  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.
