In this article, we propose numerical scheme for solving a multi-term time-fractional nonlocal parabolic partial differential equation (PDE). The scheme comprises $L2$-$1_{\sigma}$ scheme on a graded mesh in time and Galerkin finite element method (FEM) in space. We present the discrete fractional Gr$\ddot{{o}}$nwall inequality for $L2$-$1_{\sigma}$ scheme in case of multi-term time-fractional derivative, which is a multi-term analogue of~\cite[Lemma 4.1]{[r16]}. We derive \textit{a priori} bound and error estimate for the fully-discrete solution. The theoretical results are confirmed via numerical experiments. We should note that, though the way of proving the discrete fractional Gr$\ddot{{o}}$nwall inequality is similar to~\cite{[r5]}, the calculation parts are more complicated in this article.

翻译：本文提出了一种求解多项时间分数阶非局部抛物型偏微分方程的数值格式。该格式在时间上采用渐变网格上的$L2$-$1_{\sigma}$格式，在空间上采用伽辽金有限元方法。我们给出了多项时间分数阶导数情形下$L2$-$1_{\sigma}$格式的离散分数阶Grö̈nwall不等式，该不等式是~\cite[引理4.1]{[r16]}的多项类比。我们推导了全离散解的\textit{先验}界和误差估计。理论结果通过数值实验得到验证。需要指出的是，尽管离散分数阶Grö̈nwall不等式的证明方法与~\cite{[r5]}类似，但本文中的计算部分更为复杂。