Towards global SEM mantle convection simulations on polyhedral-based grids

A novel approach to the construction of three-dimensional grids in spherical geometries is described. The grids are based on a range of underlying regular polyhedra. The faces of the polyhedra are arranged into a number of diamonds in the latitudinal and longitudinal directions to facilitate the creation of a structured mesh. Each polyhedron is completely characterized in terms of the arrangement of diamonds. The approach is shown to be very flexible in terms of the meshes that can be generated. It also aids comparisons between the grids used in many mantle convection studies. A spectral element discretization of Poisson's equation is performed to demonstrate the efficacy of the grid generation technique. Rapidly convergent approximations are obtained that demonstrate that fewer degrees of freedom are required to obtain a desired level of accuracy compared with low-order finite element approximations.