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.

离散外部演算（DEC）提供了无坐标的外部演算离散化，特别适用于弯曲空间的计算。在这项工作中，我们提出了由普通多边形形成的表面网格上的DEC的扩展版本，它绕过了组合细分的需求，并且不涉及任何双重网格。从本质上讲，我们的方法引入了一种新的多边形楔形乘积，在满足莱布尼兹乘积法则的意义上，该乘积与离散的外部导数兼容。然后，基于离散楔形积，我们得出一个新颖的，从原始到原始的霍奇星算子。然后结合这三个“基本运算符”，我们定义了收缩运算符和Lie派生，共微分和Laplace运算符的新离散版本。我们讨论了每个拟议的算子的数值收敛性，并将它们与现有的DEC方法进行比较。最后，我们展示了算子在Helmholtz-Hodge分解，Laplacian表面光顺，函数和矢量场在一般多边形形成的网格上的Lie对流的简单应用。