On the application of the Fast Multipole Method to Helmholtz-like problems with complex wavenumber
A. Frangi; and M. Bonnet; 

doi:10.3970/cmes.2010.058.271
Source CMES: Computer Modeling in Engineering & Sciences, Vol. 58, No. 3, pp. 271-296, 2010
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Keywords Fast Multipole Method, Helmholtz problem, complex wavenumber, Gegenbauer addition theorem
Abstract This paper presents an empirical study of the accuracy of multipole expansions of Helmholtz-like kernels with complex wavenumbers of the form k=(a+ib)J, witha= 0,±1 andb> 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. Forb³1 it is observed that the value of N is independent ofband of the size of the octree cells employed while, forb< 1, simple empirical formulas are proposed yielding the required N in terms ofb.
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