Abstract
In this work we introduce an accurate definition of the curl operator on weighted networks that completes the discrete vector calculus developed by the authors. This allows us to define the circulation of a vector field along a curve and to characterize the conservative fields. In addition, we obtain an adequate discrete version of the De Rham cohomology of a compact manifold, giving in particular discrete analogues of the Poincar? and Hodge's decomposition theorems.
Highlights
The discrete vector calculus theory is a very fruitful area of work in many mathematical branches for its intrinsic interest and for its applications, [1, 2, 5, 8, 10, 12, 14]
The last approach deals with the mesh as the only existent space and the discrete vector calculus is described through Algebraic Topology tools since the geometric realization of the mesh is a unidimensional CW-complex, [1, 4, 9, 10, 14]
Our work falls within the last ambit but, instead of importing the tools from Algebraic Topology, we construct the discrete vector calculus from the graph structure itself following the guidelines of Differential Geometry, see [2, 3]
Summary
The discrete vector calculus theory is a very fruitful area of work in many mathematical branches for its intrinsic interest and for its applications, [1, 2, 5, 8, 10, 12, 14]. Our work falls within the last ambit but, instead of importing the tools from Algebraic Topology, we construct the discrete vector calculus from the graph structure itself following the guidelines of Differential Geometry, see [2, 3]. In this work we complete our discrete vector calculus on weighted networks by introducing the curl operator. Our approach is based on the consideration of general vector fields whereas other authors consider only symmetric or antisymmetric fields, see [1, 4, 6] They can not consider the curl operator and the Hodge Laplacian and Hodge’s decomposition do not contain all the possible information about the network. We end with the application of our developments to purely resistive networks and to the study of difference schemes on n-dimensional uniform grids
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