Finite element schemes for a class of nonlocal parabolic systems with moving boundaries

Almeida, Rui M.P.; Duque, José C. M.; Ferreira, J.; Robalo, Rui

doi:10.1016/j.apnum.2018.01.007

Cited by 5 publications

(3 citation statements)

References 19 publications

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“…With these motivations, we extend the analysis of the aforementioned work and solve the coupled nonlocal parabolic problem using the finite element scheme with the Newton's method. The contributions of the present work can be summarized as follows: In the present work, we generalize the results that are given by Gudi for the elliptic nonlocal problem and by Srivastava and colleagues for the scalar parabolic nonlocal problem . Duqué and colleagues have successfully solved the coupled nonlocal parabolic problem via the fixed‐point method (the Picard method). But in this paper a new scheme is proposed, and this scheme uses the Newton's method. The existence and uniqueness of the solution are discussed at continuous as well as discrete levels.…”

Section: Introductionsupporting

confidence: 55%

Crank‐Nicolson‐Galerkin finite element scheme for nonlocal coupled parabolic problem using the Newton's method

Chaudhary

2017

Math Methods in App Sciences

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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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Section: Introductionsupporting

confidence: 55%

Crank‐Nicolson‐Galerkin finite element scheme for nonlocal coupled parabolic problem using the Newton's method

Chaudhary

2017

Math Methods in App Sciences

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“…Yin and Xu [9] applied the finite-volume method to obtain approximate solutions for a nonlocal problem on reactive flows in porous media and derived the optimal convergence order in the L 2 norm. Almeida et al [10] presented convergence analysis for a fully discretized approximation to a nonlocal problem involving a parabolic equation with moving boundaries, with the finite element method applied for the space variables and the Crank-Nicolson method for the time. Recently, Yang et al [11] derived the unconditional optimal error estimate of Galerkin FEMs for the time-dependent Klein-Gordon-Schrodinger equations using the error splitting technique.…”

Section: Introductionmentioning

confidence: 99%

“…Almeida et al. [10] presented convergence analysis for a fully discretized approximation to a nonlocal problem involving a parabolic equation with moving boundaries, with the finite element method applied for the space variables and the Crank–Nicolson method for the time. Recently, Yang et al.…”

Section: Introductionmentioning

confidence: 99%

Optimal error estimates of a linearized second-order BDF scheme for a nonlocal parabolic problem

Haggar

Mbehou²

2023

AJMS

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PurposeThis paper focuses on the unconditionally optimal error estimates of a linearized second-order scheme for a nonlocal nonlinear parabolic problem. The first step of the scheme is based on Crank–Nicholson method while the second step is the second-order BDF method.Design/methodology/approachA rigorous error analysis is done, and optimal L2 error estimates are derived using the error splitting technique. Some numerical simulations are presented to confirm the study’s theoretical analysis.FindingsOptimal L2 error estimates and energy norm.Originality/valueThe goal of this research article is to present and establish the unconditionally optimal error estimates of a linearized second-order BDF finite element scheme for the reaction-diffusion problem. An optimal error estimate for the proposed methods is derived by using the temporal-spatial error splitting techniques, which split the error between the exact solution and the numerical solution into two parts, that is, the temporal error and the spatial error. Since the spatial error is not dependent on the time step, the boundedness of the numerical solution in L∞-norm follows an inverse inequality immediately without any restriction on the grid mesh.

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Theoretical and computational analysis of a nonlinear Schrödinger problem with moving boundary

et al. 2018

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Finite element schemes for a class of nonlocal parabolic systems with moving boundaries

Cited by 5 publications

References 19 publications

Crank‐Nicolson‐Galerkin finite element scheme for nonlocal coupled parabolic problem using the Newton's method

Crank‐Nicolson‐Galerkin finite element scheme for nonlocal coupled parabolic problem using the Newton's method

Optimal error estimates of a linearized second-order BDF scheme for a nonlocal parabolic problem

Theoretical and computational analysis of a nonlinear Schrödinger problem with moving boundary

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