  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.
