# A Large-Scale Comparison of Tetrahedral and Hexahedral Elements for Solving Elliptic PDEs with the Finite Element Method

## TL;DR

This paper systematically compares tetrahedral and hexahedral finite elements across a large set of benchmark problems involving elliptic PDEs, highlighting their practical performance differences on automatically meshed 3D geometries.

## Contribution

It introduces a comprehensive benchmark suite for evaluating FEM element types on real-world geometries, providing insights into their practical performance and system-level impacts.

## Key findings

- Hexahedral elements often outperform tetrahedral elements in certain PDE solutions.
- Mesh quality significantly influences FEM performance and accuracy.
- The benchmark enables systematic comparison of FEM pipelines from meshing to solving.

## Abstract

The Finite Element Method (FEM) is widely used to solve discrete Partial Differential Equations (PDEs) in engineering and graphics applications. The popularity of FEM led to the development of a large family of variants, most of which require a tetrahedral or hexahedral mesh to construct the basis. While the theoretical properties of FEM basis (such as convergence rate, stability, etc.) are well understood under specific assumptions on the mesh quality, their practical performance, influenced both by the choice of the basis construction and quality of mesh generation, have not been systematically documented for large collections of automatically meshed 3D geometries. We introduce a set of benchmark problems involving most commonly solved elliptic PDEs, starting from simple cases with an analytical solution, moving to commonly used test problem setups, and using manufactured solutions for thousands of real-world, automatically meshed geometries. For all these cases, we use state-of-the-art meshing tools to create both tetrahedral and hexahedral meshes, and compare the performance of different element types for common elliptic PDEs. The goal of his benchmark is to enable comparison of complete FEM pipelines, from mesh generation to algebraic solver, and exploration of relative impact of different factors on the overall system performance.