Applications of FMM to BIEM are investigated by many authors: by Fukui and Hattori  and Miyakoshi et al. to two-dimensional Laplace's equation, by Fukui and Mochida and Helsing to two-dimensional elastostatics, by Nishimura et al. and Nishida and Hayami to three-dimensional Laplace's equation, by Yoshida et al.[84,87], Takahashi et al., Fukui and Kutsumi and Fu et al.[18,19] to three-dimensional elastostatics, by Chew et al., Fujiwara and Iritani et al. to two-dimensional elastodynamics, by Fujiwara and Yoshida et al. to three-dimensional elastodynamics, and by Gomez and Power[30,31], Mammoli and Ingber[57,58], Fu and Rodin and Takahashi et al. to fluid mechanics. BIEM are particularly suitable for wave analyses in the infinite domain. Because of this advantage, applications of FM-BIEM to large-scale wave problems, particularly to acoustical and electromagnetic scattering problems, are vigorously investigated by many researchers: by Rokhlin, Lu and Chew[54,55], Song and Chew, Fukui and Katsumoto and Yoshida et al..
Recently several techniques to enhance the efficiencies of FMM are proposed by several authors. Rokhlin[70,71,72] proposed such a technique called the ``diagonal form''. We shall call this technique ``Rokhlin's diagonal form''. Epton and Dembart and Elliot and Board also proposed similar techniques. However, Rokhlin's diagonal form is known to have numerical instabilities in dealing with static problems and low frequency problems. Hence, Greengard et al.[36,11,34] and Hrycak and Rokhlin proposed techniques based on the integral representation of a fundamental solution. In this thesis we call their techniques the ``new FMM''. The new FMM resolves problems of Rokhlin's diagonal form. Besides above techniques, several other techniques to improve the efficiency are proposed by White and Head-Gordon, Hu et al., etc.
Applications of these techniques to FM-BIEM
are found in several papers mentioned above, Koc and
Chew, Hu et al., Gyure and
Stalzer, Dembart and Yip, Fukui and
Katsumoto, Nishimura et al. and Yoshida et