 ¾ ¢ ½ ‹ ¾ ÍÍ Ç È
Í
Í
ÍÍ Ô ½qÓM¾ ÍÍ Ç<È Œ S € Ú Ó
J Í
Ú Í
Ú
iløqÉ ¢
where € is defined in (3.5). Taking into account that
is the solution to (2.2), we conclude the estimate
Ž
.
Ó )– û © 7
ÍÍ Ô ½ q‹ – 7 ¾ ÍÍ <Ç È
¸
Í
‹qü Ó Í
Æ ¤0ŸE: Æ Ç<È ½lÓ – û ¾ ¢
as ýÿþ
X
, where
¢
ÍÍ ¢ ½lÓM¾ ¢ ÍÍ Ç È Œ S € Ú Ó Æ ¤0ŸE: Æ Ç È ‚
J 0
Í
Í
Ú
From here, it follows that
ÍÍ ¢ ½lÓM¾ — ÍÍ Ç<È Œ Æ ¢ — Æ Ç È & S € Ú Ó Æ ¤0ŸE: Æ Ç È ‚
J Í
Í
Ú
From here and (3.9) for the reaction-diffusion problem, and (3.11) for the convection-diffusion
problem, we prove the theorem.
3.2.3. Bakhvalov-type mesh. The reaction-diffusion problem (1.1). For the reactionç $ jJ¨•ç $ ç ( jJ¨•ç ( diffusion problem (1.1), we choose the transition points ,
and ,
in
Bakhvalov’s sense (see [2] for details), i.e.
çV$™jJ4] ù S <~ilk¨J+]+)_ça(~™jJ4] ù S £~ilk“J+]4)
and the mesh generating function ð is given in the form
ð
(3.12)
èaJ Þjõ+y ‚
#õ| qi k ´ J —
qi k The difference scheme (2.2) on the Bakhvalov-type mesh converges -uniformly to the
solution of (1.1):
Æ ¢ — Æ Ç<È ŒAú~Ž © 7 _Žˆ•ÉCøqk~D1Ž$<Ž(L
where constant
ú
is independent of
and
Ž
. The proof of this result can be found in [2].
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95
UNIFORM CONVERGENCE
The convection-diffusion problem (1.1). For the convection-diffusion problem (1.1),
jJHç)$£
J“Hça(|
we choose the transition points
and
in Bakhvalov’s sense (see [15] for
details), i.e.
ç $ rw­:]4^ 7 /ilk“J+]1/|)_ç ( ™w­]+^ !  /ilk“J+]1/|)
and the mesh generating function ð is given in the form
ð
(3.13)
/|J œ­Mõ|y ‚
#õ|f qi k ´ J HJ ÿ
qi k0/
/
The upwind difference scheme (2.2) on the Bakhvalov-type mesh converges -uniformly
to the solution of (1.1):
Æ ¢ Þ Æ Ç<È ŒAú~Ž © 7 _Žm\ÉCølk~D+Ž $ ŸŽ ( L
/
Ž
ú
where constant is independent of and . The proof of this result can be found in [15].
Similar to Theorem 3.6, for the monotone iterative method (3.2) on the log-meshes (3.12)
and (3.13), we can prove the following theorem.
¢ ½€ ¾
T HEOREM 3.7. Suppose that the initial upper or lower solution
is chosen in the
form of (3.4). Then the monotone iterative method (3.2) on the log-meshes (3.12) and (3.13)
converges parameter-uniformly to the solution of problem (1.1):
where Ú
ÍÍ ¢ ½qÓM¾ Þ ÍÍ Ç<È H
Œ ú ¥ Ž © 7 & Ó § _ŽˆAÉNøqk¨D%Ž $ ŸŽ ( L
Ú
Í
Í
J —S ]MS%
ú
/
Ž
and constant
is independent of
or and
.
4. Monotone iterative method for the parabolic problems.
4.1. Monotone convergence. For solving the nonlinear difference scheme (2.4), we
investigate uniform convergence of the monotone iterative methods constructed in [6] and
[8].
