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Electronic Transactions on Numerical Analysis.
Volume 17, pp. 206-217, 2004.
Copyright  2004, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
ON THE NUMERICAL SOLUTION OF SOME SEMILINEAR ELLIPTIC
PROBLEMS
KENDALL ATKINSON AND WEIMIN HAN
Abstract. We discuss a general framework for the numerical solution of a family of semilinear elliptic problems
whose leading differential operator is the Laplacian. A problem is first transformed to one on a standard domain via
a conformal mapping. The boundary value problem on the standard domain is then reduced to an equivalent integral
operator equation. We employ the Galerkin method to solve the integral operator equation, using the eigenfunctions
of the Laplacian on the standard domain. An error analysis of the method is given.
Key words. elliptic, nonlinear, integral equation, Galerkin method.
AMS subject classifications. 65R20, 65N99, 35J65.
1. Introduction. The purpose of the paper is to propose a general framework for the
numerical solution of a semilinear elliptic boundary value problem of the form
!
(1.1)
#"%$'& is a simply-connected open domain with a boundary ! . For the domain ,
where
assume there exists a standard open domain ( , such that and ( are conformally equivalent.
The first step of the method is to reduce the problem (1.1) to an equivalent problem on the
standard domain ( ,
)+*,*,0/ -'.+*! ( (1.2)
(21
Then, (1.2) is converted to an equivalent integral equation. Finally, we use the Galerkin
method to solve the integral equation, with the eigenfunctions of the Laplacian operator on
the standard domain as the basis functions.
An advantage of the above framework is its generality. For conformally equivalent domains, a boundary value problem (1.1) is reduced to a problem on a standard domain ,
which is usually taken to assume a simple geometry, e.g., a disk or a square. On the standard
domain, explicit forms of the eigenvalues and associated eigenfunctions of the Laplacian operator are usually available from the literature. The information on the eigenpairs can be used
in the efficient implementation of the Galerkin method for the integral operator equation, as
well as in the theoretical analysis of the proposed numerical method (convergence, convergence rate, etc.). In particular, we mention the possibility of using FFT algorithms for solving
the resulting discrete systems for a problem on the standard domain.
To see how (1.1) is reduced to (1.2), let
(
(
3547698 ;:<>= 4?6@ ACBDE=D @ 4?6GF >=H @ >=H@ denotes
( to ,*'where
be a conformal mapping from
a generic point in ( and 3 8 is
>=
H
A
C
@ 3 8 . It is then easy to verify that
the corresponding point in , and let
3 8 A 3 8 3 8 .IJ 3 8 LK,
M
(1.3)
Received February 19, 2004. Accepted for publication July 9, 2004. Recommended by M. Gutknecht.
Department of Mathematics, University of Iowa, Iowa City, IA 52242, U.S.A.
206
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On the numerical solution of some semilinear elliptic problems
<* E=D @ A;-'>=H @ .*'>=H @ .I >=H @ LK ( -'>=H @ .*'>=H @ .A : E= 4?6@ & BDE=D @ I F E=D @ I.*'>=H @ . 1
is transformed to
where
(1.4)
(
) * * C-/ ' .*! ( (
/
/
E=
D
2
3 8 for/ K>=H @ &K E ( . We assume
where
to
by the relation
@
/ KK &is& related
( . Notice
:
that since is a conformal mapping,
- ( if and only if
So without loss of generality, we only need consider the following problem on
,
(1.5)
. The problem is solvable under mild assumptions on ; see the next section.
As an example of a related approach to solving (1.1), see [2]. They too convert the
problem to an integral equation, but their approach is different from ours.
2. Some Existence Results. Concerning semilinear elliptic problems such as (1.5), one
can find general solvability results in the context of applications of nonlinear operator theory,
cf. Zeidler [15]. Here, wemention
two
such results.
A function
is called a Carathéodory function if, (1) for any
,
is measurable, and (2) for a.e.
,
is continuous. Adapting a
proof from [15, 27.4], we have following existence result.
T HEOREM 2.1. Consider the problem
*2K2$ -'.*! '- >=D @ .*! ( $
$ E=D K '- E=D @ ( @
)+* * 4 - .*! AC- & ( (2.1)
(
- E=D @ * is a Carathéodory function, - & K ( . Further assume
where - >=H @ .*! * -%/. (2.2)
!#"%$ &')(*$,+ - E=D @ * 1032 54 >=D @ 4 * 6 and
(2.3)
*2K
* @>BA 4 - E=D @ * A C=DC @
Consider the weak formulation of (2.1): find
=
(2.4)
&
?>
for some 7
;
<
=
&
K 5 8 . 94 K: & ( 1
( , such that
- & AEC=DC @ GF/AK
<
( 1
Then this has a solution.
