Accelerated Boundary Integral Solution of 3-D Maxwell’s Equations Using the Interpolated Factored Green Function Method

Abstract

This article presents an $\mathcal {O}(N\log N)$ method for the numerical solution of Maxwell’s equations for dielectric scatterers using a 3-D boundary integral equation (BIE) method. The underlying BIE method used is based on a hybrid Nyström collocation method using Chebyshev polynomials. It is well known that such an approach produces a dense linear system, which requires $\mathcal {O}(N^{2})$ operations in each step of an iterative solver. In this work, we propose an approach using the recently introduced Interpolated Factored Green Function (IFGF) acceleration strategy to reduce the cost of each iteration to $\mathcal {O}(N\log N)$ . To the best of our knowledge, this article presents the first-ever application of the IFGF method to fully vectorial 3-D Maxwell problems. The Chebyshev-based integral solver and IFGF method are first introduced, followed by the extension of the scalar IFGF to the full-vectorial Maxwell case. Several examples are presented, verifying the $\mathcal {O}(N\log N)$ computational complexity of the approach, including scattering from spheres, complex computer-aided design (CAD) models, and nanophotonic waveguiding devices. In one particular example with more than 6 million unknowns, the accelerated IFGF solver runs $42\times $ faster than the unaccelerated method.