In this paper, we consider the discretization of a parabolic nonlocal problem within the framework of the virtual element method. Using the fixed point argument, we prove that the fully discrete scheme has a unique solution. The presence of the nonlocal term makes the problem nonlinear, and the resulting nonlinear equations are solved using the Newton method. The computational cost of the Jacobian of the nonlinear scheme increases in the presence of nonlocal coefficient. To reduce the computational burden in computing the Jacobian, which otherwise is inevitable in the usual approach, in this paper, we propose an equivalent formulation. A priori error estimates in the L² and the H¹ norms are derived. Furthermore, we employ a linearized scheme without compromising the rate of convergence in the respective norms. Finally, the theoretical convergence results are verified through numerical experiments over polygonal meshes.

中文翻译：

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通用网格类型上非局部抛物线问题的虚拟元素方法

在本文中，我们考虑了在虚拟元素方法框架内的抛物线形非局部问题的离散化。使用定点参数，我们证明了完全离散的方案具有唯一的解决方案。非局部项的存在使问题成为非线性，并使用牛顿法求解所得的非线性方程。在存在非局部系数的情况下，非线性方案的雅可比行列式的计算成本增加。为了减少计算雅可比行列式的计算负担，否则在通常的方法中这是不可避免的，在本文中，我们提出了一个等效公式。L ²和H ¹中的先验误差估计规范是衍生出来的。此外，我们采用线性化方案，而不会损害各个规范的收敛速度。最后，通过对多边形网格的数值实验验证了理论收敛结果。