Spectral Matrix Conditioning in Cylindrical and Spherical Elliptic Equations
F. Auteri 1*, L. Quartapelle 11 Politecnico di Milano, Dipartimento di Ingegneria Aerospaziale, Via La Masa 34, 20156 Milano, Italy.
Received 31 July 2010; Accepted (in revised version) 16 November 2010
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.AMS subject classifications: 65M10, 78A48
Key words: Poisson equation, cylindrical and spherical coordinates, Fourier expansion, spherical harmonics, Jacobi polynomials, spectral methods, condition number.
Email: email@example.com (F. Auteri)