Efficient Spectral Methods for Eigenvalue Problems of the Integral Fractional Laplacian on a Ball of Any Dimension (Journal of Computational Mathematics, 2024)
An efficient spectral-Galerkin method for eigenvalue problems of the integral fractionalLaplacian on a unit ball of any dimension is proposed in this paper. The symmetric positivedefinite linear system is retained explicitly which plays an important role in the numericalanalysis. And a sharp estimate on the algebraic system’s condition number is establishedwhich behaves as $N^{4s}$ with respect to the polynomial degree $N,$ where $2s$ is the fractionalderivative order. The regularity estimate of solutions to source problems of the fractionalLaplacian in arbitrary dimensions is firstly investigated in weighted Sobolev spaces. Thenthe regularity of eigenfunctions of the fractional Laplacian eigenvalue problem is readilyderived. Meanwhile, rigorous error estimates of the eigenvalues and eigenvectors are obtained. Numerical experiments are presented to demonstrate the accuracy and efficiencyand to validate the theoretical results.