Accurate error estimates in the fast multipole method for electromagnetics

Fourier-Based Fast Multipole Method for the Helmholtz Equation

“…Consequently, a large number of quadrature points along θ 57 are required. 58 We propose to use a variant of the scheme by J. Sarvas in [22] whereby the 59 integration is extended from 0 to 2π and the integrand modified: (−1) n (2n + 1)h (1) n (κ |r 0 |)j n (κ |r|)P n (r ·r 0 ) (2.…”

“…The series converges absolutely and uniformly for |r 0 | ≥ 2 √ 3 |r| and has been studied 120 extensively in [2,6]. 121 Truncating the Gegenbauer series at ℓ and using an integral over the unit sphere, ı n (2n + 1)h (1) n (κ |r 0 |)P n (ŝ ·r 0 ).…”

“…This fails for small |r| when the upper bound is obtained by choosingr ·r 0 such that 166 the oscillation of P n (r·r 0 ) compensates for the oscillation of (−1) n h (1) n (κ |r 0 |)j n (κ |r|).…”

“…This is the 170 scheme we selected for this paper. 171 Carayol and Collino in [1] and [2] present an in-depth analysis of the Jacobi-Anger 172 series and the Gegenbauer series. They find the asymptotic formula…”