# Why do we need Voronoi cells and Delaunay meshes? Essential properties of the Voronoi finite volume method

## TL;DR

This paper highlights the importance of Voronoi finite volume methods and Delaunay meshes in accurately preserving the stability and qualitative properties of solutions to parabolic and elliptic problems across various resolutions and time scales.

## Contribution

It provides a concise overview of the key properties of Voronoi FVM, demonstrates their advantages, and advocates for their increased adoption in practical applications.

## Key findings

- Voronoi FVM preserves essential stability properties.
- Delaunay meshes accurately approximate problem geometry.
- Voronoi FVM maintains qualitative solution properties across resolutions.

## Abstract

Unlike other schemes that locally violate the essential stability properties of the analytic parabolic and elliptic problems, Voronoi finite volume methods (FVM) and boundary conforming Delaunay meshes provide good approximation of the geometry of a problem and are able to preserve the essential qualitative properties of the solution for any given resolution in space and time as well as changes in time scales of multiple orders of magnitude. This work provides a brief description of the essential and useful properties of the Voronoi FVM, application examples, and a motivation why Voronoi FVM deserve to be used more often in practice than they are currently.