Local discontinuous Galerkin method for the fractional diffusion equation with integral fractional Laplacian

TL;DR

This paper develops a local discontinuous Galerkin method for solving the fractional diffusion equation involving the integral fractional Laplacian in two dimensions, with proven stability and convergence.

Contribution

It introduces a novel DG scheme for the fractional Laplacian and provides theoretical proof and numerical verification of its stability and convergence.

Findings

Scheme is stable under theoretical analysis.

Numerical experiments confirm convergence rate of at least (h^{k+1/2}).

Method effectively solves fractional diffusion problems.

Abstract

In this paper, we provide a framework of designing the local discontinuous Galerkin scheme for integral fractional Laplacian $(-\Delta)^{s}$ with $s\in(0,1)$ in two dimensions. We theoretically prove and numerically verify the numerical stability and convergence of the scheme with the convergence rate no worse than $\mathcal{O}(h^{k+\frac{1}{2}})$.