Abstract Discrete Laplacians are the basis for various tasks in geometry processing. While the most desirable properties of the discretization invariably lead to the so‐called cotangent Laplacian fortrianglemeshes, applying the same principles topolygonLaplacians leaves degrees of freedom in their construction. From linear finite elements it is well‐known how the shape of triangles affects both the error and the operator's condition. We notice that shape quality can be encapsulated as the trace of the Laplacian and suggest that trace minimization is a helpful tool to improve numerical behavior. We apply this observation to the polygon Laplacian constructed from a virtual triangulation [BHKB20] to derive optimal parameters per polygon. Moreover, we devise a smoothing approach for the vertices of a polygon mesh to minimize the trace. We analyze the properties of the optimized discrete operators and show their superiority over generic parameter selection in theory and through various experiments.

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.

Abstract 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 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 paper discusses the different constructions of these three ubiquitous differential operators on arbitrary polygons and analyzes their individual advantages and properties in common computer graphics applications.

Solving partial differential equations (PDEs) on geometric domains is an important component of computer graphics, geometry processing, and many other fields. Typically, the given discrete mesh is the geometric representation and should not be altered for simulation purposes. Hence, accurately solving PDEs on general meshes is a central goal and has been considered for various differential operators over the last years. While it is known that using higher-order basis functions on simplicial meshes can substantially improve accuracy and convergence, extending these benefits to general surface or volume tessellations in an efficient fashion remains an open problem. Our work proposes variationally optimized piecewise quadratic shape functions for polygons and polyhedra, which generalize quadratic P 2 elements, exactly reproduce them on simplices, and inherit their beneficial numerical properties. To mitigate the associated cost of increased computation time, particularly for volumetric meshes, we introduce a custom two-level multigrid solver which significantly improves computational performance.

Abstract The discrete Laplace‐Beltrami operator for surface meshes is a fundamental building block for many (if not most) geometry processing algorithms. While Laplacians on triangle meshes have been researched intensively, yielding the cotangent discretization as the de‐facto standard, the case of general polygon meshes has received much less attention. We present a discretization of the Laplace operator which is consistent with its expression as the composition of divergence and gradient operators, and is applicable to general polygon meshes, including meshes with non‐convex, and even non‐planar, faces. Byvirtuallyinserting a carefully placed point we implicitly refine each polygon into a triangle fan, but then hide the refinement within the matrix assembly. The resulting operator generalizes the cotangent Laplacian, inherits its advantages, and is empirically shown to be on par or even better than the recent polygon Laplacian of Alexa and Wardetzky [AW11] — while being simpler to compute.