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Discrete Poincaré Lemma

Desbrun, Mathieu and Leok, Melvin and Marsden, Jerrold E. (2005) Discrete Poincaré Lemma. Applied Numerical Mathematics, 53 (2-4). pp. 231-248. ISSN 0168-9274. http://resolver.caltech.edu/CaltechAUTHORS:DESanm05

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Abstract

This paper proves a discrete analogue of the Poincar´e 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 Poincar´e 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 Poincar´e lemma is globally valid.


Item Type:Article
Additional Information:Author preprint
Subject Keywords:Discrete geometry, Discrete exterior calculus, Compatible discretizations
Record Number:CaltechAUTHORS:DESanm05
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:DESanm05
Alternative URL:http://dx.doi.org/10.1016/j.apnum.2004.09.035
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:321
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:27 May 2005
Last Modified:26 Dec 2012 08:39

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