Error estimates for Galerkin finite element approximations of time-fractional nonlocal diffusion equation

Abstract

This paper is concerned to study the well-posedness, the Mittag–Leffler stability of solutions of time-fractional nonlocal reaction–diffusion equation in bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n.$ We use the Faedo–Galerkin approximation method with initial data in $L^2\left(\mathrm{\Omega }\right)$ to show a solution in $u\in L^{\mathrm{\infty }}(0,T;L^2(\mathrm{\Omega }\left)\right)\cap L^2(0,T;H_0^1(\mathrm{\Omega }\left)\right).$ Further, we construct the suitable Lyapunov function to ensure that a solution of the proposed model is the Mittag–Leffler stable. Furthermore, we fully discretize the Galerkin finite element method for the proposed time-fractional model in two-space dimension. Here, time-fractional derivative is given in Caputo's sense and discretized using $L^1$ approximation scheme. Error analysis of the proposed numerical method is performed and error bounds are obtained for the error measured in $L^2$ norm. All the theoretical results are validated with several constructive numerical examples.

Keywords:

• Error estimates
• Faedo–Galerkin method
• time-fractional equation
• finite element methods
• nonlocal diffusion

2010 Mathematics Subject Classifications:

• 65M60
• 35R11
• 65N15
• 35K57

Acknowledgements

The authors wish to thank the anonymous referees for their comments and suggestions which improved the quality of the paper. The first author is thankful to the Ministry of Human Resource Development (MHRD) and National Institute of Technology Goa, India, for awarding Junior Research Fellowship.

Disclosure statement

No potential conflict of interest was reported by the author(s).