# Error Analysis of Approximate Operators for a Particle Method based on Voronoi Diagram

## TL;DR

This paper analyzes the errors of approximate gradient and Laplace operators in a Voronoi diagram-based particle method, providing error estimates and an example application.

## Contribution

It introduces and studies new approximate operators for gradient and Laplace in a Voronoi-based particle method, with derived error estimates.

## Key findings

- Error estimates for approximate operators are derived.
- The integration region is divided into specific areas for analysis.
- An example application demonstrates the main results.

## Abstract

This paper considers several approximate operators used in a particle method based on a Voronoi diagram. We introduce and study our approximate operators on gradient and Laplace operators. We derive error estimates for these approximate operators by applying our weight functions. The key idea of deriving our error estimates is to divide the integration region into a ring-shaped area and some areas. In the Appendix, we give an example application of the main results of this paper.