$L2$-$1_{\sigma}$ Scheme on a Graded Mesh for a Multi-term Time-fractional Nonlocal Parabolic Problem
Abstract
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.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2023
- DOI:
- 10.48550/arXiv.2306.03114
- arXiv:
- arXiv:2306.03114
- Bibcode:
- 2023arXiv230603114K
- Keywords:
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- Mathematics - Numerical Analysis
- E-Print:
- arXiv admin note: text overlap with arXiv:2306.02576