We present a novel approach of discretizing variable coefficient diffusion operators in the context of meshfree generalized finite difference methods. Our ansatz uses properties of derived operators and combines the discrete Laplace operator with reconstruction functions approximating the diffusion coefficient. Provided that the reconstructions are of a sufficiently high order, we prove that the order of accuracy of the discrete Laplace operator transfers to the derived diffusion operator. We show that the new discrete diffusion operator inherits the diagonal dominance property of the discrete Laplace operator. Finally, we present the possibility of discretizing anisotropic diffusion operators with the help of derived operators. Our numerical results for Poisson's equation and the heat equation show that even low-order reconstructions preserve the order of the underlying discrete Laplace operator for sufficiently smooth diffusion coefficients. In experiments, we demonstrate the applicability of the new discrete diffusion operator to interface problems with point clouds not aligning to the interface and numerically show first-order convergence.

翻译：本文提出了一种在无网格广义有限差分法中离散化变系数扩散算子的新方法。我们的方案利用派生算子的性质，将离散拉普拉斯算子与逼近扩散系数的重构函数相结合。在重构函数具有足够高阶精度的情况下，我们证明了离散拉普拉斯算子的精度阶数可转移至派生的扩散算子。我们证明新的离散扩散算子继承了离散拉普拉斯算子的对角占优特性。最后，我们展示了借助派生算子离散化各向异性扩散算子的可能性。针对泊松方程和热传导方程的数值结果表明，即使采用低阶重构，对于足够光滑的扩散系数，该方法仍能保持底层离散拉普拉斯算子的精度阶数。在实验中，我们证明了新离散扩散算子适用于点云未对齐界面的界面问题，并通过数值实验展示了一阶收敛性。