Finite element approximation of nonlocal parabolic problem

In this article, we present a finite element scheme combined with backward Euler method to solve a nonlocal parabolic problem. An important issue in the numerical solution of nonlocal problems while using Newton's method is related to its structure. In fact differently from the local case where the Jacobian matrix is sparse and banded, in the nonlocal case the Jacobian matrix is dense and computations are much more onerous compared to that for differential equations. In order to avoid this difficulty, we use the technique given by Gudi (SIAM J Numer Anal 50 (2012), 657-668) 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 semidiscrete and fully discrete formulations in L-2 and H-1 norms. Results based on the usual finite element method are provided to confirm the theoretical estimates. (c) 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 786-813, 2017