A Second-order method on graded meshes for fractional Laplacian via Riesz fractional derivative with a singular source term

TL;DR

This paper introduces a second-order numerical scheme on graded meshes for the fractional Laplacian with singular sources, achieving high accuracy even under low regularity conditions.

Contribution

It develops a grid mapping and preconditioning technique to attain second-order convergence for fractional Laplacian problems with singular sources on graded meshes.

Findings

Achieves second-order convergence on graded meshes with singular sources.

Effective for multidimensional fractional diffusion and nonlinear equations.

Numerical results confirm theoretical error estimates.

Abstract

The high-order numerical analysis for fractional Laplacian via the Riesz fractional derivative, under the low regularity solution, has presented significant challenges in the past decades. To fill in this gap, we design a grid mapping function on graded meshes to analyse the local truncation errors, which are far less than second-order convergence at the boundary layer. To restore the second-order global errors, we construct an appropriate right-preconditioner for the resulting matrix algebraic equation. We prove that the proposed scheme achieves second-order convergence on graded meshes even if the source term is singular or hypersingular. Numerical experiments illustrate the theoretical results. The proposed approach is applicable for multidimensional fractional diffusion equations, gradient flows and nonlinear equations.