Meshfree Collocation for Elliptic Problems with Discontinuous Coefficients

Abstract

We present a meshfree generalized finite difference method (GFDM) for solving Poisson's equation with coefficients containing 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 of enforcing Neumann boundary conditions that is compatible with the conservative formulation of the diffusion operator. Finally, we introduce a way to switch between the strong form and the conservative formulation to obtain a locally conservative and positivity preserving scheme. The presented numerical methods are benchmarked against four test cases with varying complexity and different jump magnitudes on point clouds 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 standard GFDM approach for high jumps in the diffusivity parameter.