Efficient spectral-Galerkin methods for polar and cylindrical geometries

A new spectral-Galerkin approach for solving the Poisson-type equation in polar geometry is introduced and analyzed. The pole singularity is treated naturally through an appropriate variational formulation. Clustering of collocation points near the pole, a problem common to the spectral-Galerkin algorithms in the literature, is prevented through a change of variable in the radial direction. The method is very efficient and gives spectral accuracy, and can be easily adopted to solve problems in cylindrical geometries and with general boundary conditions. Boundary lifting of general inhomogeneous boundary conditions is also addressed.