Gravity from surface triangulation: convergence acceleration with nested grids

The determination of the gravitational potential by the polyhedral method is revisited in the case where the surface of a body is composed of triangular facets. Based upon six test-shapes of astrophysical interest (sphere, spheroid, triaxial, lemon-shape, dumbell and torus) projected on nested grids, we verify that the convergence toward reference values is second-order in the step size of the grid, inside the body, at the surface and outside. We then show that the accuracy or computing time can be drastically enhanced by implementing the Repeated Richardson Extrapolation. This technique is especially efficient when the body's surface is smooth enough, and is therefore well adapted to the theory of figures (single and multi-layer fluids) and to dynamical studies (test-particle and mutual interactions), which require a large number of field evaluations. For real objects like asteroids that have very irregular terrains at small scales, the gain is modest. In that context, we estimate the discretization level beyond which the typical error in potential values due to altimetric uncertainties dominates over the contribution of sub-grid cavities and bumps. For bodies close to spherical, the criterion reads $T \gtrsim \frac{64 D}{3 \lambda},$ where $D$ is the diameter of the body, $\lambda$ the typical shape error and $T$ the number of triangular facets involved. The case of 433 Eros is considered as an example.