Spectral collocation methods and polar coordinate singularities

Spectral collocation schemes on the unit disc

“…For our method we observe the well known high spectral accuracy. This is demonstrated by numerical results which are compared to the results of Chen et al [4], Eisen et al [5], Huang et al [16] and Shen [21]. Only in cases where the solution is explicitly given in r (e.g., u ¼ r 4 ) the treatment with polar coordinates yields better results.…”

“…In the previous spectral literature the Poisson equation is transformed into polar coordinates and then solved by means of a combined Chebyshev (or Legendre) and Fourier expansion. Here we refer to the existing papers in [1,4,5,16,18,19,21]. Unfortunately, this transform leads to a coordinate singularity along the axis at the center r ¼ 0.…”

“…0. This has been discussed in detail by several authors [5,16,21]. An alternative is the use of Gauss-Radau collocation nodes which exclude the center r ¼ 0.…”

Spectral radial basis functions for full sphere computations

“…Firstly, we follow [3] and compute the monomial expansion of the truncated Chebyshev series approximation to the solution of Bessel's equation of order l = 21:…”

“…where the rightmost equation expresses the regularity condition and g n (r) is necessarily itself both even and smooth [1][2][3]. Similarly in the spherical case, an expansion in terms of spherical harmonics Y m l ðcos hÞe im/ where (h, /) are respectively colatitude and longitude implies that the multiplying radial function must be of the form r l g lm (r).…”