Surface Integral Equations in Computational Electromagnetics: A Comprehensive Overview of Theory, Formulations, Discretization Schemes and Implementations

Computational electromagnetics based on surface integral equations provides accurate and efficient solutions for three-dimensional electromagnetic scattering problems in the frequency domain. In this review paper, we first introduce a complete and detailed theoretical analysis of the surface integral equation method, including different properties of the corresponding integral operators and equations. Using a pedagogical approach that should appeal to electrical engineers, we provide a systematic and comprehensive derivation of the different formulations found in the literature and discuss their advantages and pitfalls. Additionally, we provide a mathematical overview of the corresponding function spaces that clarifies the importance of correctly combining basis and testing functions and we examine the various aspects of discretization schemes, such as the Green’s function singularity subtraction and the application of different testing methods. Moreover, we assess alternative formulations and discretization procedures and draw particular conclusions about them, by comparing numerous examples and results from previously published works. Finally, we provide a detailed discussion on numerical solvers and approaches.