Discrete Poincaré lemma

Desbrun, M., M. Leok, and J. E Marsden

Appl. Numer. Math., 53, (2005), 231-248

Abstract:

This paper proves a discrete analogue of the Poincaré lemma in the context of a discrete exterior calculus based on simplicial cochains. The proof requires the construction of a generalized cone operator, p : C_k(K) $\rightarrow$ C_{k + 1}(K), as the geometric cone of a simplex cannot, in general, be interpreted as a chain in the simplicial complex. The corresponding cocone operator H : C^k(K) $\rightarrow$ C^{k - 1}(K) can be shown to be a homotopy operator, and this yields the discrete Poincaré lemma.

The generalized cone operator is a combinatorial operator that can be constructed for any simplicial complex that can be grown by a process of local augmentation. In particular, regular triangulations and tetrahedralizations of $\mathbb {R}$² and $\mathbb {R}$³ are presented, for which the discrete Poincaré lemma is globally valid.