A Meshfree Point Collocation Method for Elliptic Interface Problems

TL;DR

This paper introduces a meshfree point collocation method for elliptic interface problems with discontinuous coefficients, combining strong and conservative formulations to improve accuracy and positivity preservation on unaligned point clouds.

Contribution

A novel hybrid meshfree method that switches between strong and conservative formulations for better handling of discontinuities in elliptic interface problems.

Findings

Hybrid method outperforms classical approaches for high jump discontinuities.

Method maintains positivity and local conservation.

Benchmark tests demonstrate improved accuracy across complex cases.

Abstract

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.