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Basic role of FMM in FM-BIEM

In BIEM an integral equation is given by
$\displaystyle f(\mbox{\boldmath$ x $ }) = \int_{S} K(\mbox{\boldmath$ x $ },\mb...
...{\boldmath$ y $ }) dS
\quad \mbox{\boldmath$ x $ },\mbox{\boldmath$ y $ }\in S,$     (2.1)

where $\varphi$ is an unknown function on S, f is a given function on S and K is a given kernel function on $S \times S$. When one solves the integral equation numerically one discretises it into the form of a matrix-vector product. In FM-BIEM an iterative method is used as a solver for linear equations. Namely FMM is utilised to reduce the computational complexity for the multiplication of a matrix and a candidate vector in BIEM. Indeed, FMM can reduce this complexity from O(N2) to O(N). In FM-BIEM the integral in (2.1) is evaluated with direct computation in the same manner as in the conventional BIEM only when $\mbox{\boldmath$\space x $ }$ is in the neighbourhood of $\mbox{\boldmath$\space y $ }$ because of a singularity at $\mbox{\boldmath$\space x $ }= \mbox{\boldmath$\space y $ }$ of a kernel function. Otherwise, the integral is evaluated efficiently with FMM.

Ken-ichi Yoshida