Title: On the consistency of the combinatorial codifferential
Authors: Douglas N. Arnold, Richard S. Falk, Johnny Guzman, and
Gantumur Tsogtgerel
Source: Trans. Amer. Math. Soc. 366 (2014), no. 10, pp. 5487-5502.
Status: Published
Abstract: In 1976, Dodziuk and Patodi employed Whitney forms to define a
combinatoral 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.