A Chebyshev/rational Chebyshev spectral method for the Helmholtz equation in a sector on the surface of a sphere: defeating corner singularities

When the boundaries of a domain meet at an angle, the solutions to an elliptic partial differential equation will usually be singular at the corner. Using the example of the Helmholtz equation on the surface of a sphere in a domain bounded by meridians, we show how corner singularities can be defeated by mapping the corner to infinity. By applying a Chebyshev series in longitude and a rational Chebyshev series in the “Mercator” coordinate, y = arctan h(cos(colatitude)), we obtain an exponential rate of convergence despite the corner singularities.