Galerkin finite element method for time fractional diffusion-wave equations with constant time delay

Abstract

This paper studies numerical methods for the time fractional diffusion-wave equations with the constant delay. Using a symmetric fractional-order reduction technique and introducing the auxiliary variable w, we reformulate the original problem as an equivalent coupled system and for the first time carry out a rigorous regularity analysis of w. Based on the symmetric graded time mesh, the coupled problem is discretised by the L1 scheme in time and a standard finite element method in space, and the stability result of the developed fully discrete finite element scheme is established by means of a discrete fractional Grönwall inequality. Furthermore, we derive a new comparison principle and reconstruct a new barrier function on each interval $((k-1)τ,kτ]$, 1≤ k≤ K according to the properties of the symmetric graded mesh. Based on the new barrier function, we obtain new relevant properties, which lead to a novel and crucial stability property for the discrete fractional operator. Combining the obtained local truncation error in time gives a local error estimate for the fully discrete finite element scheme, which shows that our convergence result provides a unified framework that seamlessly encompasses both pointwise-in-time error estimates on uniform meshes and global-in-time error estimates on symmetrical graded meshes. Numerical experiments confirm the theoretical results.

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BibTeX RIS

BibTeX

@misc{li2026galerkin,
  title = {{Galerkin finite element method for time fractional diffusion-wave equations with constant time delay}},
  author = {Xuan Li and Kaile Qiu and Chaobao Huang and Weiping Bu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21056},
  eprint = {2610.21056},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21056},
}

RIS

TY  - UNPB
TI  - Galerkin finite element method for time fractional diffusion-wave equations with constant time delay
AU  - Xuan Li
AU  - Kaile Qiu
AU  - Chaobao Huang
AU  - Weiping Bu
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21056
UR  - https://arxiv.org/abs/2610.21056
AB  - This paper studies numerical methods for the time fractional diffusion-wave equations with the constant delay. Using a symmetric fractional-order reduction technique and introducing the auxiliary variable w, we reformulate the original problem as an equivalent coupled system and for the first time carry out a rigorous regularity analysis of w. Based on the symmetric graded time mesh, the coupled problem is discretised by the L1 scheme in time and a standard finite element method in space, and the stability result of the developed fully discrete finite element scheme is established by means of a discrete fractional Grönwall inequality. Furthermore, we derive a new comparison principle and reconstruct a new barrier function on each interval $((k-1)τ,kτ]$, 1≤ k≤ K according to the properties of the symmetric graded mesh. Based on the new barrier function, we obtain new relevant properties, which lead to a novel and crucial stability property for the discrete fractional operator. Combining the obtained local truncation error in time gives a local error estimate for the fully discrete finite element scheme, which shows that our convergence result provides a unified framework that seamlessly encompasses both pointwise-in-time error estimates on uniform meshes and global-in-time error estimates on symmetrical graded meshes. Numerical experiments confirm the theoretical results.
ER  -

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