  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.
