POLE CONDITION FOR SINGULAR PROBLEMS - THE PSEUDOSPECTRAL APPROXIMATION

This paper deals with the pseudospectral solution of differential equations with coordinate singularities such as those which describe situations in spherical or cylindrical geometries. We use the differential equation, together with a smoothness assumption on the solution, to construct “pole conditions.” The pole conditions, which are straightforward and easily implemented, serve as numerical boundary conditions at the coordinate singularity. Standard pseudospectral methods, including fast transformation techniques, can then be applied to the singular problem. The method is illustrated using the eigenvalue problem of Bessel’s equation and a Poisson equation on the unit disk. Numerical results show that spectral convergence is achieved. © 1993 by Academic Press, Inc.