On explicit form of the FEM stiffness matrix for the integral fractional Laplacian on non-uniform meshes

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