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In this work, we propose novel discretizations 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 ofdiscretizations with order of convergence depending on the regularity of thedomain and the function on which the spectral fractional Laplacian is acting.Our method does not require the computation of the eigenpairs of the Laplacianon the considered domain, can be implemented on possibly irregular boundeddomains, and can naturally handle different types of boundary constraints.Various numerical simulations are provided to illustrate performance of theproposed method and support our theoretical results.

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