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.

翻译：离散外微积分（DEC）提供了一种无坐标的离散外微积分方法，特别适用于弯曲空间上的计算。本文提出了一种在一般多边形曲面网格上扩展的DEC版本，该版本无需组合细分且不涉及任何对偶网格。其核心在于，我们引入了一种与离散外导数兼容的新型多边形楔积，使其满足莱布尼茨乘积法则。基于该离散楔积，我们进一步推导出一种新颖的原初到原初霍奇星算子。通过结合这三个“基本算子”，我们定义了收缩算子、李导数、余微分和拉普拉斯算子的新离散版本。我们讨论了每个算子的数值收敛性，并将其与现有DEC方法进行了比较。最后，我们展示了这些算子在亥姆霍兹-霍奇分解、拉普拉斯曲面光顺以及一般多边形网格上函数与向量场的李平流中的简单应用。