Numerical Methods for a Nonlocal Parabolic Problem with Nonlinearity of Kirchhoff Type

Mbehou, M.; Chendjou, G.

doi:10.1134/s1995423919030042

Cited by 3 publications

(1 citation statement)

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“…Simsen and Ferreira [4] have discussed not only the existence and uniqueness of solutions for this problem but also continuity with respect to initial values, the exponential stability of weak solutions and important results on the existence of a global attractor. The numerical methods for the nonlocal problems have been investigated by many authors as like in Refs [5,6] and the references therein. However, they are restricted to nonlocal reaction terms or nonlocal boundary conditions.…”

Section: Optimal Error Estimates Of a Bdf Schemementioning

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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Section: Optimal Error Estimates Of a Bdf Schemementioning

confidence: 99%