Finite element analysis of nonlocal coupled parabolic problem using Newton's method

In this article, we propose finite element method to approximate the solution of a coupled nonlocal parabolic system. An important issue in the numerical solution of nonlocal problems while using the Newton's method is related to its structure. Indeed, unlike the local case the Jacobian matrix is sparse and banded, the nonlocal term makes the Jacobian matrix dense. As a consequence computations consume more time and space in contrast to local problems. To overcome this difficulty we reformulate the discrete problem and then apply the Newton's method. We discuss the well-posedness of the weak formulation at continuous as well as at discrete levels. We derive a priori error estimates for both semi-discrete and fully-discrete formulations. Results based on usual finite element method are provided to confirm the theoretical estimates.