Represent the
¡ difference equation from (2.4) in the equivalent
¡
¡ form
¢ ¤0t
w¤0t. ¢ "& ¢ ¤0t—±V ±
å Š & ±J ‚
æ
t“=3? ® ²¤0t
a time level
,
is an upper solution with a given function
}w¤0We
tÞsay
±V that on
, if it¡ satisfies
7
²¤0t;&6²¥z¤0t. § —± © }w¤0t—±V0QHEV;¤…=²? Š R¤0t Q'`,¤0t.¤™=I[a? Š ‚
Similarly,
}w¤0tÞ±V
¤0t
t=O? ®
is called a lower solution on a time level
with a given function
, if it satisfies all the reversed inequalities.
Additionally, we assume that from (1.3) satisfies the two-sided constraints
E Œ• P Œ6S _S C
‚
const
}¤0t
An iterative solution
to (2.4)½lÓMis¾ constructed in the following way. On each time
t“='? ®
¤0t.0¤‰= ?fŠ
†J:*‚%‚*‚
ä using the recurlevel
, we calculate ä iterates
,ä
¡
rence formulas
&'S Ô ½lÓ – 7 ¾ ¤0t2Õ ½qÓM¾ ¤0t.Z¤…=²? Š (4.1)
(4.2)
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Õ l½ ÓM¾ w ¤0t
¡
I. BOGLAEV
½lÓM¾ w¤0t&6 Ö ¤0t. ½lÓM¾p× Þ± © 7 ²¤0t³—±V.
Ô ½lÓ – 7 ¾ ¤0t\EV;¤…=I[V? Š ä
\E<%‚*‚*‚*
ä
HJ|
½qÓ – 7 ¾ ¤0t l½ ÓM¾ ¤0t;& Ô ½qÓ – 7 ¾ ¤0t’¤…= ? Š }w¤0t l½ Ó¾ w¤0t¤…= ? Š ²¤0EA)€:w¤“.¤™= ? Š ½ € ¾ 0
¤ t
where an initial guess
satisfies the boundary condition
½
€ ¾ ¤0tA`w¤0t.¤…=[a? Š ‚
½ € ¾ w¤0t
P ROPOSITION 4.1. Let
be an upper or a lower solution of problem (2.4) and
let satisfy (4.1). If on each time level the number of iterates ä in the iterative method (4.2)
QH­
satisfies ä
, then the following estimate on convergence rate of the iterative method (4.2)
holds
7 CAS ]wS &'± © 7 . t ¯ ¢ #t ¯ Æ ÇÈ ŒAú Ó © 7ÉR¯
Ê4Ë  Æ }
(4.3)
¢ w¤0t
ú
±
where
is the solution to (2.4),
˜ ½lÓM¾ w¤0and
t constant is independent of . Furthermore, on
each time level the sequence
converges monotonically.
The proof of the theorem for the reaction-diffusion problem (2.4) can be found in [6], the
result for the convection-diffusion problem (2.4) may be proved in a similar way.
R EMARK
the following approach for constructing initial upper and lower
½ ¾ 4.2. Consider
€ ¤0t
½ € ¾ ¤0t
t
¤0t
solutions Š
and
. Suppose that for fixed, a mesh function Û
is
?
¤0t>O`,¤0t [a? Š
defined on
and satisfies the boundary condition Û
on
. Introduce the
¡
¡
following difference
problems
(4.4)
Ô Ü ½ € ¾ w¤0tƒÝ Ò Û ¤0t&9w¤0t. Û ³Þ± © 7 ²¤0t—±V Ò ’¤…=²? Š Ò
Ò
Ô Ü ½ € ¾ w¤0tOE<’¤™=I[a? Š ZÝRrJ|*¨J:‚
½ ¾
€ ¤0t
w¤0t¡ & Ô ¡ 7 ½ € ¾ ¤0t. ½ € ¾ ¤0tf ¤0t£& Ô © ½ € 7 ¾ w¤0t
Then the functions
Û
Û
are
7
upper and ¡ lower¡ solutions, respectively.
