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

TL;DR

This paper derives an exact finite element stiffness matrix for the integral fractional Laplacian on non-uniform meshes in 1D, enabling detailed numerical analysis of its properties and transition to classical Laplacian behavior.

Contribution

It provides the first explicit form of the FEM stiffness matrix for the fractional Laplacian on non-uniform meshes, derived via Fourier transform methods.

Findings

Exact stiffness matrix formulation for fractional Laplacian on non-uniform meshes

Numerical insights into the matrix's properties on graded meshes

Analysis of the transition to classical Laplacian behavior

Abstract

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.