Geometric triangulations and discrete Laplacians on manifolds: an update (Q6579117)

The Laplacian operator \(\Delta\), the fundamental differential operator, can be defined in many geometric spaces. The paper id devoted to the definition of the Laplacian on a general Euclidean structure called as the duality triangulation. This structure allows to measure distance between points and volumes of simplices and to describe a geometric dual cell decomposition and the volume of dual cells. A comprehensive introduction to Euclidean structures is provided, by recalling the definitions of weighted and Thurston triangulations and the introduction of dual triangulations (including high quality graphs and figures which are particularly helpful for the non-specialist). Then, the definition and properties of the Laplacian on these sturctures are proved. Of exceptional interest is the paragraph devoted to the study of the relavant Laplace, Poisson and heat equations. Generalizations in the context of discrete Riemannian manifolds are also discussed in the last sections of the paper.\N\NThe paper will be a valuable resource for the subject, highlighting the potential for important applications in various physical and engineering contexts.