$L2$-$1_{\sigma }$ Scheme on a Graded Mesh for a Multi-term Time-fractional Nonlocal Parabolic Problem

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 $\mathit{\text{a}}\mathit{\text{priori}}$ 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.