A discrete wedge product on general polygonal meshes

Abstract

Discrete exterior calculus offers a coordinate--free discretization of exterior calculus especially suited for computations on meshes over curved manifolds. The discretization of the wedge product, that would be compatible with discrete exterior derivative, has been a challenging task. The cup product of cochains is traditionally considered to be an appropriate discrete wedge product. However, only the case of pure triangle or pure quadrilateral surface meshes has been studied thoroughly. In this work, we extend this tradition to general polygonal meshes. Specifically, we present explicit formulas for calculation of a cup/discrete wedge product on surface meshes that correspond to 2--dimensional pseudomanifolds, whose 2--dimensional faces are any simple polygons. We rigorously prove that the proposed product satisfies the definition of an abstract cup product; notably, we show that the product is compatible with the discrete exterior derivative in the sense that it satisfies the Leibniz product rule. Furthermore, the product is associative on the cohomology level, but not on the cochain level in general. We analyze the lack of associativity on the cochain level and prove that the error tends to zero under refinement of the mesh. We thus argue that the proposed product is an appropriate discretization of the wedge product of differential forms on general polygonal meshes.