We derive exact form of the piecewise-linear finite element stiffness matrix on general non-uniform meshes for the integral fractional Laplacian operator in one dimension, where the derivation is accomplished in the Fourier transformed space. With such an exact formulation at our disposal, we are able to numerically study some intrinsic properties of the fractional stiffness matrix on some commonly used non-uniform meshes (e.g., the graded mesh), in particular, to examine their seamless transition to those of the usual Laplacian.

中文翻译：

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非均匀网格上积分分数拉普拉斯算子的有限元刚度矩阵的显式形式

我们在一般非均匀网格上为一维积分分数拉普拉斯算子推导出分段线性有限元刚度矩阵的精确形式，其中推导是在傅立叶变换空间中完成的。有了这样一个精确的公式，我们就能够数值研究一些常用的非均匀网格（例如渐变网格）上分数刚度矩阵的一些内在属性，特别是检查它们与非均匀网格的无缝过渡。通常的拉普拉斯算子。