  Discrete exterior calculus (DEC) offers a coordinate-free discretization of
exterior calculus especially suited for computations on curved spaces. In this
work, we present an extended version of DEC on surface meshes formed by general
polygons that bypasses the need for combinatorial subdivision and does not
involve any dual mesh. At its core, our approach introduces a new polygonal
wedge product that is compatible with the discrete exterior derivative in the
sense that it satisfies the Leibniz product rule. Based on the discrete wedge
product, we then derive a novel primal-to-primal Hodge star operator. Combining
these three `basic operators' we then define new discrete versions of the
contraction operator and Lie derivative, codifferential and Laplace operator.
We discuss the numerical convergence of each one of these proposed operators
and compare them to existing DEC methods. Finally, we show simple applications
of our operators on Helmholtz-Hodge decomposition, Laplacian surface fairing,
and Lie advection of functions and vector fields on meshes formed by general
polygons.
