A fast solver for the Helmholtz equation on long, thin structures

Abstract

We present a fast solver for the Helmholtz equation on long, thin structures. It operates on an integral equation formulation of the problem, in which the solution is represented as a superposition of fields generated by sources on the structure (usually on the boundary or boundaries of the structure). It uses a standard iterative solver for linear equations, in conjunction with a novel method for applying the forward matrix, whose computational complexity is O(N), where N is the number of points on which the integral equation is solved. The algorithm is suitable for structures in either two dimensions (2-D) or three dimensions. It does not depend in any great detail on the specifics of the Helmholtz equation, and, thus, is also suitable for similar equations. We demonstrate the algorithm by using it to simulate scattering in 2-D from dielectric structures, using an integral equation formulation constructed using a combination of single-layer and double-layer potentials, yielding a second-kind integral equation. Numerical results show the algorithm to be efficient and accurate.