Discrete Exterior Calculus (DEC) aims to offer a discrete counterpart of
the exterior calculus of differential forms, which is a geometry–based
calculus independent of a coordinate system. We target general polygonal
meshes and introduce a discrete version of the wedge
product on them, called the polygonal cup product, that is fully compatible
with the discrete exterior derivative in the sense that the Leibniz product
law is maintained valid. Our cup product shares with the wedge product
defining properties such as the bilinearity, skew-commutativity, and the
Leibniz rule. The wedge product is also associative, but our cup product is
associative only on cohomology, this limitation is common for all discrete
versions of the wedge product we know. On the other hand, our discrete
wedge product is metric–independent such as its continuous counterpart,
which is a property not satisfied by many other versions of the product of
discrete forms.

Keywords: Discrete exterior calculus, cup product, discrete wedge product.