Discrete Laplacians for General Polygonal and Polyhedral Meshes.

The Laplace-Beltrami operator is one of the essential tools in geometric processing. It allows us to solve numerous partial differential equations on discrete surface and volume meshes, which is a fundamental building block in many computer graphics applications. Discrete Laplacians are typically limited to standard elements like triangles or quadrilaterals, which severely constrains the tessellation of the mesh. But in recent years, several approaches were able to generalize the Laplace Beltrami and its closely related gradient and divergence operators to more general meshes. This allows artists and engineers to work with a wider range of elements which are sometimes required and beneficial in their field. This course, which extends the state-of-the-art report by Bunge and Botsch [2023], discusses the different constructions of these three ubiquitous differential operators on arbitrary polygons and polyhedra and analyzes their individual advantages and properties in common computer graphics applications.