Accelerated 3D Maxwell Integral Equation Solver using the Interpolated Factored Green Function Method

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