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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2504.20524 |
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| _version_ | 1866913812203962368 |
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| author | Bu, Weiping Zhang, Xueqin Liao, Weizhi Zhao, Yue |
| author_facet | Bu, Weiping Zhang, Xueqin Liao, Weizhi Zhao, Yue |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20524 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finite element method with Grünwald-Letnikov type approximation in time for a constant time delay subdiffusion equation Bu, Weiping Zhang, Xueqin Liao, Weizhi Zhao, Yue Numerical Analysis 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. |
| title | Finite element method with Grünwald-Letnikov type approximation in time for a constant time delay subdiffusion equation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2504.20524 |