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In the application of FMM to BIEM our starting point is to expand the
fundamental solution
into a series of products of
functions of
and those of
.
From the expression of
(4) one finds that it is necessary to expand
into a
series. Yoshida et al.[7] obtained an expansion for
as follows:
where R_{n,m} and S_{n,m} are solid harmonic functions defined as
are the polar coordinates of the
point
,
P_{n}^{m} is the associated Legendre function and a
superposed bar indicates the complex conjugate. The use of solid
harmonics in FMM has been suggested by PerezJorda and
Yang[24]. The functions R_{n,m} and S_{n,m} satisfy the
following relations given by
Using (7), one rewrites (4) as[7]
where
F^{S}_{ij,n,m} and
G^{S}_{i,n,m} are functions defined as[7]
We now compute the integral on the right hand side of (5) over
a subset of S denoted by S_{y} for
which is away from
S_{y}. Using (10) we obtain
where
M^{1}_{j,n,m} and
M^{2}_{n,m} are the multipole moments
centred at O, expressed as
The multipole moments are translated according to the following
formulae as the centre of multipole expansion is shifted from Oto O':
M^{1}_{j,n,m}(O')= 







(15) 
M^{2}_{n,m}(O')= 







(16) 
where we have used (9), (14) and (15).
In the evaluation of the integral on the right hand side of
(5) one can use not only the multipole moments but also the
coefficients of local expansion in the following manner:
where
L^{1}_{j,n,m} and L^{2}_{n,m} are expressed with
M^{1}_{j,n,m} and M^{2}_{n,m} by
and
F^{R}_{ij,n,m} and
G^{R}_{i,n,m} are functions obtained by replacing
S_{n,m} by R_{n,m} in (11) and (12). In these formulae
we have used (8) and have assumed that the inequality
holds. The procedures given by
(19) and (20) are called M2L translation.
The coefficients of the local expansion are translated according to the
following formulae when the centre of the local expansion is shifted
from
to
where we have used (9) and (18).
Next: Algorithm for original FMM
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Kenichi Yoshida
20010326