A simple and complete discrete exterior calculus on general polygonal meshes

Discrete exterior calculus (DEC) offers a coordinate–free discretization of exterior calculus especially suited for computations on curved spaces. We present a new version of DEC on surface meshes formed by general polygons, possibly non–planar and non–convex. Unlike previous approaches in DEC, our framework 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 as it satisfies the Leibniz product rule. Based on the discrete wedge product, we derive a novel primal–to–primal Hodge star operator. Combining these three 'basic operators' we then define new discrete versions of the contraction operator, Lie derivative, codifferential, and Laplace operator. We discuss the numerical convergence of the proposed operators and compare them to existing DEC methods. Finally, we show simple applications of our operators on Helmholtz–Hodge decomposition of vector fields, Laplacian surface fairing, and Lie advection of functions and vector fields on meshes formed by general polygons.