On the consistency of the combinatorial codifferential

TL;DR

This paper investigates the convergence of the combinatorial codifferential operator to the smooth exterior codifferential, extending previous results to higher dimensions and analyzing the effects of different triangulation refinements.

Contribution

It generalizes the consistency results of the combinatorial codifferential to arbitrary dimensions under specific triangulation conditions.

Findings

Consistency holds for 1-forms in arbitrary dimensions with uniform triangulations.

Counterexample shows inconsistency with Whitney's standard subdivision.

Numerical evidence indicates inconsistency for 2-forms in three dimensions.

Abstract

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.