2
As for the uniqueness of a solution, let < be the best constant in the Poincaré-Friedrichs
inequality
(2.5)
*
* HJILK &P" GF *2KQ < ( 1
0O2 < >
H HJILK M&N"
H
We have
2<
R
S T ETNA
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208
Kendall Atkinson and Weimin Han
T
where - * * T * + ( (21
*
If either is monotone with respect to , i.e.,
- >=H @ .* - >=H @ .* & !>* * & GF>=D @ K ( H* .* & K $ *
or is differentiable with respect to and
- >=H @ .*! -C T GF>=D K <* K,$ *
@ (
*
then the solution of (2.4) is unique.
>=H @ .3 8 be the Green’s function for the problem
3. Galerkin Approximations. Let
* * C- ( (21
>=H *
@ be the harmonic function assuming the Dirichlet data / E=D @ on ( . Then a
Let
*
solution of the problem (1.5) satisfies
E=D = E=D
<
*
E=
D
*
@
@ 4 & @ 3 8 -' 3 8 .*' 3 8 C 3 C 8 >=H @ LK (21
(3.1)
A E=D @ -'E=D @ *<E=D @ . The function A is a
As in Kumar and Sloan [11], we introduce
solution of
A >=H C-G=D .* E=D = >=H
@
@ @ 4 & @ .3 8 1 A 3 8 C 3 C 8 E=D @ K (21
(3.2)
is the smallest eigenvalue of the eigenvalue problem
For a further discussion of this approach, see [9].
We use Galerkin’s method to solve (3.2). To do this, we consider the eigenvalue problem
for the Laplacian operator:
(3.3)
T
0
T
& 0 .
: :
T
: + ( (21
: : &
Let
be the sequence of the eigenvalues, and be
corresponding eigenfunctions. The existence of the eigenpair : sequence is guaranteed, and the
eigenfunctions can be chosen to form an orthogonal basis of
, cf. [12]. Then we have
: :
or,
=
(3.4)
Let &
T
&(
: ( (
: >=D @ >=H @ LK ( E=D @ 3 8 : 3 8 C 3 C 8 T R
1
R
4 : !:
A
: K
. The Galerkin method for (3.2) is to find " , such that
A # $
(3.5)
: A '- E=D @ * >=H @ 4
!
=
&
>=H @.3 8 A 3 8 LC 3 C 8 :# R 0 6 0 ETNA
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:
: # >=D @ - =D @ * >=H @ 4
209
On the numerical solution of some semilinear elliptic problems
i.e., upon using the orthogonality of
(3.6)
: # : # A
#
!
=
&
# #
,
:9>=H @ C=DC @ R 0 6 0 1
!T
The advantage of applying the Galerkin method to solve the auxiliary problem (3.2) is
that the integral
=
&
>=H @.3 8 1A 3 8 C 3 C 8
:>=H @ !T
is available and has been computed exactly. If we apply the Galerkin method directly to the
problem (3.1), in each iteration for solving the resulting nonlinear algebraic system, we will
have to evaluate double integrals
E=D @ 3 8 -' 3 8 .* " 3 8 . :#E=D @ C 3 C 8 C=DC @ R 0 6 0 where the superscript F refers to the number of iterations. Another advantage of the method
is that the left side of the system (3.6) is diagonal; this may bring in some convenience in
solving (3.6) numerically.
A
*
After we compute the Galerkin solution , we can generate an approximation of by,
e.g., (4.8) of the next section.
R
3.1. Assume the standard domain ( is the unit disk. The eigenvalues are
& G T R
R
(3.7)
1
A set of corresponding eigenfunctions is given by
* < < < 7 R !
=
=
&
&
EMARK
* " and
where
([1]),
I
7
th
is the
H* & " ) R G L 7
th
is the
Bessel function,
4 >* " * , & " A >* " * , " A >* & " * , & " A We have
R
zero of
&
! ! &
"
& 4 & ,* E=D @ (
1
1
?
)
, the eigen-
.D = .D @ I , R 1
# G !
(cf. [12]). Asymptotically
.
