A Novel Analytic Diagonalization Technique Finite Element Method for the Spectral Fractional Laplacian

The goal of this dissertation is to present a new technique for approximating solutions to problems involving the spectral fractional Laplacian. Previous works have used the Dirichlet-to-Neumann extension technique of Caffarelli and Silvestre, together with a diagonalization method to reduce computational complexity. Building on this method, a novel scheme is proposed where the analytic solution to the associated eigenvalue problem in the extended dimension is used, thus avoiding the numerical issues of ill conditioning in computing the eigenpairs. It is then shown how this new method is related to a quadrature scheme to approximate the spectral fractional Laplacian via a Balakrishnan integral formula. The quadrature scheme used in the algorithm demonstrates exponential convergence to the analytic integral. Numerical examples illustrate the theoretical convergence rates. The parallel performance of the algorithm is studied using both strong and weak scaling.