  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 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.
