Being able to accurately solve PDEs on arbitrary polygonal/polyhedral meshes is a central goal and has been considered for various differential operators over the last years. In this talk I will present a simple approach for computing (piecewise) linear and quadratic basis functions for general polygons and polyhedra, from which discrete operators for gradient, divergence and Laplacian can be derived. The central idea is to (virtually) split each polygon/polyhedron into simplices, but to hide this refinement through a special prolongation operator in the matrix assembly stage. The resulting Laplacian will be evaluated and compared to other approaches in a range of applications.
Mario Botsch is professor for Computer Science at TU Dortmund University, where he is heading the Chair of Computer Graphics. The focus of his research is geometry processing, in particular the efficient acquisition, optimization, animation, and visualization of three-dimensional geometric objects. He is currently investigating 3D-scanning and motion capturing of humans, modelling and animation of virtual characters, and real-time visualization in interactive virtual reality scenarios.