Title: Efficient Spectral-Galerkin Methods III. Polar and Cylindrical
       Geometries

Author: Jie Shen

Status: SIAM J. Sci. Comput. Vol. 18, No. 6, 1997

Abstract:

We present in this paper several extremely efficient and accurate
spectral-Galerkin methods for second- and fourth-order equations in
polar and cylindrical geometries. These methods are based on
appropriate variational formulations which incorporate naturally the
pole condition(s).  In particular, the computational complexities of
the Chebyshev-Galerkin method in a disk and of the Chebyshev-Legendre
Galerkin method in a disk or a cylinder are quasi-optimal (optimal up
to a logarithmic term).  As an indication of efficiency, the CPU time
for the Poisson solver on a disk by our Chebyshev-Galerkin method is
only about 70\% of the corresponding finite-difference code in FISHPACK.