Chebyshev-Fourier spectral methods in bipolar coordinates

Bipolar coordinates provide an efficient cartography for a variety of geometries: the exterior of two disks or cylinders, a half-plane containing a disk, an eccentric annulus with a small disk offset from the center of an outer boundary that is a large circle, and so on. A pseudospectral method that employs a tensor product basis of Fourier functions in the cyclic coordinate η and Chebyshev polynomials in the quasi-radial coordinate ξ gives easy-to-program spectral accuracy. We show, however, that as the inner disk becomes more and more offset from the center of the outer boundary circle, the grid is increasingly non-uniform, and the rate of exponential convergence increasingly slow. One-dimensional coordinate mappings significantly reduce the non-uniformity. In spite of this non-uniformity, the Chebyshev-Fourier method is quite effective in an idealized model of the wind-driven ocean circulation, resolving both internal and boundary layers. Bipolar coordinates are also a good starting point for solving problems in a domain which is not one of the “bipolar-compatible” domains listed above, but is a sufficiently small perturbation of such. This is illustrated by applying boundary collocation with bipolar harmonics to solve Laplace's equation in a perturbed eccentric annulus in which the disk-shaped island has been replaced by an island bounded by an ellipse. Similarly a perturbed bipolar domain can be mapped to an eccentric annulus by a smooth change of coordinates.