R EMARK 3.2. When the standard domain is a square,
values and associated orthonormal eigenfunctions are
T
R
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Kendall Atkinson and Weimin Han
4. Error Analysis. In this section, we perform a convergence analysis for the Galerkin
approximations defined in the last section. We will use H H for H H@I K &N" . Introduce the
Nemyckii operator
,>*!. >=H @ C-'>=D @ .*'>=D @ . (4.1)
and the linear integral operator
E=D @ 3 8 1A 3 8 LC 3 C 8 1
:L& The operator is a linear continuous mapping from
( to < ( ,
)A K : & ( 032 A F AQ
H
H &N"
H H
: & : & ( to ( .
and is compact on
We will need the following assumption.
E* 4 .D *: & ( E : & ( is completely contin(A1) The operator
uous.
The assumption (A1) is satisfied, if, e.g., the function is Carathéodory, and satisfies the
growth condition
K : & ( I 7 -3 1
-'E=D A 032 )4 >=H A 6 I > . E4 @
@ 4
(4.3)
A ,>* AG.
Indeed, using the mapping property
4
:L& of , we: & then know that the mapping
is continuous and bounded from
( to ( with (cf. [15, 26.3])
,>* 4 )A H 0 2 H 4 HA4 H * 4 )A H 6I K &N" H
* 6
A 6
0 2 4
(4.4)
H HA4 H H &N" 4 H H 1
: & : & Since
as an operator
,>* 4 is .compact
is compact
: & from: & ( to
from
( to ( . Therefore,( ,, >*it is4 easy .' to : see& ( thatP the: & map
(
(4.2)
I>=H @ L
)A
=
&
is completely continuous.
In the operator equation form, (3.1) and (3.2) can be rewritten as
* * 4 ,>*!
(4.5)
,>* 4 AG 1
* a solution of (4.5). Then A ,>* is a solution of (4.6).
We denote by
: & ( orthogonal projection of : & ( onto
Let be the
: : "
: A K
where the sequence
has been normalized. Then the Galerkin solution and
A
(4.6)
!
#"
%$
(3.5) satisfies the operator equation
" of
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On the numerical solution of some semilinear elliptic problems
A
(4.7)
!
*
We define
(4.8)
E* 4
A
A
E* 4
and the iterated solution
(4.9)
*4
!
A
1
A
211
1
Easily,
1
E*
.
: & P: & Since
4
(
( is completely continuous, the following convergence result can be found from Krasnoselskii [10, Chap. 3, 3].
T
4.1. Assume (A1) and
E* (A2) R is not an eigenvalue of
.
8 ;P: & : & Then A
A ( A
( is
. bounded, and for sufficiently large , (4.7) has a
solution such that as
.
A
!
A
HEOREM
#
E* ;
R EMARK 4.2. The assumption (A2) is equivalent to the statement that the equation
E* does not admit a nontrivial solution, or letting (4.10)
, that the equation
does not have a nonzero solution. Notice that the equation (4.10) is a linear boundary value
problem
!-'. * . *
(
+
(21
(4.11)
-'E=D @ * >=H @ .
*
The problem (4.11) does not have a nonzero solution, if, e.g.,
"
!#"%$ &
From now on, we will assume the discrete solutions
The rest of the section is devoted to error estimations.
P ROPOSITION 4.3.
H
A
A H
where
0
4 HA A A
HA A H
1
H
H
R
!
T A
!
1
are the ones converging to
A
H
A
.
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Kendall Atkinson and Weimin Han
Proof.
H
A H
A
A4
A A
H
HA4
A A
H
HA4
A R 4 H
0
A
H
!
A
H
H! !
0
!
!
A H
AH
A
A
H! A
A
H
H
A
H
1
A
& ( The
: significance of Proposition 4.3 is that the discrete solution is almost
as
accurate
as the
orthogonal projection of the exact solution to , as long as , which is
true most of the time. Indeed, it is shown in [5] that if (A2) is satisfied, then
032
(4.12)
7 E* 4
' ,>* 4 A ' A ;A H A ;
A H
H H H H 1
H
(4.14)
For the behavior of , we have the following result.
E* 4 A AG is differP
4.4. Assume (A1) and (A2). Assume further that E*
A
4 is Lipschitz
entiable at . Then, is uniformly bounded.
