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Efficient fast multipole method for low-frequency scattering. (English) Zbl 1073.65133
The solution of the Helmholtz and Maxwell equations using integral formulations requires to solve large complex linear systems. A direct solution of those problems using a Gauss elimination is practical only for very small systems with few unknowns. The use of an iterative method such as GMRES can reduce the computational expense. Most of the expense is then computing large complex matrix vector products. The cost can be further reduced by using the fast multipole method which accelerates the matrix vector product. For a linear system of size \(N\), the use of an iterative method combined with the fast multipole method reduces the total expense of the computation to \(N\log N\). There exist two versions of the fast multipole method: one which is based on a multipole expansion of the interaction kernel \(\exp\iota kr/r\) and which was first proposed by V. Rokhlin [ibid. 60, 187–207 (1985; Zbl 0629.65122)] and another based on a plane wave expansion of the kernel, first proposed by W. C. Chew, J. M. Jin, C. C. Lu, E. Michielssen and J. M. M. Song [Fast solution emthods in electromagnetics, IEEE Trans. Antenn. Progag. 45, No. 3, 533–543 (1997)]. In this paper, the authors propose a third approach, the stable plane wave expansion, which has a lower computational expense than the multipole expansion and does not have the accuracy and stability problems of the plane wave expansion. The computational complexity is \(N\log N\) as with the other methods.

MSC:
65N38 Boundary element methods for boundary value problems involving PDEs
78A45 Diffraction, scattering
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35Q60 PDEs in connection with optics and electromagnetic theory
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
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