Engineering Analysis with Boundary Elements 25 (2001) 141±145
www.elsevier.com/locate/enganabound
Research Note
BEM solution of Poisson's equation with source function satisfying
7 2r ˆ constant
R. Suciu a,*, G. De Mey a, E. De Baetselier b
a
Department of Electronics and Information Systems, Gent University, Sint Pietersnieuwstraat 41, 9000 Gent, Belgium
b
Agilent Technologies Belgium N.V./S.A., Lammerstraat 20, 9000 Gent, Belgium
Received 23 February 2000; accepted 13 September 2000
Abstract
In this paper Poisson's equation in two dimensions is studied and different solutions with a BEM are outlined depending on whether the
source function is harmonic or not. When the Laplacian of the source function is zero or a constant, the Galerkin vector and the multiplereciprocity method are applied. When the Laplacian of the source function is no longer zero, the source function is approximated by Lagrange
polynomials. This method is then improved by subtracting a parabola from the source function. q 2001 Elsevier Science Ltd. All rights
reserved.
Keywords: Poisson's equation; Galerkin vector; Lagrange interpolation; Boundary element method
the condition:
1. Introduction
The main problem that occurs in solving Poisson's
equation:
2
7 Fˆr
…1†
with a BEM is the domain integral that appears due to the
in¯uence of the source function r .
Different approaches have been developed in order to
overcome this problem, such as: the Monte Carlo method
[3, pp. 37±41], the dual reciprocity technique [3, pp. 69±
223], the Galerkin vector approach [3, pp. 43±45].
In Ref. [1], the authors of the present contribution have
focused on the Galerkin vector approach, which is valid
only for the particular case when the source term r satis®es
Laplace's equation:
7 2 r ˆ 0:
…2†
Also a Lagrange interpolation with harmonic polynomials
of r was used for the cases when the source function is not
satisfying condition (2) any more. Since the harmonic polynomials do satisfy the Laplace equation, condition (2) is
ful®lled and the Galerkin vector method can be applied.
In this contribution the Galerkin vector method will
be extended for another particular case, when r satis®es
* Corresponding author.
E-mail address: [email protected] (R. Suciu).
7 2 r ˆ a ˆ constant:
…3†
If the source function r does not satisfy Eq. (3), the
following idea will be used. First, the given source
function r is approximated by a parabola r p satisfying
Eq. (3) exactly. The remaining difference r 2 rp is
approximated by a Laplace interpolation p in the
complex plane. As a consequence 7 2 r ˆ 0; whereas
the function r 2 rp is generally not a harmonic function
7 2 r 2 rp ± 0: This approximation will give rise to less
accurate numerical results.
2. The Galerkin vector method
Considering a two-dimensional domain S with boundary
2S; Poisson's equation can be easily transformed into:
"
#
I
2G…r ur 0 †
r†
0
0 2F…
F…r † ˆ
F…r†
2 G…rur †
dl
2n
2n
2S
1
ZZ
S
r…r †G…rur 0 † dS:
…4†
In the above equation G is Green's function of the Laplace
equation in two dimensions:
G…r ur 0 † ˆ
0955-7997/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved.
PII: S 0955-799 7(00)00044-8
1
1
ln
2p ur 2 r 0 u
…5†
The remainder of this paper is not included as this paper is copyrighted material. If you wish to obtain an electronic version of this paper, please send
an email to [email protected] with a
request for publication P101.198.pdf.
1