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Green's functions for heterogeneous media potential problems
Ana/y~ with Boundary E/ement* 13 (1994) 219-221 © 1994 Ebevier $deaoe l/mlted Printed in Great Britain. All right, re~fved 09S5-7997/94/$07.00
ELSEVIER
Green's functions for heterogeneous media potential problems Richard P a u l S h a w Department of Civil Engineering, State University of New York at Buffalo, 212 Engineering West R-8, Buffalo,
New York 14260, USA
(Received 1 November 1993; accepted 10 January 1994) Procedures are discussed for obtaining Green's functions for potential problems in heterogeneous materials. Such fundamental solutions are required in the boundary element method. An exact fundamental solution for a linearly varying, layered two dimensional potential problem is given that does not appear to be available elsewhere.
Key words: Green's functions, heterogeneous media, boundary element method. INTRODUCTION
dimensional solutions, using R = [7- Fo[, are G(2) ....
o ~r, ro) = (1/(2Ko))In Ca/R); a is an arbitrary
The boundary integral/element method (BIEM) relies on the availability of fundamental solutions for the various types of equations to which it is applied. When heterogeneous materials are encountered, these Green's functions become more difficult to find. To this end, a collection of Green's functions for the heterogeneous medium, potential equation are obtained below. While this is not meant to be an exhaustive account, the approaches are fairly general.
constant, usually set to 1 ~(3),.-, - . , t16 (r, r o ) = 1/(41rKoR)
(4a) (4b)
e.g. Brebbia and Dominguez, 1 Beskos,2 etc. If the physical problem for heterogeneous media leads to either of the following two equations, the solution for the hetergeneous Green's function is trivial;
K(P')V2U(P.)
=
Q(P')
(5a)
V2{K(P.) U(P.)} = Q(P')
HETEROGENEOUS POTENTIAL EQUATION
(5b)
The corresponding Green's functions are, respectively, The homogeneous medium potential equation is KoV2U(P.) = Q(P.) in V(P.)
G(~, P'o) KoGO(F,P'o)/ K(P'o) GCP.,P'o) KoGoCP',P'o)/ K (r-)
(1)
where Ko is a constant 'conductivity' and Q(P.) is a volume distributed source. Typical boundary conditions would be
U(P.) =fl(p') on $1 (Dirichlet conditions)
(6b)
v • {KCP.)VUCP.)}
=
Q(P.)
(7)
(8)
is then the focus of attention for the remainder of this discussion. Only specific material heterogeneities have led to known solutions so far. These come through transformation of the heterogeneous equation to an equation whose fundamental solution is already known and proceed via three main paths: (a) transformation of the dependent variable, (b) transformation of the independent variable, (c) transformation of both variables. Some illustrations of each are given.
(2c)
The corresponding Green's functions to this problem are well known and satisfy
KoV2G(F, P'0) ----- 6 ( F - F0) in V(P')
=
6(P'- P'0) * FCP.) = 6(P'- P'0) * FCP'0) has been introduced. The more dimcult ease
(2a)
(2b) on $2 (flux conditions)
(6a)
where, since the delta function only operates at P.= P.0, the useful 'trick'
fi • VU(P.) =f2(r') on $2 (Neumann conditions) or
• KoVU(P') =f2(p')
=
(3)
where the constant /Co has been included for comparison with the heterogeneous case. The two and three
219
R. P. Shaw
220 (a) Tramformatlea of flke depeadmt variable
with K(x) = ax + ~. If a coordinate transformation = x + ~ / a (a is not zero or else K
Define U(V) = F(K)W(V). Equation (7) then becomes
would be constant)
V2W + [(2/F)dF/dK + ( 1 / K ) ] V K . V W
~/= Y
+ [{(I/K)dF/dK + (1/F)d2F/dK2}VK • V K + (1/F)dF/dKV2K] W = Q/(KF)
(9)
It is useful to eliminate the V W term by requiring
(2/F)dF/dK + (l/K) = 0
o~U(~, ~/)/0~2 + (1/~)OU(~, ~!)/0~
(ll)
Any other choice for F(K) would leave a VW(V) term which complicates the solution. If KU2(F) is harmonic, i.e. if V2{KI/2(V)} is zero, eqn (9) reduces further. Replacing W in terms of U leads to
V2{K1/2(F)U(F)} ----K(F)-I/2Q(r ")
(12)
If Q is replaced by - 6 ( ? ' - ?'o) and W(?') by G(~, Vo), the trick of eqn (6) leads to
V2{xl/2(~)xl/2(r'o)G(~, r'0)} = - 6 ( r ' - r'0)
(13)
thereby identifying the heterogeneous Green's function, G(V, V0), with the known homogeneous Green's function,
