Multipole translation theory for the three-dimensional Laplace and Helmholtz equations
SIAM Journal on Scientific Computing
Calculation of the addition coefficients in electromagnetic multisphere-scattering theory
Journal of Computational Physics
Fast and Efficient Algorithms in Computational Electromagnetics
Fast and Efficient Algorithms in Computational Electromagnetics
The Fast Multipole Method I: Error Analysis and Asymptotic Complexity
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific Computing
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Fast multipole method for the biharmonic equation in three dimensions
Journal of Computational Physics
Journal of Computational Physics
Efficient FMM accelerated vortex methods in three dimensions via the Lamb-Helmholtz decomposition
Journal of Computational Physics
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We develop a computational method based on the Debye scalar potential representation, which efficiently reduces the solution of Maxwell's equations to the solution of two scalar Helmholtz equations. One of the key contributions of this paper is a theory for the translation of Maxwell solutions using such a representation, since the scalar potential form is not invariant with respect to translations. The translation theory is developed by introducing ''conversion'' operators, which enable the representation of the electric and magnetic vector fields via scalar potentials in an arbitrary reference frame. Advantages of this representation include the fact that only two Helmholtz equations need be solved, and moreover, the divergence free constraints are satisfied automatically by construction. Truncation error bounds are also presented. The availability of a translation theory and error bounds for this representation can find application in methods such as the Fast Multipole Method. For illustration of the use of the representation and translation theory we implemented an algorithm for the simulation of Mie scattering off a system of spherical objects of different sizes and dielectric properties using a variant of the T-matrix method. The resulting system was solved using an iterative method based on GMRES. The results of the computations agree well with previous computational and experimental results.