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Error analysis of a collocation method on graded meshes for nonlocal diffusion problems with weakly singular kernels

Minghua Chen, Chao Min, Jiankang Shi, Jizeng Wang

TL;DR

The paper analyzes a collocation discretization on graded meshes for time-dependent nonlocal diffusion with a weakly singular kernel ($0<\alpha<1$). It shows that standard graded meshes can underperform uniform grids or even diverge for steady-state problems with smooth solutions, while anomalous graded meshes ($0<r<1$) achieve optimal convergence; for low regularity, graded meshes yield sharp but limited rates, whereas the time-dependent case attains second-order convergence in space-time, with extensions to certain multidimensional settings. The results are supported by detailed local truncation error and convergence analyses for both steady-state and time-dependent problems, and are validated by numerical experiments. These findings provide practical guidance on mesh design to balance accuracy and stability in nonlocal diffusion simulations.

Abstract

Can graded meshes yield more accurate numerical solution than uniform meshes? A time-dependent nonlocal diffusion problem with a weakly singular kernel is considered using collocation method. For its steady-state counterpart, under the sufficiently smooth solution, we first clarify that the standard graded meshes are worse than uniform meshes and may even lead to divergence; instead, an optimal convergence rate arises in so-called anomalous graded meshes. Furthermore, under low regularity solutions, it may suffer from a severe order reduction in (Chen, Qi, Shi and Wu, IMA J. Numer. Anal., 41 (2021) 3145--3174). In this case, conversely, a sharp error estimates appears in standard graded meshes, but offering far less than first-order accuracy. For the time-dependent case, however, second-order convergence can be achieved on graded meshes. The related analysis are easily extended for certain multidimensional problems. Numerical results are provided that confirm the sharpness of the error estimates.

Error analysis of a collocation method on graded meshes for nonlocal diffusion problems with weakly singular kernels

TL;DR

The paper analyzes a collocation discretization on graded meshes for time-dependent nonlocal diffusion with a weakly singular kernel (). It shows that standard graded meshes can underperform uniform grids or even diverge for steady-state problems with smooth solutions, while anomalous graded meshes () achieve optimal convergence; for low regularity, graded meshes yield sharp but limited rates, whereas the time-dependent case attains second-order convergence in space-time, with extensions to certain multidimensional settings. The results are supported by detailed local truncation error and convergence analyses for both steady-state and time-dependent problems, and are validated by numerical experiments. These findings provide practical guidance on mesh design to balance accuracy and stability in nonlocal diffusion simulations.

Abstract

Can graded meshes yield more accurate numerical solution than uniform meshes? A time-dependent nonlocal diffusion problem with a weakly singular kernel is considered using collocation method. For its steady-state counterpart, under the sufficiently smooth solution, we first clarify that the standard graded meshes are worse than uniform meshes and may even lead to divergence; instead, an optimal convergence rate arises in so-called anomalous graded meshes. Furthermore, under low regularity solutions, it may suffer from a severe order reduction in (Chen, Qi, Shi and Wu, IMA J. Numer. Anal., 41 (2021) 3145--3174). In this case, conversely, a sharp error estimates appears in standard graded meshes, but offering far less than first-order accuracy. For the time-dependent case, however, second-order convergence can be achieved on graded meshes. The related analysis are easily extended for certain multidimensional problems. Numerical results are provided that confirm the sharpness of the error estimates.
Paper Structure (16 sections, 17 theorems, 103 equations, 5 tables)

This paper contains 16 sections, 17 theorems, 103 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries: Numerical schemes
  3. Collocation method for steady-state problems \ref{['equ1.2']}
  4. Collocation method for time-dependent problems \ref{['equ1.1']}
  5. Spectral analysis for stiffness matrix $A$ in \ref{['nequ2.5']}
  6. Error analysis: Steady-state problems with smooth solution
  7. Local truncation error
  8. Convergence analysis
  9. Error analysis: Steady-state problems with low regularity solution
  10. Local truncation error
  11. Convergence analysis
  12. Error analysis: Time-dependent case with low regularity solution
  13. Local truncation error
  14. Convergence and stability analysis
  15. Numerical experiments
  16. ...and 1 more sections

Key Result

Lemma 1

Quarteroni:07 \newlabelnnlem2.10 A real matrix $A$ of order $n$ is positive definite if and only if its symmetric part $H=\frac{A+A^T}{2}$ is positive definite. Let $H \in \mathbb{R}^{n\times n}$ be symmetric. Then $H$ is positive definite if and only if the eigenvalues of $H$ are positive.

Theorems & Definitions (37)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Proof 1
  • Lemma 4
  • Proof 2
  • Lemma 5
  • Proof 3
  • Remark 2.1
  • Lemma 1
  • ...and 27 more