Discretizations of the spectral fractional Laplacian on general domains with Dirichlet, Neumann, and Robin boundary conditions

TL;DR

This paper introduces new discretization methods for the spectral fractional Laplacian on general domains using heat-semigroup integral formulations combined with finite element methods, avoiding eigenpair computations.

Contribution

It presents novel discretizations that do not require eigenpair calculations, are applicable to irregular domains, and handle various boundary conditions.

Findings

Methods achieve convergence depending on domain and function regularity

Numerical results confirm theoretical convergence rates

Applicable to domains with Dirichlet, Neumann, and Robin boundary conditions

Abstract

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.