Discrete Poincaré lemma Original Research Article

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, View the MathML sourcep:Ck(K)→Ck+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 View the MathML sourceH:Ck(K)→Ck−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 R2R2 and R3R3 are presented, for which the discrete Poincaré lemma is globally valid.