PARTITIONING, BOUNDARY INTEGRAL-EQUATIONS, AND EXACT GREENS-FUNCTIONS

Discretization of boundary integral equations leads to large full systems of algebraic equations, in practice. Partitioning is a method for solving such systems by breaking them down into smaller systems. It may be viewed merely as a technique from linear algebra. However, it is profitable to view it as arising directly from partitions of the boundary; these partitions could be natural (such as two separate boundaries) but they need not be. We investigate partitioning in the context of multiple scattering of acoustic waves by two sound-hard obstacles (the ideas extend to other physical situations). Specifically, we make a connection between partitioning and the use of the exact Green's function for a single obstacle in isolation. This suggests computing the action of this Green's function once-and-for-all, storing it (perhaps on a compact disc), and then using it to solve other problems in which the second obstacle is altered. One example of this approach is computing the stress distribution around a cavity of a standard-but-complicated shape inside a structure whose shape is varied. The theoretical foundation for these ideas is given, as well as a connection with the use of generalized Born series for multiple-scattering problems. Important distinctions between the partitioning/Green's function idea in this paper and seemingly similar ideas such as substructuring, multi-zoning, and domain decomposition are made.