On the convergence in H1-norm for the fractional laplacian

We consider the numerical solution of the fractional Laplacian of index s \in (1/2, 1) in a bounded domain Ω with homogeneous boundary conditions. Its solution a priori belongs to the fractional-order Sobolev space \widetilde Hs(Ω ). For the Dirichlet problem and under suitable assumptions on the data, it can be shown that its solution is also in H1(Ω ). In this case, if one uses the standard Lagrange finite element to discretize the problem, then both the exact and the computed solution belong to H1(Ω ). A natural question is then whether one can obtain error estimates in H1(Ω ) norm in addition to the classical ones that can be derived in the \widetilde Hs(Ω ) energy norm. We address this issue, and in particular we derive error estimates for the Lagrange finite element solutions on both quasi-uniform and graded meshes.