A Fractional Laplace Equation: Regularity of Solutions and Finite Element Approximations

In this work we deal with the Dirichlet homogeneous problem for the integral fractional Laplacian on a bounded domain Ω ⊂ R. Namely, we deal with basic analytical aspects required to convey a complete Finite Element analysis of the problem (1) { (−∆)u = f in Ω, u = 0 in Ω, where the fractional Laplacian of order s is defined by (−∆)u(x) = C(n, s) P.V. ∫ Rnu(x)− u(y)|x− y|n+2s dyand C(n, s) is a normalization constant.Independently of the Sobolev regularity of the source f , solutions of (1) are not expectedto be in a better space than Hs+min{s,1/2− }(Ω) (see [2, 4]). However, by building on Hölderestimates developed in [3], we were able to obtain further regularity results in a novel frameworkof weighted fractional Sobolev spaces, leading to a priori estimates in terms of the Hölderregularity of the data [1].After developing a suitable polynomial interpolation theory in these weighted fractionalspaces, optimal order of convergence in the energy norm for the standard linear finite ele-ment method is proved for graded meshes. Numerical experiments are in agreement with ourtheoretical predictions, and illustrate the optimality of the aforementioned estimates.