# A conservative, consistent, and scalable meshfree mimetic method

## TL;DR

This paper introduces a novel meshfree mimetic divergence operator for point clouds that ensures local conservation, is scalable, and can be instantiated with or without a background mesh, enabling robust discretization of conservation laws.

## Contribution

The work defines a new meshfree mimetic divergence operator that satisfies a discrete divergence theorem and can be implemented with or without a background mesh, improving scalability and robustness.

## Key findings

- The proposed MMD operator is first-order accurate.
- It satisfies a discrete divergence theorem and local conservation.
- Numerical results show mimetic properties similar to finite volume methods.

## Abstract

Mimetic methods discretize divergence by restricting the Gauss theorem to mesh cells. Because point clouds lack such geometric entities, construction of a compatible meshfree divergence remains a challenge. In this work, we define an abstract Meshfree Mimetic Divergence (MMD) operator on point clouds by contraction of field and virtual face moments. This MMD satisfies a discrete divergence theorem, provides a discrete local conservation principle, and is first-order accurate. We consider two MMD instantiations. The first one assumes a background mesh and uses generalized moving least squares (GMLS) to obtain the necessary field and face moments. This MMD instance is appropriate for settings where a mesh is available but its quality is insufficient for a robust and accurate mesh-based discretization. The second MMD operator retains the GMLS field moments but defines virtual face moments using computationally efficient weighted graph-Laplacian equations. This MMD instance does not require a background grid and is appropriate for applications where mesh generation creates a computational bottleneck. It allows one to trade an expensive mesh generation problem for a scalable algebraic one, without sacrificing compatibility with the divergence operator. We demonstrate the approach by using the MMD operator to obtain a virtual finite-volume discretization of conservation laws on point clouds. Numerical results in the paper confirm the mimetic properties of the method and show that it behaves similarly to standard finite volume methods.