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.
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