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Spectral Matrix Conditioning in Cylindrical and Spherical Elliptic Equations

Published online by Cambridge University Press:  28 May 2015

F. Auteri  and
L. Quartapelle
F. Auteri*
Affiliation:
Politecnico di Milano, Dipartimento di Ingegneria Aerospaziale, Via La Masa 34, 20156 Milano, Italy
L. Quartapelle
Affiliation:
Politecnico di Milano, Dipartimento di Ingegneria Aerospaziale, Via La Masa 34, 20156 Milano, Italy
*
Corresponding author.Email address:auteri@aero.polimi.it
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Abstract

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.

Keywords

65M1078A48Poisson equationcylindrical and spherical coordinatesFourier expansionspherical harmonicsJacobi polynomialsspectral methodscondition number
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 2 , May 2011 , pp. 113 - 141
Copyright
Copyright © Global Science Press Limited 2011

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