Multi-level fast multipole solution of the scattering problem

In this paper we study the multi-level fast multipole solution of Burton and Miller's hypersingular formulation for the Helmholtz equation in two space dimensions. We provide a complete and rigorous error and complexity analysis for the method. We prove the O(nln2n) computational complexity for the method and provide numerical results to support the theory.

[1]  R. Kress,et al.  Integral equation methods in scattering theory , 1983 .

[2]  R. Coifman,et al.  The fast multipole method for the wave equation: a pedestrian prescription , 1993, IEEE Antennas and Propagation Magazine.

[3]  Weng Cho Chew,et al.  A multilevel algorithm for solving a boundary integral equation of wave scattering , 1994 .

[4]  Vidar Thomée,et al.  Mathematical theory of finite and boundary element methods , 1990 .

[5]  G. F. Miller,et al.  The application of integral equation methods to the numerical solution of some exterior boundary-value problems , 1971, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[6]  Jianxin Zhou,et al.  Boundary element methods , 1992, Computational mathematics and applications.

[7]  S. Amini.  On boundary integral operators for the Laplace and the Helmholtz equations and their discretisations , 1999 .

[8]  Christopher Laubreuche.  A convergence theorem for the fast multipole method for 2 dimensional scattering problems , 1998 .

[9]  Stefan A. Sauter,et al.  Variable Order Panel Clustering , 2000, Computing.

[10]  S. Amini,et al.  Regularization of strongly singular integrals in boundary integral equations , 1996 .

[11]  G. Golub,et al.  Iterative solution of linear systems , 1991, Acta Numerica.

[12]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[13]  Michael A. Epton,et al.  Multipole Translation Theory for the Three-Dimensional Laplace and Helmholtz Equations , 1995, SIAM J. Sci. Comput..

[14]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[15]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[16]  Jussi Rahola,et al.  Diagonal forms of the translation operators in the fast multipole algorithm for scattering problems , 1995 .

[17]  S. Amini,et al.  PRECONDITIONED KRYLOV SUBSPACE METHODS FOR BOUNDARY ELEMENT SOLUTION OF THE HELMHOLTZ EQUATION , 1998 .

[18]  Weng Cho Chew,et al.  A study of wavelets for the solution of electromagnetic integral equations , 1995 .

[19]  A. Zemanian,et al.  Distribution theory and transform analysis , 1966 .

[20]  V. Rokhlin.  Rapid solution of integral equations of classical potential theory , 1985 .

[21]  S. Amini,et al.  Analysis of a Diagonal Form of the Fast Multipole Algorithm for Scattering Theory , 1999 .

[22]  Jianming Jin,et al.  Fast solution methods in electromagnetics , 1997 .

[23]  W. Hackbusch,et al.  On the fast matrix multiplication in the boundary element method by panel clustering , 1989 .

[24]  V. Rokhlin.  Rapid Solution of Integral Equations of Scattering Theory , 1990 .

[25]  S. Amini.  On the choice of the coupling parameter in boundary integral formulations of the exterior acoustic problem , 1990 .