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Finite element method with Grünwald-Letnikov type approximation in time for a constant time delay subdiffusion equation

Weiping Bu, Xueqin Zhang, Weizhi Liao, Yue Zhao

TL;DR

This work addresses a time-fractional diffusion equation with constant time delay $\tau$, focusing on regularity, stability, and convergence of a fully discrete scheme. The authors derive the exact solution structure and establish that the first time derivative may blow up at $t=0^+$ while the second time derivative may blow up at $t=0^+$ and $t=\tau^+$, enabling a decomposition into singular and regular parts. A fully discrete scheme is developed using Grünwald-Letnikov time discretization and standard Galerkin finite elements in space, with stability proven via a new discrete Gronwall inequality and convergence shown through a decomposition-based local truncation error analysis. Numerical experiments corroborate the theoretical rates, demonstrating near-$\alpha$ temporal accuracy in early times and near-quadratic spatial accuracy, validating the method's effectiveness for delay-influenced subdiffusion problems. The combination of regularity results, a robust GL-time discretization, and a rigorous error framework offers a reliable tool for simulating memory-laden diffusion processes with delays.

Abstract

In this work, a subdiffusion equation with constant time delay $τ$ is considered. First, the regularity of the solution to the considered problem is investigated, finding that its first-order time derivative exhibits singularity at $t=0^+$ and its second-order time derivative shows singularity at both $t=0^+$ and $τ^+$, while the solution can be decomposed into its singular and regular components. Then, we derive a fully discrete finite element scheme to solve the considered problem based on the standard Galerkin finite element method in space and the Grünwald-Letnikov type approximation in time. The analysis shows that the developed numerical scheme is stable. In order to discuss the error estimate, a new discrete Gronwall inequality is established. Under the above decomposition of the solution, we obtain a local error estimate in time for the developed numerical scheme. Finally, some numerical tests are provided to support our theoretical analysis.

Finite element method with Grünwald-Letnikov type approximation in time for a constant time delay subdiffusion equation

TL;DR

This work addresses a time-fractional diffusion equation with constant time delay , focusing on regularity, stability, and convergence of a fully discrete scheme. The authors derive the exact solution structure and establish that the first time derivative may blow up at while the second time derivative may blow up at and , enabling a decomposition into singular and regular parts. A fully discrete scheme is developed using Grünwald-Letnikov time discretization and standard Galerkin finite elements in space, with stability proven via a new discrete Gronwall inequality and convergence shown through a decomposition-based local truncation error analysis. Numerical experiments corroborate the theoretical rates, demonstrating near- temporal accuracy in early times and near-quadratic spatial accuracy, validating the method's effectiveness for delay-influenced subdiffusion problems. The combination of regularity results, a robust GL-time discretization, and a rigorous error framework offers a reliable tool for simulating memory-laden diffusion processes with delays.

Abstract

In this work, a subdiffusion equation with constant time delay is considered. First, the regularity of the solution to the considered problem is investigated, finding that its first-order time derivative exhibits singularity at and its second-order time derivative shows singularity at both and , while the solution can be decomposed into its singular and regular components. Then, we derive a fully discrete finite element scheme to solve the considered problem based on the standard Galerkin finite element method in space and the Grünwald-Letnikov type approximation in time. The analysis shows that the developed numerical scheme is stable. In order to discuss the error estimate, a new discrete Gronwall inequality is established. Under the above decomposition of the solution, we obtain a local error estimate in time for the developed numerical scheme. Finally, some numerical tests are provided to support our theoretical analysis.
Paper Structure (6 sections, 14 theorems, 131 equations, 4 tables)

This paper contains 6 sections, 14 theorems, 131 equations, 4 tables.

Table of Contents

  1. Introduction
  2. The regularity of the solution
  3. Fully discrete scheme and its stability
  4. Convergence analysis
  5. Numerical experiment
  6. Conclusion

Key Result

Lemma 2.1

(Sakamoto2011Initial) If $\alpha>0,\lambda>0,t>0$ and positive integer $m\in \mathbb{N}$, then Moreover, if $0<\alpha<1$ and $\eta \geq 0$, then $E_{\alpha,\alpha}(-\eta)\geq 0.$

Theorems & Definitions (14)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • ...and 4 more