$\alpha$-Robust Error Analysis of $L2$-$1_{\sigma }$ Scheme on Graded Mesh for Time-fractional Nonlocal Diffusion Equation

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 $\alpha$ $\to 1^{-},$ where $\alpha \in (0,1)$ is the order of a temporal fractional derivative. The numerical experiments are presented to justify the theoretical findings.