A monotone discretization for integral fractional Laplacian on bounded Lipschitz domains: Pointwise error estimates under H\"{o}lder regularity

TL;DR

This paper introduces a monotone discretization method for the integral fractional Laplacian on Lipschitz domains, achieving optimal pointwise error estimates based on Hölder regularity, applicable to unstructured grids.

Contribution

It develops a flexible, monotone discretization scheme on unstructured grids that accounts for boundary effects, providing sharp error estimates for fractional Laplacian problems.

Findings

Achieves optimal pointwise convergence rates.

Works on unstructured and graded grids.

Validated by numerical experiments.

Abstract

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.