标题： 有限元方法结合Grünwald-Letnikov型时间逼近在常时间延迟次扩散方程中的应用 显示英文标题

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

摘要： 在这项工作中，考虑了一个具有常数时间延迟$\tau$的次扩散方程。 首先，研究了所考虑问题解的规律性，发现其一阶时间导数在$t=0^+$处表现出奇异性，而二阶时间导数在$t=0^+$和$\tau^+$处显示出奇异性，同时解可以分解为其奇异部分和正则部分。 然后，我们基于空间上的标准伽辽金有限元方法和时间上的Grünwald-Letnikov型近似，推导出一个完全离散的有限元方案来求解所考虑的问题。 分析表明，所开发的数值格式是稳定的。 为了讨论误差估计，建立了新的离散Gronwall不等式。 在上述解的分解下，我们得到了所开发数值格式的时间局部误差估计。 最后，提供了一些数值测试以支持我们的理论分析。 显示英文摘要

摘要： 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.