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.

中文翻译：

---

具有Kirchhoff型非线性的非局部抛物线问题的数值方法

当使用有限元法和牛顿-拉夫森方法数值求解问题时，非局部问题中非局部项的存在破坏了雅可比矩阵的稀疏性。结果，与局部问题相比，计算消耗更多的时间和空间。为了克服这个困难，本文致力于分析带有非线性Kirchhoff型时变非局部问题的线性化theta-Galerkin有限元方法。在此，我们专注于时间离散基于θ与步进-时间方案θ∈ [½，1）。对于标准的Crank–Nicolson（θ =½），移位的Crank–Nicolson（θ =½+ δ，其中δ是时间步长）和一般情况（θ≠ ½+ kδ，其中k = 0，1）。最后，展示了验证理论发现的数值模拟。