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.