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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. doi:10.1016/j.apnum.2004.09.035.

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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
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URLURL TypeDescription
Desbrun, Mathieu0000-0003-3424-6079
Additional Information:Author preprint
Subject Keywords:Discrete geometry, Discrete exterior calculus, Compatible discretizations
Issue or Number:2-4
Record Number:CaltechAUTHORS:DESanm05
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:321
Deposited By: Archive Administrator
Deposited On:27 May 2005
Last Modified:08 Nov 2021 19:01

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