Second-order error analysis for FEM of fractional Laplacian on graded meshes via FDM auxiliary

On graded meshes, the superlinear convergence for fractional Laplacian problem was proved in Borthagaray et al. (2021) by finite element method (FEM). Furthermore, numerical experiments of FEM in Chen, et al. (2021) demonstrate a second-order accuracy on a suitably graded mesh for the 1D case, but a convergence analysis for the proposed scheme remains unavailable. To fill this gap, we provide a second-order error analysis for the resulting FEM algebraic system. Our analysis employs the finite difference method (FDM) as an auxiliary tool based on our previous work [ arXiv:2520.11117 ], where FEM scheme can be viewed as a modification of the FDM scheme.