Properties of Laplace Operators for Tetrahedral Meshes

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.