Furthermore, if
A
continuous in some neighborhood of ,
E*
F A A & K > H
4 A E* 4 A & H 0 H A ; A & H G
then we have the estimate:
A H R T A1
032 A H A A
,>* 4 AG at A , it
Proof. From the convergence and the differentiability of
is easy to see that 7 is uniformly bounded. Obviously, is uniformly bounded. Thus, is
uniformly bounded.
A >* is Lipschitz continuous in some neighborhood
Now
we further assume that
A
A K 4
of . For sufficiently large , , and so
A H1
032 A ;
(4.15)
7
H For the term , we have
HPH E* H 1
0
H
A
2 : , we have
With
R 2 : )A T where
7
(4.13)
H
A
!
!
ROPOSITION
!
$
$
+ )A #
!
$
R 2
: 1
T
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On the numerical solution of some semilinear elliptic problems
Hence,
H
GA H &
&
T & $
R 2
!
Thus,
H
0
and
(4.16)
0
H T
A H &
!
E* H
& 1
R
T
&
A
HT H
0
H
0
H
!
&
T
R
1
The estimate for follows from (4.15) and (4.16).
We have seen that under the assumptions stated in the first part of Proposition 4.4, the
Galerkin solution is as accurate as the orthogonal projection. Actually, we can show that
A
HA
H
(4.17)
A
A H H
!
!
which is a type of superconvergence result. Write
$
A
Then
A
H
!
A
Then
A A
!
;A A
Using Proposition 4.3, we obtain
H
A
;A H &
Hence,
(4.18)
A
H 2 !
A
H
&
2 " $
2 2 & 4H A 2 :1
:
2 " :4
2 " &
!
2 " 0
& 1
2
2 "
: 1
&
H
Also write
A
2
.
A
$
&
H
0
R
2
: 1
4 & HA 4 HA !
A
H
!
&1
A
H
&1
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Kendall Atkinson and Weimin Han
* * . We have
* * 4 ,>* ' A * * 4 )A ;A * * 4 T R 2 2 " : 4 R 2 2 " : * * T 4
, we have (4.17).
Since Finally we consider an error estimate for
* *
!
!
$
!
R 2
:
T
!
and
* * H&
H
& 4 H* * H&
2 " &
4H * * H &
4 H A A H & 4H * * H & 1
2 &
R 2 T &
0
&
R
0
T
R
T
2 " !
!
!
!
In conclusion, we have proved the following theorem.
T HEOREM
4.5. We make the assumptions stated in Proposition 4.4. For the Galerkin
A
solution , we have the error estimate
A H 0 R 4 H A A H . 1
*
For the approximation of the problem (3.1), defined by the relation (4.8), we have the error
H
estimate
H
A
* * H&
0
!
T
R
&
4 HA !
& 4 * * H&1
A
H H
!
R EMARK 4.6. The above discussion on error
@A estimations is based on the convergence
result, Theorem 4.1. Alternatively, if we assume is a bounded sequence, then Proposition
4.4 still holds. This follows
from the fact that 7 is uniformly bounded, a property
impliedA by
@A
the boundedness of , the inequality (4.4) and the A differentiability
of
at .
A
Then by Proposition 4.3, we obtain the convergence . Thus all the error estimates
derived above hold. We 0 also notice that if the function is Carathéodory and satisfies the
R , then 7 is uniformly bounded even without the boundedness
inequality (4.3) A with 7
A
0 R
assumption of . Consequently, if 7
, then (4.6) has a unique
solution
satisfying
A
the condition (A2), and any discrete solution of (4.7) converges to .
-
,>* 4 . 5. Numerical Example. We conclude the paper with an illustrative example. Further
results on the implementation of the proposed method and further numerical experiments are
given in [6].
Let be the bounded elliptical region
(5.1)
=& @&
4
4
R
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On the numerical solution of some semilinear elliptic problems
: ((
=& 4 @&
215
4 - - R.
. For the standard region
with
; , we
use the unit circle in the plane,
We construct the conformal mapping
and its derivative by using the SC Toolbox
(to be run from within Matlab) that is described in [8, Appendix].
Begin by introducing the elliptic integral of the first kind
(5.2)
with
The function
the rectangle
= & !C & & <
R
R F
F
1
C maps conformally the half-space -O , denoted by R
, onto
which we denote by
. Introduce
= (5.3)
= (5.4)
C
& ! R F & & C
& & & R
R
F
<
<
R
F R F
with
or
= K
<
F & & R
C
F&
F 1
The mapping can be produced using SC.
