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

Minghua Chen, Jianxing Han, Jiankang Shi, Fan Yu

TL;DR

The paper develops a second-order accurate difference-quadrature scheme on graded meshes for the 1D fractional Laplacian defined via the Riesz derivative, addressing boundary-layer singularities and potential singular source terms. It introduces grid-mapping functions and a natural-skew ordering to tightly control local truncation errors, and constructs a right-preconditioner to restore global second-order convergence. Theoretical results establish local truncation-error bounds and global convergence, including scenarios with singular sources, supported by numerical experiments that confirm the predicted rates and demonstrate robustness. The approach is extendable to multidimensional fractional diffusion, gradient flows, and nonlinear problems, offering improved accuracy near boundaries where low regularity typically degrades performance.

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.

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

TL;DR

The paper develops a second-order accurate difference-quadrature scheme on graded meshes for the 1D fractional Laplacian defined via the Riesz derivative, addressing boundary-layer singularities and potential singular source terms. It introduces grid-mapping functions and a natural-skew ordering to tightly control local truncation errors, and constructs a right-preconditioner to restore global second-order convergence. Theoretical results establish local truncation-error bounds and global convergence, including scenarios with singular sources, supported by numerical experiments that confirm the predicted rates and demonstrate robustness. The approach is extendable to multidimensional fractional diffusion, gradient flows, and nonlinear problems, offering improved accuracy near boundaries where low regularity typically degrades performance.

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.

Paper Structure

This paper contains 28 sections, 34 theorems, 168 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. The main results
  3. Difference-quadrature scheme
  4. Regularity of the true solution
  5. Main results
  6. Local Truncation Error
  7. Proof of \ref{['thm:truncation-error']}
  8. Grid mapping functions
  9. Case 1
  10. Case 2
  11. Case 3
  12. Error analysis of $R_i$
  13. Case 1
  14. Case 2
  15. Case 1
  16. ...and 13 more sections

Key Result

Lemma 2.2

ROSOTON2014275 \newlabelthm:Xacier1.20 Let $f\in L^\infty(\Omega)$ and $u$ be a solution of eq:equation. Then, $u\in C^{\alpha/2}(\mathbb{R})$ and $u/\delta^{\alpha/2}\in C^\sigma(\bar{\Omega})$ for some $\sigma \in (0, 1-\alpha/2)$, $\alpha\in (1,2)$ with for some positive constant $C=C(\Omega, \alpha)$.

Figures (1)

  • Figure 1: Natural-Skew ordering of $R_i$.

Theorems & Definitions (67)

  • Definition 2.1: $\delta$-dependent Hölder norms ROSOTON2014275
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proof 1
  • Theorem 2.5: Local Truncation Error
  • Theorem 2.6: Global Error
  • Lemma 3.1
  • Theorem 3.2
  • Proof 2
  • ...and 57 more