有界Lipschitz域上整数分数次Laplacian的单调离散化：Hölder正则性下的点态误差估计
A monotone discretization for integral fractional Laplacian on bounded Lipschitz domains: Pointwise error estimates under Hölder regularity
摘要：
对具有齐次Dirichlet边界条件的有界Lipschitz域上的分数阶Laplace积分方程提出了一种单调离散方法。该方法的灵感来自于基于正交的有限差分法的黄和Oberman，但定义在非结构化网格在任意尺寸的更灵活的域近似奇异积分。奇异积分域的尺度不仅取决于局部网格的大小，而且取决于到边界的距离，因为解的Hölder系数随着其接近边界而恶化。通过使用一个离散的障碍函数，也反映了到边界的距离，我们显示了最佳的逐点收敛速度的Hölder正则性的数据上的准均匀和梯度网格。几个数值例子来说明理论结果的清晰度。
英文摘要：
We propose a monotone discretization for the integral fractional Laplace equation on bounded Lipschitz domains with the homogeneous Dirichlet boundary condition. The method is inspired by a quadrature-based finite difference method of Huang and Oberman, but is defined on unstructured grids in arbitrary dimensions with a more flexible domain for approximating singular integral. The scale of the singular integral domain not only depends on the local grid size, but also on the distance to the boundary, since the Hölder coefficient of the solution deteriorates as it approaches the boundary. By using a discrete barrier function that also reflects the distance to the boundary, we show optimal pointwise convergence rates in terms of the Hölder regularity of the data on both quasi-uniform and graded grids. Several numerical examples are provided to illustrate the sharpness of the theoretical results.
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