  We present a meshfree generalized finite difference method for solving
Poisson's equation with a diffusion coefficient that contains jump
discontinuities up to several orders of magnitude. To discretize the diffusion
operator, we formulate a strong form method that uses a smearing of the
discontinuity; and a conservative formulation based on locally computed Voronoi
cells. Additionally, we propose a novel conservative formulation for enforcing
Neumann boundary conditions that is compatible with the conservative
formulation of the diffusion operator. Finally, we introduce a way to switch
from the strong form to the conservative formulation to obtain a locally
conservative and positivity preserving scheme. The presented numerical methods
are benchmarked against four test cases of varying complexity and jump
magnitude on point clouds with nodes that are not aligned to the discontinuity.
Our results show that the new hybrid method that switches between the two
formulations produces better results than the classical generalized finite
difference approach for high jumps in diffusivity.
