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Time-dependent problems with the boundary integral equation
method

*Martin Costabel*
Chapter 25 of
Encyclopedia of Computational Mechanics, pages 703 - 721

Edited by Erwin Stein, Renee de Borst and Thomas Hughes

Copyright (C) 2003 John Wiley & Sons, Ltd.

PDF (254 k)

Time-dependent problems that are modeled by initial-boundary value
problems for parabolic or hyperbolic partial differential equations
can be treated with the boundary integral equation method. The ideal
situation is when the right-hand side in the partial differential
equation and the initial conditions vanish, the data are given only
on the boundary of the domain, the equation has constant coefficients,
and the domain does not depend on time. In this situation, the
transformation of the problem to a boundary integral equation follows
the same well-known lines as for the case of stationary or
time-harmonic problems modeled by elliptic boundary value problems.
The same main advantages of the reduction to the boundary prevail:
Reduction of the dimension by one, and reduction of an unbounded
exterior domain to a bounded boundary.

There are, however, specific difficulties due to the additional time
dimension: Apart from the practical problems of increased complexity
related to the higher dimension, there can appear new stability
problems. In the stationary case, one often has unconditional
stability for reasonable approximation methods, and this stability
is closely related to variational formulations based on the
ellipticity of the underlying boundary value problem. In the
time-dependent case, instabilities have been observed in practice,
but due to the absence of ellipticity, the stability analysis is
more difficult and fewer theoretical results are available.

In this article, the mathematical principles governing the
construction of boundary integral equation methods for
time-dependent problems are presented. We describe some of the main
algorithms that are used in practice and have been analyzed in the
mathematical literature.