Discrete Poincaré lemma

  • Authors:

  • Mathieu Desbrun;Melvin Leok;Jerrold E. Marsden

  • Affiliations:

  • 256-80, Department of Computer Science, Caltech, Pasadena, CA 91125, USA;Department of Mathematics, University of Michigan, 525 East University Ave., Ann Arbor, MI 48109, USA;107-81, Control and Dynamical Systems, Caltech, Pasadena, CA 91125, USA

  • Venue:
  • Applied Numerical Mathematics
  • Year:
  • 2005

Quantified Score

Hi-index 0.00

Visualization

Abstract

This paper proves a discrete analogue of the Poincare 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)-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)-C^k^-^1(K) can be shown to be a homotopy operator, and this yields the discrete Poincare 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 R^2 and R^3 are presented, for which the discrete Poincare lemma is globally valid.