Discretizations of the spectral fractional Laplacian on general domains with Dirichlet, Neumann, and Robin boundary conditions
This work provides a practical numerical method for solving fractional PDEs on general bounded domains with various boundary conditions, addressing a computational bottleneck in fractional calculus.
The authors propose novel discretizations of the spectral fractional Laplacian on bounded domains using the heat-semigroup formalism, achieving convergence orders dependent on domain regularity. The method avoids eigenpair computation, works on irregular domains, and handles Dirichlet, Neumann, and Robin boundary conditions.
In this work, we propose novel discretizations of the spectral fractional Laplacian on bounded domains based on the integral formulation of the operator via the heat-semigroup formalism. Specifically, we combine suitable quadrature formulas of the integral with a finite element method for the approximation of the solution of the corresponding heat equation. We derive two families of discretizations with order of convergence depending on the regularity of the domain and the function on which the spectral fractional Laplacian is acting. Our method does not require the computation of the eigenpairs of the Laplacian on the considered domain, can be implemented on possibly irregular bounded domains, and can naturally handle different types of boundary constraints. Various numerical simulations are provided to illustrate performance of the proposed method and support our theoretical results.