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In contrast with domain methods such as Finite Element Method (FEM) or
Finite Differential Method (FDM),
Boundary Integral Equation Method (BIEM), also called Boundary Element
Method (BEM), discretises only the boundary of the domain.
Because of this reduction of the dimensionality BIEM was expected to be
advantageous in large-scale problems. However, the application
of this method has so far been limited to relatively small
problems. This was because coefficient matrices in BIEM are full, due to
which both the operation count and the memory requirements for the
matrix equation buildup are of the order of *O*(*N*^{2}), where *N* is the
number of unknowns. The operation count even increases to *O*(*N*^{3}) as
one attempts to solve the matrix equations with conventional direct
solvers such as Crout's method and Gaussian elimination. In particular,
the full matrix property leads to a serious exhaustion of the memory of
a computer and is an obstacle for applications of BIEM to large-scale
problems. On the other hand, coefficient matrices in domain methods are
banded and both computational complexity for matrix buildup and memory
requirements are *O*(*N*). This is why applications of domain methods to
large-scale problems are relatively easy. However, the appearance of
Fast Multipole Method (FMM), also referred to as Fast Multipole Algorithm
(FMA), drastically changed the circumstances around BIEM. The use of FMM
in conjunction with iterative solvers such as Conjugate Gradient (CG) method
and Generalised Minimal RESidual (GMRES) method has been shown to reduce
the memory requirements to *O*(*N*) and the operation count to
,
where
is a small non-negative number. Thus FMM
enables us to apply BIEM to large-scale problems. Thanks to FMM, BIEM is
no longer what it was.
FMM was initially introduced by Rokhlin[69] as a fast
solution method for integral equations for two-dimensional Laplace's
equation and then developed by Greengard[33,35] as
a fast evaluation method for the pairwise force calculation in multibody
problems with Coulombic potential. FMM has been applied to problems in
various fields such as BIEM and Molecular Dynamics (MD).

The author has been studying this topic in order to apply Fast Multipole
Boundary Integral Equation Method (FM-BIEM) to practical problems in
fracture mechanics and earthquake engineering. In this thesis
the author discusses applications of FM-BIEM to various fundamental
boundary value problems in three dimensions and show its efficiencies so
as to make the first step toward practical applications.

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*Ken-ichi Yoshida*

*2001-07-28*