On the consistency of the combinatorial codifferential

In 1976, Dodziuk and Patodi employed Whitney forms to dene a combinatorial codierential operator on cochains, and they raised the ques- tion whether it is consistent in the sense that for a smooth enough dierential form the combinatorial codierential of the associated cochain converges to the exterior codierential of the form as the triangulation is rened. In 1991, Smits proved this to be the case for the combinatorial codierential applied to 1-forms in two dimensions under the additional assumption that the ini- tial triangulation is rened 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 codierential on 1-forms is consistent if the triangulations are uniform or piecewise uniform in a cer- tain precise sense. We also show that this restriction on the triangulations is needed, giving a counterexample in which a dierent regular renement pro- cedure, namely Whitney's standard subdivision, is used. Further, we show by numerical example that for 2-forms in three dimensions, the combinatorial codierential is not consistent even for the most regular subdivision process.