Spectral collocation schemes on the unit disc

Two spectral collocation schemes on the unit disc are presented. The first one is based on the mapping of Gordon and Hall. Here the unit square is directly mapped onto the unit disc by means of an interpolation technique. Unlike other Poisson solvers on the unit disc no polar coordinates are involved. Hence the usual problems with the singularity of polar coordinates are avoided. This is also shown for more complex geometries. The second method is based on a diameter approach where the collocation nodes are no more clustering in the center. Numerical results are presented which demonstrate the high accuracy of our new spectral collocation schemes.