TY - JOUR
T1 - Spectral Matrix Conditioning in Cylindrical and Spherical Elliptic Equations
AU - F. Auteri & L. Quartapelle
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 2
SP - 113
EP - 141
PY - 2011
DA - 2011/04
SN - 4
DO - http://doi.org/10.4208/nmtma.2011.42s.1
UR - https://global-sci.org/intro/article_detail/nmtma/5961.html
KW - Poisson equation, cylindrical and spherical coordinates, Fourier expansion, spherical harmonics, Jacobi polynomials, spectral methods, condition number.
AB - <p style="text-align: justify;">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.</p>