  An error in the gravitational force that the source of gravity induces on
itself (a self-force error) violates both the conservation of linear momentum
and the conservation of energy. If such errors are present in a
self-gravitating system and are not sufficiently random to average out, the
obtained numerical solution will become progressively more unphysical with
time: the system will acquire or lose momentum and energy due to numerical
effects. In this paper, we demonstrate how self-force errors can arise in the
case where self-gravity is solved on an adaptively refined mesh when the
refinement is nonuniform. We provide the analytical expression for the
self-force error and numerical examples that demonstrate such self-force errors
in idealized settings. We also show how these errors can be corrected to an
arbitrary order by straightforward addition of correction terms at the
refinement boundaries.
