  We consider the numerical solution of the fractional Laplacian of index
$s\in(1/2,1)$ in a bounded domain $\Omega$ with homogeneous boundary
conditions. Its solution a priori belongs to the fractional order Sobolev space
${\widetilde H}^s(\Omega)$. 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 ${\widetilde H}^s(\Omega)$ 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.
