Local discontinuous Galerkin method for the integral fractional Laplacian

TL;DR

This paper introduces a novel local discontinuous Galerkin method for solving integral fractional Laplacian problems, effectively handling nonlocal interactions and boundary singularities with proven optimal error estimates.

Contribution

It develops a new LDG scheme based on a three-field formulation for fractional Laplacians, incorporating regularity analysis and stabilization techniques for improved accuracy.

Findings

Optimal a priori error estimates established

Numerical experiments confirm theoretical accuracy

Effective handling of boundary singularities

Abstract

We develop and analyze a local discontinuous Galerkin (LDG) method for solving integral fractional Laplacian problems on bounded Lipschitz domains. The method is based on a three-field mixed formulation involving the primal variable, its gradient, and the corresponding Riesz potential, yielding a flux-based structure well suited for LDG discretizations while retaining the intrinsic nonlocal interaction. A key ingredient of our analysis is a rigorous study of the weighted H\"older and Sobolev regularity of the Riesz potential, which enables accurate characterization of boundary singularities. Guided by these regularity results, we propose LDG schemes on quasi-uniform and graded meshes, with additional stabilization in the graded case to reconcile the discrepancy between the discrete spaces for the Riesz potential and flux fields. Optimal a priori error estimates are established, and numerical experiments corroborate the theoretical results.