Discrete exterior calculus offers a coordinate--free discretization of exterior calculus especially suited for computations on meshes over curved manifolds. The discretization of the wedge product, that would be compatible with discrete exterior derivative, has been a challenging task. The cup product of cochains is traditionally considered to be an appropriate discrete wedge product. However, only the case of pure triangle or pure quadrilateral surface meshes has been studied thoroughly. In this work, we extend this tradition to general polygonal meshes. Specifically, we present explicit formulas for calculation of a cup/discrete wedge product on surface meshes that correspond to 2--dimensional pseudomanifolds, whose 2--dimensional faces are any simple polygons. We rigorously prove that the proposed product satisfies the definition of an abstract cup product; notably, we show that the product is compatible with the discrete exterior derivative in the sense that it satisfies the Leibniz product rule. Furthermore, the product is associative on the cohomology level, but not on the cochain level in general. We analyze the lack of associativity on the cochain level and prove that the error tends to zero under refinement of the mesh. We thus argue that the proposed product is an appropriate discretization of the wedge product of differential forms on general polygonal meshes.

翻译：离散外微积分提供了一种无坐标的外微分离散化方法，特别适用于曲流形上的网格计算。如何构造与离散外导数相容的楔积离散化，一直是一项具有挑战性的任务。上链的杯积传统上被视为合适的离散楔积，但此前仅有纯三角形或纯四边形曲面网格得到了深入研究。本研究将这一传统推广至一般多边形网格：我们针对二维伪流形（其二维面为任意简单多边形）的曲面网格，给出了杯积/离散楔积的显式计算公式。我们严格证明了所提出的乘积满足抽象杯积的定义，尤其展示了该乘积与离散外导数相容——即满足莱布尼茨乘积法则。此外，该乘积在上同调层次具有结合性，但在一般上链层次不具有结合性。我们分析了上链层次结合性缺失的原因，并证明在网格细化条件下该误差趋近于零。因此，我们论证所提出的乘积是一般多边形网格上微分形式楔积的合适离散化。