c0(V, r0) as G(7, ro) = KoGO(F, r'o)/[Kl/2(F)xl/2(Fo)]
(14)
An interesting extension of this is the case where KU2(r) satisfies a He!mholtz equation, i.e. V2{Kl/2(r)}'-'--is -aK1/2(7). The Green's function then satisfies
V2{K1/2(F)KI/2(7o)G(7, F0)} + olKI/2(F)KI/2(Fo)G(F, F0) ------6(F-- r'0)
(15)
(16)
While other fundamental solutions may be obtained in this manner, this path does not appear to be worth pursuing further at this time. This derivation al~pears to have first been carried out by Georghitza and applied, for example, by Cheng4 to various problems. Ca) Tramformatiom of ~
+ 02U(~,y)/&rl 2 = Q/(a, ~)
(19)
This is dearly an axisymmetric three dimensional potential equation and the Green's function may be obtained from the three dimensional Green's function by integrating over 0. It is important here to keep in mind the eventual use of this function in a boundary integral equation where the 0 integration may be carried out directly. In the following derivation, ~ will be replaced by r and ~/ by z. However, ~ and ~/ are really two dimemfional cartesian coordinates which are temporarily laving treated as cylindrical coordinates for convenience. The corresponding Green's function equation and Green's function are eqn (3) and eqn (4b) respectively, V2G(o3)(r , 0, z; to, 0o, Zo) ----- 6 ( F - F0)/K0 G~3)(r, 0, z; to, 00, Zo) = 1/ (4~'KoR) where
R = [r2 + ~ - 2fro cos (0 - 00) + (z - Zo)2]1/2 Green's integral theorem takes the standard BIEM form
IVo
o, z; r0~ Oo,z0)V02 u(r0, z0)
- U(ro, Zo)V2G(r, 0, z; to, 00, z0)}r0 dro d80 dzo
i.e. the Green's function for a heterogeneous potential problem is related to that for a homogeneous Helmholtz problem, w.(h)/t.,r F0; ol)/[Kl/2(F; a)Kl/2(Fo; a)] G(F, F0) = r..or0
(lSb)
is made, the governing equation, in two dimensional form, becomes
(10)
which leads to F(K) = K -1/2. The coefficient of the W term under this condition also simplifies, yielding eqn (8) as
V2W(~) - K(~)-l/2~2(K(~)1/2) = K(F)-l/2Q(F)
(18a)
lmlepemlem variable
Here the variables x, y a n d z are transformed to have the governing heterogeneous equation resemble a known problem. This is seen most clearly with the layered linearly varying two dimensional case, V * { K ( x ) V V ( x , y ) } = Q(x,y) (17)
= [ (a(r, o, z; ro, 0o, zo)Q(ro, zo)/ J Iio
(aKoro}ro dro d0o dzo + cU(r, z)/Ko
= f {G(r,O,z;ro,Oo, zo)no * VoU(ro, zo) 3So - U(ro, Zo)~ • VoG(r, 0, z; ro, 00, Zo)}ro d0o dso
(20)
The integration variables are indicated by the subscript zero while the field variables are not subscripted. The unit outward normal (from Vo) is ~ and the del operator, V0, are both defined in tenns of integration variables. The coefficient c is 1 for 7 within V, 0 for ~' outside of V and 1/2 at a smooth point on the boundary S. The integrations over 00 may be carried out directly since U(ro,zo) is independent of 0o. The axisymmetric volume, Vo, and its surface, So, are formed by rotating an area, Ao, and its boundary, F0 with a line element ds0, about the z0 axis. The outward normal to So has no 0o component and 0o derivatives in the
Green's functions for heterogeneous media potential problems operator 8o * V0 are immaterial. The field coordinate 0 may be chosen for convenience to be ~r. Then eqn (20) becomes
221
certain relationships between these coefficients, i.e. if
a(x) =
(I/A) dA/dx; d[(A/¢ 2)d~/dx]/dx; 4/ A2
b(x) = (~IA)
Iro
* VoU(ro, zo)[J:o=odOo/[4'rKoR]]
- U(ro, zo)Ro . =I
c(x ) =
Vo[J:oodOo/[4rKoR]]}rodso
Q(ro, zo)/(aro)[J:o~=odOo/[4~rKoR]]rodrodzo
+ cU(r, z)/Ko
G2(r, z; r0, z0)
= (I/4~'Ko)
Ioor [r2 + ~ -
x
2rro cos Or - 0o) + (z - Zo)21-1/2d0o
=0
= K(k)/{lrI~[(r + to) 2 + (z - Zo)2]1/2} =
G~(r,z; r0, z0)/K0
u(x,y) =
=
-
-
-
cV(r, z) = - [ (Q(ro, zo)la)G~(r, z; ro, ZO) dro dzo JA o
* VoU(ro, zo)G~(r,z; ro, zo)
- U(ro,zo)fi o .VoG~(r,z;ro, zo)}rods o
(23)
where x = ~ - / ~ / a = r - / 3 / a and y = z = r/. This derivation is easy, but does not appear to have been previously developed elsewhere. Other transformations of the independent variable are possible, for example using ~ = ax + ~ for K = Ko[~]"y, but this does not seem to lead to a known equation for 7 # 1.
