Fast Implementation of FEM for Integral Fractional Laplacian on Rectangular Meshes (Communications in Computational Physics, 2024)
We show that the entries of the stiffness matrix, associated with the $C^0$-piecewise linear finite element discretization of the hyper-singular integral fractionalLaplacian (IFL) on rectangular meshes, can be simply expressed as one-dimensionalintegrals on a finite interval. Particularly, the FEM stiffness matrix on uniform mesheshas a block-Toeplitz structure, so the matrix-vector multiplication can be implementedby FFT efficiently. The analytic integral representations not only allow for accurateevaluation of the entries, but also facilitate the study of some intrinsic properties ofthe stiffness matrix. For instance, we can obtain the asymptotic decay rate of the entries, so the “dense” stiffness matrix turns out to be “sparse” with an $\mathcal{O}(h^3)$ cutoff.We provide ample numerical examples of PDEs involving the IFL on rectangular or $L$-shaped domains to demonstrate the optimal convergence and efficiency of this semi-analytical approach. With this, we can also offer some benchmarks for the FEM ongeneral meshes implemented by other means (e.g., for accuracy check and comparisonwhen triangulation reduces to rectangular meshes).