Spectral methods for fractional Laplacian on bounded domains

Date: 5/13/25

Abstract: It is well-known that the fractional Laplacian introduces a mild singularity near the boundary of the bounded domain when a fractional elliptic problem is considered. Thus, numerical methods often suffer in accuracy. However, the weak boundary singularity for such problems can is explicitly characterized. If a fractional Poisson problem is considered, higher-order methods can be achieved by using the technique of singularity separation on smooth domains. The technique fails to be highly accurate if higher or lower-order differential operators appear in fractional elliptic problems. We demonstrate that this occurs with spectral methods that use weighted Jacobi polynomials for computations on an interval. We also find similar effects for spectral Galerkin methods on a disk. Additionally, we conjecture that the pattern of singularities in these solutions is similar to the structure in Mittag-Leffler functions.