(e) Transformations of both dependent and independent variables This is the most general of all of these approaches and also the most difficult. Classically, this has been the area of greatest mathematical study and most details will be left to the references. Discussions of this approach have been given by Clements 6'7 in several papers. For example, the equation
02u(x, y)/Ox 2 + a(x)Ou(x, y)/Ox + b(x)u(x,y) = c(x)O2u(x, y)/Oy 2
(24)
with c(x) < 0 to have an elliptic system, may be transformed into the standard Laplace equation under
[[¢2(x)lA(x)]dx; = y (26)
transform eqn (24) into
02u*/O~2 + Ou*/Orl2 = 0
(27)
for which the Green's function is known, for arbitrary A(x) and ~(x), but with the three conditions of eqns (25) requiring some forced relationship between A and ~. Unfortunately, the form of eqn (8) places a severe restriction on the form of K(x), i.e. A ( x ) = K(x),
#h(x) = K(x) 1/2, (22)
where m = k 2 = 4rro/[(r + r0) 2 -k (z Z0)2]1/2] and K(k) or K(m) is the complete elliptic integral of the first kind (K(k) or K(m) is a standard notation that has no relation to the conductivity K0), see, for example, Wrobel and Brebbia. s Then the BIE is
+ L, I t o
the transformations of both dependent and independent variables,
(21)
The integral over Oo is straightforward and gives G2(r, z; ro, Zo) as
(25)
and
K(x)-l/2 d2(K(x)ll2)/dx 2 = 0
which leads to the same result as in Section (a). Other transformations are possible which lead to series solutions with a finite number of terms for K ( x ) = (ax +/~)2~ for integer n. These are not as useful for realistic media as the linear variation described above. One of the major drawbacks to two dimensional approaches that are directly based on complex variable theory is that they do not readily generali~,e to three dimensions. In all cases, the boundary conditions follow the same transformation.
REFERENCES 1. Brebbia, C. A. & Dominguez, J. Boundary Elements: An Introductory Course, 2nd edn, Computational Mechanics Publications/McGraw-Hill, New York, 1992. 2. Beskos, D. E. Boundary Element Methods in Mechanics, ed. by D. E. Beskos, Chapter 2, North-Holland/Elsevier, New York, 1987. 3. Georghitza, St I. On the plane steady flow of water through inhomogeneous porous media, First Symp. on the Fundamentals of Transport Phenomena in Porous Media, IAHR, Haifa, 1969. 4. Cheng, A. H-D. Darcy flow with variable permeability, Water Resourc. Res., 1984, 20, 980-4. 5. Wrobel, L. C. & Brebbia, C. A. Axisymmetric potential problems, in New Developments in Boundary Element Methods, ed. C. A. Brebbia, Butterworths, London, 1980, pp. 77-89. 6. Clements,D. L. A boundary integral equation method for the numerical solution of a second order elliptic equation with variable coefficients, J. Austral. Math. Soc., 1980, 22(B), 218-28. 7. Clements, D. L. Green's functions for the boundary element method, in Boundary Elements IX, Vol. 1, ed. C. A. Brebbia, W. L. Wendland & G. Kulm, Computational Mechanics Publications/Springer-Verlag, New York, 1987, pp. 13-20.
