Spherical geodesic grids offer several attractive features for geophysical fluid dynamics modeling: quasi-isotropic discretizations and quasi-uniform resolution (which eliminates the pole problem of conventional latitude-longitude grids). Modeling fluid flow on such a grid results in various elliptic problems which must be solved efficiently, e.g., the Poisson problem relating streamfunction to vorticity and modified Helmholtz problems with variable coefficients arising in semi-implicit time integration.

Heikes and Randall (1995) introduced a multigrid algorithm for the two-dimensional Poisson problem on a spherical geodesic grid. In this paper we: (1) provide a smoothing analysis for this method and discuss possible algorithmic improvements, (2) extend the algorithm to variable-coefficient problems in two and three dimensions (using an isentropic vertical coordinate), and (3) provide numerical experiments showing performance consistent with the analysis. We also illustrate the use of these multigrid solvers in a three-dimensional dynamical core for a climate model.