文档介绍：DISCRETE EXTERIOR CALCULUS
MATHIEU DESBRUN, ANIL N. HIRANI, MELVIN LEOK, AND JERROLD E. MARSDEN
Abstract. We present a theory and applications of discrete exterior calculus on plexes of
arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not
only discrete differential forms but also discrete vector fields and the operators acting on these objects. This
allows us to address the various interactions between forms and vector fields (such as Lie derivatives) which
are important in applications. Previous attempts at discrete exterior calculus have addressed only differential
forms. We also introduce the notion of a circumcentric dual of a plex. The importance of dual
complexes in this field has been well understood, but previous researchers have used barycentric subdivision
or barycentric duals. We show that the use of circumcentric duals is crucial in arriving at a theory of discrete
exterior calculus that admits both vector fields and forms.
Contents
1. Introduction 1
2. History and Previous Work 4
3. Primal plex and Dual plex 4
4. Local and Global Embeddings 10
5. Differential Forms and Exterior Derivative 12
6. Hodge Star and Codifferential 14
7. Maps between 1-Forms and Vector Fields 15
8. Wedge Product 17
9. Divergence and Laplace–Beltrami 22
10. Contraction and Lie Derivative 24
11. Discrete Poincar´eLemma 27
12. Discrete Variational Mechanics and DEC 37
13. Extensions to Dynamic Problems 44
. Groupoid Interpretation of Discrete Variational Mechanics 44
. Discrete Diffeomorphisms and Discrete Flows 45
. Push-Forward and Pull-Back of Discrete Vector Fields and Discrete Forms 47
14. Remeshing Cochains and Multigrid Extensions 49
15. Conclusions and Future Work 50
References 51
1. Introduction
This work presents a theory of discrete exterior calculus (DEC) motivated by potential applications
putational methods for field theories such as elasticity, flu