Discretizations of the spectral fractional Laplacian on general 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.

arXiv comment31 pages, 6 figures. Published in SIAM Journal on Numerical Analysis. Minor change in the title. This version includes many improvements in both the results and techniques. In particular the regularity assumptions are weaker and error estimates are now presented in the $L^2$. We also include a new tail control that reduces significantly the complexity of the numerical method