This paper considers the finite element method to solve a generalized constant delay diffusion equation. The regularity of the solution of the considered model is investigated, which is the first time to discover that the solution has non-uniform multi-singularity in time compared with Tan et al. (2022). To overcome the multi-singularity, a symmetrical graded mesh is used to devise the fully discrete finite element scheme for the considered problem based on L1 formula of the Caputo fractional derivative and fractional trapezoidal formula of the Riemann–Liouville fractional integral. Then we investigate the unconditional stability of this scheme. Next, the local truncation errors of the L1 formula and the fractional trapezoidal formula are analyzed in detail, especially the later one is discussed at the first time, under the multi-singularity of the solution and the symmetrical graded mesh. Using these error results, we obtain the convergence of the proposed numerical scheme. Finally, some numerical tests are provided to verify the obtained theoretical results.

中文翻译：

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广义常延迟扩散方程的有限元法

本文考虑用有限元方法求解广义常延迟扩散方程。研究了所考虑模型的解的规律性，与Tan等人相比，首次发现解具有时间上的非均匀多重奇异性。 （2022）。为了克服多重奇异性，基于 Caputo 分数阶导数的 L1 公式和 Riemann-Liouville 分数阶积分的分数梯形公式，使用对称分级网格为所考虑的问题设计完全离散的有限元方案。然后我们研究该方案的无条件稳定性。接下来，详细分析了L1公式和分数梯形公式的局部截断误差，特别是首次讨论了后者在解的多重奇异性和对称渐变网格下的局部截断误差。利用这些误差结果，我们获得了所提出的数值方案的收敛性。最后，提供了一些数值试验来验证所获得的理论结果。