PDEs in spherical geometries arise in many important areas of application such as meteorology, geophysics, and astrophysics. A fundamental problem for most discretization techniques is that it is impossible to cover a sphere with grids that are both dense and uniform. We begin by describing some numerical approaches designed to address or bypass this problem.

Approximately uniform grids over the sphere. One might start by laying out a coarse, perfectly uniform grid based on one of the five platonic bodies (in particular the icosahedron with 20 equilateral triangular faces), and then carry out subdivisions within each face. Variations on this theme include using grids reminiscent of

• the dimple pattern on golf balls;

• “Buckminster Fullerenes” – or the patterns of carbon atoms in “Buckey balls”; or

• approximations found in biology, such as the pattern of composite eyes in some insects or the silica skeletons of some radiolaria.

Grids of this type can be well suited for low-order FD and FE methods (see e.g. Bunge and Baumgardner 1995), but even their relatively minor irregularities cause considerable algebraic complications in connection with higher-order FD and PS methods.

Spherical (surface) harmonics. These form an infinite set of analytic basis functions with a completely uniform approximation ability over all parts of a sphere. Galerkin techniques are particularly attractive for linear constant-coefficient problems. Equations with variable coefficients and nonlinearities are best handled via (repeated) transformations to and from a grid-based physical representation. Drawbacks include algebraic complexity and lack of very fast transforms.