Fully discrete finite element scheme for nonlocal parabolic problem involving the Dirichlet energy

In this article we present a finite element scheme for solving a nonlocal parabolic problem involving the Dirichlet energy. For time discretization, we use backward Euler method. The nonlocal term causes difficulty while using Newton’s method. Indeed, after applying Newton’s method we get a full Jacobian matrix due to the nonlocal term. In order to avoid this difficulty we use the technique given by Gudi (SIAM J Numer Anal 50(2):657–668, 2012) for elliptic nonlocal problem of Kirchhoff type. We discuss the well-posedness of the weak formulation at continuous as well as at discrete levels. We also derive a priori error estimates for both semi-discrete and fully discrete formulations. Results based on the usual finite element method are provided to confirm the theoretical estimates.