Š &6± ©
can be found in [6] and this result for
The proof of this result for
-Š &'± © 7 (2.4) with
(2.4) with
may be proved in a similar way.
R EMARK 4.3. On each time level the initial iteration in the monotone iterative method
(4.2) is either an upper or a lower solution, which can be constructed directly from the difference equation without any knowledge of the solution as we have suggested in the previous
remark, hence, this algorithm eliminates the search for the initial iteration as is often needed
in Newton’s method. This elimination gives a practical advantage in the computation of
numerical solutions.
`}KE
Without loss of generality, we assume that the boundary condition
. This
assump ½ € ¾ w¤0t
tion can always be obtained via a ½ change
of
variables.
On
each
time
level,
let
be
¡ of (4.4), i.e. € ¾ w¤0t is the solution of the following difference problem
chosen in the form
(4.5)
7
½ ¾
Ü € ¤0tƒÝ ÒÒ w¤0t.E’Þ± © ²¤0t’—±V ÒÒ ;¤…=? Š ETNA
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97
UNIFORM CONVERGENCE
¤0t“éE
½ ¾
Ü € w¤0tOE<’¤™=I[a? Š ZÝRrJ|%¨J|
7 ½ € ¾ w¤0t. ½ € 7 ¾ w¤0t
©
where Û
. Then the functions
are the upper and the lower
solutions.
T HEOREM 4.4. Let initial upper or lower solutions be chosen in the formQof­ (4.5), and
let satisfy (4.1). Suppose that on each time level the number of iterates ä
. Then the
monotone iterative method (4.2) converges parameter-uniformly, and the estimate (4.3) holds
ú
±
/
true with constant which is independent of , the perturbation parameter ( or ) and the
nonuniform mesh.
Ô ½lÓ|¾ , we have
Proof.
Using the mean-value theorem and the equation for
¡
(4.6)
¤0t’—±V
½lÓM¾ w¤0t;&6µÖ1¤0t. ½lÓM¾p× ²
r « S œ P q½ ÓM¾ ¤0t ¬ Ô q½ ÓM¾ ¤0t.
±
where
P l½ ÓM¾ ¤0t\ P « 0
¤ t. l½ Ó © 7 ¾ ¤0t;& q½ ÓM¾ ¤0t Ô l½ ÓM¾ w ¤0t ¬ EF l½ Ó|¾ ¤0t3F J
Ô ½qÓ – 7 ¾ ¤0t
and
. From here and (4.2), it follows that
¡
difference equation
&9S Ô ½lÓ – 7 ¾ w¤0t Ö S 3 P ½lÓM¾ × Ô ½lÓM¾ ¤0t.³¤™=? Š ‚
Using (2.6) and (4.1), we conclude
(4.7)
ÍÍ Ô ½lÓ – 7 ¾ # t ÍÍ Ç<È Œ
Í
Í
Introduce the notation
}w¤0t ½lÓ ¾ ¤0t
where
·¤0¡ ±V
that
satisfies
satisfies the
S% ‚
Ó ÍÍ Ô ½ 7 ¾ # t ÍÍ Ç<È N
S '
& ± © 7
Í
Í
·w¤0t ¢ w¤0t}w¤0t.
