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Numerical Analysis for a Nonlocal Parabolic Problem

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  19 October 2016

M. Mbehou ,
R. Maritz  and
P.M.D. Tchepmo
M. Mbehou*
Affiliation:
Department of Mathematical Sciences, University of South Africa, Pretoria 0003, South Africa Department of Mathematics, University of Yaounde I, Cameroon
R. Maritz
Affiliation:
Department of Mathematical Sciences, University of South Africa, Pretoria 0003, South Africa
P.M.D. Tchepmo
Affiliation:
Department of Mathematical Sciences, University of South Africa, Pretoria 0003, South Africa
*
*Corresponding author. Email address:mohamm@unisa.ac.za (M. Mbehou)
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Abstract

This article is devoted to the study of the finite element approximation for a nonlocal nonlinear parabolic problem. Using a linearised Crank-Nicolson Galerkin finite element method for a nonlinear reaction-diffusion equation, we establish the convergence and error bound for the fully discrete scheme. Moreover, important results on exponential decay and vanishing of the solutions in finite time are presented. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

Keywords

Convergencenumerical simulationCrank-Nicolson schemesGalerkin finite element methodnonlinear parabolic equationnonlocal diffusion term

MSC classification

Secondary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 6 , Issue 4 , November 2016 , pp. 434 - 447
Copyright
Copyright © Global-Science Press 2016 

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References

[1] Chipot, M. and Lovat, B., Some remarks on non local elliptic and parabolic problems, Nonlinear Analysis: Theory, Methods & Applications 30, 4619–4627 (1997).Google Scholar
[2] Chipot, M., The diffusion of a population partly driven by its preferences, Archive Rational Mech. Analysis 155, 237–259 (2000).Google Scholar
[3] Hamik, C. T. and Steinbock, O., Excitation waves in reaction-diffusion media with non-monotonic dispersion relations, New J. Physics 5, 58 (2003).Google Scholar
[4] Anderson, A. R. and Chaplain, M., Continuous and discrete mathematical models of tumor-induced angiogenesis, Bulletin Math. Biology 60, 857–899 (1998).CrossRefGoogle ScholarPubMed
[5] Corrêa, F. J. S., Menezes, S. D., and Ferreira, J., On a class of problems involving a nonlocal operator, Appl. Math. Comp. 147, 475–489 (2004).Google Scholar
[6] Simsen, J. and Ferreira, J., A global attractor for a nonlocal parabolic problem, Nonlinear Stud. 21, 405–416 (2014).Google Scholar
[7] Yin, Z. and Xu, Q., A fully discrete symmetric finite volume element approximation of nonlocal reactive flows in porous media, Math. Problems Eng. 2013, 1–7 (2013).Google Scholar
[8] Ammi, M. R. S. and Torres, D. F., Numerical analysis of a nonlocal parabolic problem resulting from thermistor problem, Math. & Computers in Simulation 77, 291–300 (2008).CrossRefGoogle Scholar
[9] Almeida, R. M., Duque, J., Ferreira, J., and Robalo, R. J., The Crank–Nicolson–Galerkin finite element method for a nonlocal parabolic equation with moving boundaries, Num. Meth. Partial Differential Equations 31, 1515–1533 (2015).Google Scholar
[10] Thomée, V., Galerkin Finite Element Methods for Parabolic Problems, Springer (1984).Google Scholar
[11] Ciarlet, P., Finite Element Method for Elliptic Problems, North Holland (1978).CrossRefGoogle Scholar
[12] Barrett, J. W. and Liu, W., Finite element approximation of the p-Laplacian, Math. Comp. 61, 523–537 (1993).Google Scholar