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

Abstract

Abstract. We present a new class of C ∞ -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.

On particles and splines in bounded domains

“…It turns out that splines and particles seem to ideally complement each other and that it is also possible to solve the problem of particle initialization in the bounded setting. This problem has not been addressed in our previous work [19] and our results are summarized in Theorem 3.1. We recall evidence that these results are essentially optimal and cannot be expected to be considerably improved.…”

“…In this work we are going to build on and extend the results from our previous work on particle regularization [19]. In there, we constructed spaces of ∞ -smooth trial functions, which however have the disadvantage that an explicit representation of their basis functions is unavailable.…”

“…This results in poor approximations near its discontinuities at the boundaries. The stabilized 2 -projection (brown) [19] yields an approximation of a smooth extension. It is not only accurate on the entire interval but also extrapolates well after its ends.…”

“…Recently, we proposed another approach to the regularization problem, which is based on the 2 -projection and allows a rigorous analysis [19]. First, ∞ -smooth finite-element spaces on simple uniform Cartesian grids are created, where denotes the length of the cells.…”

On Particles and Splines in Bounded Domains

“…This results in poor approximations near its discontinuities at the boundaries. The stabilized L 2 -projection (brown) [17] yields an approximation of a smooth extension. It is not only accurate on the entire interval but also extrapolates well after its ends.…”