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

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