Spectral Matrix Conditioning in Cylindrical and Spherical Elliptic Equations

In the spectral solution of 3-D Poisson equations in cylindrical and spherical coordinates including the axis or the center, it is convenient to employ radial basis functions that depend on the Fourier wavenumber or on the latitudinal mode. This idea has been adopted by Matsushima and Marcus and by Verkley for planar problems and pursued by the present authors for spherical ones. For the Dirichlet boundary value problem in both geometries, original bases have been introduced built upon Jacobi polynomials which lead to a purely diagonal representation of the radial second-order differential operator of all spectral modes. This note details the origin of such a diagonalization which extends to cylindrical and spherical regions the properties of the Legendre basis introduced by Jie Shen for Cartesian domains. Closed form expressions are derived for the diagonal elements of the stiffness matrices as well as for the elements of the tridiagonal mass matrices occurring in evolutionary problems. Furthermore, the bound on the condition number of the spectral matrices associated with the Helmholtz equation are determined, proving in a rigorous way one of the main advantages of the proposed radial bases.