Robust Discrete Differential Operators for Wild Geometry

Many geometry processing algorithms rely on solving PDEs on discrete surface meshes. Their accuracy and robustness crucially depend on the mesh quality, which oftentimes cannot be guaranteed — in particular when automatically processing geometries extracted from arbitrary implicit representations. Through extensive numerical experiments, we evaluate the robustness of various Laplacian implementations across geometry processing libraries on synthetic and “in-the-wild” surface meshes with degenerate or near-degenerate elements, revealing their strengths, weaknesses, and failure cases. To improve numerical stability, we extend the recently proposed tempered finite elements method (TFEM) to meshes with strongly varying element sizes, to arbitrary polygonal elements, and to gradient and divergence operators. Our resulting differential operators are simple to implement, efficient to compute, and robust even in the presence of fully degenerate mesh elements.