ELSEVIER
Green's functions for heterogeneous media potential problems Richard P a u l S h a w Department of Civil Engineering, State University of New York at Buffalo, 212 Engineering West R-8, Buffalo,
New York 14260, USA
(Received 1 November 1993; accepted 10 January 1994) Procedures are discussed for obtaining Green's functions for potential problems in heterogeneous materials. Such fundamental solutions are required in the boundary element method. An exact fundamental solution for a linearly varying, layered two dimensional potential problem is given that does not appear to be available elsewhere.
Key words: Green's functions, heterogeneous media, boundary element method. INTRODUCTION
dimensional solutions, using R = [7- Fo[, are G(2) ....
o ~r, ro) = (1/(2Ko))In Ca/R); a is an arbitrary
The boundary integral/element method (BIEM) relies on the availability of fundamental solutions for the various types of equations to which it is applied. When heterogeneous materials are encountered, these Green's functions become more difficult to find. To this end, a collection of Green's functions for the heterogeneous medium, potential equation are obtained below. While this is not meant to be an exhaustive account, the approaches are fairly general.
constant, usually set to 1 ~(3),.-, - . , t16 (r, r o ) = 1/(41rKoR)
(4a) (4b)
e.g. Brebbia and Dominguez, 1 Beskos,2 etc. If the physical problem for heterogeneous media leads to either of the following two equations, the solution for the hetergeneous Green's function is trivial;
K(P')V2U(P.)
=
Q(P')
(5a)
V2{K(P.) U(P.)} = Q(P')
HETEROGENEOUS POTENTIAL EQUATION
(5b)
The corresponding Green's functions are, respectively, The homogeneous medium potential equation is KoV2U(P.) = Q(P.) in V(P.)
G(~, P'o) KoGO(F,P'o)/ K(P'o) GCP.,P'o) KoGoCP',P'o)/ K (r-)
(1)
where Ko is a constant 'conductivity' and Q(P.) is a volume distributed source. Typical boundary conditions would be
U(P.) =fl(p') on $1 (Dirichlet conditions)
(6b)
v • {KCP.)VUCP.)}
=
Q(P.)
(7)
(8)
is then the focus of attention for the remainder of this discussion. Only specific material heterogeneities have led to known solutions so far. These come through transformation of the heterogeneous equation to an equation whose fundamental solution is already known and proceed via three main paths: (a) transformation of the dependent variable, (b) transformation of the independent variable, (c) transformation of both variables. Some illustrations of each are given.
(2c)
The corresponding Green's functions to this problem are well known and satisfy
KoV2G(F, P'0) ----- 6 ( F - F0) in V(P')
=
6(P'- P'0) * FCP.) = 6(P'- P'0) * FCP'0) has been introduced. The more dimcult ease
(2a)
(2b) on $2 (flux conditions)
(6a)
where, since the delta function only operates at P.= P.0, the useful 'trick'
fi • VU(P.) =f2(r') on $2 (Neumann conditions) or
• KoVU(P') =f2(p')
=
(3)
where the constant /Co has been included for comparison with the heterogeneous case. The two and three
219
R. P. Shaw
220 (a) Tramformatlea of flke depeadmt variable
with K(x) = ax + ~. If a coordinate transformation = x + ~ / a (a is not zero or else K
Define U(V) = F(K)W(V). Equation (7) then becomes
would be constant)
V2W + [(2/F)dF/dK + ( 1 / K ) ] V K . V W
~/= Y
+ [{(I/K)dF/dK + (1/F)d2F/dK2}VK • V K + (1/F)dF/dKV2K] W = Q/(KF)
(9)
It is useful to eliminate the V W term by requiring
(2/F)dF/dK + (l/K) = 0
o~U(~, ~/)/0~2 + (1/~)OU(~, ~!)/0~
(ll)
Any other choice for F(K) would leave a VW(V) term which complicates the solution. If KU2(F) is harmonic, i.e. if V2{KI/2(V)} is zero, eqn (9) reduces further. Replacing W in terms of U leads to
V2{K1/2(F)U(F)} ----K(F)-I/2Q(r ")
(12)
If Q is replaced by - 6 ( ? ' - ?'o) and W(?') by G(~, Vo), the trick of eqn (6) leads to
V2{xl/2(~)xl/2(r'o)G(~, r'0)} = - 6 ( r ' - r'0)
(13)
thereby identifying the heterogeneous Green's function, G(V, V0), with the known homogeneous Green's function,
c0(V, r0) as G(7, ro) = KoGO(F, r'o)/[Kl/2(F)xl/2(Fo)]
(14)
