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

摘要

In this work, we propose novel discretisations of the spectral fractionalLaplacian on bounded domains based on the integral formulation of the operatorvia the heat-semigroup formalism. Specifically, we combine suitable quadratureformulas of the integral with a finite element method for the approximation ofthe solution of the corresponding heat equation. We derive two families ofdiscretisations with order of convergence depending on the regularity of thedomain and the function on which the fractional Laplacian is acting. Unlikeother existing approaches in literature, our method does not require thecomputation of the eigenpairs of the Laplacian on the considered domain, can beimplemented on possibly irregular bounded domains, and can naturally handledifferent types of boundary constraints. Various numerical simulations areprovided to illustrate performance of the proposed method and support ourtheoretical results.