SPECTRAL COLLOCATION METHODS AND POLAR COORDINATE SINGULARITIES

This paper considers the numerical solution of elliptic differential equations on the unit disk. Using polar coordinates, the disk is mapped onto a rectangle. The resulting transformed problem is solved by a method related to collocation. Since the origin is a coordinate singularity, some natural trial functions are singular there and a special technique is applied to use zero as a collocation point. For Poisson and Helmholtz equations, a fast algorithm with an operation count of O(N2 log N) is presented. Numerical results show the different stability and convergence properties of the algorithms. © 1991.