An interesting extension of this is the case where KU2(r) satisfies a He!mholtz equation, i.e. V2{Kl/2(r)}'-'--is -aK1/2(7). The Green's function then satisfies
V2{K1/2(F)KI/2(7o)G(7, F0)} + olKI/2(F)KI/2(Fo)G(F, F0) ------6(F-- r'0)
(15)
(16)
While other fundamental solutions may be obtained in this manner, this path does not appear to be worth pursuing further at this time. This derivation al~pears to have first been carried out by Georghitza and applied, for example, by Cheng4 to various problems. Ca) Tramformatiom of ~
+ 02U(~,y)/&rl 2 = Q/(a, ~)
(19)
This is dearly an axisymmetric three dimensional potential equation and the Green's function may be obtained from the three dimensional Green's function by integrating over 0. It is important here to keep in mind the eventual use of this function in a boundary integral equation where the 0 integration may be carried out directly. In the following derivation, ~ will be replaced by r and ~/ by z. However, ~ and ~/ are really two dimemfional cartesian coordinates which are temporarily laving treated as cylindrical coordinates for convenience. The corresponding Green's function equation and Green's function are eqn (3) and eqn (4b) respectively, V2G(o3)(r , 0, z; to, 0o, Zo) ----- 6 ( F - F0)/K0 G~3)(r, 0, z; to, 00, Zo) = 1/ (4~'KoR) where
R = [r2 + ~ - 2fro cos (0 - 00) + (z - Zo)2]1/2 Green's integral theorem takes the standard BIEM form
IVo
o, z; r0~ Oo,z0)V02 u(r0, z0)
- U(ro, Zo)V2G(r, 0, z; to, 00, z0)}r0 dro d80 dzo
i.e. the Green's function for a heterogeneous potential problem is related to that for a homogeneous Helmholtz problem, w.(h)/t.,r F0; ol)/[Kl/2(F; a)Kl/2(Fo; a)] G(F, F0) = r..or0
(lSb)
is made, the governing equation, in two dimensional form, becomes
(10)
which leads to F(K) = K -1/2. The coefficient of the W term under this condition also simplifies, yielding eqn (8) as
V2W(~) - K(~)-l/2~2(K(~)1/2) = K(F)-l/2Q(F)
(18a)
lmlepemlem variable
Here the variables x, y a n d z are transformed to have the governing heterogeneous equation resemble a known problem. This is seen most clearly with the layered linearly varying two dimensional case, V * { K ( x ) V V ( x , y ) } = Q(x,y) (17)
= [ (a(r, o, z; ro, 0o, zo)Q(ro, zo)/ J Iio
(aKoro}ro dro d0o dzo + cU(r, z)/Ko
= f {G(r,O,z;ro,Oo, zo)no * VoU(ro, zo) 3So - U(ro, Zo)~ • VoG(r, 0, z; ro, 00, Zo)}ro d0o dso
(20)
The integration variables are indicated by the subscript zero while the field variables are not subscripted. The unit outward normal (from Vo) is ~ and the del operator, V0, are both defined in tenns of integration variables. The coefficient c is 1 for 7 within V, 0 for ~' outside of V and 1/2 at a smooth point on the boundary S. The integrations over 00 may be carried out directly since U(ro,zo) is independent of 0o. The axisymmetric volume, Vo, and its surface, So, are formed by rotating an area, Ao, and its boundary, F0 with a line element ds0, about the z0 axis. The outward normal to So has no 0o component and 0o derivatives in the
Green's functions for heterogeneous media potential problems operator 8o * V0 are immaterial. The field coordinate 0 may be chosen for convenience to be ~r. Then eqn (20) becomes
221
certain relationships between these coefficients, i.e. if
a(x) =
(I/A) dA/dx; d[(A/¢ 2)d~/dx]/dx; 4/ A2
b(x) = (~IA)
Iro
* VoU(ro, zo)[J:o=odOo/[4'rKoR]]
- U(ro, zo)Ro . =I
c(x ) =
Vo[J:oodOo/[4rKoR]]}rodso
Q(ro, zo)/(aro)[J:o~=odOo/[4~rKoR]]rodrodzo
+ cU(r, z)/Ko
G2(r, z; r0, z0)
= (I/4~'Ko)
Ioor [r2 + ~ -
x
2rro cos Or - 0o) + (z - Zo)21-1/2d0o
=0
= K(k)/{lrI~[(r + to) 2 + (z - Zo)2]1/2} =
G~(r,z; r0, z0)/K0
u(x,y) =
=
-
-
-
cV(r, z) = - [ (Q(ro, zo)la)G~(r, z; ro, ZO) dro dzo JA o
* VoU(ro, zo)G~(r,z; ro, zo)
- U(ro,zo)fi o .VoG~(r,z;ro, zo)}rods o
(23)
where x = ~ - / ~ / a = r - / 3 / a and y = z = r/. This derivation is easy, but does not appear to have been previously developed elsewhere. Other transformations of the independent variable are possible, for example using ~ = ax + ~ for K = Ko[~]"y, but this does not seem to lead to a known equation for 7 # 1.
