A spectral collocation method for the anisotropic tempered fractional Laplacian

Abstract

The accurate and efficient numerical evaluation of the anisotropic tempered nonlocal Laplacian in multiple dimensions is highly challenging, primarily due to the singularities in its integral kernel and difficulties of high-dimensional integration. Existing numerical methods are mainly confined to low-dimensional and isotropic settings and are difficult to extend directly to anisotropic cases; they often suffer from high computational complexity, challenging implementation, and a lack of systematic convergence analysis. This paper proposes a simple sinc collocation method-based spectral discretization scheme for the anisotropic tempered fractional Laplacian. Leveraging the tensor product structure of sinc basis functions, the method naturally generalizes to multi-dimensional problems and allows for fast algorithm implementation. We rigorously prove the optimal convergence rate of the proposed scheme in the discrete energy norm and develop a preconditioned iterative algorithm to efficiently solve the resulting nonlocal partial differential equations. Numerical experiments validate the accuracy, flexibility, and excellent capability of the method in capturing anisotropic diffusion behaviors.