Convergence Analysis of Spectral Collocation Methods for a Singular Differential Equation

Authors:
Weizhang Huang;Heping Ma;Weiwei Sun
Affiliations:
-;-;-
Venue:
SIAM Journal on Numerical Analysis
Year:
2003

Citing 0
Cited 3

Efficient spectral-Galerkin methods for polar and cylindrical geometries

Applied Numerical Mathematics
A Legendre Petrov-Galerkin method for fourth-order differential equations

Computers & Mathematics with Applications
Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan-Shepp ridge polynomials, Chebyshev-Fourier Series, cylindrical Robert functions, Bessel-Fourier expansions, square-to-disk conformal mapping and radial basis functions

Journal of Computational Physics

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Abstract

Solutions of partial differential equations with coordinate singularities often have special behavior near the singularities, which forces them to be smooth. Special treatment for these coordinate singularities is necessary in spectral approximations in order to avoid degradation of accuracy and efficiency. It has been observed numerically in the past that, for a scheme to attain high accuracy, it is unnecessary to impose all the pole conditions, the constraints representing the special solution behavior near singularities. In this paper we provide a theoretical justification for this observation. Specifically, we consider an existing approach, which uses a pole condition as the boundary condition at a singularity and solves the reformulated boundary value problem with a commonly used Gauss--Lobatto collocation scheme. Spectral convergence of the Legendre and Chebyshev collocation methods is obtained for a singular differential equation arising from polar and cylindrical geometries.