# Numerical solution of 2D boundary value problems on merged Voronoi-Delaunay meshes

## TL;DR

This paper presents a numerical method for solving 2D boundary value problems using merged Voronoi-Delaunay meshes, enabling efficient approximation of differential operators on irregular grids.

## Contribution

It introduces a novel mesh structure combining Voronoi and Delaunay elements with orthodiagonal quadrilaterals for improved numerical solutions.

## Key findings

- Effective approximation of gradient and divergence operators.
- Successful application to steady-state diffusion-reaction problems.
- Mesh structure facilitates handling anisotropic media.

## Abstract

Computational technologies for the approximate solution of multidimensional boundary value problems often rely on irregular computational meshes and finite-volume approximations. In this framework, the discrete problem represents the corresponding conservation law for control volumes associated with the nodes of the mesh. This approach is most naturally and consistently implemented using Delaunay triangulations together with Voronoi diagrams as control volumes. In this paper, we employ meshes with nodes located both at the vertices of Delaunay triangulations and at the generators of Voronoi partitions. The cells of the merged Voronoi-Delaunay mesh are orthodiagonal quadrilaterals. On such meshes, scalar and vector functions, as well as invariant gradient and divergence operators of vector calculus, can be conveniently approximated. We illustrate the capabilities of this approach by solving a steady-state diffusion-reaction problem in an anisotropic medium.