(e) Transformations of both dependent and independent variables This is the most general of all of these approaches and also the most difficult. Classically, this has been the area of greatest mathematical study and most details will be left to the references. Discussions of this approach have been given by Clements 6'7 in several papers. For example, the equation
02u(x, y)/Ox 2 + a(x)Ou(x, y)/Ox + b(x)u(x,y) = c(x)O2u(x, y)/Oy 2
(24)
with c(x) < 0 to have an elliptic system, may be transformed into the standard Laplace equation under
[[¢2(x)lA(x)]dx; = y (26)
transform eqn (24) into
02u*/O~2 + Ou*/Orl2 = 0
(27)
for which the Green's function is known, for arbitrary A(x) and ~(x), but with the three conditions of eqns (25) requiring some forced relationship between A and ~. Unfortunately, the form of eqn (8) places a severe restriction on the form of K(x), i.e. A ( x ) = K(x),
#h(x) = K(x) 1/2, (22)
where m = k 2 = 4rro/[(r + r0) 2 -k (z Z0)2]1/2] and K(k) or K(m) is the complete elliptic integral of the first kind (K(k) or K(m) is a standard notation that has no relation to the conductivity K0), see, for example, Wrobel and Brebbia. s Then the BIE is
+ L, I t o
the transformations of both dependent and independent variables,
(21)
The integral over Oo is straightforward and gives G2(r, z; ro, Zo) as
(25)
and
K(x)-l/2 d2(K(x)ll2)/dx 2 = 0
which leads to the same result as in Section (a). Other transformations are possible which lead to series solutions with a finite number of terms for K ( x ) = (ax +/~)2~ for integer n. These are not as useful for realistic media as the linear variation described above. One of the major drawbacks to two dimensional approaches that are directly based on complex variable theory is that they do not readily generali~,e to three dimensions. In all cases, the boundary conditions follow the same transformation.
REFERENCES 1. Brebbia, C. A. & Dominguez, J. Boundary Elements: An Introductory Course, 2nd edn, Computational Mechanics Publications/McGraw-Hill, New York, 1992. 2. Beskos, D. E. Boundary Element Methods in Mechanics, ed. by D. E. Beskos, Chapter 2, North-Holland/Elsevier, New York, 1987. 3. Georghitza, St I. On the plane steady flow of water through inhomogeneous porous media, First Symp. on the Fundamentals of Transport Phenomena in Porous Media, IAHR, Haifa, 1969. 4. Cheng, A. H-D. Darcy flow with variable permeability, Water Resourc. Res., 1984, 20, 980-4. 5. Wrobel, L. C. & Brebbia, C. A. Axisymmetric potential problems, in New Developments in Boundary Element Methods, ed. C. A. Brebbia, Butterworths, London, 1980, pp. 77-89. 6. Clements,D. L. A boundary integral equation method for the numerical solution of a second order elliptic equation with variable coefficients, J. Austral. Math. Soc., 1980, 22(B), 218-28. 7. Clements, D. L. Green's functions for the boundary element method, in Boundary Elements IX, Vol. 1, ed. C. A. Brebbia, W. L. Wendland & G. Kulm, Computational Mechanics Publications/Springer-Verlag, New York, 1987, pp. 13-20.