Geodesic
Numerical methods for a nonlocal parabolic problem with nonlinearity of Kirchhoff type
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 22 (2019) no. 3, pp. 301-313.

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The presence of the nonlocal term in the nonlocal problems 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, 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 $\theta$-time stepping scheme with $\theta\in [1/2, 1)$. Some a error estimates are derived for the standard Crank–Nicolson ($\theta = 1/2$), the shifted Crank–Nicolson ($\theta = 1/2 + \delta$, $\delta$ is the time-step) and the general case ($\theta\ne 1/2 + k\delta$, $k = 0, 1$). Finally, numerical simulations that validate the theoretical findings are exhibited. 
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     author = {M. Mbehou and G. Chendjou},
     title = {Numerical methods for a nonlocal parabolic problem with nonlinearity of {Kirchhoff} type},
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M. Mbehou; G. Chendjou. Numerical methods for a nonlocal parabolic problem with nonlinearity of Kirchhoff type. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 22 (2019) no. 3, pp. 301-313. http://geodesic.mathdoc.fr/item/SJVM_2019_22_3_a3/

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