Three‐dimensional weak‐form conjugate‐ and biconjugate‐gradient FFT methods for volume integral equations

A large‐scale three‐dimensional volume integral equation solution for electromagnetic radiation and scattering problems remains a great challenge in spite of many ongoing research efforts. The conventional method of moments, although accurate and flexible, is limited to small‐scale problems because of its large requirement of computer memory and computation time. In this paper, we develop two fast methods, the weak‐form conjugate‐ and biconjugate‐gradient FFT methods, to solve the Fredholm integral equation of the second kind arising from Maxwell's equations in three dimensions. The weak form is a modified version of the Zwamborn–van den Berg formulation, where the singularity is circumvented by employing the weak‐form discretization by rooftop vectorial basis and testing functions. Both weak‐ form CG–FFT and BCG–FFT methods require O ( N log 2 N ) CPU time, and O ( N ) computer memory, but the latter converges three–six times faster than the CG–FFT method. We validate the numerical results by comparing them with analytical solutions to multilayer spherical media, and with other published results.