Jacobson, AlecHu, RuizhenLefebvre, Sylvain2024-06-202024-06-2020241467-8659https://doi.org/10.1111/cgf.15133https://diglib.eg.org/handle/10.1111/cgf15133Constructing well-behaved Laplacian and mass matrices is essential for tetrahedral mesh processing. Unfortunately, the de facto standard linear finite elements exhibit bias on tetrahedralized regular grids, motivating the development of finite-volume methods. In this paper, we place existing methods into a common construction, showing how their differences amount to the choice of simplex centers. These choices lead to satisfaction or breakdown of important properties: continuity with respect to vertex positions, positive semi-definiteness of the implied Dirichlet energy, positivity of the mass matrix, and unbiased-ness on regular grids. Based on this analysis, we propose a new method for constructing dual-volumes which explicitly satisfy all of these properties via convex optimization.10.1111/cgf.151339 pages