Numerical methods for a nonlocal parabolic problem with nonlinearity of Kirchhoff type

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.