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### Analytical computation of the multipole moment

Noting that Rn,m is a homogeneous polynomial of degree n, one can compute the right-hand side in (3.39) analytically when SL is a polygon as follows:

 = = = (3.40)

where we have used (3.21) in (3.40), NL is the number of the plane elements in Sy, are the vertices of SL and PLK+1=PL1 (See Fig.3.4). In FM-BIEM 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(3NLKn2).

On the other hand, we compute the integral in (3.23) numerically with Gaussian quadrature as follows:
 (3.41)

where wI and are the Gaussian weight and Gaussian point on SL, Ng is the number of the Gaussian points and is the area of SL. Therefore, the computational cost for Mn,m(O)for a given pair of n and m is O(Ng NL ). As n increases, the computational cost for for a given pair of n and m becomes large whereas, that for Mn,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
Ken-ichi Yoshida
2001-07-28