. Using the mean-value theorem, from (2.4) and (4.6), conclude
·w¤0±V;&64P)¤0±V·¤0±V « S 3 P l½ Ó ¾ w ¤0±V ¬ Ô ½lÓ ¾ w ¤0±V¤…=? Š ·w¤0±V>\EV;¤…=[a? Š P ½lӟ¾ ¤0±V~4P ¤0±< ¢ w¤0±V’&Vw¤0±V·¤0±VpyVfEdFV¤0±VFéJ
, and we have
¢ ¤0ŸE:
. By (2.6), (4.1) and (4.7),
Æ ·#±V Æ Ç<È ŒHS ± Ó © 7 ÍÍ Ô ½ 7 ¾ ±V ÍÍ Ç È ‚
Í
Í
7
½ ¾
¡ theorem, estimate Ô ¤0±V from (4.2) by (2.6),
Using (4.5) and the mean-value
ÍÍ Ô ½ 7 ¾ #±V ÍÍ Ç<È Œ6± ÍÍ ½ € ¾ ±V ÍÍ Ç<È &9S ± ÍÍ ½ € ¾ #±V ÍÍ Ç<È
Í
Í
Í
Í
Í
Í
&Y±CÍÍ Gz¤0±<ŸE:’—± © 7 € ÍÍ Ç<È
Œ ¥ ­4±“&9S ± ! § ÍÍ Gw¤0±<ŸE:—± © 7 )€<ÍÍ Ç È
ŒKz­2&'S ±V Ì ± Æ Gw¤0±<ŸE: Æ Ç È & ÍÍ a€ ÍÍ Ç<È Ï Œ•ú 7 ´
where
²¤0ŸE:
taken into account that
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I. BOGLAEV
where
Thus,
ú27
±
/
is independent of , the perturbation parameter ( or ) and the nonuniform mesh.
Æ ·±V Æ Ç<È Œ•S ú 7 ± Ó © 7 ‚
(4.8)
Similarly,
from (2.4) and (4.6), it follows that
¡
·¤0±V
±
«
& S œ P q½ Ó ¾ ¤0Ÿ­M±V ¬ Ô ½lÓ ¾ ¤08­4±V‚
·¤08­4±V’&9 P ¤08­4±V·¤0Ÿ­M±Vf
Using (4.7), by (2.6),
(4.9)
Æ ·w­M±V Æ Ç<È Œ Æ ·±V Æ ÇÈ &'S ± Ó © 7 ÍÍ Ô ½ 7 ¾ w ­M±V ÍÍ ÇÈ ‚
Í
Í
Ô ½ 7 ¾ ¤08­4±V
Using (4.5), estimate
from (4.2) by (2.6),
ÍÍ Ô ½ 7 ¾ z­4±V ÍÍ Ç£È ŒKz­2&'S ±V ± Æ Gw¤08­4±<ŸE: Æ Ç<È & Æ R#±V Æ Ç<È yŒAú !
´
Í
Í
½lÓ|¾ w ¤0±V Ù
Ó Ÿ¾ w¤0±V
A¤0±Vf ½l
where
. As follows from [6], the monotone sequences Ø
Ù
½ ¾
½lÓM¾ w¤0±V
€ w¤0±V
½ € ¾ z ¤0±V
and Ø
are bounded from above by
and from below by
.
t•±
Applying (2.6) to the problem (4.5) at
, we have
ÍÍ ½ € ¾ #±V ÍÍ Ç<È Œ9± Í ¤0±VE:’—± © 7 )€:¤“ Í ÇÈ Œ 7 Í
Í
Í
Í
7
±
/
where constant
is independent
ú ! of , the perturbation
± parameter ( or ) and the nonuni /
form mesh. Thus, we prove that
is independent of , the perturbation parameter ( or )
and the nonuniform mesh. From (4.8) and (4.9), we conclude
Æ ·w­M±V Æ Ç<È Œ•S wú 7 &6ú ! £ ± Ó © 7 ‚
°
By induction on , we prove
Æ ·e#t¯| Æ Ç<È Œ•S
ú
¸
¯
ü 7
ú
±
± Ó © 7 °RJ|*‚%‚*‚ŸŽ ® /
where all constants
are independent of , the perturbation parameter ( or ) and the
Ž ± x
nonuniform mesh. Taking into account that ®
, we prove the estimate (4.3) with
ú\S%xµÉRÊ4Ë 7 ú
.
