Resolution of Dynamic Singularities with Analytical Regularization and Static Singularities with Conformal Mapping Approach in Electromagnetic Transmission Problems

In this work it is shown that Boundary Integral Equation (BIE) methods may become attractive to solve Electromagnetic Scattering and Electromagnetic Transmission Problems when techniques to deal with singularities of the integral kernels are found corresponding to both kinds of them: static and dynamic. The techniques to deal with dynamic singularities, which appear due to the choice of Green’s function, are rewired with particular attention to ideas of dealing with them by means of special choice of basic functions combined with analytical regularization if needed, in particular those based on singularity subtraction, allowing fast calculations in addition to good accuracy. It is shown that spectral functions of corresponding BIE operators may be chosen as basic functions leading to solving of the problem with the best possible spectral accuracy, due to that resulting methods are called as Spectral Boundary Integral Equation (SBIE) methods. In this work it is demonstrated, that when static singularities are present together with dynamic singularities, the problem may be solved by SBIE method combined with conformal mapping trick, due to that resulting algorithm started to be known as Conformal Mapping Enhanced Spectral Fourier-Galerkin Boundary Integral Equation method with singularity subtraction improved by Fast Fourier Transform. It is demonstrated, that accuracy should be controlled by global error, that includes several local errors, among those are not only truncation error, but also discretization and integration errors. The efficiency of resulting numerical simulation algorithm is verified by calculating errors while solving Electromagnetic Transmission Problem (ETP), which is applicable for studies of electromagnetic wave propagation in complex media with obstacles made on dielectric materials, including those with strong absorption and/or gain, with possibilities of having dispersion and negative diffraction, including also high nearfield enhancements caused by appearance of resonances.