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In this paper, we develop fast multipole methods for 3D Helmholtz kernel inlayered media. Two algorithms based on different forms of Taylor expansion oflayered media Green's function are developed. A key component of the firstalgorithm is an efficient algorithm based on discrete complex imageapproximation and recurrence formula for the calculation of the layered mediaGreen's function and its derivatives, which are given in terms of Sommerfeldintegrals. The second algorithm uses symmetric derivatives in the Taylorexpansion to reduce the size of precomputed tables for the derivatives oflayered media Green's function. Numerical tests in layered media have validatedthe accuracy and O(N) complexity of the proposed algorithms.

Taylor expansion based fast Multipole Methods for 3-D Helmholtz equations in Layered Media