An Analytical Diagonalization Technique for Approximating the Spectral Fractional Laplacian

Problems involving non-local operators have recently attracted increasing interest in many diverse fields. However, non-locality necessarily increases the computational complexity to approximate solutions to these problems.

We study the spectral fractional Laplacian \( (-\Delta)^s u =f \) in a bounded domain \(\Omega \subset \mathbb{R}^d\). Previous works have used the Caffarelli-Silvestre extension to convert the fractional Laplacian into a Dirichlet-to-Neumann mapping in \(\mathbb{R}_+^{d+1}\). A diagonalization scheme is used to reduce the computational complexity by exposing the inherent parallelizability of the method.

We refine the diagonalization scheme by proposing an analytic approach to compute the eigenpairs of the eigenvalue problem in the extended dimension, avoiding the numerical instability in approximating the eigenpairs with a finite element method. We demonstrate that this new analytical approach is related to certain quadrature schemes used to approximate the spectral fractional Laplacian. We further show that this novel algorithm maintains exponential convergence. Numerical examples in two dimension demonstrate the performance of the method.