Particle methods discretize the solutions of PDEs onto zero-dimensional,
connectivity-less colocation points called ""particles"". Due to their
Lagrangian nature and a common, unifying algorithmic framework, they have a
set of unique properties. We briefly review the basics of particle-based
function and operator approximation and highlight some of the current
limitations for consistency and convergence on parabolic problems. We then
present a formal discretization-correction framework that restores
consistency on almost all particle distributions. Furthermore, we discuss
ongoing work in using these numerical schemes for multi-resolution
simulations on distributed-memory parallel computers.


[Invited by J-P. Gabriel and A. Janka]