Finite element error analysis of a time-fractional nonlocal diffusion equation with the Dirichlet energy

Abstract

A time-fractional diffusion equation involving the Dirichlet energy is considered with nonlocal diffusion operator in the space which has dimension d∈{2,3} and the Caputo sense fractional derivative in time. Further, nonlocal term in diffusion operator is of Kirchhoff type. We discretize the space using the Galerkin finite elements and time using the finite difference scheme on a uniform mesh. First, we prove the existence and uniqueness of a fully discrete numerical solution of the problem using the Brouwer fixed point theorem. Then, we give a priori bounds and convergence estimates in L2 and L∞ norms for fully-discrete problem. A more delicate analysis in the error provides the second order convergence for the proposed scheme. Numerical results are provided to validate the theoretical analysis.