Discrete Poincaré lemma

  • Authors:

  • Mathieu Desbrun;Melvin Leok;Jerrold E. Marsden

  • Affiliations:

  • Department of Computer Science, Caltech, Pasadena, CA;Department of Mathematics, University of Michigan, Ann Arbor, MI;Control and Dynamical Systems, Caltech, Pasadena, CA

  • Venue:
  • Applied Numerical Mathematics - Tenth seminar on and differential-algebraic equations (NUMDIFF-10)
  • Year:
  • 2005

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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: 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 H: 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 R2 and R3 are presented, for which the discrete Poincaré lemma is globally valid.