Discrete Laplacians for triangle meshes are a fundamental tool in geometry processing. The so‐called cotan Laplacian is widely used since it preserves several important properties of its smooth counterpart. It can be derived from different principles: either considering the piecewise linear nature of the primal elements or associating values to the dual vertices. Both approaches lead to the same operator in the two‐dimensional setting. In contrast, for tetrahedral meshes, only the primal construction is reminiscent of the cotan weights, involving dihedral angles. We provide explicit formulas for the lesser‐known dual construction. In both cases, the weights can be computed by adding the contributions of individual tetrahedra to an edge. The resulting two different discrete Laplacians for tetrahedral meshes only retain some of the properties of their two‐dimensional counterpart. In particular, while both constructions have linear precision, only the primal construction is positive semi‐definite and only the dual construction generates positive weights and provides a maximum principle for Delaunay meshes. We perform a range of numerical experiments that highlight the benefits and limitations of the two constructions for different problems and meshes.

中文翻译：

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四面体网格的拉普拉斯算子的性质

三角形网格的离散拉普拉斯算子是几何处理的基本工具。所谓的 cotan Laplacian 被广泛使用，因为它保留了其平滑对应物的几个重要属性。它可以源自不同的原则：要么考虑原始元素的分段线性性质，要么将值与对偶顶点相关联。两种方法都导致二维设置中的相同算子。相比之下，对于四面体网格，只有原始构造会让人联想到 cotan 权重，涉及二面角。我们为鲜为人知的对偶结构提供了明确的公式。在这两种情况下，权重都可以通过将单个四面体的贡献添加到边缘来计算。由此产生的用于四面体网格的两个不同的离散拉普拉斯算子仅保留了其二维对应物的一些属性。特别是，虽然两种构造都具有线性精度，但只有原始构造是半正定的，并且只有对偶构造产生正权重，并为 Delaunay 网格提供了最大值原理。我们进行了一系列数值实验，突出了两种结构对于不同问题和网格的优点和局限性。