R EMARK 4.5. The implicit two-level difference schemes (2.4) are of the first order with
±
}ŒHS1±
O­
respect to . From here and since
, one may choose ä
to keep the global error of
the monotone iterative method (4.2) consistent with the global error of the difference schemes
(2.4).
4.2. Uniform convergence of the monotone iterative method (4.2). Here we analyze
a convergence rate of the monotone iterative method (4.2) defined on the spatial meshes of
Shishkin-type (3.8), (3.10) and on the spatial meshes of Bakhvalov-type (3.12), (3.13).
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UNIFORM CONVERGENCE
4.2.1. Shishkin-type mesh. The difference scheme (4.2) on the spatial meshes of Shishkintype (3.8), (3.10) converges parameter-uniformly to the solution of problem (1.3):
ú ¥ Ž © !,iqk ! â
Ž &9± § 7,ÉR¯
Ê4Ë  Æ ¢ #t¯M³Þ#t¯4 Æ Ç<È Œ ú ¥ Ž © 7 qi k ! Žâ9
& ± § ú
/ Ž
±
'\ -
for
'\
for
where constant is independent of or , and . The proof of these results can be found
in [12]. From here and Theorem 4.4, we conclude the following
theorem.
½ € ¾ w¤0t ¯ T HEOREM 4.6. Let initial upper or lower solutions
Qƒbe
­ chosen in the form
of (4.5). Suppose that on each time level the number of iterates ä
. Then the monotone
iterative method (4.2) on the piecewise uniform meshes (3.8) and (3.10) converges parameteruniformly to the solution of problem (1.3):
ú ¥ Ž © !"ilk ! Žé&'±G&
É ¯
ÊM Ë  Æ }
#t¯|Þ#t¯4 Æ Ç<È Œ 7,C
ú ¥ Ž © 7 li k ! é
Ž &'±G&
C\S1%] S%&'±V
where
ú
and constant
Ó© 7§ Ó© 7§ / Ž
is independent of
or ,
9A -
±
for
9A
for
and .
R EMARK 4.7. In the case of the parabolic reaction-diffusion problem (4.2), Theorem 4.6
E
holds true on the piecewise uniform mesh (3.8) with an arbitrary fixed constant ! T
instead
S
ù
of
in the transition points.
4.2.2. Bakhvalov-type mesh. The difference scheme (4.2) on the spatial meshes of
Bakhvalov-type (3.12), (3.13) converges parameter-uniformly to the solution of problem
(1.3):
7
7,ÉR¯
ÊM Ë  Æ ¢ #t ¯ —’t ¯ Æ Ç È Œ•ú ¥ Ž © &'± § ú
/ Ž
±
where constant is independent of or , and . The proof of this result for the reactiondiffusion problem can be found in [4] and for the convection-diffusion problem in [3]. From
here and Theorem 4.4, we conclude the following theorem. ½ ¾
€ w¤0t ¯ T HEOREM 4.8. Let initial upper or lower solutions
Qƒbe
­ chosen in the form
. Then the monotone
of (4.5). Suppose that on each time level the number of iterates ä
iterative method (4.2) on the log-meshes (3.12) and (3.13) converges parameter-uniformly to
the solution of problem (1.3):
7 &'±& Ó © 7 § #t¯M—#t¯M Æ ÇÈ Œ•ú ¥ Ž ©
7,ÉR¯
ÊM Ë  Æ ²
C\S1%] S%&'±V
ú
/ Ž
±
where
and constant is independent of or , and .
R EMARK 4.9. In the case of the parabolic reaction-diffusion problem (4.2), Theorem 4.8
E
ù S in
holds true on the log-mesh (3.12) with an arbitrary fixed constant ! T
instead of
the transition points.
5. Numerical experiments. It is found that in all the numerical experiments the basic
feature of monotone convergence of the upper and lower sequences is observed. In fact, the
monotone property of the sequences holds at every mesh point in the domain. This is, of
course, to be expected from the analytical consideration.
