Convergence analysis of pseudospectral collocation methods for a singular differential equation

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Issue Date
2003-01-02Author
Huang, Weizhang
Ma, Heping
Sun, Weizwei
Publisher
Society for Industrial and Applied Mathematics
Type
Article
Article Version
Scholarly/refereed, publisher version
Metadata
Show full item recordAbstract
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.
Description
This is the published version, also available here: http://dx.doi.org/10.1137/S0036142902381024.
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Citation
Huang, Weizhang., Ma, Heping., Sun, Weiwei. "Convergence analysis of pseudospectral collocation methods for a singular differential equation." SIAM J. Numer. Anal., 41(6), 2333–2349. (17 pages). http://dx.doi.org/10.1137/S0036142902381024.
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