For a conformal mapping of the closed unit disk onto the closure of the ellipse in (5.1),
use
;;:H A 2 S F
S
4
=
=
&
&
& & , 4 6 @ , 4 @ 0 R . The definition extends to the lower half of the
2
with
unit disk by using
:< A :< 3 1
2 9 are the foci of the ellipse in (5.1). The constant F can be produced as
The points
follows, using the construction of Szegö [14]. Introduce
4
- R
S 4 & &
!
4! ! & R R 1
(5.5)
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Kendall Atkinson and Weimin Han
Then
F
R
S
4 & ! R 4 ! R 4 !A R 1
R 4
R 4
R 4
The derivative is given by
: A S2 S F R
F
R & R F& &1
S F
:
According to (1.4), we need only the absolute value of the derivative of , and therefore we
do not need to choose a particular branch to the square root function. More precisely, we need
to evaluate
: &
(5.6)
0
S
F
& R 2
R
R . Note that
S
F &
R
& R F & F
, and therefore the apparent singularity in (5.6) at
F
is actually removable.
As a simple illustration of our numerical method from 3, we use the true solution
>=D @ A =4 & 4 @ & R >=H @ LK2 1
(5.7)
We solve the problem
*
/ E=D @ I >=>=HH @ KK2 E=D @ 2
4
(5.8)
@ 1
/
The function is to be so chosen that the of (5.7) is the true solution of the problem.
The details of the implementation of our numerical method are given in the paper [6],
along with further theoretical results on the rate of convergence. * Here we simply give some
numerical results for this one sample problem. From the solution
defined on the unit disk,
for
we
obtain the solution
defined on the elliptical region . For our example, we double
repeatedly, where is the number of distinct eigenvalues taken in increasing order
C from
- (3.7).C Note that the multiplicity can vary, and thus the number of basis functions ,
with , roughly. In Table 5.1, we give two measures of error:
&
9>= @ F 1 1 %'&(&
R
>= @ <? E= @ ;? E= @ 7 >= @ & S R 4 C? & 1
$
0"!/0
$#
R
, large, are
The points
to cover well the closed ellipse .
)4 chosen
The numerical results are given for the case of
. For the construction of in (5.2),
.
These results in Table 5.1 are converging slowly to zero; but the cost of setting up and
solving the nonlinear system is also fairly small. We plan on exploring extrapolation methods
for accelerating the convergence, using techniques described in the book of Sidi [13]. We
are also looking into alternative polynomial bases for approximating the solution using
collocation.
*
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On the numerical solution of some semilinear elliptic problems
217
TABLE 5.1
Errors in solving a problem over an ellipse
#
&
# R %
R
#R 11 EE R
# 1 E R 1 % E #
G 11 E #
E #
R 1R % E
%
&
&
1 11 %% EEE & 1 ## E ##
1 E #
R 1 & E 1& E REFERENCES
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Appl. Math. Stochastic Anal., 9 (1996), pp. 57–67.
[3] K. ATKINSON , The Numerical Solution of Integral Equations of the Second Kind, Cambridge University
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[4] K. ATKINSON AND W. H AN , Theoretical Numerical Analysis, Springer-Verlag, New York, 2001.
[5] K. ATKINSON AND F. A. P OTRA , Projection and iterated projection methods for nonlinear integral equations, SIAM J. Numer. Anal., 24 (1987), pp. 1352–1373.
[6] K. ATKINSON AND A. S OMMARIVA , On the numerical solution of some semilinear elliptic problems – II,
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[7] H. B R ÉZIS , Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier,
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[8] T. D RISCOLL AND L. T REFETHEN , Schwarz-Christoffel Mapping, Cambridge University Press, 2002.
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[10] M. A. K RASNOSELSKII, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon
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[11] S. K UMAR AND I. S LOAN , A new collocation-type method for Hammerstein integral equations, Math. Comp.,
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[12] J. R. K UTTLER AND V. G. S IGILLITO , Eigenvalues of the Laplacian in two dimensions, SIAM Rev., 26
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[13] A. S IDI , Practical Extrapolation Methods, Cambridge University Press, Cambridge, 2003.
[14] G. S ZEG Ö , Conformal mapping of the interior of an ellipse onto a circle, Amer. Math. Monthly, 57 (1950),
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