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 (SINUM, 2014, pp. 3056–3084), 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. 

Shuonan Wu received his B.S. and Ph.D. degrees at Peking University in 2009 and 2014, respectively. After graduate school, he was a postdoctoral fellow at Pennsylvania State University working with Prof. Jinchao Xu during 2014-2018, then joined the School of Mathematical Sciences at Peking University as a tenure-track assistant professor. His general research interest is in computational and applied mathematics with an emphasis on numerical methods for partial differential equations, and the research topics include the stable discretization of magnetic advection and finite element methods for nonlinear and high-order PDEs. Recent progress has been made on monotone discretization and fast solvers for integral fractional Laplacian. His research works have been published in top journals in computational mathematics such as Math. Comp., Numer. Math. and SINUM.