Error estimates in the fast multipole method for scattering problems Part 1: Truncation of the Jacobi-Anger series

¹ Dassault Aviation, 78, quai Marcel Dassault, Cedex 300, 92552 Saint-Cloud Cedex, France, quentin.carayol@dassault-aviation.fr.
² CERFACS, 42 avenue G. Coriolis, 31057 Toulouse, France, Collino@cerfacs.fr.

Received: 1 July 2003

Abstract

We perform a complete study of the truncation error of the Jacobi-Anger series. This series expands every plane wave in terms of spherical harmonics . We consider the truncated series where the summation is performed over the 's satisfying . We prove that if is large enough, the truncated series gives rise to an error lower than ϵ as soon as L satisfies where W is the Lambert function and are pure positive constants. Numerical experiments show that this asymptotic is optimal. Those results are useful to provide sharp estimates for the error in the fast multipole method for scattering computation.