On the application of the fast multipole method to helmholtz-like problems with complex wavenumber

This paper presents an empirical study of the accuracy of multipole expansions of Helmholtz-like kernels with complex wavenumbers of the form k = (α +i β) ν, with α = 0;±1 and β > 0, which, the paucity of available studies notwithstanding, arise for a wealth of different physical problems. It is suggested that a simple point-wise error indicator can provide an a-priori indication on the number N of terms to be employed in the Gegenbauer addition formula in order to achieve a prescribed accuracy when integrating single layer potentials over surfaces. For β ≥ 1 it is observed that the value of N is independent of b and of the size of the octree cells employed while, for β < 1, simple empirical formulas are proposed yielding the required N in terms of β.