Efficient Spectral-Galerkin Methods III: Polar and Cylindrical Geometries

We present in this paper several extremely efficient and accurate spectral-Galerkin methods for secondand 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 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.