ON THE CONSISTENCY OF THE COMBINATORIAL CODIFFERENTIAL

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 the result of Smits 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.

1976年，多齐乌克（Dodziuk）和帕托迪（Patodi）利用惠特尼形式（Whitney forms）在上链（cochains）上定义了一个组合余微分算子，并提出了一个问题：从某种意义上说，它是否具有一致性，即对于一个足够光滑的微分形式，当三角剖分细化时，相关上链的组合余微分是否收敛到该形式的外余微分。1991年，斯米茨（Smits）在额外假设初始三角剖分以一种完全规则的方式细化（即通过将每个三角形分成四个相似三角形）的情况下，证明了对于二维的1 - 形式应用组合余微分时是这种情况。在本文中，我们将斯米茨的结果推广到任意维度，表明如果三角剖分在某种精确意义下是均匀的或分段均匀的，那么1 - 形式上的组合余微分是一致的。我们还表明对三角剖分的这种限制是必要的，并给出了一个反例，其中使用了一种不同的规则细化过程，即惠特尼标准细分。此外，我们通过数值例子表明，对于三维的2 - 形式，即使对于最规则的细分过程，组合余微分也不一致。