On the errors arising in surface integral equations due to the discretization of the identity operator

Surface integral equations (SIEs) are commonly used to formulate scattering and radiation problems involving three-dimensional metallic and homogeneous dielectric objects with arbitrary shapes. For numerical solutions, equivalent electric and/or magnetic currents defined on surfaces are discretized and expanded in a series of basis functions. Then, the boundary conditions are tested on surfaces via a set of testing functions. Solutions of the resulting dense matrix equations provide the expansion coefficients of the equivalent currents, which can be used to compute the scattered or radiated electromagnetic fields. This study consists of two parts. In the first part, the authors show that the identity operator is truly a major error source in normal and mixed formulations that are discretized with low-order functions, e.g., Rao-Wilton-Glisson (RWG) functions. In the second part, the authors investigate the incompatibility of SIE formulations in the context of iterative solutions. The authors show that a compatibility test can be used to determine the breakpoint, where the accuracy of the solution is saturated and cannot be enhanced any more.