5.1. The elliptic problems. The stopping criterion for the monotone iterative method
(3.2) is defined by
ÍÍ ¢ l½ Ó – 7 ¾ ¢ l½ ÓM¾ ÍÍ Ç È Œ#")‚
Í
Í
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I. BOGLAEV
RJ%E ©%$
ŽG$“\ŽG(
"
and
.
In our numerical experiments we use
Ar —é#
The reaction-diffusion problem. Consider the problem (1.1) with
,
èŸ]'&¨3,
`déJ
)(M¤“Yè
and
.
We
mention
that
is
the
solution
to
the
reduced
problem.
S J+]M­& S%
J
,
,
This problem gives
¢ ½ € ¾ ¤“rJ|’¤™= ? Š (5.1)
¤™=? Š 
¢ ½ € ¾ ¤“ èa:J ˆ
ˆ¤™=µ[a? Š ¢ ½ € ¾ ¤“
¢ ½ € ¾ ¤“
where
and
are the lower and upper solutions to (2.2).
All the discrete linear systems are solved by GMRES-solver [1].
Introduce the notation: ä and ä are numbers of iterative steps required for the
¢ ½ monotone
€ ¾ w¤“
"
iterative
method
(3.2)
to
reach
the
prescribed
accuracy
with
the
initial
guesses
and
½ ¾
¢ € ¤“
, respectively.
TABLE 5.1
Numbers of iterations for method (3.2) on the piecewise uniform mesh (3.8).
1J E © 7
ŒKJ%E © !
Ž$
20; 15
20; 15
64

ä ä
20; 15
20; 15
128
20; 15
20; 15
256
ŽG$
20; 15
20; 14
Q*&J+­
In Tables 5.1 and 5.2, for various numbers of
and , we give the numbers of iterations
ä and ä , required to satisfy the stopping criterion, for the monotone method (3.2) on the
piecewise uniform mesh (3.8) and on the log-mesh (3.12), respectively. From the data, we
conclude that the numbers of iterations are independent of the perturbation parameter .
These numerical results confirm our theoretical results stated in Theorems 3.6 and 3.7.
J%E © !í
%J E © ì
%J E ©
Ž$
J%E
%J E
%J E
%J E
Ž
TABLE 5.2
Numbers of iterations for method (3.2) on the log-mesh (3.12).
20; 15
20; 15
20; 14
64
20; 15
20; 15
20; 15
128

ä ä
20; 14
20; 15
20; 15
256
20; 15
20; 14
20; 15
512
20; 15
20; 14
20; 14
1024
20; 15
20; 14
20; 15
2048
TABLE 5.3
Numbers of iterations for the Newton method on the piecewise uniform mesh (3.8).
© 7
© !í
© ì
©
$
38; 10; 13
8; 8; 8
8; 8; 8
6; 8; 8
64
36; 68; 18
7; 15; 7
7; 11; 7
6; 8; 7
128
Ó,+  ,Ó +  Ó,+
ä € ä ! ä ì
58; 187; *
15; 14; 6
7; 9; 6
6; 8; 6
256
*; *; *
10; 23; 6
7; 11; 6
6; 9; 6
512
Ó-+
*; *; *
13; 35; *
7; 10; 8
7; 8; 6
1024
*; *; *
13; *; *
8; 15; *
7; 9; *
2048
Table 5.3 presents the number of iterations ä¢ ½ ¾ for solving the test problem by the
€ w¤“RmE<8­èa>¤Â=\? Š . We denote
Newton iterative method with the initial guesses
by an ‘*’ if more then 200 iterations is needed to satisfy the stopping criterion, or if the
ETNA
Kent State University
etna@mcs.kent.edu
101
UNIFORM CONVERGENCE
Newton method diverge. The experimental results show that the Newton method cannot be
successfully used for this test problem.
ré - 5 7.b ! ,
problem. Consider the problem (1.1) with
E<‚qJ The
A convection-diffusion
#dHèŸ]'&H,
`3 J
( w¤“N è
,
and
. We mention that
is the solution to the
S J+]|­-& S J
reduced problem. This problem gives
,
, and the initial lower and upper
solutions are defined by (5.1).
