ON THE CONVERGENCE IN H1-NORM FOR THE FRACTIONAL LAPLACIAN

We consider the numerical solution of the fractional Laplacian of index s is an element of (1/2, 1) in a bounded domain Omega with homogeneous boundary conditions. Its solution a priori belongs to the fractional-order Sobolev space (H) over tilde (s) (Omega). For the Dirichlet problem and under suitable assumptions on the data, it can be shown that its solution is also in H-1(Omega). 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 H-1(Omega). A natural question is then whether one can obtain error estimates in H-1(Omega) norm in addition to the classical ones that can be derived in the (H) over tilde (s)(Omega) 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.