  We prove non-asymptotic rates of convergence in the $W^{s,2}(\mathbb
R^d)$-norm for the solution of the fractional Dirichlet problem to the solution
of the local Dirichlet problem as $s\uparrow 1$. For regular enough boundary
values we get a rate of order $\sqrt{1-s}$, while for less regular data the
rate is of order $\sqrt{(1-s)|\log(1-s)|}$. We also obtain results when the
right hand side depends on $s$, and our error estimates are true for all
$s\in(0,1)$. The proofs use variational arguments to deduce rates in the
fractional Sobolev norm from energy estimates between the fractional and the
standard Dirichlet energy.
