体积约束的非局部扩散问题基于新的技巧的有限元方法

体积约束的非局部扩散问题在复合材料的断裂、多晶体的断裂、纳米纤维网络、裂缝的不稳定、图像处理等领域有重要应用，现存的数值方法精度不高。因此，设计一种高阶的有限元方法来求解二维体积约束的非局部扩散问题是十分必要的，但需克服维数增加带来的自由度骤增的困难。为此，采用了一种新技巧计算线性元的刚度矩阵，该数值方法的刚度矩阵是从一个新的矩阵$B$中提取的，该矩阵易于计算，并给出了单元的编码原理和数值计算节点的编码表达式，并通过数值算例验证了该方法对二维体积约束的非局部扩散问题具有几乎最优收敛阶。值得一提的是，求解二维体积约束的非局部扩散问题并不是平凡的。

Finite Element Method Based on a New Technique for the Nonlocal Diffusion Problem with Volume Constraints

The nonlocal diffusion problem with volume constraints has been widely applied in many fields, such as fracture of composites, fracture of polycrystal, nanofiber networks, image analysis and financial engineering, the accuracy of the existing numerical methods, including the finite element method for the piecewise constant element, finite difference method, quadrature and particle methods are low. To obtain a high order numerical method, a high order finite element method is proposed for a 2D nonlocal diffusion problem with volume constraints, and the stiffness matrix of the numerical method is extracted from a new matrix $B$, which is easily computed. And the coding principle of the elements and the expression of code in the numerical calculation nodes are given. Also, the numerical example illustrates the convergent order of the new finite element method for 2D nonlocal diffusion problem with volume constraints. It is worth pointing out that it is not trivial to solve the 2D nonlocal diffusion problem with volume constraints.