Boundary Element Methods (BEM) play an important role in modern
numerical computations in the applied and engineering sciences. These
methods turn out to be powerful tools for numerical studies of various
physical phenomena which can be described mathematically by partial
differential equations.
The most prominent example is the potential equation (Laplace equation),
which is used to model physical phenomena in electromagnetism,
gravitation theory, and in perfect fluids. A further application leading
to the Laplace equation is the model of steady state heat flow. One of
the most popular applications of the BEM is the system of linear
elastostatics, which can be considered in both bounded and unbounded
domains. A simple model for a fluid flow, the Stokes system, can also be
solved by the use of the BEM. The most important examples for the
Helmholtz equation are the acoustic scattering and the sound radiation.
The Fast Solution of Boundary Integral Equations provides a detailed
description of fast boundary element methods which are based on rigorous
mathematical analysis. In particular, a symmetric formulation of
boundary integral equations is used, Galerkin discretisation is
discussed, and the necessary related stability and error estimates are
derived. For the practical use of boundary integral methods, efficient
algorithms together with their implementation are needed. The authors
therefore describe the Adaptive Cross Approximation Algorithm, starting
from the basic ideas and proceeding to their practical realization.
Numerous examples representing standard problems are given which
underline both theoretical results and the practical relevance of
boundary element methods in typical computations.
