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The Helmholtz equation arises in the study of electromagnetic radiation,optics, acoustics, etc. In spherical coordinates, its general solution can bewritten as a spherical harmonic series which satisfies the radiation conditionat infinity, ensuring that the wave is outgoing. The boundary condition atinfinity is hard to enforce with a finite element method since a suitableapproximation needs to be made within reasonable distance from scatterers.  Luckily, the Helmholtz equation can be represented as a Lippmann-Schwingerintegral equation which removes the necessity of the boundary approximationsand its Green's function can be expanded as a spherical harmonic series whichleads to our numerical scheme based on spherical harmonic polynomial transform.In this paper, we present an efficient solver for the Helmholtz equation whichcosts $O(N\log N)$ operations, where $N$ is the number of the discretizationpoints. We use the fast spherical harmonic transforms which are originallydeveloped in \cite{suda}. The convergence order of the method is tied to theglobal regularity of the solution. At the lower end, it is second orderaccurate for discontinuous material properties. The order increases withincreasing regularity leading to spectral convergence for globally smoothsolutions.

An efficient solver for volumetric scattering based on fast spherical harmonics transforms