# On Particles and Splines in Bounded Domains

## TL;DR

This paper introduces numerical schemes combining particle methods with splines and fictitious domain techniques to solve advection problems in bounded domains, requiring only a boundary-fitted mesh and ensuring stability and consistency.

## Contribution

The paper presents a novel particle-spline scheme for bounded domains that simplifies implementation by using an unfitted Cartesian grid and proves its stability and consistency.

## Key findings

- Scheme is stable in $W^{s,p}$-norms.
- Scheme is consistent in $W^{s,p}$-norms.
- Implementation only needs a boundary-fitted mesh.

## Abstract

We propose numerical schemes that enable the application of particle methods for advection problems in general bounded domains. These schemes combine particle fields with Cartesian tensor product splines and a fictitious domain approach. Their implementation only requires a fitted mesh of the domain's boundary, and not the domain itself, where an unfitted Cartesian grid is used. We establish the stability and consistency of these schemes in $W^{s,p}$-norms, $s\in\mathbb{R}$, $1<p\leq\infty$.