All the discrete linear systems are solved by GMRES-solver [1] with the diagonal preconditioner as in [14].
Ž $
/
In Tables 5.4 and 5.5, for various numbers of
and , we present the numbers of
iterations ä and ä for the monotone method (3.2) on the piecewise uniform mesh (3.10)
and on the log-mesh (3.13), respectively. From the data, we conclude that the numbers of
/
iterations are independent of the perturbation parameter . These numerical results confirm
our theoretical results stated in Theorems 3.6 and 3.7.
TABLE 5.4
Numbers of iterations for method (3.2) on the piecewise uniform mesh (3.10).
/
© 7
© !í
© ì
©
$
J1E
1J E
1J E
1J E
Ž
J1E
1J E
1J E
1J E
Ž
/
16; 13
22; 20
20; 19
19; 19
64
16; 16
22; 20
19; 18
18; 18
128

ä ä
16; 16
22; 20
18; 18
17; 17
256
16; 16
22; 20
17; 18
16; 17
512
16; 16
22; 20
17; 17
16; 17
1024
16; 16
22; 20
17 ; 17
16 ; 16
2048
TABLE 5.5
Numbers of iterations for method (3.2) on the log-mesh (3.13).
© 7
© !í
© ì
©
$
16; 16
23; 20
20; 20
19; 19
64
16; 16
22; 20
19; 19
18; 18
128

ä ä
16; 16
22; 20
18; 18
17; 17
256
16; 16
22; 20
17; 18
16; 17
512
16; 16
22; 20
17; 17
16; 17
1024
16; 16
22; 20
17 ; 17
16 ; 16
2048
Similar to Table 5.3, the numerical results presented in Table 5.6 indicate that the Newton
method cannot be successfully used for this test problem.
/
TABLE 5.6
Numbers of iterations for the Newton method on the piecewise uniform mesh (3.10).
J1E © 7
J1E© !í
J1E ©
J1E © ì
Ž $
4; 3; 5
9; 6; 6
9; 8; 6
7; 9; 6
64
4; 3; 5
8; 6; 6
9; 11; 6
9; 12; 6
128
Ó,+  ,Ó +  Ó-+
ä € ä ! ä ì
4; 4; 5
9; 7; 5
13; 13; 7
13; 9; 6
256
4; 4; 5
42; *; *
10; 24; 16
8; 12; 7
512
5.2. The parabolic problems. On each time level
in the form
tŸ¯
5; 4; 6
*; *; *
39; 34; 31
30; 20; *
1024
6; 5; 7
*; *; *
*; *; *
27; *; *
2048
, the stopping criterion is chosen
ÍÍ l½ ÓM¾ # t¯:’. l½ Ó © 7 ¾ t¯M ÍÍ Ç È Œ#"
Í
Í
ETNA
Kent State University
etna@mcs.kent.edu
102
I. BOGLAEV
J%E ©%$
ŽN$\ ŽG(
"
. All the discrete linear systems (
) in the algorithm (4.2) are
where
solved by GMRES-solver [1] for the reaction-diffusion problem and by GMRES-solver with
the diagonal preconditioner as in [14] for the convection-diffusion problem.
f 9ñJU
problem (1.3) with
,
Ë/ 0The
reaction-diffusion
`CJ
€ J problem. B Consider S1the
2r
J
,
and
. This
problem gives í
.
7
± 7 J1E © ± ! & J%E © !
± rJ%E © !
,
and
and for various values of and
Ž $ In Table 5.7, for
, we give the average (over ten time levels) numbers of iterations ä ®1 ä ®32 ä ®34 , required
to satisfy the stopping criterion, for the monotone method (4.2) on the piecewise uniform
mesh (3.8). From the data, we conclude that the numbers of iterations are independent of
the perturbation parameter . We mention that the numerical experiments with the monotone
method (4.2) on the log-mesh (3.12) give the same numerical results as in Table 5.7. These
numerical results confirm our theoretical results stated in Theorems 4.6 and 4.8.
