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arXiv · 2504.20524

Finite element method with Gr\"unwald-Letnikov type approximation in time for a constant time delay subdiffusion equation

Weiping Bu·Xueqin Zhang·Weizhi Liao·Yue Zhao
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Abstract

In this work, a subdiffusion equation with constant time delay $\tau$ 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 $\tau^+$, 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 Gr\"unwald-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.

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Weiping Bu, Xueqin Zhang, Weizhi Liao, Yue Zhao. 2025-04-29. Finite element method with Gr\"unwald-Letnikov type approximation in time for a constant time delay subdiffusion equation. https://arxiv.org/abs/2504.20524

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