On the application of the Fast Multipole Method to Helmholtz-like problems with complex wavenumber

Abstract

This paper presents an empirical study of the accuracy of multipole expansions of Helmholtz-like kernels with complex wavenumbers of the form $k=(\alpha+\rmi\beta)\vartheta$, with $\alpha=0,\pm1$ and $\beta>0$, 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 $\beta\geq 1$ it is observed that the value of $N$ is independent of $\beta$ and of the size of the octree cells employed while, for $\beta<1$, simple empirical formulas are proposed yielding the required $N$ in terms of $\beta$.