A Smooth Partition of Unity Finite Element Method for Vortex Particle Regularization
For computational fluid dynamics researchers using vortex methods, this provides a more efficient regularization technique with theoretical guarantees.
The paper introduces a new class of smooth finite element spaces for regularizing vortex particle methods, achieving high-order accuracy and conservation. Numerical experiments show that the grid size should scale with the square root of particle spacing, yielding significant computational speed-ups.
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 $σ$ should be taken proportional to the square-root of the particle spacing $h$, resulting in significant speed-ups in vortex methods.