Numerical Analysis for a Nonlocal Parabolic Problem

Mbehou, M.; Maritz, Riette; Tchepmo, P.M.D.

doi:10.4208/eajam.260516.150816a

Cited by 5 publications

(5 citation statements)

References 11 publications

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“…Chaudhary et al [7] studied the convergence analysis of the Crank-Nicolson finite element method for the nonlocal problem involving the Dirichlet energy. Mbehou et al [8] studied (1.1) using the Crank-Nicolson Galerkin finite element method. The main focus on this paper was to present the exponential decay and vanishing of the solutions in finite time.…”

Section: Optimal Error Estimates Of a Bdf Schemementioning

confidence: 99%

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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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Section: Optimal Error Estimates Of a Bdf Schemementioning

confidence: 99%

“…Mbehou et al. [8] studied (1.1) using the Crank–Nicolson Galerkin finite element method. The main focus on this paper was to present the exponential decay and vanishing of the solutions in finite time.…”

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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“…Theorem 3.1. Let (u, ) and (u h , h ) be the solutions of (5) to (7) and (19) to (21), respectively. Then there exists a constant C which does not depend on h such that…”

Section: Spatial Discretizationmentioning

confidence: 99%

“…In this paper, a linearized Theta‐Galerkin finite element method is proposed for solving the coupled system to . The focus is on time discretization based on the so‐called θ ‐scheme with

θ \in false[\frac{1}{2}, 1 false)

including the standard Crank‐Nicolson (see, for instance, Mbehou et al and references therein) and the shifted Crank‐Nicolson as discussed in Mbehou et al and Luskin . Optimal error estimates in L 2 and H 1 ‐norms are presented for the spatial discrete problems.…”

Section: Introductionmentioning

confidence: 99%

The theta‐Galerkin finite element method for coupled systems resulting from microsensor thermistor problems

Mbehou

2017

Math Methods in App Sciences

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This paper is devoted to the analysis of a linearized theta‐Galerkin finite element method for the time‐dependent coupled systems resulting from microsensor thermistor problems. Hereby, we focus on time discretization based on θ‐time stepping scheme with θ∈false[12,1false) including the standard Crank‐Nicolson ( θ=12) and the shifted Crank‐Nicolson ( θ=12+δ, where δ is the time‐step) schemes. The semidiscrete formulation in space is presented and optimal error bounds in L2‐norm and the energy norm are established. For the fully discrete system, the optimal error estimates are derived for the standard Crank‐Nicolson, the shifted Crank‐Nicolson, and the general case where θ≠12+kδ with k=0,1 . Finally, numerical simulations that validate the theoretical findings are exhibited.

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“…One of the justifications of such models lies in the fact that in reality the measurements are not made pointwise but through some local average. Some interesting features of nonlocal problems and more motivation are described in [1,5,6,9,19] and in the references therein.…”

Section: Introductionmentioning

confidence: 99%

Crank-Nicolson finite element methods for nonlocal problems with p -Laplace-type operator

Haggar,

Mbehou,

Njifenjou

2023

J. Math. Computer Sci.

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A theoretical analysis of a Crank-Nicolson Galerkin finite element method for a class of nonlinear nonlocal diffusion problems associated with p -Laplace-type operator is presented here. It is shown, by a rigorous analysis that the unconditionally optimal error estimates for the fully discrete scheme are established. The presence of the nonlocal term in the models destroys the sparsity of the Jacobian matrices when solving the problem numerically using finite element method and Newton-Raphson method. As a consequence, computations consume more time and space in contrast to local problems. To overcome this difficulty, a new algorithm is proposed to avoid the full Jacobian matrix. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

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Numerical Analysis for a Nonlocal Parabolic Problem

Cited by 5 publications

References 11 publications

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

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

The theta‐Galerkin finite element method for coupled systems resulting from microsensor thermistor problems

Crank-Nicolson finite element methods for nonlocal problems with p -Laplace-type operator

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