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 L2 and H1 norms. Results based on the usual finite element method are provided to confirm the theoretical estimates. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 786–813, 2017

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In this article, a finite element scheme based on the Newton's method is proposed to approximate the solution of a nonlocal coupled system of parabolic problem. The Crank-Nicolson method is used for time discretization. Well-posedness of the problem is discussed at continuous and discrete levels. We derive a priori error estimates for both semidiscrete and fully discrete formulations. Results based on usual finite element method are provided to confirm the theoretical estimates. KEYWORDSCrank-Nicolson method, finite element method, nonlocal problem, Newton iteration methodwhere Ω ⊂ R d (d ≥ 1) is a bounded domain with Lipschitz boundary Ω,|| · || denotes the L 2 (Ω) norm and f ∶ Ω × R → R, M 1 ∶ R → R + are given functions. Gudi 1 made a successful attempt to study finite element analysis for the problem (1.2). The author also addresses the key issue of sparsity of Jacobian matrix obtained while applying the Newton's method to this nonlocal problem. Chipot and colleagues 6,7 studied the existence and uniqueness of the solution for the following nonlocal problem:where M 1 is some function from R → R + and l ∶ L 2 (Ω) → R is continuous linear form in u. Chipot and Lavot 8 and Srivastava et al 9 have considered the same problem with l(u) as a nonlinear function defined by l(u) = ∫ Ω |∇u| 2 dx. Srivastava and colleagues 9-11 made the first successful attempt to study a finite element scheme together with the Newton's method for the numerical approximation to the parabolic nonlocal problem (1.3). Raposo et al 12 studied the existence and uniqueness of the solution for a coupled nonlocal system of the following form:where a(·) > 0, l is in continuous linear form, > 0 is a parameter, and f is a Lipschitz continuous function. Recently, Almeida and colleagues 1,2 have considered the following coupled nonlocal parabolic system:where 1 , 2 > 0, p ≥ 2, and a 1 , a 2 are nonlocal functions acting on two continuous linear functions l 1 and l 2 . They discussed about the existence and uniqueness of the strong global solution and have given important results on polynomial and exponential decay of the solution in finite time. For obtaining a numerical solution, Duqué et al 2 have used the Euler-Galerkin finite element method and have provided optimal rates of convergence in the L 2 (Ω) norm only. The aim of this paper is to obtain a numerical solution to the coupled nonlocal parabolic problem using the finite element scheme with the Crank-Nicolson method. The Newton's method is used for linearizing the discrete nonlinear problem. The robustness and higher rate of convergence of the Newton's method make it an attractive alternative to the Picard method, particularly for strongly nonlinear problems. 13,14 However, the main issue in solving nonlocal problems with the Newton's method is that if we apply it directly to the discrete weak formulation, then we get a full Jacobian matrix. As a consequence, computations consume more time and space in contrast to the local problem. To overcome this difficulty, we reformulate the discrete pr...

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