In this paper, we compute stiffness matrix of the nonlocal Laplacian discretized by the piecewise linear finite element on nonuniform meshes, and implement the FEM in the Fourier transformed domain. We derive useful integral expressions of the entries that allow us to explicitly or semi-analytically evaluate the entries for various interaction kernels. Moreover, the limiting cases of the nonlocal stiffness matrix when the interactional radius $\delta\rightarrow0$ or $\delta\rightarrow\infty$ automatically lead to integer and fractional FEM stiffness matrices, respectively, and the FEM discretisation is intrinsically compatible. We conduct ample numerical experiments to study and predict some of its properties and test on different types of nonlocal problems. To the best of our knowledge, such a semi-analytic approach has not been explored in literature even in the one-dimensional case.

翻译：本文计算了在非均匀网格上采用分段线性有限元离散的非局部拉普拉斯算子的刚度矩阵，并在傅里叶变换域中实现了有限元方法。我们推导了矩阵元素的有用积分表达式，使得能够对各种相互作用核函数进行显式或半解析的数值计算。此外，当相互作用半径 $\delta\rightarrow0$ 或 $\delta\rightarrow\infty$ 时，非局部刚度矩阵的极限情况分别自动导出整数阶和分数阶有限元刚度矩阵，且有限元离散本质上是兼容的。我们进行了大量数值实验以研究并预测其部分性质，并在不同类型的非局部问题上进行了测试。据我们所知，即使在一维情况下，此类半解析方法在文献中尚未被探索。