Comparison of three FMM techniques for solving hybrid FE-BI systems

By virtue of its low operation count, the application of the Fast Multipole Method (FMM) results in substantial speed-up of the boundary-integral (BI) portion of the hybrid Finite-Element/Boundary-Integral technique, independent of the shape of the BI contour. Recently, various versions of the Fast Multipole Method have been proposed, each introducing a different approximation to the implementation of the boundary integral. The main goal of this paper is to provide a comparison of the various FMM approaches on the basis of implementation, CPU time, and accuracy. To gain an appreciation of the differences among the various FMM methodologies, a large portion of the paper is devoted to a discussion of the algorithms at a tutorial lever. Flow charts and pseudo-code are also given, at sufficient detail to facilitate their implementation. We present quantitative CPU and memory requirements, using the scattering by a groove as the basis for comparison, and conclude that the FMM can accelerate the BI computation without any significant deterioration in accuracy. A simpler FMM-based algorithm results in a much smaller execution time but has larger error. However, it turns out that a third algorithm, designated the ''windowed'' FMM, provides a very good compromise with respect to error and execution time. The paper concludes with the presentation of some three-dimensional applications for which a hybrid FE-BI technique, in conjunction with a fast-integral algorithm, is well suited.