Error Controllable Solutions of Large-Scale Problems in Electromagnetics: MLFMA-Accelerated Locally Corrected Nyström Solutions of CFIE in 3D [Open Problems in CEM]

This work provides an overview of a parallel, high-order, error-controllable framework for solving large-scale scattering problems in electromagnetics, as well as open problems pertinent to such solutions. The method is based on the higher-order locally corrected Nyström (LCN) discretization of the combined-field integral equation (CFIE), accelerated with the error-controlled Multi-Level Fast Multipole Algorithm (MLFMA). Mechanisms for controlling the accuracy of calculations are discussed, including geometric representation, stages of the locally corrected Nyström method, and the MLFMA. Also presented are the key attributes of parallelization for the developed numerical framework. Numerical results validate the proposed numerical scheme by demonstrating higher-order error convergence for smooth scatterers. For the problem of scattering from a sphere, the developed numerical solution is shown to have the ability to produce a solution with a maximum relative error of the order 10^-9. Open-ended problems, such as treatment of general scatterers with geometric singularities, construction of well-conditioned operators, and current challenges in development of fast iterative and direct algorithms, are also discussed.