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**Fast integral equation solver for Maxwell’s equations in layered media with FMM for Bessel functions.**
*(English)*
Zbl 1306.78010

Summary: The paper presents a new fast integral equation solver for Maxwell’s equations in 3-D layered media. First, the spectral domain dyadic Green’s function is derived, and the 0-th and the 1-st order Hankel transforms or Sommerfeld-type integrals are used to recover all components of the dyadic Green’s function in real space. The Hankel transforms are performed with the adaptive generalized Gaussian quadrature points and window functions to minimize the computational cost. Subsequently, a fast integral equation solver with \(O(N_z^2N_xN_y\log(N_xN_y))\) in layered media is developed by rewriting the layered media integral operator in terms of Hankel transforms and using the new fast multipole method for the \(n\)-th order Bessel function in 2-D. Computational cost and parallel efficiency of the new algorithm are presented.

### MSC:

78M16 | Multipole methods applied to problems in optics and electromagnetic theory |

65F10 | Iterative numerical methods for linear systems |

65R20 | Numerical methods for integral equations |

65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

65T50 | Numerical methods for discrete and fast Fourier transforms |

65N80 | Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs |

78A48 | Composite media; random media in optics and electromagnetic theory |

65D30 | Numerical integration |

### Keywords:

Maxwell’s equations; Helmholtz equation; layered media; Green’s function; fast multipole method### Software:

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\textit{M. H. Cho} and \textit{W. Cai}, Sci. China, Math. 56, No. 12, 2561--2570 (2013; Zbl 1306.78010)

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