STABILITY ASSESSMENT OF A UNIFIED VARIATIONAL BOUNDARY INTEGRAL METHOD APPLICABLE TO THIN SCATTERERS AND SCATTERERS WITH CORNERS

The objective of this study is to assess the stability, especially at and near critical frequencies, of a new unified, stable, symmetric, do main finite element and boundary integral methodology for solving time -harmonic interface problems for scatterers of arbitrary shape. The va lidity of this energy-based variational procedure for rigid and penetr able smooth scatterers, including the existence, uniqueness and optima l convergence of the corresponding numerical approximations, has been proved rigorously in the context of fluid-structure interaction proble ms. The emphasis here is on investigating the applicability of this pr ocedure to the case of scatterers with corners, using square and thin rectangular two-dimensional rigid obstacles as prototypes. We also est ablish that the apparently distinct direct boundary integral formulati on due to Burton and Miller and the combined single- and double-layer indirect formulation due to Leis, Brakhage and Werner, and Panich, int roduced to insure that boundary integral equations for scattering prob lems are uniquely solvable for all wavenumbers, are entirely equivalen t within our variational setting. By examining the condition number of the matrix of coefficients of the discretized equations, as well as t he resulting solutions, both directly on the surface of the scatterers and in the far field, it is demonstrated that the new methodology is robust, and completely insensitive to critical frequencies, even for t hin objects. Most significantly, the precise singular behaviour of the pressure field at corners is also predicted quite accurately by using standard finite elements along the boundary of the scatterer, without having to introduce special singularity functions, which for general interface problems, may not be known in advance.