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

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 Omega subset of R-n. We use the Faedo-Galerkin approximation method with initial data in L-2(Omega) to show a solution in u is an element of L-infinity(0, T; L-2(Omega)) boolean AND L-2(0, T; H-0(1)(Omega)). 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-(fr)actional 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.

Automated summary

This paper investigates the well-posedness and Mittag-Leffler stability of solutions of time-fractional nonlocal reaction-diffusion equation in a bounded domain. The Faedo-Galerkin approximation method is utilized, and a suitable Lyapunov function is constructed for stability. The proposed numerical method is validated with error analysis and constructive numerical examples.