TABLE 5.7
Numbers of iterations for method (4.2) on the piecewise uniform mesh (3.8).
1J E © 7
ŒOJ1E © !
Ž $
74

7
ä €65 ä €75 € $ ä €65 €
4; 4; 3
4.1; 4; 3
64
4.1; 4; 3
4.1; 4; 3
QKJ1­-8
Consider the problem (1.3) with
µThe
J / convection-diffusion
Ë0j , `CJ
problem.
S J
€ rJ
,
and
. This problem gives
.
'\ - 5 7.b ! rJ
,
,
TABLE 5.8
Numbers of iterations for method (4.2) on the piecewise uniform mesh (3.8).
/
1J E © 7
ŒOJ1E © !
Ž$

7
7
ä €65 ä €75 € $ ä €65 €
4; 3.8; 3
4; 4; 3
64
4; 3.9; 3
4; 4; 3
QKJ1­-8
Similar to Table 5.7, Table 5.8 presents the numerical results for the monotone method
(4.2) on the piecewise uniform mesh (3.8). From the data, we conclude that the numbers of
/
iterations are independent of the perturbation parameter . We mention that the numerical
experiments with the monotone method (4.2) on the log-mesh (3.12) give the same numerical
results as in Table 5.8. These numerical results confirm our theoretical results stated in
Theorems 4.6 and 4.8.
REFERENCES
[1] R. B ARRETT ET AL , Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods,
SIAM, Philadelphia, 1994.
[2] I. P. B OGLAEV , A numerical method for a quasilinear singular perturbation problem of elliptic type, USSR
Comput. Maths. Math. Phys., 28 (1988), pp. 492–502.
[3]
, Numerical solution of a quasi-linear parabolic equation with a boundary layer, USSR Comput.
Maths. Math. Phys., 30 (1990), pp. 55–63.
[4]
, Finite difference domain decomposition algorithms for a parabolic problem with boundary layers,
Comput. Math. Applic., 36 (1998), pp. 25–40.
[5]
, On monotone iterative methods for a nonlinear singularly perturbed reaction-diffusion problem, J.
Comput. Appl. Math., 162 (2004), pp. 445–466.
[6]
, Monotone iterative algorithms for a nonlinear singularly perturbed parabolic problem, J. Comput.
Appl. Math., 172 (2004), pp. 313-335.
ETNA
Kent State University
etna@mcs.kent.edu
UNIFORM CONVERGENCE
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
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, A monotone Schwarz algorithm for a semilinear convection-diffusion problem, Numer. Maths., 12
(2004), pp. 169-191.
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O. A. L ADY ŽENSKAJA AND N. N. U RAL’ CEVA , Linear and Quasi-Linear Elliptic Equations, Academic
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O. A. L ADY ŽENSKAJA , V. A. S OLONNIKOV, AND N. N. U RAL’ CEVA , Linear and Quasi-Linear Equations
of Parabolic Type, Academic Press, New York, 1968.
J. J. H. M ILLER , E. O’R IORDAN , AND G. I. S HISHKIN , Fitted Numerical Methods for Singular Perturbation
Problems, World Scientific, Singapore, 1996.
C. V. PAO , Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.
H.-G. R OOS , A note on the conditioning of upwind scheme on Shishkin meshes, IMA J. Numer. Anal., 16
(1996), pp. 529–538.
H.-G. R OOS AND T. L INSS , Sufficient conditions for uniform convergence on layer adapted grids, Computing, 64 (1999), pp. 27–45.
H.-G. R OOS , M. S TYNES , AND L. T OBISKA , Numerical Methods for Singularly Perturbed Differential
Equations, Springer, Heidelberg, 1996.
A. S AMARSKII , The Theory of Difference Schemes, Marcel Dekker Inc., New York-Basel, 2001.