  An algorithm for the generation of non-uniform unstructured grids on
ellipsoidal geometries is described. This technique is designed to generate
high quality triangular and polygonal meshes appropriate for general
circulation modelling on the sphere, including applications to atmospheric and
ocean simulation, and numerical weather predication. Using a recently developed
Frontal-Delaunay-refinement technique, a method for the construction of
high-quality unstructured ellipsoidal Delaunay triangulations is introduced. A
dual polygonal grid, derived from the associated Voronoi diagram, is also
optionally generated as a by-product. Compared to existing techniques, it is
shown that the Frontal-Delaunay approach typically produces grids with
near-optimal element quality and smooth grading characteristics, while imposing
relatively low computational expense. Initial results are presented for a
selection of uniform and non-uniform ellipsoidal grids appropriate for
large-scale geophysical applications. The use of user-defined mesh-sizing
functions to generate smoothly graded, non-uniform grids is discussed.
