Approximation based methods, such as the cubetree algorithm, have proved to be significantly faster than traditional methods for complex force evaluations near small irregular bodies. Such methods also hold the promise of simplifying the inclusion of experimental data to update the force model. However, the cubetree algorithm does not preserve intrinsic properties of gravitational force such as continuity, divergence freedom or exactness. These properties may be needed for trajectory optimization, for the use of geometric (e.g. symplectic) integrators for long term propagation and for other trajectory design problems. This paper presents several adaptive schemes preserving global continuity, exactness or divergence-freedom and discusses the difficulties involved in preserving all of these properties globally.