On the consistency of the combinatorial codifferential
This work addresses a long-standing question in discrete differential geometry about the consistency of combinatorial codifferential operators, providing both positive and negative results that clarify the conditions under which consistency holds.
The paper extends Smits's 1991 result on the consistency of the combinatorial codifferential for 1-forms from 2D to arbitrary dimensions under uniform or piecewise uniform triangulations, and provides a counterexample showing inconsistency for Whitney's standard subdivision. It also shows numerically that for 2-forms in 3D, the combinatorial codifferential is inconsistent even for regular subdivisions.
In 1976, Dodziuk and Patodi employed Whitney forms to define a combinatorial codifferential operator on cochains, and they raised the question whether it is consistent in the sense that for a smooth enough differential form the combinatorial codifferential of the associated cochain converges to the exterior codifferential of the form as the triangulation is refined. In 1991, Smits proved this to be the case for the combinatorial codifferential applied to 1-forms in two dimensions under the additional assumption that the initial triangulation is refined in a completely regular fashion, by dividing each triangle into four similar triangles. In this paper we extend Smits's result to arbitrary dimensions, showing that the combinatorial codifferential on 1-forms is consistent if the triangulations are uniform or piecewise uniform in a certain precise sense. We also show that this restriction on the triangulations is needed, giving a counterexample in which a different regular refinement procedure, namely Whitney's standard subdivision, is used. Further, we show by numerical example that for 2-forms in three dimensions, the combinatorial codifferential is not consistent even for the most regular subdivision process.