THE INSTITUTE OF MATHEMATICS
AND ITS APPLICATIONS

Minutes of Discussion Session: Future Research Directions in Boundary Integral Equation Methods

Prof Graham opened the session with some brief introductory remarks and by introducing the first speaker, Prof Wolfgang Dahmen. Prof Dahmen firstly stressed that anything he could present on this subject was only to be understood as his subjective opinion. In his presentation he focused on the two issues of on the one hand choosing subjects or problems (What to do?) and on the other hand on the question of (How to do?) this research (see also this slide).

As a list of possible problems he mentioned time dependent problems, stochastic problems, complex geometries and oscillatory problems. As a common feature, all these problems deal with multi-scale phenomena and hence require some sort of adaptivity. In this context it is important to distinguish between approximation with respect to a norm or the evalutation of a functional of the solution.

On the second question, Prof Dahmen stressed the importance of the question "What are we computing? in the sense of balancing accuracy and efficiency of computations. This issue requires reliable benchmarking to be answered satisfyingly. Furthermore, successful future research in BIEM will strongly require the combination of methods from analysis and numerics. He closed by pointing out that he is a strong believer in complimentary research by mathematicians and engineers.

The second speaker was Dr Marc Bonnet who stated his views on requirements for future research in BIEM (see also these slides). As key issues he stated the further development of fast solution techniques and of time domain solution techniques. It is furthermore important to reduce redundant work in order to accelerate the attaining of goals, for example by establishing reference techniques or by publishing implementations of key methods for simple cases.

A major obstacle to the development of BIE methods is the fact that the Finite Element Method (FEM) is the dominant method for solving engineering applications. However, in specific areas BIE methods have advantages and research should concentrate on development for such problems.

First to contribute to the general discussion was Prof Mazy'a, pointing out the possibility to solve non-linear problems using an iterative boundary integral equation approach. He also stressed that while huge achievements have been made on numerical issues, less development has been seen in theoretical aspects. As an example of an important problem from the view point of applications, on which hardly anything is known theoretically, Prof Mazy'a mentioned the representation of the flow around a ship moving in water. True progress is only possible by a synthesis of numerics and asymptotics.

Prof Wendland answered that the theory of boundary integral equations is often decisive for numerical achievements as well. This fact has been proved, for example, by the development of the Fast Multipole Method in recent years.

Prof Graham next commented on the contribution of Dr Bonnet. He emphasized that it would be very beneficial for the community to have a LAPACK like software package used by everyone to speed up the development of new methods.

Some deliberately controversial remarks were made by Prof Mikhailov in saying that boundary integral equation techniques must move from treating linear, constant coefficient problems to variable coefficient or nonlinear problems if they are to have a scientific future. Prof Michielsen added that the boundary integral equation community is still significantly smaller than the finite element community and probably less proactive in putting out results. Also very few commercial BEM codes are available on the market. Prof Dahmen conceded that FE methods have the advantage of being easier to handle while on the other hand certain problem classes are much better suited to boundary integral equation formulations.

The discussion was closed by some remarks by Prof Wendland who observed that he has experiences with industry cooperations where FE methods did not work but the BEM succeeded. He added some probable future topics to the list initially proposed by Prof Dahmen, namely domain decomposition techniques, localized problem formulations which can produce error estimators through BIE techniques.