The multilevel fast multipole algorithm (MLFMA) is used in computing acoustic and electromagnetic fields with integral equation The multilevel fast multipole algorithm (MLFMA) is used in computing acoustic and electromagnetic fields with integral equation methods. The traditional MLFMA, however, suffers from a low-frequency breakdown that effectively limits the minimum division cube side length to approximately one wavelength. To overcome this low-frequency breakdown and get a broadband MLFMA, we propose an efficient and relatively straightforward implementation of the field translations based on the spectral representation of the Green's function. As an alternative we also consider the so called uniform MLFMA, which has a lower computational cost but limited accuracy. We consider the essential implementation details and finally provide numerical examples to demonstrate the error controllability of the translations.
2. Chew, W. C., J. M. Jin, E. Michielssen, and J. M. Song (eds.), Fast and Efficient Algorithms in Computational Electromagnetics, Artech House, 2001.
3. Rokhlin, V., "Diagonal form of translation operators for the Helmholtz equation in three dimensions," Applied and Computational Harmonic Analysis, Vol. 1, No. 12, 82-93, 1993.
4. Coifman, R., V. Rokhlin, and S. Wandzura, "The fast multipole method for the wave equation: a pedestrian prescription," IEEE Ant. Prop. Mag., Vol. 35, No. 6, 7-12, 1993.
5. Michielssen, E. and W. C. Chew, "Fast steepest descent path algorithm for analyzing scattering from two-dimensional ob jects," Radio Science, Vol. 31, No. 10, 1215-1224, 1996.
6. Greengard, L., J. Huang, V. Rokhlin, and S. Wadzura, "Accelerating fast multipole methods for the Helmholtz equation at low frequencies," IEEE Comput. Sci. Eng., Vol. 5, No. 9, 32-38, 1998.
7. Jiang, L. J. and W. C. Chew, "Low-frequency fast inhomogeneous plane-wave algorithm (LF-FIPWA)," Microwave Opt. Tech. Lett., Vol. 40, No. 1, 117-122, 2004.
8. Darve, E. and P. Havé, "A fast multipole method for Maxwell equations stable at all frequencies," Phil. Trans. R. Soc. Lond. A, Vol. 362, No. 3, 603-628, 2004.
9. Yarvin, N. and V. Rokhlin, "Generalized Gaussian quadratures and singular value decompositions of integral operators," SIAM J. Sci. Comput., Vol. 20, No. 2, 699-718, 1998.
10. Xuan, L., A. Zhu, R. J. Adams, and S. D. Gedney, "A broadband multilevel fast multipole algorithm," IEEE AP-S International Symposium and USNC/URSI National Radio Science Meeting, Vol. 2, No. 6, 1195-1198, 2004.
11. Hastriter, M. L., S. Ohnuki, and W. C. Chew, "Error control of the translation operator in 3D MLFMA," Microwave Opt. Tech. Lett., Vol. 37, No. 5, 184-188, 2003.
12. Sarvas, J., "Performing interpolation and anterpolation entirely by fast fourier transform in the 3D multilevel fast multipole algorithm," SIAM J. Numer. Anal., Vol. 41, No. 6, 2180-2196, 2003.
13. Bucci, O. M. and G. Franceschetti, "On the spatial bandwidth of scattered fields," IEEE Trans. Antennas Propagat., Vol. AP-35, No. 12, 1445-1455, 1987.
14. Kress, R., Linear Integral Equations, Vol. 82, Applied Mathematical Sciences, second ed., Vol. 82, Springer-Verlag, New York, 1999.