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

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Abstract

The presence of the nonlocal term in the nonlocal problems destroys the sparsity of the Jacobian matrices when solving the problem numerically using finite elementmethod and Newton–Raphson method. As a consequence, computations consume more time and space in contrast to local problems. To overcome this difficulty, this paper is devoted to the analysis of a linearized theta-Galerkin finite element method for the time-dependent nonlocal problem with nonlinearity of Kirchhoff type. Hereby, we focus on time discretization based on θ-time stepping scheme with θ ∈ [½, 1). Some error estimates are derived for the standard Crank–Nicolson (θ = ½), the shifted Crank–Nicolson (θ = ½ + δ, where δ is the time-step) and the general case (θ ≠ ½ + kδ, where k = 0, 1). Finally, numerical simulations that validate the theoretical findings are exhibited.

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Authors and Affiliations

  1. Department of Mathematics, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon

    M. Mbehou & G. Chendjou

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  1. M. Mbehou
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  2. G. Chendjou
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Correspondence to M. Mbehou or G. Chendjou.

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Russian Text © The Author(s), 2019, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2019, Vol. 22, No. 3, pp. 295–307.

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Mbehou, M., Chendjou, G. Numerical Methods for a Nonlocal Parabolic Problem with Nonlinearity of Kirchhoff Type. Numer. Analys. Appl. 12, 251–262 (2019). https://doi.org/10.1134/S1995423919030042

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  • DOI: https://doi.org/10.1134/S1995423919030042

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