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Noting that R_{n,m} is a homogeneous polynomial of degree n,
one can compute the righthand side in (3.39) analytically
when S_{L} is a polygon as follows:
where we have used (3.21) in (3.40), N_{L} is the
number of the plane elements in S_{y},
are the
vertices of S_{L} and
P^{L}_{K+1}=P^{L}_{1} (See Fig.3.4). In
FMBIEM with the regularised integral equation we can use
(3.40) for the computation of the multipole moment in
(3.31). The computational cost for
for a
given pair of n and m is
O(3N_{L}Kn^{2}).
Figure 3.4:
Line integration for analytical computation of

On the other hand, we compute the integral in (3.23)
numerically with Gaussian quadrature as follows:



(3.41) 
where w_{I} and
are the Gaussian weight and Gaussian point on
S_{L}, N_{g} is the number of the Gaussian points and
is
the area of S_{L}. Therefore, the computational cost for
M_{n,m}(O)for a given pair of n and m is
O(N_{g} N_{L} ). As n increases, the
computational cost for
for a given pair of n and
m becomes large whereas, that for
M_{n,m}(O) does not depend on
n. Since the computational cost for the multipole moments is not so
dominant in FMM, this is not so serious as we shall see. As we have
mentioned above, however, the difference in the cost for M2L
translations is considerably more serious than that in the cost for the
multipole moments.
Next: Algorithm
Up: Numerical procedure
Previous: Discretisation of the regularised
Kenichi Yoshida
20010728