论文标题
用于椭圆界面问题的网状点搭配方法
A Meshfree Point Collocation Method for Elliptic Interface Problems
论文作者
论文摘要
我们提出了一种无线通用的有限差方法，该方法可通过扩散系数求解泊松方程，其中包含跳跃不连续性的最高几个数量级。为了离散扩散操作员，我们制定了一种强大的形式方法，该方法使用不连续性的涂抹。以及基于局部计算的Voronoi细胞的保守配方。此外，我们提出了一种新型的保守配方，以实施与扩散算子的保守配方兼容的Neumann边界条件。最后，我们介绍了一种从强大的形式转换为保守配方的方法，以获得当地保守和积极的保存方案。在具有不同的复杂性和跳跃幅度的四个测试用例上，对所提供的数值方法进行了基准测试，并在点云上具有与不连续性不符的节点。我们的结果表明，在两种配方之间切换的新混合方法比经典的广义有限差异方法可以产生更好的结果。
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.
