Polyhedral Discretizations for Elliptic PDEs

TL;DR

This paper investigates polyhedral discretizations for solving heat diffusion and elastodynamic problems in computer graphics, comparing their effectiveness to traditional methods and providing benchmarks for future development.

Contribution

It introduces a finite element approach using barycentric coordinates and virtual finite elements for polyhedral meshes, offering insights into their advantages and limitations.

Findings

Polyhedral meshes can be effective for certain graphics applications.

The virtual finite element method adapts well to polyhedral discretizations.

Benchmark results guide future polyhedral meshing techniques.

Abstract

We study the use of polyhedral discretizations for the solution of heat diffusion and elastodynamic problems in computer graphics. Polyhedral meshes are more natural for certain applications than pure triangular or quadrilateral meshes, which thus received significant interest as an alternative representation. We consider finite element methods using barycentric coordinates as basis functions and the modern virtual finite element approach. We evaluate them on a suite of classical graphics problems to understand their benefits and limitations compared to standard techniques on simplicial discretizations. Our analysis provides recommendations and a benchmark for developing polyhedral meshing techniques and corresponding analysis techniques.