Error estimates for Galerkin finite element approximations of time-fractional nonlocal diffusion equation. Issue 7 (3rd July 2021)

• Record Type: Journal Article
• Authors: Manimaran, J.; Shangerganesh, L.
• Is Part Of: International journal of computer mathematics. Volume 98:Issue 7(2021)
• Page Start: 1365
• Page End: 1384
• Publication Date: 2021-07-03
• Subjects: Error estimates -- Faedo–Galerkin method -- time-fractional equation -- finite element methods -- nonlocal diffusion; 65M60 -- 35R11 -- 65N15 -- 35K57; Computers -- Periodicals; Numerical analysis -- Periodicals; Automation -- Periodicals; 004.0151
• DOI: 10.1080/00207160.2020.1820492
• Languages: English
• ISSNs: 0020-7160

Abstract

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 Ω ⊂ R n . We use the Faedo–Galerkin approximation method with initial data in L 2 ( Ω ) to show a solution in 1 u ∈ L ∞ ( 0, T ; L 2 ( Ω ) ) ∩ L 2 ( 0, T ; H 0 1 ( Ω ) ) . 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.