变系数扩散算子的高阶广义有限差分方法
Higher-Order Generalized Finite Differences for Variable Coefficient Diffusion Operators
AI总结：
该研究提出无网格广义有限差分框架下离散变系数扩散算子的新方法，结合离散拉普拉斯算子与扩散系数重构函数，证明精度阶传递性与对角占优性，数值验证其在光滑系数、界面问题中的有效性与一阶收敛性。
摘要：
我们提出了一种在无网格广义有限差分方法框架下离散变系数扩散算子的新方法。我们的方案利用导出算子的性质，将离散拉普拉斯算子与逼近扩散系数的重构函数相结合。在重构具有足够高阶的前提下，我们证明了离散拉普拉斯算子的精度阶可传递到导出的扩散算子。我们表明，新的离散扩散算子继承了离散拉普拉斯算子的对角占优性质。最后，我们提出了借助导出算子离散各向异性扩散算子的可能性。我们针对泊松方程和热方程的数值结果表明，对于足够光滑的扩散系数，即使低阶重构也能保持底层离散拉普拉斯算子的阶数。在实验中，我们验证了新的离散扩散算子适用于点云不与界面对齐的界面问题，并通过数值证明了一阶收敛性。
英文摘要：
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.
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