A meshfree point collocation method for elliptic interface problems

We present a meshfree generalized finite difference method for solving Poisson's equa-tion 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 com-puted Voronoi cells. Additionally, we propose a novel conservative formulation for enforc-ing 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 com-plexity and jump magnitude on point clouds with nodes that are not aligned to the discon-tinuity. Our results show that the new hybrid method that switches between the two for-mulations produces better results than the classical generalized finite difference approach for high jumps in diffusivity.(c) 2022 Elsevier Inc. All rights reserved.

Automated summary

This paper presents a meshfree generalized finite difference method for solving Poisson's equation with a diffusion coefficient that contains jump discontinuities. The diffusion operator is discretized using a strong form method and a conservative formulation based on locally computed Voronoi cells. Additionally, a novel conservative formulation for enforcing Neumann boundary conditions is proposed, and a switching method from the strong form to the conservative formulation is introduced. The results of benchmark tests show that the new hybrid method produces better results compared to the classical generalized finite difference approach for high jumps in diffusivity.