Conforming testing of surface integral equations for penetrable objects

Electromagnetic scattering by a homogeneous penetrable object is a fundamental problem in many electromagnetic and electrical engineering radar, scattering and antenna applications. Surface integral equation method provides an elegant way to find solutions for such problems. For homogeneous penetrable bodies, the surface integral equations, however, can be formulated in many alternative ways. Different formulations are shown to have different numerical properties. For example, the accuracy of the solution and conditioning of the matrix depend strongly on the choice of the formulation (e.g., P. Ylä-Oijala, M. Taskinen and S. Järvenpää, Radio Science, 40, RS6002, 2005). Recently, it has been observed that the conventional discretization technique based on the Galerkin´s method and divergence conforming basis and testing functions (e.g., Rao-Wilton-Glisson functions), does not necessarily provide the most optimal testing procedure for all formulations. In particular, for the integral equations of the second Galerkin´s method may not lead to optimally converging solutions.