  The advantage of particle Lagrangian methods in computational fluid dynamics
is that advection is accurately modeled. However, this complicates the
calculation of space derivatives. If a mesh is employed, it must be updated at
each time step. On the other hand, fixed mesh, Eulerian, formulations benefit
from the mesh being defined at the beginning of the simulation, but feature
non-linear advection terms. It therefore seems natural to combine the two
approaches, using a fixed mesh to perform calculations related to space
derivatives, and using the particles to advect the information with time. The
idea of combining Lagrangian particles and a fixed mesh goes back to
Particle-in-Cell methods, and is here considered within the context of the
finite element method (FEM) for the fixed mesh, and the particle FEM (pFEM) for
the particles.
  Our results, in agreement with recent works, show that interpolation
("projection") errors, especially from particles to mesh, are the culprits of
slow convergence of the method if standard, linear, FEs, are employed. By
including quadratically consistent shape functions for the particles, the rate
of convergence is restored to values competitive with purely Lagrangian
simulations.
  The procedure is validated on the Zalesak's disk problem, the Taylor-Green
vortex sheet flow, and the Rayleigh-Taylor instability.
