- Chapter 2
In chapter 2 we briefly describe the history of FMM and other fast solution methods such as tree method, panel clustering method and wavelet-based method. Also, we explain the framework of FM-BIEM and the algorithm and computational cost of FMM.

- Chapter 3
In chapter 3 we apply the original FM-BIEM to three-dimensional problems. In particular, we deal with hypersingular integral equations for crack problems. Crack has the singularity at the tip and this property sometimes leads to failures of structures. This is why crack is worthy to be considered in engineering. Furthermore, techniques for the hypersingular integral equation can be easily extented to the single-layer and double-layer potentials. In this chapter we consider the following cases:

- Laplace's equation: Laplace's equation is a very important and basic PDE. Laplace's equation can be interpreted as the stationary heat equation, the electrostatic equation, etc. We apply FM-BIEM to crack problems for Laplace's equation. For example a crack may mean a thin insulation in the stationary heat equation or in the electrostatic equation.
- Elastostatics:
Crack is an interesting object because its behaviours have
much influence on the stress field. We apply FMM to BIE
analyses of crack problems for three-dimensional
elastostatics with collocation method and Galerkin's method.
- Helmholtz's equation (wave equation in the frequency domain):
Because BIEM is particularly suitable for wave analyses,
BIEM is often used for numerical analyses of elastodynamic
problems in non-destructive evaluation,
earthquake engineering or computational seismology.
Bearing applications of FM-BIEM to elastodynamics in mind,
we apply FM-BIEM to three-dimensional scattering of scalar
waves by cracks since techniques developed for Helmholtz's
equation can be easily extended to elastodynamics.
- Elastodynamics in the frequency domain: Using techniques proposed for Helmholtz's equation, we apply FM-BIEM to three-dimensional scattering of elastic waves by a crack.

- Chapter 4
In FMM the computational cost for the M2L translation (details of
the M2L translation will be given later) dominates the performance
especially in three-dimensional problems or problems dealing with
Helmholtz's equation. In order to reduce the computational cost for
the M2L translation Greengard and Rokhlin[36] proposed
some techniques based on an integral representation for a
fundamental solution. In this thesis we call FMM and FM-BIEM
connected with these techniques ``new FMM'' and ``new FM-BIEM'',
respectively. In chapter 4 we apply new FMM to BIE analyses of
crack problems for three-dimensional Laplace's equation and
three-dimensional elastostatics and compare results obtained with
new FM-BIEM with those obtained in chapter 3.
- Chapter 5
In chapter 5 we state conclusions.

- Appnedix
In appendix we give derivations of some formulae and equations in this thesis.