In this work, a time-fractional nonlocal diffusion equation is considered. Based on the $L2$-$1_{\sigma}$ scheme on a graded mesh in time and the standard finite element method (FEM) in space, the fully-discrete $L2$-$1_{\sigma}$ finite element method is investigated for a time-fractional nonlocal diffusion problem. We prove the existence and uniqueness of fully-discrete solution. The $\alpha$-robust error bounds are derived, i.e. bounds remain valid as $\alpha$ $\rightarrow {1}^{-},$ where $\alpha \ \in (0,1)$ is the order of a temporal fractional derivative. The numerical experiments are presented to justify the theoretical findings.

翻译：本文考虑了一类时间分数阶非局部扩散方程。基于时间方向渐变网格上的$L2$-$1_{\sigma}$格式和空间方向的标准有限元方法（FEM），研究了时间分数阶非局部扩散问题的全离散$L2$-$1_{\sigma}$有限元方法。我们证明了全离散解的存在唯一性，并推导了$\alpha$-鲁棒误差界，即当$\alpha \rightarrow {1}^{-}$时该误差界仍然有效，其中$\alpha \in (0,1)$为时间分数阶导数的阶数。数值实验验证了理论结果的正确性。