Convergence Analysis of Spectral Collocation Methods for a Singular Differential Equation

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.

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