# A Smooth Partition of Unity Finite Element Method for Vortex Particle Regularization

## TL;DR

This paper introduces a novel smooth finite element method on Cartesian grids for vortex particle regularization, enhancing stability and efficiency in vortex simulations.

## Contribution

It develops a new $C^ty$-smooth finite element space with a partition of unity approach and a fictitious domain formulation for general domains.

## Key findings

- Stable and convergent scheme with conservation properties
- Numerical results confirm theoretical analysis
- Optimal grid-size proportional to square root of particle spacing

## Abstract

We present a new class of $C^\infty$-smooth finite element spaces on Cartesian grids, based on a partition of unity approach. We use these spaces to construct smooth approximations of particle fields, i.e., finite sums of weighted Dirac deltas. In order to use the spaces on general domains, we propose a fictitious domain formulation, together with a new high-order accurate stabilization. Stability, convergence, and conservation properties of the scheme are established. Numerical experiments confirm the analysis and show that the Cartesian grid-size $\sigma$ should be taken proportional to the square-root of the particle spacing $h$, resulting in significant speed-ups in vortex methods.