Generic polyhedron grid generation for solving partial differential equations on spherical surfaces

A new method for generating a numerical grid on a spherical surface is presented. This method allows the grid to be based on several different regular polyhedrons (including octahedron, cube, icosahedron, and rhombic dodecahedron). The type of polyhedron on which the grid is based can be changed by altering only a few input parameters. Each polygon face can then be subdivided using a mapping technique that is described. An advantage of this new grid is that it gives increased flexibility in terms of the total number of nodes in the system. It also makes comparison between different numerical grids easier and simplifies the transfer of code/data between numerical simulators with different grids. This generic grid is then used to solve Poissonʹs equation on a spherical surface using a spectral element implementation for a range of actual grids. The generic grid allows us to quickly compare the actual grids and illustrates its utility.