Vector Identities In Discrete Exterior Calculus

Exterior calculus,an important ingredient of calculus,is principally concerned with calculus on manifolds.During the 20th century,great progress has been made in this field,and it has been widely applied to electromagnetism,fluids dynamics,as well as other fields.With the emergence of computer and the development of the computer science,most continuous problems need to be discreted.The theory of discrete exterior calculus appears in this environment.In comparison with other methods of discretization,the discrete exterior calculus could remain the topological and geometrical features of the continuous problems.Therefore, discrete exterior calculus,serving as a mathematical and computational tool,has revealed its enormous potential in many fields.Because of the essential distinction between smooth manifold and discrete mesh,it does not exist any discrete scheme,which could satisfy all properties of continuous scheme. Different format should be chose according to different research demand.This paper analyses the existing discrete scheme.Using some property in the smooth theory,we describe a new discrete method of operations about exterior calculus.Meanwhile,we proof that these discrete operators,as same as continuous operators,could satisfy some basic vector identities,these vector identities provide a perfect theoretical foundation for mesh processing and vector field editing.