We propose a monotone discretization for the integral fractional Laplace equation on bounded Lipschitz domains with the homogenous 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 H\"{o}lder 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 H\"{o}lder regularity of the data on both quasi-uniform and graded grids. Several numerical examples are provided to illustrate the sharpness of the theoretical results.

翻译：我们建议对连接的利普西茨域域域的分形拉普特方程式采用单色分解。 这种方法受黄与欧伯曼基于四方基的有限差异法的启发, 但定义在任意的无结构网格上, 以更灵活的域进行近似单一整体。 单整体域的规模不仅取决于本地网格大小, 也取决于与边界的距离, 因为解决方案的H\"{o}lder 系数在接近边界时会恶化。 通过使用一个离散屏障函数来反映与边界的距离, 我们从准统一和分级网格的数据的H\"{ o}lder规律性来看, 我们显示了最佳的点向趋同率。 提供了几个数字例子来说明理论结果的清晰性 。