An Efficient and Accurate Numerical Method for the Spectral Fractional Laplacian Equation

Abstract

We propose in this paper an efficient and accurate numerical method for the spectral fractional Laplacian equation using the Caffarelli–Silvestre extension. In particular, we propose several strategies to deal with the singularity and the additional dimension associated with the extension problem: (i) reducing the $$d+1$$ dimensional problem to a sequence of d-dimension Poisson-type problems by using the matrix diagonalizational method; (ii) resolving the singularity by applying the enriched spectral method in the extended dimension. We carry out rigorous analysis for the proposed numerical method, and provide abundant numerical examples to verify the theoretical results and illustrate effectiveness of the proposed method.