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.

我们提出了一种无网格广义有限差分法，用于求解具有高达几个数量级的跳跃不连续性的扩散系数的泊松方程。为了离散化扩散算子，我们制定了一种使用不连续性涂抹的强形式方法；以及基于局部计算的 Voronoi 单元的保守公式。此外，我们提出了一种新的保守公式，用于执行与扩散算子的保守公式兼容的 Neumann 边界条件。最后，我们介绍了一种从强形式切换到保守公式的方法，以获得局部保守和正性保持方案。所提出的数值方法针对四个具有不同复杂性和跳跃幅度的测试用例进行了基准测试，点云的节点未与不连续性对齐。我们的结果表明，在两种公式之间切换的新混合方法比经典的广义有限差分方法产生更好的结果，用于扩散率的高跃迁。